Chapter 6

 

Analytical techniques and solutions for plastic solids

 

 

 

6.2 Bounding theorems in plasticity and their applications

 

To set the background for plastic limit analysis, it is helpful to review the behavior of an elastic-plastic solid or structure subjected to mechanical loading.  The solution to an internally-pressurized elastic-perfectly plastic sphere given in Section 4.2 provides a representative example.  All elastic-perfectly plastic structures will exhibit similar behavior.  In particular

 An inelastic solid will reach yield at some critical value of applied load.

 If the load exceeds yield, a plastic region starts to spread through the solid. As an increasing area of the solid reaches yield, the displacements in the structure progressively increase.

 At a critical load, the plastic region becomes large enough to allow unconstrained plastic flow in the solid. The load cannot be increased beyond this point. The solid is said to collapse. 

 

Strain hardening will influence the results quantitatively, but if the solid has a limiting yield stress (a stress beyond which it can never harden) its behavior will be qualitatively similar.

 

In a plasticity calculation, often the two most interesting results are (a) the critical load where the solid starts to yield; and (b) the critical load where it collapses.  Of course, we don’t need to solve a plasticity problem to find the yield point MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we only need the elastic fields.  In many design problems this is all we need, since plastic flow must be avoided more often than not.  But there are situations where some plasticity can be tolerated in a structure or component; and there are even some situations where it’s desirable (e.g. in designing crumple zones in cars).  In this situation, we usually would like to know the collapse load for the solid.  It would be really nice to find some way to get the collapse load without having to solve the full boundary value problem.

 

This is the motivation for plastic limit analysis.  The limit theorems of plasticity provide a quick way to estimate collapse loads, without needing any fancy calculations.  In fact, collapse loads are often much easier to find than the yield point!

 

In this section, we derive several useful theorems of plastic limit analysis and illustrate their applications.

 

 

6.2.1 Definition of the plastic dissipation

 

Consider a rigid perfectly plastic solid, which has mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@357C@ , and a Von-Mises yield surface with yield stress in uniaxial tension Y. (By definition, the elastic strains are zero in a rigid plastic material). The solid is subjected to tractions t * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bWaaWbaaSqabeaacaGGQaaaaa aa@34AE@  on the its boundary.  The solid may also be subjected to a body force b (per unit mass) acting on the interior of the solid.  Assume that the loading is sufficient to cause the solid to collapse.

 

 

 

 

Velocity discontinuities: Note that the velocity and stress fields in a collapsing rigid plastic solid need not necessarily be continuous.  The solution often has shear discontinuities, as illustrated on the right.  In the picture, the top part of the solid slides relative to the bottom part.  We need a way to describe this kind of deformation.  To do so,

 

1.      We assume that the velocity field u ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWH1bGbaiaaaaa@33CD@  at collapse may have a finite set of such shear discontinuities, which occur over a collection of surfaces S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36BE@ .  Let m be a unit vector normal to the surface at some point , and let u ˙ ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaca WaaWbaaSqabeaacqGHXcqSaaaaaa@3A1F@    σ ij ± MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaGaeyySaelaaaaa@3CB8@  denote the limiting values of velocity and stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaaaaa@3AC9@  on the two sides of the surface.

2.      To ensure that no holes open up in the material, the velocity discontinuity must satisfy

( u ˙ + u ˙ )m=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiqahw hagaGaamaaCaaaleqabaGaey4kaScaaOGaeyOeI0IabCyDayaacaWa aWbaaSqabeaacqGHsislaaGccaGGPaGaeyyXICTaaCyBaiabg2da9i aaicdaaaa@4177@

3.      The solids immediately adjacent to the discontinuity exert equal and opposite forces on each other.  Therefore

σ ij + m i = σ ij m i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaGaey4kaScaaOGaamyBamaaBaaaleaa caWGPbaabeaakiabg2da9iabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiabgkHiTaaakiaad2gadaWgaaWcbaGaamyAaaqabaaaaa@45A2@

4.      We will use the symbol [[ u ˙ ]] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaai4waiqadwhagaGaaiaac2 facaGGDbaaaa@3749@  to denote the relative velocity of sliding across the discontinuity, i.e.

[[ u ˙ ]]=| u ˙ + u ˙ |= ( u ˙ i + u ˙ i )( u ˙ i + u ˙ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaacU faceWG1bGbaiaacaGGDbGaaiyxaiabg2da9maaemaabaGabCyDayaa caWaaWbaaSqabeaacqGHRaWkaaGccqGHsislceWH1bGbaiaadaahaa WcbeqaaiabgkHiTaaaaOGaay5bSlaawIa7aiabg2da9maakaaabaWa aeWaaeaaceWG1bGbaiaadaqhaaWcbaGaamyAaaqaaiabgUcaRaaaki abgkHiTiqadwhagaGaamaaDaaaleaacaWGPbaabaGaeyOeI0caaaGc caGLOaGaayzkaaWaaeWaaeaaceWG1bGbaiaadaqhaaWcbaGaamyAaa qaaiabgUcaRaaakiabgkHiTiqadwhagaGaamaaDaaaleaacaWGPbaa baGaeyOeI0caaaGccaGLOaGaayzkaaaaleqaaaaa@572B@

5.      The yield criterion and plastic flow rule require that σ ij m j ( u ˙ i + u ˙ i )=Y[[u]]/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiaad2gadaWgaaWcbaGaamOAaaqa baGccaGGOaGabmyDayaacaWaa0baaSqaaiaadMgaaeaacqGHRaWkaa GccqGHsislceWG1bGbaiaadaqhaaWcbaGaamyAaaqaaiabgkHiTaaa kiaacMcacqGH9aqpcaWGzbGaai4waiaacUfacaWG1bGaaiyxaiaac2 facaGGVaWaaOaaaeaacaaIZaaaleqaaaaa@4D38@  on any surfaces of velocity discontinuity. 

 

Kinematically admissible collapse mechanism: The kinematically admissible collapse mechanism is analogous to the kinematically admissible displacement field that was introduced to define the potential energy of an elastic solid.  By definition, a kinematically admissible collapse mechanism is any velocity field v satisfying v i / x i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcaWG2bWaaSbaaSqaaiaadM gaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaGc cqGH9aqpcaaIWaaaaa@3C45@  (i.e. v is volume preserving)

Like u, the virtual velocity v may have a finite set of discontinuities across surfaces S ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4uayaaja aaaa@36CE@  with normal m ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyBayaaja aaaa@36FC@  (these are not necessarily the discontinuity surfaces for the actual collapse mechanism).  We use

[[v]]=| v + v |= ( v i + v i )( v i + v i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaacU facaWG2bGaaiyxaiaac2facqGH9aqpdaabdaqaaiaahAhadaahaaWc beqaaiabgUcaRaaakiabgkHiTiaahAhadaahaaWcbeqaaiabgkHiTa aaaOGaay5bSlaawIa7aiabg2da9maakaaabaWaaeWaaeaacaWG2bWa a0baaSqaaiaadMgaaeaacqGHRaWkaaGccqGHsislcaWG2bWaa0baaS qaaiaadMgaaeaacqGHsislaaaakiaawIcacaGLPaaadaqadaqaaiaa dAhadaqhaaWcbaGaamyAaaqaaiabgUcaRaaakiabgkHiTiaadAhada qhaaWcbaGaamyAaaqaaiabgkHiTaaaaOGaayjkaiaawMcaaaWcbeaa aaa@56F3@

to denote the magnitude of the velocity discontinuity. We also define the virtual strain rate

ε ˙ ^ ij = 1 2 ( v i x j + v j x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai GbaKaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaa caaIXaaabaGaaGOmaaaacaGGOaWaaSaaaeaacqGHciITcaWG2bWaaS baaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWG QbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kaadAhadaWgaaWcba GaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqa aaaakiaacMcaaaa@4DD1@

 (note that ε ˙ ^ kk =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGagaqcamaaBaaaleaaca WGRbGaam4AaaqabaGccqGH9aqpcaaIWaaaaa@385B@  ) and the effective virtual plastic strain rate

ε ¯ ˙ ^ p = 2 ε ˙ ^ ij ε ˙ ^ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaqegaGagaqcamaaCaaale qabaGaamiCaaaakiabg2da9maakaaabaGaaGOmaiqbew7aLzaacyaa jaWaaSbaaSqaaiaadMgacaWGQbaabeaakiqbew7aLzaacyaajaWaaS baaSqaaiaadMgacaWGQbaabeaakiaac+cacaaIZaaaleqaaaaa@40B9@

 

Plastic Dissipation: Finally, we define the plastic dissipation associated with the virtual velocity field v as

Φ(v)= R Y ε ¯ ˙ ^ p dV+ S ^ Y 3 [[v]]dA R ρ 0 b i v i dA R t i * v i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiaahAhacaGGPaGaeyypa0Zaa8quaeaacaWGzbGafqyTduMbaeHb aiGbaKaadaahaaWcbeqaaiaadchaaaaabaGaamOuaaqab0Gaey4kIi pakiaadsgacaWGwbGaey4kaSYaa8quaeaadaWcaaqaaiaadMfaaeaa daGcaaqaaiaaiodaaSqabaaaaOGaai4waiaacUfacaWG2bGaaiyxai aac2facaWGKbGaamyqaaWcbaGabm4uayaajaaabeqdcqGHRiI8aOGa eyOeI0Yaa8quaeaacqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIb WaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaa kiaadsgacaWGbbaaleaacaWGsbaabeqdcqGHRiI8aOGaeyOeI0Yaa8 quaeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamODamaa BaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacqGHciITcaWGsb aabeqdcqGHRiI8aaaa@6956@

The terms in this expression have the following physical interpretation:

1.      The first integral represents the work dissipated in plastically straining the solid;

2.      The second integral represents the work dissipated due to plastic shearing on the velocity discontinuities;

3.      The third integral is the rate of mechanical work done by body forces

4.      The fourth integral is the rate of mechanical work done by the prescribed surface tractions.

 

 

 

6.2.2. The Principle of Minimum Plastic Dissipation

 

Let u ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWH1bGbaiaaaaa@33DD@  denote the actual velocity field that causes a rigid plastic solid to collapse under a prescribed loading.  Let v be any kinematically admissible collapse mechanism.  Let Φ(v) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGaaCODaiaacMcaaa a@36A8@  denote the plastic dissipation, as defined in the preceding section.  Then

1.      Φ(v)Φ( u ˙ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGaaCODaiaacMcacq GHLjYScqqHMoGrcaGGOaGabCyDayaacaGaaiykaaaa@3C48@

2.      Φ( u ˙ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGabCyDayaacaGaai ykaiabg2da9iaaicdaaaa@3870@

Thus, Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@3770@  is an absolute minimum for v=u MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCODaiabg2 da9iaahwhaaaa@38F9@  - in other words, the actual velocity field at collapse minimizes Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@3770@ .  Moreover, Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@3770@  is zero for the actual collapse mechanism.

 

Derivation: Begin by summarizing the equations governing the actual collapse solution. Let [ u ˙ i , ε ˙ ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiqadw hagaGaamaaBaaaleaacaWGPbaabeaakiaacYcacuaH1oqzgaGaamaa BaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaakiaac2faaaa@43ED@  denote the actual velocity, strain rate and stress in the solid at collapse. Let S ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHsislcqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq 2aaSbaaSqaaiaadMgacaWGQbaabeaakiaac+cacaaIZaaaaa@448B@  denote the deviatoric stress. The fields must satisfy governing equations and boundary conditions

 Strain-displacement relation ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaacIcacqGHciITcaWG 1bWaaSbaaSqaaiaadMgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaa WcbaGaamOAaaqabaGccqGHRaWkcqGHciITcaWG1bWaaSbaaSqaaiaa dQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqaba GccaGGPaGaai4laiaaikdaaaa@4EF5@

 Stress equilibrium σ ij /x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhacaWLa8+aaSbaaSqa aiaadMgaaeqaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaO GaamOyamaaBaaaleaacaWGQbaabeaakiabg2da9iaaicdaaaa@4532@

 Plastic flow rule and yield criterion

ε ˙ ij ={ ε ¯ ˙ p 3 2 S ij Y 3 2 S ij S ij =Y 0 3 2 S ij S ij <Y       MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0Zaaiqaaeaafaqa beGabaaabaGafqyTduMbaeHbaiaadaahaaWcbeqaaiaadchaaaGcda WcaaqaaiaaiodaaeaacaaIYaaaamaalaaabaGaam4uamaaBaaaleaa caWGPbGaamOAaaqabaaakeaacaWGzbaaaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaakaaabaWaaSaaaeaacaaI ZaaabaGaaGOmaaaacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaaki aadofadaWgaaWcbaGaamyAaiaadQgaaeqaaaqabaGccqGH9aqpcaWG zbaabaGaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7daGcaaqaamaalaaabaGaaG4maaqaaiaaikdaaaGaam4u amaaBaaaleaacaWGPbGaamOAaaqabaGccaWGtbWaaSbaaSqaaiaadM gacaWGQbaabeaaaeqaaOGaeyipaWJaamywaaaaaiaawUhaaiaabcca caqGGaGaaeiiaiaabccacaqGGaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVdaa@D29E@

On velocity discontinuities, these conditions require that σ ij m j ( u ˙ i + u ˙ i )=Y[[ u ˙ ]]/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiaad2gadaWgaaWcbaGaamOAaaqa baGccaGGOaGabmyDayaacaWaa0baaSqaaiaadMgaaeaacqGHRaWkaa GccqGHsislceWG1bGbaiaadaqhaaWcbaGaamyAaaqaaiabgkHiTaaa kiaacMcacqGH9aqpcaWGzbGaai4waiaacUfaceWG1bGbaiaacaGGDb Gaaiyxaiaac+cadaGcaaqaaiaaiodaaSqabaaaaa@4D41@

 Boundary conditions

σ ij n i = t j * x i R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiaad6gadaWgaaWcbaGaamyAaaqa baGccqGH9aqpcaWG0bWaa0baaSqaaiaadQgaaeaacaGGQaaaaOGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyicI4 SaeyOaIyRaamOuaaaa@5798@

 

We start by showing that Φ( u ˙ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGabCyDayaacaGaai ykaiabg2da9iaaicdaaaa@3860@

1.      By definition

Φ( u ˙ )= R Y ε ¯ ˙ p dV+ S Y 3 [[ u ˙ ]]dA R ρ 0 b i u ˙ i dA R t i * u ˙ i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiqahwhagaGaaiaacMcacqGH9aqpdaWdrbqaaiaadMfacuaH1oqz gaqegaGaamaaCaaaleqabaGaamiCaaaaaeaacaWGsbaabeqdcqGHRi I8aOGaamizaiaadAfacqGHRaWkdaWdrbqaamaalaaabaGaamywaaqa amaakaaabaGaaG4maaWcbeaaaaGccaGGBbGaai4waiqadwhagaGaai aac2facaGGDbGaamizaiaadgeaaSqaaiaadofaaeqaniabgUIiYdGc cqGHsisldaWdrbqaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadk gadaWgaaWcbaGaamyAaaqabaGcceWG1bGbaiaadaWgaaWcbaGaamyA aaqabaGccaWGKbGaamyqaaWcbaGaamOuaaqab0Gaey4kIipakiabgk HiTmaapefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaakiqa dwhagaGaamaaBaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacq GHciITcaWGsbaabeqdcqGHRiI8aaaa@6957@

2.      Note that, using (i) the flow rule, (ii) the condition that S ii =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGPb aabeaakiabg2da9iaaicdaaaa@3770@  and (iii) the yield criterion

σ ij ε ˙ ij p = σ ij ε ¯ ˙ p 3 2 S ij Y =( S ij + σ kk δ ij ) ε ¯ ˙ p 3 2 S ij Y = ε ¯ ˙ p 3 2 S ij S ij Y =Y ε ¯ ˙ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGafqyTduMbaiaadaqhaaWcbaGaamyAaiaadQgaaeaacaWG WbaaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaki qbew7aLzaaryaacaWaaWbaaSqabeaacaWGWbaaaOWaaSaaaeaacaaI ZaaabaGaaGOmaaaadaWcaaqaaiaadofadaWgaaWcbaGaamyAaiaadQ gaaeqaaaGcbaGaamywaaaacqGH9aqpcaGGOaGaam4uamaaBaaaleaa caWGPbGaamOAaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiaa cMcacuaH1oqzgaqegaGaamaaCaaaleqabaGaamiCaaaakmaalaaaba GaaG4maaqaaiaaikdaaaWaaSaaaeaacaWGtbWaaSbaaSqaaiaadMga caWGQbaabeaaaOqaaiaadMfaaaGaeyypa0JafqyTduMbaeHbaiaada ahaaWcbeqaaiaadchaaaGcdaWcaaqaaiaaiodaaeaacaaIYaaaamaa laaabaGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaGccaWGtbWaaS baaSqaaiaadMgacaWGQbaabeaaaOqaaiaadMfaaaGaeyypa0Jaamyw aiqbew7aLzaaryaacaWaaWbaaSqabeaacaWGWbaaaaaa@6FAB@

3.      Note that σ ij ε ˙ ij = σ ij ( u i / x j + u j / x i )/2= σ ij u j / x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGafqyTduMbaiaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaacIcacq GHciITcaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaai4laiabgkGi2kaa dIhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcqGHciITcaWG1bWaaS baaSqaaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGa amyAaaqabaGccaGGPaGaai4laiaaikdacqGH9aqpcqaHdpWCdaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyOaIyRaamyDamaaBaaaleaacaWG Qbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaa aa@5EF4@  from the symmetry of σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@ . Hence

R Y ε ¯ ˙ p dV= R σ ij ε ˙ ij dV = R σ ij u ˙ j / x i dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGafqyTduMbaeHbaiaadaahaaWcbeqaaiaadchaaaaabaGaamOu aaqab0Gaey4kIipakiaadsgacaWGwbGaeyypa0Zaa8quaeaacqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGafqyTduMbaiaadaWgaaWc baGaamyAaiaadQgaaeqaaOGaamizaiaadAfaaSqaaiaadkfaaeqani abgUIiYdGccqGH9aqpdaWdrbqaaiabeo8aZnaaBaaaleaacaWGPbGa amOAaaqabaGccqGHciITceWG1bGbaiaadaWgaaWcbaGaamOAaaqaba GccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiaadsga caWGwbaaleaacaWGsbaabeqdcqGHRiI8aaaa@5EBF@

4.      Note that σ ij u j / x i =( σ ij u j )/ x i ( σ ij / x i ) u ˙ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyOaIyRaamyDamaaBaaaleaacaWGQbaabeaakiaac+ca cqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeyOaIy Raaiikaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWG1bWa aSbaaSqaaiaadQgaaeqaaOGaaiykaiaac+cacqGHciITcaWG4bWaaS baaSqaaiaadMgaaeqaaOGaeyOeI0YaaeWaaeaacqGHciITcqaHdpWC daWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhada WgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaceWG1bGbaiaadaWg aaWcbaGaamOAaaqabaaaaa@5A59@ .  Substitute into the expression for Φ(u) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGaaCyDaiaacMcaaa a@3697@ , combine the two volume integrals and recall (equilibrium) that σ ij / x i + ρ 0 b j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamyA aaqabaGccqGHRaWkcqaHbpGCdaWgaaWcbaGaaGimaaqabaGccaWGIb WaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaaGimaaaa@439A@  to see that

Φ( u ˙ )= R ( σ ij u ˙ j )/ x i dV+ S Y 3 [[ u ˙ ]]dA R t i * u ˙ i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiqahwhagaGaaiaacMcacqGH9aqpdaWdrbqaaiabgkGi2oaabmaa baGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqadwhagaGaam aaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaiaac+cacqGHciIT caWG4bWaaSbaaSqaaiaadMgaaeqaaaqaaiaadkfaaeqaniabgUIiYd GccaWGKbGaamOvaiabgUcaRmaapefabaWaaSaaaeaacaWGzbaabaWa aOaaaeaacaaIZaaaleqaaaaakiaacUfacaGGBbGabmyDayaacaGaai yxaiaac2facaWGKbGaamyqaaWcbaGaam4uaaqab0Gaey4kIipakiab gkHiTmaapefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaaki qadwhagaGaamaaBaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaa cqGHciITcaWGsbaabeqdcqGHRiI8aaaa@6604@

5.      Apply the divergence theorem to the volume integral in this result.  When doing so, note that we must include contributions from the velocity discontinuity across S as follows

Φ( u ˙ )= R σ ij u ˙ j n i dA+ S σ ij u ˙ j + n i + dA+ S σ ij u ˙ j n i dA+ S Y 3 [[ u ˙ ]]dA R t i * u ˙ i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiqahwhagaGaaiaacMcacqGH9aqpdaWdrbqaaiabeo8aZnaaBaaa leaacaWGPbGaamOAaaqabaGcceWG1bGbaiaadaWgaaWcbaGaamOAaa qabaGccaWGUbWaaSbaaSqaaiaadMgaaeqaaaqaaiabgkGi2kaadkfa aeqaniabgUIiYdGccaWGKbGaamyqaiabgUcaRmaapefabaGaeq4Wdm 3aaSbaaSqaaiaadMgacaWGQbaabeaakiqadwhagaGaamaaDaaaleaa caWGQbaabaGaey4kaScaaOGaamOBamaaDaaaleaacaWGPbaabaGaey 4kaScaaaqaaiaadofaaeqaniabgUIiYdGccaWGKbGaamyqaiabgUca RmaapefabaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqadw hagaGaamaaDaaaleaacaWGQbaabaGaeyOeI0caaOGaamOBamaaDaaa leaacaWGPbaabaGaeyOeI0caaaqaaiaadofaaeqaniabgUIiYdGcca WGKbGaamyqaiabgUcaRmaapefabaWaaSaaaeaacaWGzbaabaWaaOaa aeaacaaIZaaaleqaaaaakiaacUfacaGGBbGabmyDayaacaGaaiyxai aac2facaWGKbGaamyqaaWcbaGaam4uaaqab0Gaey4kIipakiabgkHi TmaapefabaGaamiDamaaDaaaleaacaWGPbaabaGaaiOkaaaakiqadw hagaGaamaaBaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacqGH ciITcaWGsbaabeqdcqGHRiI8aaaa@815F@

6.      Finally, recall that σ ij n i = t j * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaadsha daqhaaWcbaGaamOAaaqaaiaacQcaaaaaaa@3C7C@  on the boundary, and note that the outward normals to the solids adjacent to S are related to m  by n i + = m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaa0baaSqaaiaadMgaaeaacq GHRaWkaaGccqGH9aqpcqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqa aaaa@39BF@   n i = m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaa0baaSqaaiaadMgaaeaacq GHsislaaGccqGH9aqpcaWGTbWaaSbaaSqaaiaadMgaaeqaaaaa@38DD@ .  Thus

Φ( u ˙ )= S σ ij ( u ˙ j + u ˙ j ) m i dA+ S Y 3 [[ u ˙ ]]dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiqahwhagaGaaiaacMcacqGH9aqpcqGHsisldaWdrbqaaiabeo8a ZnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGOaGabmyDayaacaWaa0 baaSqaaiaadQgaaeaacqGHRaWkaaGccqGHsislceWG1bGbaiaadaqh aaWcbaGaamOAaaqaaiabgkHiTaaakiaacMcacaWGTbWaaSbaaSqaai aadMgaaeqaaaqaaiaadofaaeqaniabgUIiYdGccaWGKbGaamyqaiab gUcaRmaapefabaWaaSaaaeaacaWGzbaabaWaaOaaaeaacaaIZaaale qaaaaakiaacUfacaGGBbGabmyDayaacaGaaiyxaiaac2facaWGKbGa amyqaaWcbaGaam4uaaqab0Gaey4kIipaaaa@5BE9@

Since σ ij m j ( u ˙ i + u ˙ i )=Y[[u]]/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiaad2gadaWgaaWcbaGaamOAaaqa baGccaGGOaGabmyDayaacaWaa0baaSqaaiaadMgaaeaacqGHRaWkaa GccqGHsislceWG1bGbaiaadaqhaaWcbaGaamyAaaqaaiabgkHiTaaa kiaacMcacqGH9aqpcaWGzbGaai4waiaacUfacaWG1bGaaiyxaiaac2 facaGGVaWaaOaaaeaacaaIZaaaleqaaaaa@4D38@ , we find that Φ(u)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGaaCyDaiaacMcacq GH9aqpcaaIWaaaaa@3857@  as required.

 

Next, we show that Φ(v)0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHMoGrcaGGOaGaaCODaiaacMcacq GHLjYScaaIWaaaaa@3918@ .  To this end,

1.      Let v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgaaeqaaa aa@34EB@  be a kinematically admissible velocity field as defined in the preceding section, with strain rate

ε ˙ ^ ij = 1 2 ( v i x j + v j x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai GbaKaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaa caaIXaaabaGaaGOmaaaacaGGOaWaaSaaaeaacqGHciITcaWG2bWaaS baaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWG QbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kaadAhadaWgaaWcba GaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqa aaaakiaacMcaaaa@4DD1@

2.      Let S ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGtbGbaKaadaWgaaWcbaGaamyAai aadQgaaeqaaaaa@35B7@  be the stress necessary to drive the kinematically admissible collapse mechanism, which must satisfy the plastic flow rule and the yield criterion

ε ˙ ^ ij = ε ¯ ˙ ^ 3 2 S ^ ij Y 3 S ^ ij S ^ ij /2 =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGagaqcamaaBaaaleaaca WGPbGaamOAaaqabaGccqGH9aqpcuaH1oqzgaqegaGagaqcamaalaaa baGaaG4maaqaaiaaikdaaaWaaSaaaeaaceWGtbGbaKaadaWgaaWcba GaamyAaiaadQgaaeqaaaGcbaGaamywaaaacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVp aakaaabaGaaG4maiqadofagaqcamaaBaaaleaacaWGPbGaamOAaaqa baGcceWGtbGbaKaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4lai aaikdaaSqabaGccqGH9aqpcaWGzbaaaa@5A0A@

3.      Recall that the plastic strains and stresses associated with the kinematically admissible field must satisfy the Principle of Maximum Plastic Resistance (Section 3.7.10), which in the present context implies that

( σ ^ ij σ ij ) ε ˙ ^ ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiqbeo8aZzaajaWaaSbaaS qaaiaadMgacaWGQbaabeaakiabgkHiTiabeo8aZnaaBaaaleaacaWG PbGaamOAaaqabaaakiaawIcacaGLPaaacuaH1oqzgaGagaqcamaaBa aaleaacaWGPbGaamOAaaqabaGccqGHLjYScaaIWaaaaa@434A@

To see this, note that σ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@36B2@  is the stress required to cause the plastic strain rate  ε ˙ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGagaqcamaaBaaaleaaca WGPbGaamOAaaqabaaaaa@369E@ , while the actual stress state at collapse σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@36A2@  must satisfy 3 S ij S ij /2 Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiaaiodacaWGtbWaaSbaaS qaaiaadMgacaWGQbaabeaakiaadofadaWgaaWcbaGaamyAaiaadQga aeqaaOGaai4laiaaikdaaSqabaGccqGHKjYOcaWGzbaaaa@3D80@ .

4.      Note that σ ^ ij ε ˙ ^ ij =( S ^ ij + σ ^ kk δ ij )d ε ¯ ˙ ^ p 3 S ^ ij /2Y=Yd ε ¯ ˙ ^ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaGccuaH1oqzgaGagaqcamaaBaaaleaacaWGPbGaamOA aaqabaGccqGH9aqpdaqadaqaaiqadofagaqcamaaBaaaleaacaWGPb GaamOAaaqabaGccqGHRaWkcuaHdpWCgaqcamaaBaaaleaacaWGRbGa am4AaaqabaGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGcca GLOaGaayzkaaGaamizaiqbew7aLzaaryaacyaajaWaaWbaaSqabeaa caWGWbaaaOGaaG4maiqadofagaqcamaaBaaaleaacaWGPbGaamOAaa qabaGccaGGVaGaaGOmaiaadMfacqGH9aqpcaWGzbGaamizaiqbew7a LzaaryaacyaajaWaaWbaaSqabeaacaWGWbaaaaaa@5840@ .  Substituting into the principle of maximum plastic resistance and integrating over the volume of the solid shows that

R Y ε ¯ ˙ ^ p dV R σ ij ε ˙ ^ ij dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiaadMfacuaH1oqzgaqega GagaqcamaaCaaaleqabaGaamiCaaaakiaadsgacaWGwbaaleaacaWG sbaabeqdcqGHRiI8aOGaeyOeI0Yaa8quaeaacqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaOGafqyTduMbaiGbaKaadaWgaaWcbaGaamyA aiaadQgaaeqaaOGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYd GccqGHLjYScaaIWaaaaa@4B99@

5.      Next, note that

σ ij ε ˙ ^ ij = σ ij ( v i / x j + v j / x i )/2= σ ij v j / x i =( σ ij v j )/ x i ( σ ij / x i ) v j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGafqyTduMbaiGbaKaadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaabm aabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaakiaac+cacqGH ciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaey4kaSIaeyOaIyRaam ODamaaBaaaleaacaWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikdacqGH9a qpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOaIyRaamOD amaaBaaaleaacaWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaS qaaiaadMgaaeqaaOGaeyypa0JaeyOaIy7aaeWaaeaacqaHdpWCdaWg aaWcbaGaamyAaiaadQgaaeqaaOGaamODamaaBaaaleaacaWGQbaabe aaaOGaayjkaiaawMcaaiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaa dMgaaeqaaOGaeyOeI0YaaeWaaeaacqGHciITcqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGa amyAaaqabaaakiaawIcacaGLPaaacaWG2bWaaSbaaSqaaiaadQgaae qaaaaa@7B67@

6.      The equilibrium equation shows that σ ij / x i = ρ 0 b j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamyA aaqabaGccqGH9aqpcqGHsislcqaHbpGCdaWgaaWcbaGaaGimaaqaba GccaWGIbWaaSbaaSqaaiaadQgaaeqaaaaa@42E1@ .  Substituting this into the result of (5) and then substituting into the result of (4) shows that

R Y ε ¯ ˙ ^ p dV R ( σ ij v j )/ x i dV R ρ 0 b j v j dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiaadMfacuaH1oqzgaqega GagaqcamaaCaaaleqabaGaamiCaaaakiaadsgacaWGwbaaleaacaWG sbaabeqdcqGHRiI8aOGaeyOeI0Yaa8quaeaacqGHciITdaqadaqaai abeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWG2bWaaSbaaSqa aiaadQgaaeqaaaGccaGLOaGaayzkaaGaai4laiabgkGi2kaadIhada WgaaWcbaGaamyAaaqabaGccaWGKbGaamOvaaWcbaGaamOuaaqab0Ga ey4kIipakiabgkHiTmaapefabaGaeqyWdi3aaSbaaSqaaiaaicdaae qaaOGaamOyamaaBaaaleaacaWGQbaabeaakiaadAhadaWgaaWcbaGa amOAaaqabaGccaWGKbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipaki abgwMiZkaaicdaaaa@5DC8@

7.      Apply the divergence theorem to the second integral. When doing so, note that we must include contributions from the velocity discontinuity across S ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadofagaqcaaaa@3150@  as follows

R Y ε ¯ ˙ ^ p dV S ^ σ ij v j n i + dA S ^ σ ij v j n i dA R σ ij v j n i dA R ρ 0 b j v j dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiaadMfacuaH1oqzgaqega GagaqcamaaCaaaleqabaGaamiCaaaakiaadsgacaWGwbaaleaacaWG sbaabeqdcqGHRiI8aOGaeyOeI0Yaa8quaeaacqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaOGaamODamaaBaaaleaacaWGQbaabeaakiaa d6gadaqhaaWcbaGaamyAaaqaaiabgUcaRaaakiaadsgacaWGbbaale aaceWGtbGbaKaaaeqaniabgUIiYdGccqGHsisldaWdrbqaaiabeo8a ZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWG2bWaaSbaaSqaaiaadQ gaaeqaaOGaamOBamaaDaaaleaacaWGPbaabaGaeyOeI0caaOGaamiz aiaadgeaaSqaaiqadofagaqcaaqab0Gaey4kIipakiabgkHiTmaape fabaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaadAhadaWg aaWcbaGaamOAaaqabaGccaWGUbWaaSbaaSqaaiaadMgaaeqaaOGaam izaiaadgeaaSqaaiabgkGi2kaadkfaaeqaniabgUIiYdGccqGHsisl daWdrbqaaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaadkgadaWgaa WcbaGaamOAaaqabaGccaWG2bWaaSbaaSqaaiaadQgaaeqaaOGaamiz aiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHLjYScaaIWaaaaa@77A1@

8.      Recall that σ ij n i = t j * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaamOBamaaBaaaleaacaWGPbaabeaakiabg2da9iaadsha daqhaaWcbaGaamOAaaqaaiaacQcaaaaaaa@3C7C@  on the boundary, and note that the outward normals to the solids adjacent to S are related to m  by n i + = m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaa0baaSqaaiaadMgaaeaacq GHRaWkaaGccqGH9aqpcqGHsislcaWGTbWaaSbaaSqaaiaadMgaaeqa aaaa@39BF@   n i = m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaa0baaSqaaiaadMgaaeaacq GHsislaaGccqGH9aqpcaWGTbWaaSbaaSqaaiaadMgaaeqaaaaa@38DD@ .  Thus

R Y ε ¯ ˙ ^ p dV + S ^ σ ij m ^ j ( v i + v i )dA R b i v i dV 2 R t i * v i dA 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGafqyTduMbaeHbaiGbaKaadaahaaWcbeqaaiaadchaaaGccaWG KbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipakiabgUcaRmaapefaba Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqad2gagaqcamaa BaaaleaacaWGQbaabeaakiaacIcacaWG2bWaa0baaSqaaiaadMgaae aacqGHRaWkaaGccqGHsislcaWG2bWaa0baaSqaaiaadMgaaeaacqGH sislaaGccaGGPaGaamizaiaadgeacqGHsisldaWdrbqaaiaadkgada WgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGa amizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHsisldaWdrb qaaiaadshadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaWG2bWaaSba aSqaaiaadMgaaeqaaOGaamizaiaadgeaaSqaaiabgkGi2oaaBaaame aacaaIYaaabeaaliaadkfaaeqaniabgUIiYdaaleaaceWGtbGbaKaa aeqaniabgUIiYdGccqGHLjYScaaIWaaaaa@6D73@

9.      Finally, note that on S ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4uayaaja aaaa@36DE@

σ ij m ^ j ( v i + v i )Y[[v]]/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaakiqad2gagaqcamaaBaaaleaacaWG QbaabeaakiaacIcacaWG2bWaa0baaSqaaiaadMgaaeaacqGHRaWkaa GccqGHsislcaWG2bWaa0baaSqaaiaadMgaaeaacqGHsislaaGccaGG PaGaeyizImQaamywaiaacUfacaGGBbGaamODaiaac2facaGGDbGaai 4lamaakaaabaGaaG4maaWcbeaaaaa@4DE8@

since the shear stress acting on any plane in the solid cannot exceed Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGzbGaai4lamaakaaabaGaaG4maa Wcbeaaaaa@352F@ .  Thus

R Y ε ¯ ˙ ^ p dV + S ^ Y[[v]]/ 3 dA R b i v i dV 2 R t i * v i dA 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGafqyTduMbaeHbaiGbaKaadaahaaWcbeqaaiaadchaaaGccaWG KbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipakiabgUcaRmaapefaba GaamywaiaacUfacaGGBbGaamODaiaac2facaGGDbGaai4lamaakaaa baGaaG4maaWcbeaakiaadsgacaWGbbGaeyOeI0Yaa8quaeaacaWGIb WaaSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaa kiaadsgacaWGwbaaleaacaWGsbaabeqdcqGHRiI8aOGaeyOeI0Yaa8 quaeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamODamaa BaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacqGHciITdaWgaa adbaGaaGOmaaqabaWccaWGsbaabeqdcqGHRiI8aaWcbaGabm4uayaa jaaabeqdcqGHRiI8aOGaeyyzImRaaGimaaaa@660F@

 proving that Φ(v)0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai ikaiaahAhacaGGPaGaeyyzImRaaGimaaaa@3C38@  as required.

 

 

6.2.3 The Upper Bound Plastic Collapse Theorem

 

Consider a rigid plastic solid, subjected to some distribution of tractions t i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaacaWGPbaabaGaaiOkaaaaaaa@39BF@  and body forces b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaaaaa@38FE@ .  We will attempt to estimate the factor β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  by which the loading can be increased before the solid collapses ( β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  is effectively the factor of safety).  We suppose that the solid will collapse for loading β t i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iDamaaDaaaleaacaWGPbaabaGaaiOkaaaaaaa@3B60@ , β b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam OyamaaBaaaleaacaWGPbaabeaaaaa@3A9F@ .

 

To estimate β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@ , we guess the mechanism of collapse.  The collapse mechanism will be an admissible velocity field, which may have a finite set of discontinuities across surfaces S ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4uayaaja aaaa@36CE@  with normal m ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyBayaaja aaaa@36EC@ , as discussed in 6.2.1.

 

The principle of minimum plastic dissipation then states that

R Y ε ¯ ˙ ^ p dV + S ^ σ ij m ^ j ( v i + v i )dA R β b i v i dV R β t i * v i dA 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGafqyTduMbaeHbaiGbaKaadaahaaWcbeqaaiaadchaaaGccaWG KbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipakiabgUcaRmaapefaba Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiqad2gagaqcamaa BaaaleaacaWGQbaabeaakiaacIcacaWG2bWaa0baaSqaaiaadMgaae aacqGHRaWkaaGccqGHsislcaWG2bWaa0baaSqaaiaadMgaaeaacqGH sislaaGccaGGPaGaamizaiaadgeacqGHsisldaWdrbqaaiabek7aIj aadkgadaWgaaWcbaGaamyAaaqabaGccaWG2bWaaSbaaSqaaiaadMga aeqaaOGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHsi sldaWdrbqaaiabek7aIjaadshadaqhaaWcbaGaamyAaaqaaiaacQca aaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaadgeaaSqaai abgkGi2kaadkfaaeqaniabgUIiYdaaleaaceWGtbGbaKaaaeqaniab gUIiYdGccqGHLjYScaaIWaaaaa@6FC1@

for any collapse mechanism, with equality for the true mechanism of collapse. Therefore

β R Y ε ¯ ˙ ^ p dV+ S ^ Y[[v]]/ 3 dA R b i v i dA + R t i * v i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey izIm6aaSaaaeaadaWdrbqaaiaadMfacuaH1oqzgaqegaGagaqcamaa CaaaleqabaGaamiCaaaaaeaacaWGsbaabeqdcqGHRiI8aOGaamizai aadAfacqGHRaWkdaWdrbqaaiaadMfacaGGBbGaai4waiaadAhacaGG DbGaaiyxaiaac+cadaGcaaqaaiaaiodaaSqabaGccaWGKbGaamyqaa WcbaGabm4uayaajaaabeqdcqGHRiI8aaGcbaWaa8quaeaacaWGIbWa aSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaaki aadsgacaWGbbaaleaacaWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8qu aeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamODamaaBa aaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacqGHciITcaWGsbaa beqdcqGHRiI8aaaaaaa@64DF@

Expressed in words, this equation states that we can obtain an upper bound to the collapse loads by postulating a collapse mechanism, and computing the ratio of the plastic dissipation associated with this mechanism to the work done by the applied loads. 

 

So, we can choose any collapse mechanism, and use it to estimate a safety factor.  The actual safety factor is likely to be lower than our estimate (it will be equal if we guessed right).  This method is evidently inherently unsafe, since it overestimates the safety factor MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but it is usually possible guess the collapse mechanism quite accurately, and so with practice you can get excellent estimates.

 

 

6.2.4 Examples of applications of the upper bound theorem

 

Example 1: collapse load for a uniaxial bar. We will illustrate the bounding theorems using a few examples.  First, we will compute  bounds to the collapse load for a uniaxial bar.  Assume the bar has unit out of plane thickness, for simplicity.

 

To get an upper bound, we guess a collapse mechanism as shown below.  The top and bottom half of the bar slide past each other as rigid blocks, as shown, with a velocity discontinuity across the line shown in red.

 

The upper bound theorem gives

β R Y ε ¯ ˙ ^ p dV+ S ^ Y[[v]]/ 3 dA R b i v i dA + R t i * v i dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey izIm6aaSaaaeaadaWdrbqaaiaadMfacuaH1oqzgaqegaGagaqcamaa CaaaleqabaGaamiCaaaaaeaacaWGsbaabeqdcqGHRiI8aOGaamizai aadAfacqGHRaWkdaWdrbqaaiaadMfacaGGBbGaai4waiaadAhacaGG DbGaaiyxaiaac+cadaGcaaqaaiaaiodaaSqabaGccaWGKbGaamyqaa WcbaGabm4uayaajaaabeqdcqGHRiI8aaGcbaWaa8quaeaacaWGIbWa aSbaaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaaki aadsgacaWGbbaaleaacaWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8qu aeaacaWG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaamODamaaBa aaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacqGHciITcaWGsbaa beqdcqGHRiI8aaaaaaa@64DF@

In this problem the strain rate vanishes, since we assume the two halves of the bar are rigid.  The plastic dissipation is

S ^ Y[[v]]/ 3 dA =Y( h ˙ /sinθ)(L/cosθ)/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGaai4waiaacUfacaWG2bGaaiyxaiaac2facaGGVaWaaOaaaeaa caaIZaaaleqaaOGaamizaiaadgeaaSqaaiqadofagaqcaaqab0Gaey 4kIipakiabg2da9iaadMfacaGGOaGabmiAayaacaGaai4laiGacoha caGGPbGaaiOBaiabeI7aXjaacMcacaGGOaGaamitaiaac+caciGGJb Gaai4BaiaacohacqaH4oqCcaGGPaGaai4lamaakaaabaGaaG4maaWc beaaaaa@553B@

The body force vanishes, and

R t i * u ˙ i dA=pL h ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGabmyDayaacaWaaSba aSqaaiaadMgaaeqaaaqaaiabgkGi2kaadkfaaeqaniabgUIiYdGcca WGKbGaamyqaiabg2da9iaadchacaWGmbGabmiAayaacaaaaa@45DD@

where h ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiAayaaca aaaa@36EC@  is the vertical component of the velocity of the top block.  Thus

βp2Y/( 3 sin2θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iCaiabgsMiJkaaikdacaWGzbGaai4laiaacIcadaGcaaqaaiaaioda aSqabaGcciGGZbGaaiyAaiaac6gacaaIYaGaeqiUdeNaaiykaaaa@451A@

The best upper bound occurs for θ=π/4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaeqiWdaNaai4laiaaisdaaaa@3BD0@ , giving βp2Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iCaiabgsMiJkaaikdacaWGzbGaai4lamaakaaabaGaaG4maaWcbeaa aaa@3D56@  for the collapse load.

 

 

 

Example 2: Collapse load for a bar containing a hole. For a slightly more interesting problem, consider the effect of inserting a hole with radius a in the center of the column.  This time we apply a force to the top of the column, rather than specify the traction distribution in detail.  We will accept any solution that has traction acting on the top surface that is statically equivalent to the applied force.

 

A possible collapse mechanism is shown.  The plastic dissipation is

S ^ Y[[v]]/ 3 dA =( Y/ 3 )( h ˙ /sinθ)(L2a)/cosθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGaai4waiaacUfacaWG2bGaaiyxaiaac2facaGGVaWaaOaaaeaa caaIZaaaleqaaOGaamizaiaadgeaaSqaaiqadofagaqcaaqab0Gaey 4kIipakiabg2da9maabmaabaGaamywaiaac+cadaGcaaqaaiaaioda aSqabaaakiaawIcacaGLPaaacaGGOaGabmiAayaacaGaai4laiGaco hacaGGPbGaaiOBaiabeI7aXjaacMcacaGGOaGaamitaiabgkHiTiaa ikdacaWGHbGaaiykaiaac+caciGGJbGaai4BaiaacohacqaH4oqCaa a@595D@

The rate of work done by applied loading is

R t i * u ˙ i dA=P h ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WG0bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGabmyDayaacaWaaSba aSqaaiaadMgaaeqaaaqaaiabgkGi2kaadkfaaeqaniabgUIiYdGcca WGKbGaamyqaiabg2da9iaadcfaceWGObGbaiaaaaa@44EC@

Our upper bound follows as

βP2Y(L2a)/( 3 sin2θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iuaiabgsMiJkaaikdacaWGzbGaaiikaiaadYeacqGHsislcaaIYaGa amyyaiaacMcacaGGVaGaaiikamaakaaabaGaaG4maaWcbeaakiGaco hacaGGPbGaaiOBaiaaikdacqaH4oqCcaGGPaaaaa@489C@

and the best upper bound solution is βP2Y(L2a)/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iuaiabgsMiJkaaikdacaWGzbGaaiikaiaadYeacqGHsislcaaIYaGa amyyaiaacMcacaGGVaWaaOaaaeaacaaIZaaaleqaaaaa@41EF@

 

Example 3: Force required to indent a rigid platic surface. For our next example, we attempt to find upper and lower bounds to the force required to push a flat plane punch into a rigid plastic solid.  This problem is interesting because we have an exact slip-line field solution, so we can assess the accuracy of the bounding calculations.

 

A possible collapse mechanism is shown above.  In each semicircular region we assume a constant circumferential velocity v θ = h ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacqaH4oqCaeqaaOGaeyypa0JabmiAayaacaaaaa@3BE0@ .  To compute the plastic dissipation in one of the regions, adopt a cylindrical-polar coordinate system with origin at the edge of the contact.  The strain distribution follows as

ε ˙ rr = ε ˙ θθ =0 ε ˙ rθ = 1 2 v θ r = h ˙ 2r ε ¯ ˙ ^ p = 2 3 ε ˙ ^ ij ε ˙ ^ ij = h ˙ r 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaH1o qzgaGaamaaBaaaleaacaWGYbGaamOCaaqabaGccqGH9aqpcuaH1oqz gaGaamaaBaaaleaacqaH4oqCcqaH4oqCaeqaaOGaeyypa0JaaGimai aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlqbew7aLzaacaWaaSbaaSqaaiaadkhacqaH4oqCaeqaaOGaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiaadAhadaWg aaWcbaGaeqiUdehabeaaaOqaaiaadkhaaaGaeyypa0ZaaSaaaeaace WGObGbaiaaaeaacaaIYaGaamOCaaaaaeaacqGHshI3cuaH1oqzgaqe gaGagaqcamaaCaaaleqabaGaamiCaaaakiabg2da9maakaaabaWaaS aaaeaacaaIYaaabaGaaG4maaaacuaH1oqzgaGagaqcamaaBaaaleaa caWGPbGaamOAaaqabaGccuaH1oqzgaGagaqcamaaBaaaleaacaWGPb GaamOAaaqabaaabeaakiabg2da9maalaaabaGabmiAayaacaaabaGa amOCamaakaaabaGaaG4maaWcbeaaaaaaaaa@731D@

Thus the plastic dissipation is

R Y ε ¯ ˙ ^ p dV+ S ^ Y[[v]]/ 3 dA =2{ 0 π 0 a/2 Y h ˙ r 3 rdrdθ+ Y 2 3 h ˙ πa }= 2π 3 h ˙ Ya MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaaca WGzbGafqyTduMbaeHbaiGbaKaadaahaaWcbeqaaiaadchaaaaabaGa amOuaaqab0Gaey4kIipakiaadsgacaWGwbGaey4kaSYaa8quaeaaca WGzbGaai4waiaacUfacaWG2bGaaiyxaiaac2facaGGVaWaaOaaaeaa caaIZaaaleqaaOGaamizaiaadgeaaSqaaiqadofagaqcaaqab0Gaey 4kIipakiabg2da9iaaikdadaGadaqaamaapehabaWaa8qCaeaacaWG zbWaaSaaaeaaceWGObGbaiaaaeaacaWGYbWaaOaaaeaacaaIZaaale qaaaaaaeaacaaIWaaabaGaamyyaiaac+cacaaIYaaaniabgUIiYdaa leaacaaIWaaabaGaeqiWdahaniabgUIiYdGccaWGYbGaamizaiaadk hacaWGKbGaeqiUdeNaey4kaSYaaSaaaeaacaWGzbaabaGaaGOmamaa kaaabaGaaG4maaWcbeaaaaGcceWGObGbaiaacqaHapaCcaWGHbaaca GL7bGaayzFaaGaeyypa0ZaaSaaaeaacaaIYaGaeqiWdahabaWaaOaa aeaacaaIZaaaleqaaaaakiqadIgagaGaaiaadMfacaWGHbaaaa@7217@

(note that there’s a velocity discontinuity at r=a). The work done by applied loading is just h ˙ P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiAayaaca Gaamiuaaaa@37B1@  giving the upper bound

βP2πYa/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iuaiabgsMiJkaaikdacqaHapaCcaWGzbGaamyyaiaac+cadaGcaaqa aiaaiodaaSqabaaaaa@3FD9@

This should be compared to the exact slip-line field solution

βP=(π+2)Ya/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iuaiabg2da9iaacIcacqaHapaCcqGHRaWkcaaIYaGaaiykaiaadMfa caWGHbGaai4lamaakaaabaGaaG4maaWcbeaaaaa@4165@

computed in section 6.1.  The error is 17% - close enough for government work.

 

 

 

Example 4: Orthogonal metal cutting. The picture shows a simple model of machining.  The objective is to determine the horizontal force P acting on the tool (or workpiece) in terms of the depth of cut h, the tool rake angle α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37D8@  and the shear yield stress of the material Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGzbaaaa@33B3@

 

To perform the calculation, we adopt a reference frame that moves with the tool.  Thus, the tool appears stationary, while the workpiece moves at speed V w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAfadaWgaa WcbaGaam4Daaqabaaaaa@383C@  to the right.  The collapse mechanism consists of shear across the red line shown in the picture.

 

Elementary geometry gives the chip thickness d as

d=h cos( ϕ+α ) sinϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeyypa0JaamiAamaalaaaba Gaci4yaiaac+gacaGGZbWaaeWaaeaacqaHvpGzcqGHRaWkcqaHXoqy aiaawIcacaGLPaaaaeaaciGGZbGaaiyAaiaac6gacqaHvpGzaaaaaa@42F6@

Mass conservation (material flowing into slip discontinuity = material flowing out of slip discontinuity) gives the velocity of material in the chip V c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaadogaaeqaaa aa@34B4@  as

V c = V w h d = V w sinϕ cos( ϕ+α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaamOvamaaBaaaleaacaWG3baabeaakmaalaaabaGaamiA aaqaaiaadsgaaaGaeyypa0JaamOvamaaBaaaleaacaWG3baabeaakm aalaaabaGaci4CaiaacMgacaGGUbGaeqy1dygabaGaci4yaiaac+ga caGGZbWaaeWaaeaacqaHvpGzcqGHRaWkcqaHXoqyaiaawIcacaGLPa aaaaaaaa@4A1F@

The velocity discontinuity across the shear band is

| V ba |= V w 2 + V c 2 +2 V c V w sinα = V w 1+ sin 2 ϕ cos 2 ( ϕ+α ) +2sinα sinϕ cos( ϕ+α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaaemaabaGaamOvamaaBaaale aacaWGIbGaamyyaaqabaaakiaawEa7caGLiWoacqGH9aqpdaGcaaqa aiaadAfadaqhaaWcbaGaam4DaaqaaiaaikdaaaGccqGHRaWkcaWGwb Waa0baaSqaaiaadogaaeaacaaIYaaaaOGaey4kaSIaaGOmaiaadAfa daWgaaWcbaGaam4yaaqabaGccaWGwbWaaSbaaSqaaiaadEhaaeqaaO Gaci4CaiaacMgacaGGUbGaeqySdegaleqaaaGcbaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaeyypa0JaamOvamaaBaaaleaacaWG3baabeaakmaa kaaabaGaaGymaiabgUcaRmaalaaabaGaci4CaiaacMgacaGGUbWaaW baaSqabeaacaaIYaaaaOGaeqy1dygabaGaci4yaiaac+gacaGGZbWa aWbaaSqabeaacaaIYaaaaOWaaeWaaeaacqaHvpGzcqGHRaWkcqaHXo qyaiaawIcacaGLPaaaaaGaey4kaSIaaGOmaiGacohacaGGPbGaaiOB aiabeg7aHnaalaaabaGaci4CaiaacMgacaGGUbGaeqy1dygabaGaci 4yaiaac+gacaGGZbWaaeWaaeaacqaHvpGzcqGHRaWkcqaHXoqyaiaa wIcacaGLPaaaaaaaleqaaaaaaa@8468@

The plastic dissipation follows as

W ˙ P = h sinϕ | V ab | Y 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGxbGbaiaadaahaaWcbeqaaiaadc faaaGccqGH9aqpdaWcaaqaaiaadIgaaeaaciGGZbGaaiyAaiaac6ga cqaHvpGzaaWaaqWaaeaacaWGwbWaaSbaaSqaaiaadggacaWGIbaabe aaaOGaay5bSlaawIa7amaalaaabaGaamywaaqaamaakaaabaGaaG4m aaWcbeaaaaaaaa@431F@

The upper bound theorem gives

P V w h sinϕ | V ab | Y 3 P hY 3 sinϕ 1+ sin 2 ϕ cos 2 ( ϕ+α ) +2sinα sinϕ cos( ϕ+α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadcfacaWGwbWaaSbaaSqaai aadEhaaeqaaOGaeyizIm6aaSaaaeaacaWGObaabaGaci4CaiaacMga caGGUbGaeqy1dygaamaaemaabaGaamOvamaaBaaaleaacaWGHbGaam OyaaqabaaakiaawEa7caGLiWoadaWcaaqaaiaadMfaaeaadaGcaaqa aiaaiodaaSqabaaaaaGcbaGaeyO0H4TaamiuaiabgsMiJoaalaaaba GaamiAaiaadMfaaeaadaGcaaqaaiaaiodaaSqabaGcciGGZbGaaiyA aiaac6gacqaHvpGzaaWaaOaaaeaacaaIXaGaey4kaSYaaSaaaeaaci GGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa ciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGcdaqadaqaai abew9aMjabgUcaRiabeg7aHbGaayjkaiaawMcaaaaacqGHRaWkcaaI YaGaci4CaiaacMgacaGGUbGaeqySde2aaSaaaeaaciGGZbGaaiyAai aac6gacqaHvpGzaeaaciGGJbGaai4Baiaacohadaqadaqaaiabew9a MjabgUcaRiabeg7aHbGaayjkaiaawMcaaaaaaSqabaaaaaa@7571@

To obtain the best estimate for P, we need to minimize the right hand side of this expression with respect to ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew9aMbaa@3801@ .  This gives

ϕ= tan 1 (1tan(α)) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpciGG0bGaaiyyai aac6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaGGOaGaaGymaiab gkHiTiGacshacaGGHbGaaiOBaiaacIcacqaHXoqycaGGPaGaaiykaa aa@430E@

The resulting upper bound to the machining force is plotted on the figure to the right.

 

 

 

 

6.2.5 The Lower Bound Plastic Collapse Theorem

 

The lower bound theorem provides a safe estimate of the collapse loads for a rigid plastic solid.

 

Consider a rigid plastic solid, subjected to some distribution of tractions t i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaDa aaleaacaWGPbaabaGaaiOkaaaaaaa@39BF@  and body forces b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGPbaabeaaaaa@38FE@ .  We will attempt to estimate the factor β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  by which the loading can be increased before the solid collapses ( β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  is effectively the factor of safety).  We suppose that the solid will collapse for loading β t i * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam iDamaaDaaaleaacaWGPbaabaGaaiOkaaaaaaa@3B60@ , β b i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaam OyamaaBaaaleaacaWGPbaabeaaaaa@3A9F@ .

 

To estimate β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@ , we guess the distribution of stress in the solid at collapse.

We will denote the guess for the stress distribution by σ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8ak0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@3691@ .  The stress distribution must

1.      Satisfy the boundary conditions σ ^ ij n j = β L t i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8ak0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0Ja eqOSdi2aaSbaaSqaaiaadYeaaeqaaOGaamiDamaaDaaaleaacaWGPb aabaGaaiOkaaaaaaa@3F23@ , where β L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGydaWgaaWcbaGaamitaaqaba aaaa@3564@  is a lower bound to β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@

2.      Satisfy the equations of equilibrium σ ^ ij / x j + β L b i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcuaHdpWCgaqcamaaBaaale aacaWGPbGaamOAaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaa caWGQbaabeaakiabgUcaRiabek7aInaaBaaaleaacaWGmbaabeaaki aadkgadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIWaaaaa@43A2@  within the solid,

3.      Must not violate the yield criterion anywhere within the solid, f( σ ^ ij )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiqbeo8aZzaajaWaaS baaSqaaiaadMgacaWGQbaabeaakiaacMcacqGHKjYOcaaIWaaaaa@3B5F@

 

The lower bound theorem states that if any such stress distribution can be found, the solid will not collapse, i.e. β L β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGydaWgaaWcbaGaamitaaqaba GccqGHKjYOcqaHYoGyaaa@38C4@ .

 

Derivation

1.      Let [ u ˙ i , ε ˙ ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGabmyDayaacaWaaSbaaSqaai aadMgaaeqaaOGaaiilaiqbew7aLzaacaWaaSbaaSqaaiaadMgacaWG QbaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaO Gaaiyxaaaa@3FA6@  denote the actual velocity field in the solid at collapse.  These must satisfy the field equations and constitutive equations listed in Section 6.4.4.

2.      Let σ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8ak0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@3691@  denote the guess for the stress field.

3.      The Principle of Maximum Plastic Resistance (see Section 3.7.10) shows that ( σ ij σ ^ ij ) ε ˙ ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiabeo8aZnaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcuaHdpWCgaqcamaaBaaaleaacaWG PbGaamOAaaqabaaakiaawIcacaGLPaaacuaH1oqzgaGaamaaBaaale aacaWGPbGaamOAaaqabaGccqGHLjYScaaIWaaaaa@433B@ , since σ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8ak0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@3691@  is at or below yield.

4.      Integrating this equation over the volume of the solid, and using the principle of virtual work on the two terms shows that

V ( σ ij σ ^ ij ) ε ˙ ij dV= V σ ij n j u ˙ i dA V σ ^ ij n j u ˙ i dA 0 β V t i u ˙ i dA β L V t i u ˙ i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaapefabaWaaeWaaeaacqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0Iafq4WdmNbaKaa daWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaayzkaaGafqyTdu MbaiaadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamizaiaadAfacqGH 9aqpdaWdrbqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGcca WGUbWaaSbaaSqaaiaadQgaaeqaaOGabmyDayaacaWaaSbaaSqaaiaa dMgaaeqaaOGaamizaiaadgeaaSqaaiabgkGi2kaadAfaaeqaniabgU IiYdGccqGHsisldaWdrbqaaiqbeo8aZzaajaWaaSbaaSqaaiaadMga caWGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGcceWG1bGbai aadaWgaaWcbaGaamyAaaqabaaabaGaeyOaIyRaamOvaaqab0Gaey4k IipakiaadsgacaWGbbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyyzIm RaaGimaaqaaiabgkDiElabek7aInaapefabaGaamiDamaaBaaaleaa caWGPbaabeaakiqadwhagaGaamaaBaaaleaacaWGPbaabeaaaeaacq GHciITcaWGwbaabeqdcqGHRiI8aOGaamizaiaadgeacqGHLjYScqaH YoGydaWgaaWcbaGaamitaaqabaGcdaWdrbqaaiaadshadaWgaaWcba GaamyAaaqabaGcceWG1bGbaiaadaWgaaWcbaGaamyAaaqabaGccaWG KbGaamyqaaWcbaGaeyOaIyRaamOvaaqab0Gaey4kIipaaaaa@8432@

 This proves the theorem.

 

 

 

 

6.2.6 Examples of applications of the lower bound plastic collapse theorem

 

Example 1: Collapse load for a plate containing a hole.  A plate with width L contains a hole of radius a at its center.  The plate is subjected to a tensile force P as shown (the traction distribution is not specified in detail we will accept any solution that has traction acting on the top surface that is statically equivalent to the applied force).

 

For a statically admissible stress distribution, we consider the stress field shown in the figure, with σ 22 =0| x 1 |<a σ 22 =Y| x 1 |>a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVpaaemaabaGaamiEamaaBaaale aacaaIXaaabeaaaOGaay5bSlaawIa7aiabgYda8iaadggacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaeyypa0JaamywaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8+aaqWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqa aaGccaGLhWUaayjcSdGaeyOpa4Jaamyyaaaa@7803@ , and all other stress components zero.

 

The estimate for the applied load at collapse follows as β L P=2Y(La) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGydaWgaaWcbaGaamitaaqaba GccaWGqbGaeyypa0JaaGOmaiaadMfacaGGOaGaamitaiabgkHiTiaa dggacaGGPaaaaa@3CE0@

 

 

 

Example 2: Rigid indenter in contact with a half-space.  We consider a flat indenter with width a that is pushed into the surface of a half-space by a force P.  The stress state illustrated in the figure will be used to obtain a lower bound to the collapse load in the solid.   Note that

1.      Regions C, E, F are stress free

2.      The stress in regions A and D consists of a state of uniaxial stress, with direction parallel to the boundaries between AC (or AE) and CD (or DF) respectively.  We will denote this stress by σ A mm MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaahaaWcbeqaaiaadgeaaa GccaWHTbGaey4LIqSaaCyBaaaa@397B@ , where m is a unit vector parallel to the direction of the uniaxial stress.

3.      The stress state in the triangular region B has principal directions of stress parallel to e α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiabeg7aHbqaba aaaa@357F@ . We will write this stress state as σ 11 B e 1 e 1 + σ 22 B e 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaaGymaiaaig daaeaacaWGcbaaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgEPi elaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHdpWCdaqhaa WcbaGaaGOmaiaaikdaaeaacaWGcbaaaOGaaCyzamaaBaaaleaacaaI YaaabeaakiabgEPielaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@479E@

The stresses in each region must be chosen to satisfy equilibrium, and to ensure that the stress is below yield everywhere.   The stress is constant in each region, so equilibrium is satisfied locally.  However, the stresses are discontinuous across AC, AB, etc.  To satisfy equilibrium, equal and opposite tractions must act on the material surfaces adjacent to the discontinuity, which requires, e.g. that σ ij A n j = σ ij B n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGbbaaaOGaamOBamaaBaaaleaacaWGQbaabeaakiabg2da 9iabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiaadkeaaaGccaWGUb WaaSbaaSqaaiaadQgaaeqaaaaa@412D@ , where n is a unit vector normal to the boundary between A and B as indicated in the figure.  We enforce this condition as follows:

1.      Note that m=cosθ e 1 +sinθ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbGaeyypa0Jaci4yaiaac+gaca GGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiGa cohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqaba aaaa@4270@   n=sinθ e 1 +cosθ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyypa0Jaci4CaiaacMgaca GGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiGa cogacaGGVbGaai4CaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqaba aaaa@4271@

2.      Equilibrium across the boundary between A and B requires

σ A ( mm )n=( σ 11 B e 1 e 1 + σ 22 B e 2 e 2 )n σ A ( cosθ e 1 +sinθ e 2 )2sinθcosθ= σ 11 B e 1 sinθ+ σ 22 B e 2 cosθ σ 11 B =2 σ A cos 2 θ σ 22 B =2 σ A sin 2 θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaCaaaleqabaGaam yqaaaakmaabmaabaGaaCyBaiabgEPielaah2gaaiaawIcacaGLPaaa cqGHflY1caWHUbGaeyypa0ZaaeWaaeaacqaHdpWCdaqhaaWcbaGaaG ymaiaaigdaaeaacaWGcbaaaOGaaCyzamaaBaaaleaacaaIXaaabeaa kiabgEPielaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaHdp WCdaqhaaWcbaGaaGOmaiaaikdaaeaacaWGcbaaaOGaaCyzamaaBaaa leaacaaIYaaabeaakiabgEPielaahwgadaWgaaWcbaGaaGOmaaqaba aakiaawIcacaGLPaaacqGHflY1caWHUbaabaGaeyO0H4Taeq4Wdm3a aWbaaSqabeaacaWGbbaaaOWaaeWaaeaaciGGJbGaai4Baiaacohacq aH4oqCcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4Caiaa cMgacaGGUbGaeqiUdeNaaCyzamaaBaaaleaacaaIYaaabeaaaOGaay jkaiaawMcaaiaaikdaciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGa ai4BaiaacohacqaH4oqCcqGH9aqpcqaHdpWCdaqhaaWcbaGaaGymai aaigdaaeaacaWGcbaaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiGa cohacaGGPbGaaiOBaiabeI7aXjabgUcaRiabeo8aZnaaDaaaleaaca aIYaGaaGOmaaqaaiaadkeaaaGccaWHLbWaaSbaaSqaaiaaikdaaeqa aOGaci4yaiaac+gacaGGZbGaeqiUdehabaGaeyO0H4Taeq4Wdm3aa0 baaSqaaiaaigdacaaIXaaabaGaamOqaaaakiabg2da9iaaikdacqaH dpWCdaahaaWcbeqaaiaadgeaaaGcciGGJbGaai4Baiaacohadaahaa WcbeqaaiaaikdaaaGccqaH4oqCcaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7cqaHdpWCdaqhaaWcbaGaaGOmaiaaikdaaeaacaWGcb aaaOGaeyypa0JaaGOmaiabeo8aZnaaCaaaleqabaGaamyqaaaakiGa cohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeI7aXjaayk W7aaaa@B7F9@

3.      We must now choose σ A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaCaaaleqabaGaamyqaaaaaa a@333E@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  to maximize the collapse load, while ensuring that the stresses do not exceed yield in regions A or B.   Clearly, this requires σ A =Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaCaaaleqabaGaamyqaaaaki abg2da9iabgkHiTiaadMfaaaa@3619@ ; while θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  must be chosen to ensure that | σ 22 B σ 11 B |<Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaemaabaGaeq4Wdm3aa0baaSqaaiaaik dacaaIYaaabaGaamOqaaaakiabgkHiTiabeo8aZnaaDaaaleaacaaI XaGaaGymaaqaaiaadkeaaaaakiaawEa7caGLiWoacqGH8aapcaWGzb aaaa@3EE9@ .  This requires 1/2<cosθ< 3 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaGGVaGaaGOmaiabgYda8iGaco gacaGGVbGaai4CaiabeI7aXjabgYda8maakaaabaGaaG4maaWcbeaa kiaac+cacaaIYaaaaa@3B94@ .   The largest value for θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  maximizes the bound.

4.      Finally, substituting for θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  gives σ 22 B =3Y/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaa qaaiaadkeaaaGccqGH9aqpcqGHsislcaaIZaGaamywaiaac+cacaaI Yaaaaa@39BE@ .  We see that the lower bound is P=3Ya/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadcfacqGH9aqpcqGHsislcaaIZaGaam ywaiaadggacaGGVaGaaGOmaaaa@3740@ .

  

 

 

 

 

6.2.7 The lower bound shakedown theorem

 

In this and the next section we derive two important theorems that can be used to estimate the maximum cyclic loads that can be imposed on a component without exceeding yield.  The concept of shakedown in a solid subjected to cyclic loads was introduced in Section 4.2.4, which discusses the behavior of a spherical shell subjected to cyclic internal pressure.   It was shown that, if the first cycle of pressure exceeds yield, residual stresses are introduced into the shell, which may prevent further plastic deformation under subsequent load cycles.  This process is known as shakedown, and the maximum load for which it can occur is known as the shakedown limit

 

We proceed to derive a theorem that can be used to obtain a safe estimate to the maximum cyclic load that can be applied to a structure without inducing cyclic plastic deformation.

 

We consider an elastic-perfectly plastic solid with Von-Mises yield surface, associated flow law, and uniaxial tensile yield stress Y. Assume that

1.      The displacement u=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWHWaaaaa@3345@  on part of the boundary of the solid 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIXaaabeaaki aadkfaaaa@33B6@

2.      The remainder of the boundary  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@  is subjected to a prescribed cycle of traction t * (t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshadaahaaWcbeqaaiaacQcaaaGcca GGOaGaamiDaiaacMcaaaa@34BC@ .   The history of traction is periodic, with a period T.

 

Define the following quantities:

1.      Let [ u i , ε ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWG1bWaaSbaaSqaaiaadMgaae qaaOGaaiilaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaGccaGG SaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaac2faaaa@3D56@  denote the actual history of displacement, strain and stress induced in the solid by the applied loading.  The strain is partitioned into elastic and plastic parts as ε ij = ε ij e + ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG LbaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam iCaaaaaaa@3F75@

2.      Let [ u i ε , ε ij ε , σ ij ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWG1bWaa0baaSqaaiaadMgaae aacqaH1oqzaaGccaGGSaGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaa baGaeqyTdugaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiabew7aLbaakiaac2faaaa@424E@  denote the history of displacement, strain and stress induced by the prescribed traction in a perfectly elastic solid with identical geometry.

3.      We introduce (time dependent) residual stress ρ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3451@  and residual strain γ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3438@  fields, which (by definition) satisfy

σ ij = σ ij ε + ρ ij ε ij = ε ij e + ε ij p = ε ij ε + γ ij + ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaHdpWCdaqhaaWcbaGaamyAaiaadQgaaeaacqaH 1oqzaaGccqGHRaWkcqaHbpGCdaWgaaWcbaGaamyAaiaadQgaaeqaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7cqaH1oqzdaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaamyz aaaakiabgUcaRiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadc haaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacqaH 1oqzaaGccqGHRaWkcqaHZoWzdaWgaaWcbaGaamyAaiaadQgaaeqaaO Gaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaa aaa@7013@

Note that, (i) because σ ij ε n j = σ ij n j = t i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiabew7aLbaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqp cqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamOBamaaBaaale aacaWGQbaabeaakiabg2da9iaadshadaqhaaWcbaGaamyAaaqaaiaa cQcaaaaaaa@42DA@  on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@ , it follows that ρ ij n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaaGimaaaa @3833@  on  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@ ; and (ii) because σ ij ε / x j = σ ij / x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabeo8aZnaaDaaaleaacaWGPb GaamOAaaqaaiabew7aLbaakiaac+cacqGHciITcaWG4bWaaSbaaSqa aiaadQgaaeqaaOGaeyypa0JaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaOGaeyypa0JaaGimaaaa@47E4@  it follows that ρ ij / x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabeg8aYnaaBaaaleaacaWGPb GaamOAaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaakiabg2da9iaaicdaaaa@3BBC@

 

The lower bound shakedown theorem can be stated as follows: The solid is guaranteed to shake down if any time independent residual stress field ρ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaaaaa@3469@  can be found which satisfies:

 The equilibrium equation ρ ¯ ij / x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRafq yWdiNbaebadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiabgkGi 2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@4249@ ;

 The boundary condition ρ ¯ ij n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaI Waaaaa@384B@  on  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@ ;

 When the residual stress is combined with the elastic solution, the combined stress does not exceed yield f( σ ij ε + ρ ¯ ij )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiabeo8aZnaaDaaale aacaWGPbGaamOAaaqaaiabew7aLbaakiabgUcaRiqbeg8aYzaaraWa aSbaaSqaaiaadMgacaWGQbaabeaakiaacMcacqGHKjYOcaaIWaaaaa@41C4@  at any time during the cycle of load.

 

The theorem is valuable because shakedown limits can be estimated using the elastic solution, which is much easier to calculate than the elastic-plastic solution. 

 

Proof of the lower bound theorem:  The proof is one of the most devious in all of solid mechanics. 

1.      Consider the strain energy associated with the difference between the actual residual stress field ρ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3451@ , and the guess for the residual stress field ρ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaaaaa@3469@ , which can be calculated as

W= 1 2 R S ijkl ( ρ ij ρ ¯ ij )( ρ kl ρ ¯ kl ) dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGxbGaeyypa0ZaaSaaaeaacaaIXa aabaGaaGOmaaaadaWdrbqaaiaadofadaWgaaWcbaGaamyAaiaadQga caWGRbGaamiBaaqabaGccaGGOaGaeqyWdi3aaSbaaSqaaiaadMgaca WGQbaabeaakiabgkHiTiqbeg8aYzaaraWaaSbaaSqaaiaadMgacaWG QbaabeaakiaacMcacaGGOaGaeqyWdi3aaSbaaSqaaiaadUgacaWGSb aabeaakiabgkHiTiqbeg8aYzaaraWaaSbaaSqaaiaadUgacaWGSbaa beaakiaacMcaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaa aa@53FA@

where S ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3788@  is the elastic compliance tensor.   For later reference note that W has to be positive, because strain energy density is always positive or zero.

2.      The rate of change of W can be calculated as

dW dt = R S ijkl ( ρ ij ρ ¯ ij ) d ρ kl dt dV0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGxbaabaGaam izaiaadshaaaGaeyypa0Zaa8quaeaacaWGtbWaaSbaaSqaaiaadMga caWGQbGaam4AaiaadYgaaeqaaOGaaiikaiabeg8aYnaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcuaHbpGCgaqeamaaBaaaleaacaWG PbGaamOAaaqabaGccaGGPaWaaSaaaeaacaWGKbGaeqyWdi3aaSbaaS qaaiaadUgacaWGSbaabeaaaOqaaiaadsgacaWG0baaaaWcbaGaamOu aaqab0Gaey4kIipakiaadsgacaWGwbGaeyyzImRaaGimaaaa@5474@

(to see this, recall that S ijkl = S klij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaOGaeyypa0Jaam4uamaaBaaaleaacaWGRbGa amiBaiaadMgacaWGQbaabeaaaaa@3D5A@  )

3.      Note that S ijkl ρ kl = γ ij = ε ij ε ij p ε ij ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaOGaeqyWdi3aaSbaaSqaaiaadUgacaWGSbaa beaakiabg2da9iabeo7aNnaaBaaaleaacaWGPbGaamOAaaqabaGccq GH9aqpcqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0Ia eqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaakiabgkHiTi abew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiabew7aLbaaaaa@50CB@ . Consequently, we see that

dW dt = R ( ρ ij ρ ¯ ij ) d ε ij p dt dV+ R ( ρ ij ρ ¯ ij )( d ε ij dt d ε ij ε dt ) dV0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGxbaabaGaam izaiaadshaaaGaeyypa0JaeyOeI0Yaa8quaeaacaGGOaGaeqyWdi3a aSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiqbeg8aYzaaraWaaS baaSqaaiaadMgacaWGQbaabeaakiaacMcadaWcaaqaaiaadsgacqaH 1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaaGcbaGaamizai aadshaaaaaleaacaWGsbaabeqdcqGHRiI8aOGaamizaiaadAfacqGH RaWkdaWdrbqaaiaacIcacqaHbpGCdaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyOeI0IafqyWdiNbaebadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaaiykamaabmaabaWaaSaaaeaacaWGKbGaeqyTdu2aaSbaaSqaai aadMgacaWGQbaabeaaaOqaaiaadsgacaWG0baaaiabgkHiTmaalaaa baGaamizaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiabew7aLb aaaOqaaiaadsgacaWG0baaaaGaayjkaiaawMcaaaWcbaGaamOuaaqa b0Gaey4kIipakiaadsgacaWGwbGaeyyzImRaaGimaaaa@728B@

4.      Using the principle of virtual work, the second integral can be expressed as an integral over the boundary of the solid

R ( ρ ij ρ ¯ ij ) n j ( d u i dt d u i ε dt ) dV=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiaacIcacqaHbpGCdaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyOeI0IafqyWdiNbaebadaWgaaWc baGaamyAaiaadQgaaeqaaOGaaiykaiaad6gadaWgaaWcbaGaamOAaa qabaGcdaqadaqaamaalaaabaGaamizaiaadwhadaWgaaWcbaGaamyA aaqabaaakeaacaWGKbGaamiDaaaacqGHsisldaWcaaqaaiaadsgaca WG1bWaa0baaSqaaiaadMgaaeaacqaH1oqzaaaakeaacaWGKbGaamiD aaaaaiaawIcacaGLPaaaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKb GaamOvaiabg2da9iaaicdaaaa@53A1@

To see this, note that ( ρ ij ρ ¯ ij ) n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeqyWdi3aaSbaaSqaaiaadM gacaWGQbaabeaakiabgkHiTiqbeg8aYzaaraWaaSbaaSqaaiaadMga caWGQbaabeaakiaacMcacaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaey ypa0JaaGimaaaa@40A2@  on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@ , while u ˙ i u ˙ i ε =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWG1bGbaiaadaWgaaWcbaGaamyAaa qabaGccqGHsislceWG1bGbaiaadaqhaaWcbaGaamyAaaqaaiabew7a Lbaakiabg2da9iaaicdaaaa@3B69@  on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@

5.      The remaining integral in (3) can be re-written as

dW dt = R ( ρ ij ρ ¯ ij ) d ε ij p dt dV= R [ σ ij ( σ ij ε + ρ ¯ ij ) ] d ε ij p dt dV0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGxbaabaGaam izaiaadshaaaGaeyypa0JaeyOeI0Yaa8quaeaacaGGOaGaeqyWdi3a aSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiqbeg8aYzaaraWaaS baaSqaaiaadMgacaWGQbaabeaakiaacMcadaWcaaqaaiaadsgacqaH 1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaaGcbaGaamizai aadshaaaaaleaacaWGsbaabeqdcqGHRiI8aOGaamizaiaadAfacqGH 9aqpcqGHsisldaWdrbqaamaadmaabaGaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiabgkHiTiaacIcacqaHdpWCdaqhaaWcbaGaamyA aiaadQgaaeaacqaH1oqzaaGccqGHRaWkcuaHbpGCgaqeamaaBaaale aacaWGPbGaamOAaaqabaGccaGGPaaacaGLBbGaayzxaaWaaSaaaeaa caWGKbGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaaaO qaaiaadsgacaWG0baaaaWcbaGaamOuaaqab0Gaey4kIipakiaadsga caWGwbGaeyyzImRaaGimaaaa@7234@

6.      Finally, recall that σ ij ε + ρ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacqaH1oqzaaGccqGHRaWkcuaHbpGCgaqeamaaBaaaleaacaWG PbGaamOAaaqabaaaaa@3D07@  lies at or below yield, while σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  is at yield and is the stress corresponding to the plastic strain rate ε ˙ ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGaamaaDaaaleaacaWGPb GaamOAaaqaaiaadchaaaaaaa@3775@ .   The principle of maximum plastic resistance therefore shows that [ σ ij ( σ ij ε + ρ ¯ ij ) ] ε ˙ ij p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabeo8aZnaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcaGGOaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaeqyTdugaaOGaey4kaSIafqyWdiNbaebadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaaiykaaGaay5waiaaw2faaiqbew7a LzaacaWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaakiabgwMiZk aaicdaaaa@4C58@ .  This inequality and  dW/dt0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaam4vaiaac+cacaWGKbGaam iDaiabgwMiZkaaicdaaaa@39A0@  can only be satisfied simultaneously if [ σ ij ( σ ij ε + ρ ¯ ij ) ] ε ˙ ij p =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabeo8aZnaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcaGGOaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaeqyTdugaaOGaey4kaSIafqyWdiNbaebadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaaiykaaGaay5waiaaw2faaiqbew7a LzaacaWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaakiabg2da9i aaicdaaaa@4B98@ .  We conclude that either the plastic strain rate vanishes, or [ σ ij ( σ ij ε + ρ ¯ ij ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabeo8aZnaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcaGGOaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaeqyTdugaaOGaey4kaSIafqyWdiNbaebadaWgaa WcbaGaamyAaiaadQgaaeqaaOGaaiykaaGaay5waiaaw2faaiabg2da 9iaaicdaaaa@46DF@ .  In either case the solid must shake down to an elastic state.

 

 

 

6.2.8 Examples of applications of the lower bound shakedown theorem

 

Example 1: A simple 3 bar problem.  It is traditional to illustrate the concept of shakedown using this problem.  Consider a structure made of three parallel elastic-plastic bars, with Young’s modulus E and cross sectional are A, as shown in the figure.  The two bars labeled 1 and 2 have yield stress Y; the central bar (labeled 3) has yield stress 2Y.  The structure is subjected to a cyclic load with mean value P ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGqbGbaebaaaa@33B3@  and amplitude ΔP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGqbaaaa@3501@ .

 

The elastic limit for the structure is P ¯ ±ΔP=3AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGqbGbaebacqGHXcqScqqHuoarca WGqbGaeyypa0JaaG4maiaadgeacaWGzbaaaa@3B43@ ; the collapse load is P ¯ ±ΔP=4AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGqbGbaebacqGHXcqScqqHuoarca WGqbGaeyypa0JaaGinaiaadgeacaWGzbaaaa@3B44@ .

 

To obtain a lower bound to the shakedown limit, we must

1.      Calculate the elastic stresses in the structure MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the axial stress in each bar is σ ε =P/3A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaahaaWcbeqaaiabew7aLb aakiabg2da9iaadcfacaGGVaGaaG4maiaadgeaaaa@3A78@

2.      Find a residual stress distribution in the structure, which satisfies equilibrium and boundary conditions, and which can be added to the elastic stresses to bring them below yield.   A suitable residual stress distribution consists of an axial stress ρ (1) = ρ (2) = ρ 0 ρ (3) =2 ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaahaaWcbeqaaiaacIcaca aIXaGaaiykaaaakiabg2da9iabeg8aYnaaCaaaleqabaGaaiikaiaa ikdacaGGPaaaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 cqaHbpGCdaahaaWcbeqaaiaacIcacaaIZaGaaiykaaaakiabg2da9i abgkHiTiaaikdacqaHbpGCdaWgaaWcbaGaaGimaaqabaaaaa@5553@  in bars 1, 2 and 3. To prevent yield at the maximum and minimum load in all three bars, we require

Y<( P ¯ ΔP)/3A+ ρ 0 ( P ¯ +ΔP)/3A+ ρ 0 <Y 2Y<( P ¯ ΔP)/3A2 ρ 0 ( P ¯ +ΔP)/3A2 ρ 0 <2Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabgkHiTiaadMfacqGH8aapca GGOaGabmiuayaaraGaeyOeI0IaeuiLdqKaamiuaiaacMcacaGGVaGa aG4maiaadgeacqGHRaWkcqaHbpGCdaWgaaWcbaGaaGimaaqabaGcca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaacIcaceWGqbGbaebacqGHRaWkcqqHuoarcaWGqbGaai ykaiaac+cacaaIZaGaamyqaiabgUcaRiabeg8aYnaaBaaaleaacaaI WaaabeaakiabgYda8iaadMfacaaMc8UaaGPaVlaaykW7caaMc8oaba GaeyOeI0IaaGOmaiaadMfacqGH8aapcaGGOaGaaGPaVlqadcfagaqe aiabgkHiTiabfs5aejaadcfacaGGPaGaai4laiaaiodacaWGbbGaey OeI0IaaGOmaiabeg8aYnaaBaaaleaacaaIWaaabeaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaacIcaceWGqbGbaebacqGHRaWkcqqHuo arcaWGqbGaaiykaiaac+cacaaIZaGaamyqaiabgkHiTiaaikdacqaH bpGCdaWgaaWcbaGaaGimaaqabaGccqGH8aapcaaIYaGaamywaaaaaa@BC33@

The first two equations show that ΔP<Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGqbGaeyipaWJaamywaa aa@36E3@ , irrespective of ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@ .  To avoid yield in all bars at the maximum load, we must choose ρ 0 =Y/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba GccqGH9aqpcqGHsislcaWGzbGaai4laiaaiodaaaa@39B7@ , which gives P+ΔP<4AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbGaey4kaSIaeuiLdqKaamiuai abgYda8iaaisdacaWGbbGaamywaaaa@3A1E@ . Similarly, to avoid yield in all bars at the minimum load, we must choose ρ 0 =Y/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba GccqGH9aqpcaWGzbGaai4laiaaiodaaaa@38CA@ , showing that 4AY<( P ¯ ΔP) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaaI0aGaamyqaiaadMfacq GH8aapcaGGOaGabmiuayaaraGaeyOeI0IaeuiLdqKaamiuaiaacMca aaa@3C87@ .

 

 

The various regimes of behavior are summarized in the figure.

 

 

 

 

 

 

Example 2: Shakedown limit for a pressurized spherical shell.  We consider an elastic-perfectly plastic thick-walled shell, with inner radius a and outer radius b.  The inner wall of the shell is subjected to a cyclic pressure, with minimum value zero, and maximum value p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@

 

To estimate the shakedown limit we must

1.      Calculate the stresses induced by the pressure in an elastic shell.  The solution can be found in Section 4.1.4. 

σ RR = p a a 3 ( b 3 a 3 ) ( 1 b 3 R 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadggaaeqa aOGaamyyamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyam aaCaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaaaaGaaGPaVpaabmaabaGaaGymaiabgk HiTmaalaaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaadkfa daahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaGaaGPaVdaa@4B96@   σ θθ = σ ϕϕ = p a a 3 ( b 3 a 3 ) ( 1+ b 3 2 R 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaamyyamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyamaa CaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbeqaaiaaio daaaaakiaawIcacaGLPaaaaaGaaGPaVpaabmaabaGaaGymaiabgUca RmaalaaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdaca WGsbWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaiaaykW7 aaa@5494@

2.      Find a self-equilibrating residual stress field, which satisfies traction free boundary conditions on R=a, R=b, and which can be added to the elastic stresses to prevent yield in the sphere.  The equilibrium equation for the residual stress can be written

d ρ RR dR + 1 R ( 2 ρ RR ρ θθ ρ ϕϕ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacqaHbpGCdaWgaa WcbaGaamOuaiaadkfaaeqaaaGcbaGaamizaiaadkfaaaGaey4kaSYa aSaaaeaacaaIXaaabaGaamOuaaaadaqadaqaaiaaikdacqaHbpGCda WgaaWcbaGaamOuaiaadkfaaeqaaOGaeyOeI0IaeqyWdi3aaSbaaSqa aiabeI7aXjabeI7aXbqabaGccqGHsislcqaHbpGCdaWgaaWcbaGaeq y1dyMaeqy1dygabeaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@5012@

We can satisfy this equation by choosing any suitable distribution for ρ RR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaamOuaiaadk faaeqaaaaa@3660@  and calculating the corresponding ρ θθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaaaaa@381E@ .  For example, we can choose ρ RR = ρ 0 (1a/R)(1b/R) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0JaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaaiik aiaaigdacqGHsislcaWGHbGaai4laiaadkfacaGGPaGaaiikaiaaig dacqGHsislcaWGIbGaai4laiaadkfacaGGPaaaaa@4503@ , which corresponds to ρ θθ = ρ 0 (1(b+a)/2R) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeg8aYnaaBaaaleaacaaIWaaabeaakiaa cIcacaaIXaGaeyOeI0IaaiikaiaadkgacqGHRaWkcaWGHbGaaiykai aac+cacaaIYaGaamOuaiaacMcaaaa@452D@ .  To avoid yield at maximum load, we must ensure that | σ RR σ θθ + ρ RR ρ θθ |Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiabeo8aZnaaBaaaleaaca WGsbGaamOuaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaeqiUdeNa eqiUdehabeaakiabgUcaRiabeg8aYnaaBaaaleaacaWGsbGaamOuaa qabaGccqGHsislcqaHbpGCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaa aOGaay5bSlaawIa7aiabgsMiJkaadMfaaaa@4D49@ , while to avoid yield at zero load, | ρ RR ρ θθ |Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaabdaqaaiabeg8aYnaaBaaaleaaca WGsbGaamOuaaqabaGccqGHsislcqaHbpGCdaWgaaWcbaGaeqiUdeNa eqiUdehabeaaaOGaay5bSlaawIa7aiabgsMiJkaadMfaaaa@426E@  throughout the shell.   The critically stressed material element lies at R=a at both the maximum and zero loads, which shows that

3 p a b 3 ( b 3 a 3 ) + ρ 0 (ba) 2a <YY< ρ 0 (ba) 2a <Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaaiodacaWGWbWaaSbaaS qaaiaadggaaeqaaOGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaa cIcacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamyyamaaCa aaleqabaGaaG4maaaakiaacMcaaaGaey4kaSIaeqyWdi3aaSbaaSqa aiaaicdaaeqaaOWaaSaaaeaacaGGOaGaamOyaiabgkHiTiaadggaca GGPaaabaGaaGOmaiaadggaaaGaeyipaWJaamywaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7cqGHsislcaWGzbGaeyipaWJaeqyWdi3aaSbaaSqaaiaaicdaae qaaOWaaSaaaeaacaGGOaGaamOyaiabgkHiTiaadggacaGGPaaabaGa aGOmaiaadggaaaGaeyipaWJaamywaaaa@7329@

Clearly, the best choice of ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba aaaa@356C@  is ρ 0 =2Ya/(ba) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHbpGCdaWgaaWcbaGaaGimaaqaba GccqGH9aqpcqGHsislcaaIYaGaamywaiaadggacaGGVaGaaiikaiaa dkgacqGHsislcaWGHbGaaiykaaaa@3EAF@

 

The estimate for the shakedown limit therefore follows as p a /Y<4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH8aapcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4103@ .  This is equal to the exact solution derived (with considerably more effort) in Section 4.1.4.

 

 

 

 

6.2.9 The Upper Bound Shakedown Theorem

 

In this section we derive a theorem that can be used to obtain an over-estimate to the maximum cyclic load that can be applied to a structure without inducing cyclic plastic deformation.  Although the estimate is inherently unsafe, the theorem is easier to use than the lower bound theorem.

 

We consider an elastic-perfectly plastic solid with Von-Mises yield surface, associated flow law, and uniaxial tensile yield stress Y. Assume that

1.      The displacement u=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWHWaaaaa@3345@  on part of the boundary of the solid 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIXaaabeaaki aadkfaaaa@33B6@

2.      The remainder of the boundary  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@  is subjected to a prescribed cycle of traction t * (t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshadaahaaWcbeqaaiaacQcaaaGcca GGOaGaamiDaiaacMcaaaa@34BC@ .   The history of traction is periodic, with a period T.

 

Define the following quantities:

1.      Let [ u i , ε ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWG1bWaaSbaaSqaaiaadMgaae qaaOGaaiilaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaGccaGG SaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaac2faaaa@3D56@  denote the actual history of displacement, strain and stress induced in the solid by the applied loading.  The strain is partitioned into elastic and plastic parts as ε ij = ε ij e + ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG LbaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam iCaaaaaaa@3F75@

2.      Let [ u i ε , ε ij ε , σ ij ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWG1bWaa0baaSqaaiaadMgaae aacqaH1oqzaaGccaGGSaGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaa baGaeqyTdugaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiabew7aLbaakiaac2faaaa@424E@  denote the history of displacement, strain and stress induced by the prescribed traction in a perfectly elastic solid with identical geometry.

To apply the upper bound theorem, we guess a mechanism of cyclic plasticity that might occur in the structure under the applied loading.  We denote the cycle of strain by ε ˙ ^ ij p (t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGagaqcamaaDaaaleaaca WGPbGaamOAaaqaaiaadchaaaGccaGGOaGaamiDaiaacMcaaaa@39E0@ , and define the change in strain per cycle as

Δ ε ^ ij p = 0 T ε ˙ ^ ij p (t)dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcuaH1oqzgaqcamaaDaaale aacaWGPbGaamOAaaqaaiaadchaaaGccqGH9aqpdaWdXbqaaiqbew7a LzaacyaajaWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaakiaacI cacaWG0bGaaiykaiaadsgacaWG0baaleaacaaIWaaabaGaamivaaqd cqGHRiI8aaaa@46EA@

To be a kinematically admissible cycle,

  Δ ε ^ ij p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcuaH1oqzgaqcamaaDaaale aacaWGPbGaamOAaaqaaiaadchaaaaaaa@38E2@  must be compatible, i.e. Δ ε ^ ij p =(Δ u ^ i / x j +Δ u ^ j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcuaH1oqzgaqcamaaDaaale aacaWGPbGaamOAaaqaaiaadchaaaGccqGH9aqpcaGGOaGaeyOaIyRa euiLdqKabmyDayaajaWaaSbaaSqaaiaadMgaaeqaaOGaai4laiabgk Gi2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcqGHciITcqqH uoarceWG1bGbaKaadaWgaaWcbaGaamOAaaqabaGccaGGVaGaeyOaIy RaamiEamaaBaaaleaacaWGPbaabeaakiaacMcacaGGVaGaaGOmaaaa @5006@   for some a displacement field Δ u ^ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarceWG1bGbaKaadaWgaaWcba GaamyAaaqabaaaaa@3650@ .  Note that only the change in strain per cycle needs to be compatible, the plastic strain rate need not be compatible at every instant during the cycle.

 The compatible displacement field must satisfy Δ u ^ i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarceWG1bGbaKaadaWgaaWcba GaamyAaaqabaGccqGH9aqpcaaIWaaaaa@381A@  on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIXaaabeaaki aadkfaaaa@33B6@ .

 

The upper bound shakedown theorem can then be stated as follows.  If there exists any kinematically admissible cycle of strain that satisfies

0 T R σ ij ε (t) ε ˙ ^ ij p (t)dVdt 0 T R Y ε ˙ ^ e p (t)dVdt ε ˙ ^ e p = 2 ε ˙ ^ ij p ε ˙ ^ ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaapefabaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaGaeqyTdugaaOGaaiikaiaadshacaGG PaGafqyTduMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWb aaaOGaaiikaiaadshacaGGPaGaamizaiaadAfacaWGKbGaamiDaaWc baGaamOuaaqab0Gaey4kIipaaSqaaiaaicdaaeaacaWGubaaniabgU IiYdGccqGHLjYSdaWdXbqaamaapefabaGaamywaiqbew7aLzaacyaa jaWaa0baaSqaaiaadwgaaeaacaWGWbaaaOGaaiikaiaadshacaGGPa GaamizaiaadAfacaWGKbGaamiDaaWcbaGaamOuaaqab0Gaey4kIipa aSqaaiaaicdaaeaacaWGubaaniabgUIiYdGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlqbew7aLzaacyaajaWaa0baaSqa aiaadwgaaeaacaWGWbaaaOGaeyypa0ZaaOaaaeaacaaIYaGafqyTdu MbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGafqyT duMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaai 4laiaaiodaaSqabaaaaa@9444@

the solid will not shake down to an elastic state.

 

Proof: The upper bound theorem can be proved by contradiction. 

  1. Suppose that the solid does shake down.  Then, from the lower bound shakedown theorem, we know that there exists a time independent residual stress field ρ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHbpGCgaqeamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@36A7@ , which satisfies equilibrium ρ ¯ ij / x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRafq yWdiNbaebadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiabgkGi 2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaIWaaaaa@4249@ ; the boundary conditions   ρ ¯ ij n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaI Waaaaa@384B@  on  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@ , and is such that σ ij ε (t)+ ρ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacqaH1oqzaaGccaGGOaGaamiDaiaacMcacqGHRaWkcuaHbpGC gaqeamaaBaaaleaacaWGPbGaamOAaaqabaaaaa@3F59@  lies below yield throughout the cycle.
  2. The principle of maximum plastic resistance then shows that ( σ ^ ij ( σ ij ε + ρ ¯ ij )) ε ˙ ^ ij p =Y ε ˙ ^ e p ( σ ij ε + ρ ¯ ij ) ε ˙ ^ ij p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGafq4WdmNbaKaadaWgaaWcba GaamyAaiaadQgaaeqaaOGaeyOeI0Iaaiikaiabeo8aZnaaDaaaleaa caWGPbGaamOAaaqaaiabew7aLbaakiabgUcaRiqbeg8aYzaaraWaaS baaSqaaiaadMgacaWGQbaabeaakiaacMcacaGGPaGafqyTduMbaiGb aKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaeyypa0Jaam ywaiqbew7aLzaacyaajaWaa0baaSqaaiaadwgaaeaacaWGWbaaaOGa eyOeI0Iaaiikaiabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiabew 7aLbaakiabgUcaRiqbeg8aYzaaraWaaSbaaSqaaiaadMgacaWGQbaa beaakiaacMcacuaH1oqzgaGagaqcamaaDaaaleaacaWGPbGaamOAaa qaaiaadchaaaGccqGHLjYScaaIWaaaaa@62F0@ .  Integrating this expression over the volume of the solid, and the cycle of loading gives

0 T R Y ε ˙ ^ e p (t)dVdt 0 T R σ ij ε (t) ε ˙ ^ ij p (t)dVdt 0 T R ρ ¯ ij (t) ε ˙ ^ ij p (t)dVdt 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaapefabaGaamywaiqbew 7aLzaacyaajaWaa0baaSqaaiaadwgaaeaacaWGWbaaaOGaaiikaiaa dshacaGGPaGaamizaiaadAfacaWGKbGaamiDaaWcbaGaamOuaaqab0 Gaey4kIipaaSqaaiaaicdaaeaacaWGubaaniabgUIiYdGccqGHsisl daWdXbqaamaapefabaGaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaaba GaeqyTdugaaOGaaiikaiaadshacaGGPaGafqyTduMbaiGbaKaadaqh aaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaaiikaiaadshacaGGPa GaamizaiaadAfacaWGKbGaamiDaaWcbaGaamOuaaqab0Gaey4kIipa aSqaaiaaicdaaeaacaWGubaaniabgUIiYdGccqGHsisldaWdXbqaam aapefabaGafqyWdiNbaebadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiikaiaadshacaGGPaGafqyTduMbaiGbaKaadaqhaaWcbaGaamyAai aadQgaaeaacaWGWbaaaOGaaiikaiaadshacaGGPaGaamizaiaadAfa caWGKbGaamiDaaWcbaGaamOuaaqab0Gaey4kIipaaSqaaiaaicdaae aacaWGubaaniabgUIiYdGccqGHLjYScaaIWaaaaa@7ACD@

  1. Finally, reversing the order of integration in the last integral and using the principle of virtual work, we see that

R 0 T ρ ¯ ij (t) ε ˙ ^ ij p (t)dtdV = R ρ ¯ ij Δ ε ^ ij p dV = R ρ ¯ ij n j Δ u i p dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaamaapehabaGafqyWdiNbae badaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiikaiaadshacaGGPaGa fqyTduMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaO GaaiikaiaadshacaGGPaGaamizaiaadshacaWGKbGaamOvaaWcbaGa aGimaaqaaiaadsfaa0Gaey4kIipaaSqaaiaadkfaaeqaniabgUIiYd GccqGH9aqpdaWdrbqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgacaWG Qbaabeaakiabfs5aejqbew7aLzaajaWaa0baaSqaaiaadMgacaWGQb aabaGaamiCaaaakiaadsgacaWGwbaaleaacaWGsbaabeqdcqGHRiI8 aOGaeyypa0Zaa8quaeaacuaHbpGCgaqeamaaBaaaleaacaWGPbGaam OAaaqabaGccaWGUbWaaSbaaSqaaiaadQgaaeqaaOGaeuiLdqKaamyD amaaDaaaleaacaWGPbaabaGaamiCaaaakiaadsgacaWGbbaaleaacq GHciITcaWGsbaabeqdcqGHRiI8aOGaeyypa0JaaGimaaaa@6E73@

To see this, note that  Δ u ^ i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarceWG1bGbaKaadaWgaaWcba GaamyAaaqabaGccqGH9aqpcaaIWaaaaa@381A@  on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIXaaabeaaki aadkfaaaa@33B6@  while   ρ ¯ ij n j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcaaI Waaaaa@384B@  on  2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaBaaaleaacaaIYaaabeaaki aadkfaaaa@33B7@ .

  1. Substituting this result back into (2) gives a contradiction, so proving the upper bound theorem.

 

 

 

6.2.10 Examples of applications of the upper bound shakedown theorem

 

Example 1: A simple 3 bar problem.  We re-visit the demonstration problem illustrated in Section 6.2.8.  Consider a structure made of three parallel elastic-plastic bars, with Young’s modulus E, length L, and cross sectional are A, as shown in the figure.  The two bars labeled 1 and 2 have yield stress Y; the central bar (labeled 3) has yield stress 2Y.  The structure is subjected to a cyclic load with mean value P ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGqbGbaebaaaa@33B3@  and amplitude ΔP MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGqbaaaa@3501@ .

 

To obtain an upper bound to the shakedown limit, we must devise a suitable mechanism of plastic flow in the solid.  We could consider three possible mechanisms:

  1. An increment of plastic strain d ε 22 p =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aa0baaSqaaiaaik dacaaIYaaabaGaamiCaaaakiabg2da9iaadsgacqaH1oqzaaa@3B90@  in bars (1) and (2) at the instant of maximum load, followed by d ε 22 p =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aa0baaSqaaiaaik dacaaIYaaabaGaamiCaaaakiabg2da9iabgkHiTiaadsgacqaH1oqz aaa@3C7D@  in bars (1) and (2) at the instant of minimum load.  Since the strain at the end of the cycle vanishes, it is automatically compatible.
  2. An equal increment of plastic strain d ε 22 p =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aa0baaSqaaiaaik dacaaIYaaabaGaamiCaaaakiabg2da9iaadsgacqaH1oqzaaa@3B90@  in all three bars at each instant of maximum load
  3. An equal increment of plastic strain d ε 22 p =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWGKbGaeqyTdu2aa0baaS qaaiaaikdacaaIYaaabaGaamiCaaaakiabg2da9iaadsgacqaH1oqz aaa@3C7D@  at each instant of minimum load.

 

By finding the combination of loads for which

0 T R σ ij ε (t) ε ˙ ^ ij p (t)dVdt 0 T R Y ε ˙ ^ e p (t)dVdt ε ˙ ^ e p = 2 ε ˙ ^ ij p ε ˙ ^ ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaapefabaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaGaeqyTdugaaOGaaiikaiaadshacaGG PaGafqyTduMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWb aaaOGaaiikaiaadshacaGGPaGaamizaiaadAfacaWGKbGaamiDaaWc baGaamOuaaqab0Gaey4kIipaaSqaaiaaicdaaeaacaWGubaaniabgU IiYdGccqGHLjYSdaWdXbqaamaapefabaGaamywaiqbew7aLzaacyaa jaWaa0baaSqaaiaadwgaaeaacaWGWbaaaOGaaiikaiaadshacaGGPa GaamizaiaadAfacaWGKbGaamiDaaWcbaGaamOuaaqab0Gaey4kIipa aSqaaiaaicdaaeaacaWGubaaniabgUIiYdGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlqbew7aLzaacyaajaWaa0baaSqa aiaadwgaaeaacaWGWbaaaOGaeyypa0ZaaOaaaeaacaaIYaGafqyTdu MbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGafqyT duMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaai 4laiaaiodaaSqabaaaaa@9444@

we obtain conditions where shakedown is guaranteed not to occur. Note that the elastic stresses in all three bars are equal, and are given by σ 22 =P(t)/3A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaamiuaiaacIcacaWG0bGaaiykaiaac+cacaaI ZaGaamyqaaaa@3C9A@ . Thus

  1. For mechanism (1): 2( P ¯ +ΔP)Ldε/32( P ¯ ΔP)Ldε/32YLAdε+2YLAdεΔP3AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaaiikaiqadcfagaqeaiabgU caRiabfs5aejaadcfacaGGPaGaamitaiaadsgacqaH1oqzcaGGVaGa aG4maiabgkHiTiaaikdacaGGOaGabmiuayaaraGaeyOeI0IaeuiLdq KaamiuaiaacMcacaWGmbGaamizaiabew7aLjaac+cacaaIZaGaeyyz ImRaaGOmaiaadMfacaWGmbGaamyqaiaadsgacqaH1oqzcqGHRaWkca aIYaGaamywaiaadYeacaWGbbGaamizaiabew7aLjabgkDiElabfs5a ejaadcfacqGHLjYScaaIZaGaamyqaiaadMfaaaa@6087@
  2. For mechanism (2): ( P ¯ +ΔP)Ldε4YLAdε P ¯ +ΔP4AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGabmiuayaaraGaey4kaSIaeu iLdqKaamiuaiaacMcacaWGmbGaamizaiabew7aLjabgwMiZkaaisda caWGzbGaamitaiaadgeacaWGKbGaeqyTduMaeyO0H4Tabmiuayaara Gaey4kaSIaeuiLdqKaamiuaiabgwMiZkaaisdacaWGbbGaamywaaaa @4DA2@
  3. For mechanism (3): ( P ¯ ΔP)Ldε4YLAdε P ¯ ΔP4AY MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaGGOaGabmiuayaaraGaey OeI0IaeuiLdqKaamiuaiaacMcacaWGmbGaamizaiabew7aLjabgwMi ZkaaisdacaWGzbGaamitaiaadgeacaWGKbGaeqyTduMaeyO0H4Tabm iuayaaraGaeyOeI0IaeuiLdqKaamiuaiabgsMiJkabgkHiTiaaisda caWGbbGaamywaaaa@4F81@

 

These agree with the lower bound calculated in Section 6.2.8, and are therefore the exact solution.

 

 

Example 2: Shakedown limit for a pressurized spherical shell.  We consider an elastic-perfectly plastic thick-walled shell, with inner radius a and outer radius b.  The inner wall of the shell is subjected to a cyclic

pressure, with minimum value zero, and maximum value p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@

 

To estimate the shakedown limit we must

1.      Calculate the stresses induced by the pressure in an elastic shell.  The solution can be found in Section 4.1.4. 

σ RR = p a a 3 ( b 3 a 3 ) ( 1 b 3 R 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOuaiaadk faaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadggaaeqa aOGaamyyamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyam aaCaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbeqaaiaa iodaaaaakiaawIcacaGLPaaaaaGaaGPaVpaabmaabaGaaGymaiabgk HiTmaalaaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaadkfa daahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaGaaGPaVdaa@4B96@   σ θθ = σ ϕϕ = p a a 3 ( b 3 a 3 ) ( 1+ b 3 2 R 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqiUdeNaeq iUdehabeaakiabg2da9iabeo8aZnaaBaaaleaacqaHvpGzcqaHvpGz aeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO GaamyyamaaCaaaleqabaGaaG4maaaaaOqaamaabmaabaGaamOyamaa CaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWcbeqaaiaaio daaaaakiaawIcacaGLPaaaaaGaaGPaVpaabmaabaGaaGymaiabgUca RmaalaaabaGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaaikdaca WGsbWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaiaaykW7 aaa@5494@

  1. Postulate a mechanism of steady-state plastic deformation in the shell.  For example, consider a mechanism consisting of a uniform plastic strain increment d ε rr =2dεd ε ϕϕ =d ε θθ =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aaSbaaSqaaiaadk hacaWGYbaabeaakiabg2da9iabgkHiTiaaikdacaWGKbGaeqyTduMa aGPaVlaaykW7caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeq y1dyMaeqy1dygabeaakiabg2da9iaadsgacqaH1oqzdaWgaaWcbaGa eqiUdeNaeqiUdehabeaakiabg2da9iaadsgacqaH1oqzcaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7aaa@5D4B@  which occurs in a spherical shell with radius a very small thickness dt at the instant of maximum pressure, followed by a strain d ε rr =2dεd ε ϕϕ =d ε θθ =dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aaSbaaSqaaiaadk hacaWGYbaabeaakiabg2da9iaaikdacaWGKbGaeqyTduMaaGPaVlaa ykW7caaMc8UaaGPaVlaadsgacqaH1oqzdaWgaaWcbaGaeqy1dyMaeq y1dygabeaakiabg2da9iaadsgacqaH1oqzdaWgaaWcbaGaeqiUdeNa eqiUdehabeaakiabg2da9iabgkHiTiaadsgacqaH1oqzcaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7aaa@5D4B@  at the instant of minimum load.

The upper bound theorem states that shakedown will not occur if

0 T R σ ij ε (t) ε ˙ ^ ij p (t)dVdt 0 T R Y ε ˙ ^ e p (t)dVdt ε ˙ ^ e p = 2 ε ˙ ^ ij p ε ˙ ^ ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaapefabaGaeq4Wdm3aa0 baaSqaaiaadMgacaWGQbaabaGaeqyTdugaaOGaaiikaiaadshacaGG PaGafqyTduMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWb aaaOGaaiikaiaadshacaGGPaGaamizaiaadAfacaWGKbGaamiDaaWc baGaamOuaaqab0Gaey4kIipaaSqaaiaaicdaaeaacaWGubaaniabgU IiYdGccqGHLjYSdaWdXbqaamaapefabaGaamywaiqbew7aLzaacyaa jaWaa0baaSqaaiaadwgaaeaacaWGWbaaaOGaaiikaiaadshacaGGPa GaamizaiaadAfacaWGKbGaamiDaaWcbaGaamOuaaqab0Gaey4kIipa aSqaaiaaicdaaeaacaWGubaaniabgUIiYdGccaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlqbew7aLzaacyaajaWaa0baaSqa aiaadwgaaeaacaWGWbaaaOGaeyypa0ZaaOaaaeaacaaIYaGafqyTdu MbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGafqyT duMbaiGbaKaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaai 4laiaaiodaaSqabaaaaa@9444@

Substituting the elastic stress field and the strain rate shows that

4π a 2 t 3 p a b 3 ( b 3 a 3 ) dε4π a 2 tY2dε+4π a 2 tY2dε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaI0aGaeqiWdaNaamyyamaaCaaale qabaGaaGOmaaaakiaadshadaWcaaqaaiaaiodacaWGWbWaaSbaaSqa aiaadggaaeqaaOGaamOyamaaCaaaleqabaGaaG4maaaaaOqaaiaacI cacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamyyamaaCaaa leqabaGaaG4maaaakiaacMcaaaGaamizaiabew7aLjabgwMiZkaais dacqaHapaCcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaamiDaiaadMfa caaIYaGaamizaiabew7aLjabgUcaRiaaisdacqaHapaCcaWGHbWaaW baaSqabeaacaaIYaaaaOGaamiDaiaadMfacaaIYaGaamizaiabew7a Lbaa@5AED@

This gives p a /Y<4(1 a 3 / b 3 )/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaO Gaai4laiaadMfacqGH8aapcaaI0aGaaiikaiaaigdacqGHsislcaWG HbWaaWbaaSqabeaacaaIZaaaaOGaai4laiaadkgadaahaaWcbeqaai aaiodaaaGccaGGPaGaai4laiaaiodaaaa@4103@  for the shakedown limit.  Again, this agrees with the lower bound, and is therefore the exact solution.