Chapter 6

 

Analytical techniques and solutions for plastic solids

 

 

 

Plasticity problems are much more difficult to solve than linear elastic problems.  In general, a numerical method must be used, as discussed in Chapters 7 and 8. Nevertheless, there are several powerful mathematical techniques that can be used to find both exact and approximate solutions.  In this chapter we outline two particularly effective methods: slip-line field theory, which gives exact solutions for plane strain boundary value problems for rigid plastic solids; and bounding theorems, which provide a quick way to estimate collapse loads for plastic solids and structures.

 

 

6.1 Slip-line field theory

 

The largest class of solutions to boundary value problems in plasticity exploits a technique known as slip line field theory.  The theory simplifies the governing equations for plastic solids by making several restrictive assumptions:

1.      Plane strain deformation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  i.e. displacement components in the basis shown satisfy u 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0JaaGimaaaa@3683@   and u 1 , u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadwhadaWgaaWcbaGaaGOmaaqabaaaaa@3753@  are functions of x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaa aa@34BA@  and x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaa aa@34BB@  only

2.      Quasi-static loading

3.      No temperature changes

4.      No body forces

5.      The solid is idealized as a rigid-perfectly plastic Mises solid. The uniaxial stress-strain curve for this material is illustrated in the figure.  The material properties are characterized by the yield stress in uniaxial tension Y.  Alternatively, the material is sometimes characterized by its yield stress in shear k=Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbGaeyypa0Jaamywaiaac+cada GcaaqaaiaaiodaaSqabaaaaa@3725@ .

 

Otherwise, the technique can be used to solve any arbitrary 2D boundary value problem for a rigid plastic solid.  It is quite difficult to apply in practice, because it is not easy to find the slip-line field that solves a particular problem.  Nevertheless, a wide range of important solutions have been found.  The main intent of this section is to illustrate how to interpret these solutions, and to outline the basis for slip-line field theory.

 

 

 

6.1.1 Interpreting a slip-line field

 

An example of a slip-line field solution is shown in the picture on the right. (This is Hill’s solution to a rigid punch indenting a rigid-plastic half-space).  

 

The slip lines consist of a curvilinear mesh of two families of lines, which always cross each other at right angles.  By convention, one set of lines are named α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  slip-lines (shown in red); the other are called β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  lines (blue).  The velocity distribution and stress state in the solid can always be determined from the geometry of these lines.

 

 

Stress state at a point in the slip-line field

 

By definition, the slip-lines are always parallel to axes of principal shear stress in the solid.  This means that the stress components in a basis oriented with the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  directions have the form

σ αα = σ ¯ σ ββ = σ ¯ σ αβ =k=Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqySdeMaeq ySdegabeaakiabg2da9iqbeo8aZzaaraGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZn aaBaaaleaacqaHYoGycqaHYoGyaeqaaOGaeyypa0Jafq4WdmNbaeba caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacqaHXoqycqaHYoGy aeqaaOGaeyypa0Jaam4Aaiabg2da9iaadMfacaGGVaWaaOaaaeaaca aIZaaaleqaaaaa@6DFF@

where σ ¯ =( σ αα + σ ββ )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeaiabg2da9maabmaaba Gaeq4Wdm3aaSbaaSqaaiabeg7aHjabeg7aHbqabaGccqGHRaWkcqaH dpWCdaWgaaWcbaGaeqOSdiMaeqOSdigabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@43F3@  is the hydrostatic stress (determined using the equations given below), k is the yield stress of the material in shear, and Y is its yield stress in uniaxial tension.  This stress state is sketched in the figure.  Since the shear stress is equal to the shear yield stress, the material evidently deforms by shearing parallel to the slip-lines: this is the reason for their name.

 

If ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@349E@  denotes the angle between the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@  slip-line and the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@34AB@  direction, the stress components in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis can be calculated as

σ 11 = σ ¯ ksin2ϕ σ 22 = σ ¯ +ksin2ϕ σ 12 =kcos2ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHdp WCdaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0Jafq4WdmNbaeba cqGHsislcaWGRbGaci4CaiaacMgacaGGUbGaaGOmaiabew9aMbqaai abeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpcuaHdpWC gaqeaiabgUcaRiaadUgaciGGZbGaaiyAaiaac6gacaaIYaGaeqy1dy gabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaa dUgaciGGJbGaai4BaiaacohacaaIYaGaeqy1dygaaaa@5CCB@

The Mohr’s circle construction (shown in the picture to the right) is a convenient way to remember these results.

 

 

Relations governing hydrostatic stress along slip-lines (Hencky equations)

 

The hydrostatic stress can be shown to satisfy the following relations along slip-lines

σ ¯ 2kϕ=constantα slip line σ ¯ +2kϕ=constantβ slip line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaHdp WCgaqeaiabgkHiTiaaikdacaWGRbGaeqy1dyMaeyypa0Jaae4yaiaa b+gacaqGUbGaae4CaiaabshacaqGHbGaaeOBaiaabshacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeqySdeMaaeiiaiaabohacaqGSbGaaeyAaiaabchacaqGGaGaae iBaiaabMgacaqGUbGaaeyzaaqaaiqbeo8aZzaaraGaey4kaSIaaGOm aiaadUgacqaHvpGzcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaae iDaiaabggacaqGUbGaaeiDaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHYoGycaqGGaGaae 4CaiaabYgacaqGPbGaaeiCaiaabccacaqGSbGaaeyAaiaab6gacaqG Lbaaaaa@8712@

If the hydrostatic stress can be determined at any one point on a slip-line (for example at a boundary), it can be deduced everywhere else. Note that if there is a region in the field where both slip lines are straight, the stress is constant.

 

 

The velocity field (Geiringer equations)

 

The velocity field can be expressed as components in a fixed { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiyFaaaa @3945@  basis, or as components parallel and perpendicular to the slip lines. 

d v 1 ds +tanϕ d v 2 ds =0 d v α ds = v β dϕ ds }αslip line d v 1 ds cotϕ d v 2 ds =0 d v β ds = v α dϕ ds }βslip line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaafa qabeGabaaabaWaaSaaaeaacaWGKbGaamODamaaBaaaleaacaaIXaaa beaaaOqaaiaadsgacaWGZbaaaiabgUcaRiGacshacaGGHbGaaiOBai abew9aMnaalaaabaGaamizaiaadAhadaWgaaWcbaGaaGOmaaqabaaa keaacaWGKbGaam4CaaaacqGH9aqpcaaIWaaabaWaaSaaaeaacaWGKb GaamODamaaBaaaleaacqaHXoqyaeqaaaGcbaGaamizaiaadohaaaGa eyypa0JaamODamaaBaaaleaacqaHYoGyaeqaaOWaaSaaaeaacaWGKb Gaeqy1dygabaGaamizaiaadohaaaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7aaaacaGL9baacaaMc8UaeqySdeMaaGPaVlaabohacaqGSb GaaeyAaiaabchacaqGGaGaaeiBaiaabMgacaqGUbGaaeyzaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7daGacaqaauaabeqaceaaaeaadaWc aaqaaiaadsgacaWG2bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizai aadohaaaGaeyOeI0Iaci4yaiaac+gacaGG0bGaeqy1dy2aaSaaaeaa caWGKbGaamODamaaBaaaleaacaaIYaaabeaaaOqaaiaadsgacaWGZb aaaiabg2da9iaaicdaaeaadaWcaaqaaiaadsgacaWG2bWaaSbaaSqa aiabek7aIbqabaaakeaacaWGKbGaam4CaaaacqGH9aqpcqGHsislca WG2bWaaSbaaSqaaiabeg7aHbqabaGcdaWcaaqaaiaadsgacqaHvpGz aeaacaWGKbGaam4CaaaacaaMc8UaaGPaVlaaykW7caaMc8oaaaGaay zFaaGaaGPaVlabek7aIjaaykW7caqGZbGaaeiBaiaabMgacaqGWbGa aeiiaiaabYgacaqGPbGaaeOBaiaabwgaaaa@B3B8@

 

 

 

Application to the Hill slip-line field

 

The stress state throughout a slip-line field can be deduced by working step-by-step along the slip lines.  We illustrate the procedure using Hill’s indentation solution.

 

Consider first the state of stress at point a.  Clearly, ϕ a =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadggaaeqaaOGaeyypa0JaeqiWdaNaai4laiaaisdaaaa@3E15@  at this point. The stress state can be transformed from a basis aligned with the slip-lines to the fixed { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiyFaaaa @3945@  basis using the Mohr’s circle construction shown in the figure.  Recall (or use the Mohr’s circle to see) that

σ 11 = σ ¯ ksin2ϕ σ 22 = σ ¯ +ksin2ϕ σ 12 =kcos2ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iqbeo8aZzaaraGaeyOe I0Iaam4AaiGacohacaGGPbGaaiOBaiaaikdacqaHvpGzcaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqa aiaaikdacaaIYaaabeaakiabg2da9iqbeo8aZzaaraGaey4kaSIaam 4AaiGacohacaGGPbGaaiOBaiaaikdacqaHvpGzcaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaaGymaiaaik daaeqaaOGaeyypa0Jaam4AaiGacogacaGGVbGaai4CaiaaikdacqaH vpGzaaa@70D2@

where σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@  is the hydrostatic component of stress.  The boundary conditions at a require that σ 12 = σ 22 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIYaaabeaakiabg2da9iabeo8aZnaaBaaaleaa caaIYaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@40A4@ .  The first condition is clearly satisfied, since the slip-lines intersect the boundary at ϕ=π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcqaHapaCcaGGVa GaaGinaaaa@38C2@ .  We can satisfy the second condition by setting σ ¯ =k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae bacqGH9aqpcqGHsislcaWGRbaaaa@3AA4@ .  Finally this gives the stress parallel to the surface as σ 11 =2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iabgkHiTiaaikdacaWG Rbaaaa@3E0B@

 

The stress must be constant in the triangular region ABC, as the slip lines in this region are straight.

 

Next, consider the stress state at b.  Here, we see that ϕ b =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dy2aaS baaSqaaiaadkgaaeqaaOGaeyypa0JaeyOeI0IaeqiWdaNaai4laiaa isdaaaa@3F03@ .  We can use the Hencky equation to determine σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@  at b.  Recall that

σ ¯ 2kϕ=constantα slip line σ ¯ +2kϕ=constantβ slip line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaHdp WCgaqeaiabgkHiTiaaikdacaWGRbGaeqy1dyMaeyypa0Jaae4yaiaa b+gacaqGUbGaae4CaiaabshacaqGHbGaaeOBaiaabshacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeqySdeMaaeiiaiaabohacaqGSbGaaeyAaiaabchacaqGGaGaae iBaiaabMgacaqGUbGaaeyzaaqaaiqbeo8aZzaaraGaey4kaSIaaGOm aiaadUgacqaHvpGzcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaae iDaiaabggacaqGUbGaaeiDaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHYoGycaqGGaGaae 4CaiaabYgacaqGPbGaaeiCaiaabccacaqGSbGaaeyAaiaab6gacaqG Lbaaaaa@8702@

so following one of the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  slip lines we get

σ ¯ b 2k ϕ b = σ ¯ a 2k ϕ a σ ¯ b =kπk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaHdp WCgaqeamaaBaaaleaacaWGIbaabeaakiabgkHiTiaaikdacaWGRbGa eqy1dy2aaSbaaSqaaiaadkgaaeqaaOGaeyypa0Jafq4WdmNbaebada WgaaWcbaGaamyyaaqabaGccqGHsislcaaIYaGaam4Aaiabew9aMnaa BaaaleaacaWGHbaabeaaaOqaaiabgkDiElqbeo8aZzaaraWaaSbaaS qaaiaadkgaaeqaaOGaeyypa0JaeyOeI0Iaam4AaiabgkHiTiabec8a WjaadUgaaaaa@54C6@

Using the basis-change equation we then get

σ 11 =πk σ 22 =(π+2)k σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iabgkHiTiabec8aWjaa dUgacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpcqGH sislcaGGOaGaeqiWdaNaey4kaSIaaGOmaiaacMcacaWGRbGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaa leaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@6669@

The pressure under the punch turns out to be uniform (the stress is constant in the triangular region of the slip-line field below the punch) and so the total force (per unit out of plane length) on the punch can be computed as

P=w(2+π)k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabg2 da9iaadEhacaGGOaGaaGOmaiabgUcaRiabec8aWjaacMcacaWGRbaa aa@3E71@

where w is the width of the punch.

 

 

 

How to distinguish the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  families of slip lines

 

Usually, slip-line fields are presented without specifying which set of slip-lines should be taken as the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  and which should be the β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  set MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it is up to you to work out which is which.  In fact, the slip-lines are interchangeable MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  switching α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  will simply change the sign of all the stresses.

 

You can see this clearly using the Hill solution.  The figure on the right shows the solution with α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3795@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  lines switched over.  At point a, ϕ=3π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcaaIZaGaeqiWda Naai4laiaaisdaaaa@398F@ , and therefore to satisfy σ 22 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaaGimaaaa@3807@  we must now choose σ ¯ =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeaiabg2da9iaadUgaaa a@36A7@ .  To find the stress under the contact, we can trace a β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@  slip line to point b. Here, we see that ϕ=π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcqGH9aqpcqaHapaCcaGGVa GaaGinaaaa@38D2@ , so the Hencky equation

σ ¯ b +2k ϕ b = σ ¯ a +2k ϕ a σ ¯ b =k+πk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaHdp WCgaqeamaaBaaaleaacaWGIbaabeaakiabgUcaRiaaikdacaWGRbGa eqy1dy2aaSbaaSqaaiaadkgaaeqaaOGaeyypa0Jafq4WdmNbaebada WgaaWcbaGaamyyaaqabaGccqGHRaWkcaaIYaGaam4Aaiabew9aMnaa BaaaleaacaWGHbaabeaaaOqaaiabgkDiElqbeo8aZzaaraWaaSbaaS qaaiaadkgaaeqaaOGaeyypa0Jaam4AaiabgUcaRiabec8aWjaadUga aaaa@53B8@

Using the basis-change equation we then get

σ 11 =πk σ 22 =(π+2)k σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iabec8aWjaadUgacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcba GaaGOmaiaaikdaaeqaaOGaeyypa0Jaaiikaiabec8aWjabgUcaRiaa ikdacaGGPaGaam4AaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq 4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaaicdaaaa@5E63@

at point b.  The normal stress acts upwards on the surface MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  so that this represents the stress induced by a rigid punch that is bonded to the surface, and pulled upwards.

 

 

 

 

6.1.2 Derivation of the slip-line field method.

 

Consider a rigid-perfectly plastic solid, with a von-Mises yield surface characterized by yield stress in uniaxial tension Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGzbaaaa@33B4@  or its yield stress in shear k=Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbGaeyypa0Jaamywaiaac+cada GcaaqaaiaaiodaaSqabaaaaa@3735@ .   Let u i , ε ij , σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaiabew7aLnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaaaaa@3DDA@  denote the components of displacement, strain and stress in the solid. The solid is assumed to be a long cylinder with its axis parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@34AD@  direction, which is constrained to deform in plane strain, with u 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0JaaGimaaaa@3683@  and u 1 , u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadwhadaWgaaWcbaGaaGOmaaqabaaaaa@3753@  independent of x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34BC@ .  It is loaded by subjecting part of its boundary 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@3604@  to a prescribed velocity, and the remainder 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@3605@  to a prescribed traction, so that

u ˙ α = v α ( x 1 , x 2 )on  1 R σ αβ n α = t β ( x 1 , x 2 )on  2 R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaaceWG1bGbaiaadaWgaaWcbaGaeq ySdegabeaakiabg2da9iaadAhadaqhaaWcbaGaeqySdegabaGaey4f IOcaaOGaaiikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam iEamaaBaaaleaacaaIYaaabeaakiaacMcacaaMcSUaaGPaRlaaykW6 caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRl aaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUa aGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6ca aMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caqGVbGaaeOBaiaa bccacqGHciITdaWgaaWcbaGaaGymaaqabaGccaWGsbaabaGaeq4Wdm 3aaSbaaSqaaiabeg7aHjabek7aIbqabaGccaWGUbWaaSbaaSqaaiab eg7aHbqabaGccqGH9aqpcaWG0bWaa0baaSqaaiabek7aIbqaaiabgE HiQaaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa dIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaaGPaRlaaykW6caaMcS UaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6 caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRl aaykW6caaMcSUaaGPaRlaaykW6caaMcSUaae4Baiaab6gacaqGGaGa eyOaIy7aaSbaaSqaaiaaikdaaeqaaOGaamOuaaaaaa@B37E@

where the Greek subscripts α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGSaGaeqOSdigaaa@36C6@  can have values of 1 or 2. In practice we will compute the velocity field v i = u ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGPbaabeaakiabg2da9iqadwhagaGaamaaBaaaleaacaWG Pbaabeaaaaa@3C4F@  rather than the displacement field.

 

 

Summary of governing equations

 

1.      Strain-rate MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  velocity relation ε ˙ αβ = 1 2 ( v α / x β + v β / x α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaeqySdeMaeqOSdigabeaakiabg2da9maalaaabaGa aGymaaqaaiaaikdaaaWaaeWaaeaacqGHciITcaWG2bWaaSbaaSqaai abeg7aHbqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaHYoGy aeqaaOGaey4kaSIaeyOaIyRaamODamaaBaaaleaacqaHYoGyaeqaaO Gaai4laiabgkGi2kaadIhadaWgaaWcbaGaeqySdegabeaaaOGaayjk aiaawMcaaaaa@5361@

2.      The plastic flow rule ε ˙ ij =3 ε ¯ ˙ p S ij /2Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaG4maiqbew7a LzaaryaacaWaaWbaaSqabeaacaWGWbaaaOGaam4uamaaBaaaleaaca WGPbGaamOAaaqabaGccaGGVaGaaGOmaiaadMfaaaa@44AE@

Plane strain deformation then requires

ε ˙ 33 =3 ε ¯ ˙ p [ σ 33 ( σ 11 + σ 22 + σ 33 )/3]/2Y=0 σ 33 =( σ 11 + σ 22 )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaH1o qzgaGaamaaBaaaleaacaaIZaGaaG4maaqabaGccqGH9aqpcaaIZaGa fqyTduMbaeHbaiaadaahaaWcbeqaaiaadchaaaGccaGGBbGaeq4Wdm 3aaSbaaSqaaiaaiodacaaIZaaabeaakiabgkHiTiaacIcacqaHdpWC daWgaaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaS qaaiaaikdacaaIYaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaI ZaGaaG4maaqabaGccaGGPaGaai4laiaaiodacaGGDbGaai4laiaaik dacaWGzbGaeyypa0JaaGimaaqaaiabgkDiElabeo8aZnaaBaaaleaa caaIZaGaaG4maaqabaGccqGH9aqpcaGGOaGaeq4Wdm3aaSbaaSqaai aaigdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGa aGOmaaqabaGccaGGPaGaai4laiaaikdaaaaa@6987@

whereupon the flow rule shows that the remaining components of plastic strain rate satisfy

ε ˙ 11 =3 ε ¯ ˙ p ( σ 11 ( σ 11 + σ 22 + σ 33 )/3 )/2Y=3 ε ¯ ˙ p ( σ 11 σ 22 )/4Y ε ˙ 22 =3 ε ¯ ˙ p ( σ 22 ( σ 11 + σ 22 + σ 33 )/3 )/2Y=3 ε ¯ ˙ p ( σ 22 σ 11 )/4Y ε ˙ 12 =3 ε ¯ ˙ p σ 12 /2Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaH1o qzgaGaamaaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaaIZaGa fqyTduMbaeHbaiaadaahaaWcbeqaaiaadchaaaGcdaqadaqaaiabeo 8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsislcaGGOaGaeq4W dm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBa aaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGa aG4maiaaiodaaeqaaOGaaiykaiaac+cacaaIZaaacaGLOaGaayzkaa Gaai4laiaaikdacaWGzbGaeyypa0JaaG4maiqbew7aLzaaryaacaWa aWbaaSqabeaacaWGWbaaaOGaaiikaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaaGOmaiaaikda aeqaaOGaaiykaiaac+cacaaI0aGaamywaaqaaiqbew7aLzaacaWaaS baaSqaaiaaikdacaaIYaaabeaakiabg2da9iaaiodacuaH1oqzgaqe gaGaamaaCaaaleqabaGaamiCaaaakmaabmaabaGaeq4Wdm3aaSbaaS qaaiaaikdacaaIYaaabeaakiabgkHiTiaacIcacqaHdpWCdaWgaaWc baGaaGymaiaaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaik dacaaIYaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIZaGaaG4m aaqabaGccaGGPaGaai4laiaaiodaaiaawIcacaGLPaaacaGGVaGaaG OmaiaadMfacqGH9aqpcaaIZaGafqyTduMbaeHbaiaadaahaaWcbeqa aiaadchaaaGccaGGOaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabe aakiabgkHiTiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccaGG PaGaai4laiaaisdacaWGzbaabaGafqyTduMbaiaadaWgaaWcbaGaaG ymaiaaikdaaeqaaOGaeyypa0JaaG4maiqbew7aLzaaryaacaWaaWba aSqabeaacaWGWbaaaOGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabe aakiaac+cacaaIYaGaamywaaaaaa@A377@

We observe that these conditions imply that

ε ˙ 11 + ε ˙ 22 =0 v 1 / x 1 + v 2 / x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIafqyTduMbaiaa daWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaaGimaiaaykW7ca aMc8UaaGPaVlabgkDiElabgkGi2kaadAhadaWgaaWcbaGaaGymaaqa baGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabgU caRiabgkGi2kaadAhadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaeyOa IyRaamiEamaaBaaaleaacaaIYaaabeaakiabg2da9iaaicdaaaa@58AD@

ε ˙ 11 ε ˙ 22 σ 11 σ 22 = ε ˙ 12 σ 12 ( ε ˙ 11 ε ˙ 22 ) σ 12 = ε ˙ 12 ( σ 11 σ 22 ) ( v 1 / x 1 v 2 / x 2 ) σ 12 = 1 2 ( v 1 / x 2 + v 2 / x 1 )( σ 11 σ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiqbew7aLzaacaWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHi Tiqbew7aLzaacaWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabeo 8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsislcqaHdpWCdaWg aaWcbaGaaGOmaiaaikdaaeqaaaaakiabg2da9maalaaabaGafqyTdu MbaiaadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeq4Wdm3aaSba aSqaaiaaigdacaaIYaaabeaaaaGccaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiEpaabmaabaGafqyTduMb aiaadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOeI0IafqyTduMbai aadaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeq4W dm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iqbew7aLzaaca WaaSbaaSqaaiaaigdacaaIYaaabeaakmaabmaabaGaeq4Wdm3aaSba aSqaaiaaigdacaaIXaaabeaakiabgkHiTiabeo8aZnaaBaaaleaaca aIYaGaaGOmaaqabaaakiaawIcacaGLPaaaaeaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7cqGHshI3caaMc8+aaeWaaeaacq GHciITcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiabgkGi2kaa dIhadaWgaaWcbaGaaGymaaqabaGccqGHsislcqGHciITcaWG2bWaaS baaSqaaiaaikdaaeqaaOGaai4laiabgkGi2kaadIhadaWgaaWcbaGa aGOmaaqabaaakiaawIcacaGLPaaacqaHdpWCdaWgaaWcbaGaaGymai aaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqa daqaaiabgkGi2kaadAhadaWgaaWcbaGaaGymaaqabaGccaGGVaGaey OaIyRaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiabgkGi2kaa dAhadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaeyOaIyRaamiEamaaBa aaleaacaaIXaaabeaaaOGaayjkaiaawMcaamaabmaabaGaeq4Wdm3a aSbaaSqaaiaaigdacaaIXaaabeaakiabgkHiTiabeo8aZnaaBaaale aacaaIYaGaaGOmaaqabaaakiaawIcacaGLPaaaaaaa@E92A@

3.      Yield criterion

3 2 S ij S ij Y=0 1 4 ( σ 11 σ 22 ) 2 + σ 12 2 = k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada WcaaqaaiaaiodaaeaacaaIYaaaaiaadofadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaaabeaaki abgkHiTiaadMfacqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaeyO0 H4TaaGPaVlaaykW7caaMc8+aaSaaaeaacaaIXaaabaGaaGinaaaada qadaqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsisl cqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaa WaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeq4Wdm3aa0baaSqaaiaa igdacaaIYaaabaGaaGOmaaaakiabg2da9iaadUgadaahaaWcbeqaai aaikdaaaaaaa@6156@

where k=Y/ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 da9iaadMfacaGGVaWaaOaaaeaacaaIZaaaleqaaaaa@3A45@  is the shear yield stress of the material, and we have used the condition that σ 33 =( σ 11 + σ 22 )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaiodacaaIZaaabeaakiabg2da9iaacIcacqaHdpWCdaWg aaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaai aaikdacaaIYaaabeaakiaacMcacaGGVaGaaGOmaaaa@4600@

4.      Equilibrium conditions

σ ij / x i =0 σ 11 / x 1 + σ 12 / x 2 =0 σ 22 / x 2 + σ 21 / x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqGHci ITcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4laiabgkGi 2kaadIhadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIWaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaeyOaIyRaeq4Wdm3aaSbaaSqaai aaigdacaaIXaaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaa igdaaeqaaOGaey4kaSIaaGPaVlabgkGi2kabeo8aZnaaBaaaleaaca aIXaGaaGOmaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaaI Yaaabeaakiabg2da9iaaicdaaeaacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaeyOaIyRaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabe aakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4k aSIaaGPaVlabgkGi2kabeo8aZnaaBaaaleaacaaIYaGaaGymaaqaba GccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabg2da 9iaaicdaaaaa@C9E5@

 

Solution of governing equations by method of characteristics

 

From the preceding section, we observe that we must calculate a velocity field v α ( x α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacqaHXoqyaeqaaOGaaiikaiaadIhadaWgaaWcbaGaeqySdega beaakiaacMcaaaa@3DF8@  and stress field σ αβ ( x α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiabeg7aHjabek7aIbqabaGccaGGOaGaamiEamaaBaaaleaa cqaHXoqyaeqaaOGaaiykaaaa@4061@  satisfying governing equations

1 4 ( σ 11 σ 22 ) 2 + σ 12 2 = k 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVlaayk W7caaMc8+aaSaaaeaacaaIXaaabaGaaGinaaaadaqadaqaaiabeo8a ZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaa WcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaaba GaaGOmaaaakiabg2da9iaadUgadaahaaWcbeqaaiaaikdaaaaaaa@4F5E@

v 1 / x 1 + v 2 / x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaam ODamaaBaaaleaacaaIXaaabeaakiaac+cacqGHciITcaWG4bWaaSba aSqaaiaaigdaaeqaaOGaey4kaSIaeyOaIyRaamODamaaBaaaleaaca aIYaaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0JaaGimaaaa@4853@

( v 1 / x 1 v 2 / x 2 ) σ 12 = 1 2 ( v 1 / x 2 + v 2 / x 1 )( σ 11 σ 22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGPaVpaabm aabaGaeyOaIyRaamODamaaBaaaleaacaaIXaaabeaakiaac+cacqGH ciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaeyOaIyRaam ODamaaBaaaleaacaaIYaaabeaakiaac+cacqGHciITcaWG4bWaaSba aSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeq4Wdm3aaSbaaSqaai aaigdacaaIYaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaWaaeWaaeaacqGHciITcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaai 4laiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqGH ciITcaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaai4laiabgkGi2kaadI hadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaadaqadaqaaiab eo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHsislcqaHdpWCda WgaaWcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@6A24@

σ 11 / x 1 + σ 12 / x 2 =0 σ 22 / x 2 + σ 21 / x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaaMc8 UaeyOaIyRaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiaac+ca cqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaGPaVl abgkGi2kabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaaIYaaabeaakiabg2da9iaaicdaae aacaaMc8UaeyOaIyRaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaa kiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaS IaaGPaVlabgkGi2kabeo8aZnaaBaaaleaacaaIYaGaaGymaaqabaGc caGGVaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaakiabg2da9i aaicdaaaaa@65EA@

together with appropriate boundary conditions.

 

We focus first on a general solution to the governing equations.  It is convenient to start by eliminating some of the stress components using the yield condition.  Since the material is at yield, we note that at each point in the solid we could find a basis in which the stress state consists of a shear stress of magnitude k (the shear yield stress), together with an unknown component of hydrostatic stress σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@ .  The stress state is sketched on the right.

 

Instead of solving for the stress components σ αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiabeg7aHjabek7aIbqabaaaaa@3C3C@ , we will calculate the hydrostatic stress σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@  and the angle ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BE@  between the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaahgdaaeqaaa aa@34AA@  direction and the m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyBamaaBa aaleaacaaIXaaabeaaaaa@37D3@  direction.   Recall that we can relate σ αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiabeg7aHjabek7aIbqabaaaaa@3C3C@  to σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@ , ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BE@  and k using Mohr’s circle of stress: from the picture, we see that

σ 11 = σ ¯ ksin2ϕ σ 22 = σ ¯ +ksin2ϕ σ 12 =kcos2ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iqbeo8aZzaaraGaeyOe I0Iaam4AaiGacohacaGGPbGaaiOBaiaaikdacqaHvpGzcaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGaaGOm aaqabaGccqGH9aqpcuaHdpWCgaqeaiabgUcaRiaadUgaciGGZbGaai yAaiaac6gacaaIYaGaeqy1dyMaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqaba GccqGH9aqpcaWGRbGaci4yaiaac+gacaGGZbGaaGOmaiabew9aMbaa @6F47@

 

We now re-write the governing equations in terms of σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37D1@ , ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BE@  and k.  The yield criterion is satisfied automatically.  The remaining four equations are most conveniently expressed in matrix form

A ij q j x 1 + B ij q j x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa aaleaacaWGPbGaamOAaaqabaGcdaWcaaqaaiabgkGi2kaadghadaWg aaWcbaGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaig daaeqaaaaakiabgUcaRiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aOWaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadQgaaeqaaaGcba GaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGccqGH9aqpcaaI Waaaaa@4D1D@

where A and B are 4-dimensional symmetric matrices and q is a 1x4 vector, defined as

q=[ ϕ v 1 v 2 σ ¯ ]A=[ 0 2kcos2ϕ 2ksin2ϕ 0 2kcos2ϕ 0 0 1 2ksin2ϕ 0 0 0 0 1 0 0 ]B=[ 0 2ksin2ϕ 2kcos2ϕ 0 2ksin2ϕ 0 0 0 2kcos2ϕ 0 0 1 0 0 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2 da9maadmaabaqbaeqabqqaaaaabaGaeqy1dygabaGaamODamaaBaaa leaacaaIXaaabeaaaOqaaiaadAhadaWgaaWcbaGaaGOmaaqabaaake aacuaHdpWCgaqeaaaaaiaawUfacaGLDbaacaaMc8UaaGPaVlaaykW7 caaMc8Uaamyqaiabg2da9maadmaabaqbaeqabqabaaaaaeaacaaIWa aabaGaeyOeI0IaaGOmaiaadUgaciGGJbGaai4BaiaacohacaaIYaGa eqy1dygabaGaeyOeI0IaaGOmaiaadUgaciGGZbGaaiyAaiaac6gaca aIYaGaeqy1dygabaGaaGimaaqaaiabgkHiTiaaikdacaWGRbGaci4y aiaac+gacaGGZbGaaGOmaiabew9aMbqaaiaaicdaaeaacaaIWaaaba GaaGymaaqaaiabgkHiTiaaikdacaWGRbGaci4CaiaacMgacaGGUbGa aGOmaiabew9aMbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaic daaeaacaaIXaaabaGaaGimaaqaaiaaicdaaaaacaGLBbGaayzxaaGa aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGcbGaeyypa0ZaamWaae aafaqabeabeaaaaaqaaiaaicdaaeaacqGHsislcaaIYaGaam4AaiGa cohacaGGPbGaaiOBaiaaikdacqaHvpGzaeaacaaIYaGaam4AaiGaco gacaGGVbGaai4CaiaaikdacqaHvpGzaeaacaaIWaaabaGaeyOeI0Ia aGOmaiaadUgaciGGZbGaaiyAaiaac6gacaaIYaGaeqy1dygabaGaaG imaaqaaiaaicdaaeaacaaIWaaabaGaaGOmaiaadUgaciGGJbGaai4B aiaacohacaaIYaGaeqy1dygabaGaaGimaaqaaiaaicdaaeaacaaIXa aabaGaaGimaaqaaiaaicdaaeaacaaIXaaabaGaaGimaaaaaiaawUfa caGLDbaaaaa@A746@

 

This is a quasi-linear hyperbolic system of PDEs, which may be solved by the method of characteristics. 

 

The first step is to find eigenvalues μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AC@  and eigenvectors r i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaaaaa@391E@  that satisfy

r i A ij =μ r i B ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadgeadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaeyypa0JaeqiVd0MaamOCamaaBaaaleaacaWGPbaabeaakiaadk eadaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@43A8@

A straightforward exercise (set det(AμB)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciizaiaacw gacaGG0bGaaiikaiaadgeacqGHsislcqaH8oqBcaWGcbGaaiykaiab g2da9iaaicdaaaa@400A@  to find the eigenvalues, and substitute back to get eigenvectors, or if you’re lazy use a symbolic manipulation program…) shows that there are two repeated eigenvalues, with corresponding eigenvectors

μ=cotϕ{ r=[1,0,0,2k] r=[0,1,tanϕ,0] μ=tanϕ{ r=[1,0,0,2k] r=[0,1,cotϕ,0] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0Jaci4yaiaac+gacaGG0bGaeqy1dyMaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7daGabaqaauaabeqaceaaaeaacaWGYbGaeyypa0Jaai 4waiaaigdacaGGSaGaaGimaiaacYcacaaIWaGaaiilaiabgkHiTiaa ikdacaWGRbGaaiyxaaqaaiaadkhacqGH9aqpcaGGBbGaaGimaiaacY cacaaIXaGaaiilaiGacshacaGGHbGaaiOBaiabew9aMjaacYcacaaI WaGaaiyxaaaaaiaawUhaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqiVd0Maeyyp a0JaeyOeI0IaciiDaiaacggacaGGUbGaeqy1dyMaaGPaVlaaykW7ca aMc8+aaiqaaeaafaqabeGabaaabaGaamOCaiabg2da9iaacUfacaaI XaGaaiilaiaaicdacaGGSaGaaGimaiaacYcacaaIYaGaam4Aaiaac2 faaeaacaWGYbGaeyypa0Jaai4waiaaicdacaGGSaGaaGymaiaacYca cqGHsislciGGJbGaai4BaiaacshacqaHvpGzcaGGSaGaaGimaiaac2 faaaaacaGL7baaaaa@9386@

We can now eliminate A from the governing matrix equation

r i B ij ( μ q j x 1 + q j x 2 )=0 1+ μ 2 r i B ij ( μ 1+ μ 2 q j x 1 + 1 1+ μ 2 q j x 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aOWaaeWaaeaacqaH8oqBdaWcaaqaaiabgkGi2kaadghadaWgaaWcba GaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqa aaaakiabgUcaRmaalaaabaGaeyOaIyRaamyCamaaBaaaleaacaWGQb aabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaGc caGLOaGaayzkaaGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlabgkDiElaaykW7caaMc8UaaGPaVlaaykW7caaM c8+aaOaaaeaacaaIXaGaey4kaSIaeqiVd02aaWbaaSqabeaacaaIYa aaaaqabaGccaWGYbWaaSbaaSqaaiaadMgaaeqaaOGaamOqamaaBaaa leaacaWGPbGaamOAaaqabaGcdaqadaqaamaalaaabaGaeqiVd0gaba WaaOaaaeaacaaIXaGaey4kaSIaeqiVd02aaWbaaSqabeaacaaIYaaa aaqabaaaaOWaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadQgaae qaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaGccqGH RaWkdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaigdacqGHRaWkcqaH8o qBdaahaaWcbeqaaiaaikdaaaaabeaaaaGcdaWcaaqaaiabgkGi2kaa dghadaWgaaWcbaGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaS qaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaaa@8972@

Finally, if we set

x 1 s = μ 1+ μ 2 x 2 s = 1 1+ μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaam4C aaaacqGH9aqpdaWcaaqaaiabeY7aTbqaamaakaaabaGaaGymaiabgU caRiabeY7aTnaaCaaaleqabaGaaGOmaaaaaeqaaaaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7daWcaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaakeaa cqGHciITcaWGZbaaaiabg2da9maalaaabaGaaGymaaqaamaakaaaba GaaGymaiabgUcaRiabeY7aTnaaCaaaleqabaGaaGOmaaaaaeqaaaaa aaa@5F2F@

and note that

( q j x 1 x 1 s + q j x 2 x 2 s )= d q j ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiabgkGi2kaadghadaWgaaWcbaGaamOAaaqabaaakeaacqGH ciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakmaalaaabaGaeyOaIy RaamiEamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadohaaaGa ey4kaSYaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadQgaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGcdaWcaaqa aiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciITca WGZbaaaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamizaiaadgha daWgaaWcbaGaamOAaaqabaaakeaacaWGKbGaam4Caaaaaaa@58B3@

we find that

r i B ij d q j ds =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aOWaaSaaaeaacaWGKbGaamyCamaaBaaaleaacaWGQbaabeaaaOqaai aadsgacaWGZbaaaiabg2da9iaaicdaaaa@42A7@

along characteristic lines in the solid that satisfy

x 1 s = μ 1+ μ 2 x 2 s = 1 1+ μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaam4C aaaacqGH9aqpdaWcaaqaaiabeY7aTbqaamaakaaabaGaaGymaiabgU caRiabeY7aTnaaCaaaleqabaGaaGOmaaaaaeqaaaaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7daWcaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaakeaa cqGHciITcaWGZbaaaiabg2da9maalaaabaGaaGymaaqaamaakaaaba GaaGymaiabgUcaRiabeY7aTnaaCaaaleqabaGaaGOmaaaaaeqaaaaa aaa@5F2F@

The special characteristic lines in the solid can be identified more easily if we note that

d x 2 d x 1 = 1 μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaadsgacaWG4bWa aSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaabaGaaGymaaqaai abeY7aTbaaaaa@4043@

which shows that the slope of the characteristic lines satisfies

d x 2 d x 1 =tanϕ d x 2 d x 1 =cotϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaadsgacaWG4bWa aSbaaSqaaiaaigdaaeqaaaaakiabg2da9iGacshacaGGHbGaaiOBai abew9aMjaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8+aaSaaaeaacaWGKbGaamiEamaaBaaaleaaca aIYaaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa kiabg2da9iabgkHiTiGacogacaGGVbGaaiiDaiabew9aMbaa@76C7@

for the two possible values of the eigenvalue μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0gaaa@37AC@ .  This shows that

a.       There are two sets of characteristic lines (one for each eigenvalue)

b.      The two sets of characteristics are orthogonal (they therefore define a set of orthogonal curvilinear coordinates in the solid)

c.       The characteristic lines are trajectories of maximum shear (to see this, recall the definition of ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BE@  ).  For this reason, the characteristics are termed slip lines MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the material slips (deforms in shear) along these lines.

 

Conventionally the characteristics satisfying d x 2 /d x 1 =tanϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiaadI hadaWgaaWcbaGaaGOmaaqabaGccaGGVaGaamizaiaadIhadaWgaaWc baGaaGymaaqabaGccqGH9aqpciGG0bGaaiyyaiaac6gacqaHvpGzaa a@42FE@  are designated α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@  slip lines, while the orthogonal set are designated β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3787@  slip lines

 

A representative set of characteristic lines is sketched on the right.

 

When solving a particular boundary value problem, the central issue will be to identify a set of characteristic lines that will satisfy the boundary conditions.  Field equations reduce to simple ODEs that govern variations of hydrostatic pressure and velocity along each slip line.

 

 

Relations along slip-lines

 

To complete the theory, we need to find equations relating the field variables q=[ϕ, v 1 / x 1 , v 2 / x 2 , σ ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2 da9iaacUfacqaHvpGzcaGGSaGaeyOaIyRaamODamaaBaaaleaacaaI Xaaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabgkGi2kaadAhadaWgaaWcbaGaaGOmaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaaIYaaabeaakiaacYcacuaHdpWCga qeaiaac2faaaa@4F20@  along the slip-lines.  To do so we return to the governing equation

r i B ij d q j ds =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aOWaaSaaaeaacaWGKbGaamyCamaaBaaaleaacaWGQbaabeaaaOqaai aadsgacaWGZbaaaiabg2da9iaaicdaaaa@42A7@

and substitute for B and r.  For the four separate eigenvectors, we find that r i B ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aaaa@3AE1@  reduce to

[0,2ksin2ϕ,2k(cos2ϕ1),0] [2ksin2ϕ+2ktanϕcos2ϕ,0,0,tanϕ] }μ=cotϕ(α slip-line) [0,2ksinϕ,2k(cosϕ+1),0] [2ksin2ϕ2kcotϕcos2ϕ,0,0,tanϕ] }μ=tanϕ(β slip-line) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaGaca qaauaabeqaceaaaeaacaGGBbGaaGimaiaacYcacqGHsislcaaIYaGa am4AaiGacohacaGGPbGaaiOBaiaaikdacqaHvpGzcaGGSaGaaGOmai aadUgacaGGOaGaci4yaiaac+gacaGGZbGaaGOmaiabew9aMjabgkHi TiaaigdacaGGPaGaaiilaiaaicdacaGGDbaabaGaai4waiabgkHiTi aaikdacaWGRbGaci4CaiaacMgacaGGUbGaaGOmaiabew9aMjabgUca RiaaikdacaWGRbGaciiDaiaacggacaGGUbGaeqy1dyMaci4yaiaac+ gacaGGZbGaaGOmaiabew9aMjaacYcacaaIWaGaaiilaiaaicdacaGG SaGaciiDaiaacggacaGGUbGaeqy1dyMaaiyxaaaaaiaaw2haaiabeY 7aTjabg2da9iGacogacaGGVbGaaiiDaiabew9aMjaaykW7caaMc8Ua aGPaVlaacIcacqaHXoqycaqGGaGaae4CaiaabYgacaqGPbGaaeiCai aab2cacaqGSbGaaeyAaiaab6gacaqGLbGaaeykaaqaamaaciaabaqb aeqabiqaaaqaaiaacUfacaaIWaGaaiilaiabgkHiTiaaikdacaWGRb Gaci4CaiaacMgacaGGUbGaeqy1dyMaaiilaiaaikdacaWGRbGaaiik aiGacogacaGGVbGaai4Caiabew9aMjabgUcaRiaaigdacaGGPaGaai ilaiaaicdacaGGDbaabaGaai4waiabgkHiTiaaikdacaWGRbGaci4C aiaacMgacaGGUbGaaGOmaiabew9aMjabgkHiTiaaikdacaWGRbGaci 4yaiaac+gacaGG0bGaeqy1dyMaci4yaiaac+gacaGGZbGaaGOmaiab ew9aMjaacYcacaaIWaGaaiilaiaaicdacaGGSaGaciiDaiaacggaca GGUbGaeqy1dyMaaiyxaaaaaiaaw2haaiabeY7aTjabg2da9iabgkHi TiGacshacaGGHbGaaiOBaiabew9aMjaaykW7caaMc8UaaGPaVlaacI cacqaHYoGycaqGGaGaae4CaiaabYgacaqGPbGaaeiCaiaab2cacaqG SbGaaeyAaiaab6gacaqGLbGaaeykaaaaaa@D248@

 

Computing r i B ij d q j /ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbaabeaakiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaamizaiaadghadaWgaaWcbaGaamOAaaqabaGccaGGVaGaamizai aadohaaaa@418A@  and simplifying the trig formulas then yields

d v 1 ds +tanϕ d v 2 ds =0 2k dϕ ds + d σ ¯ ds =0 }αslip line d v 1 ds cotϕ d v 2 ds =0 2k dϕ ds + d σ ¯ ds =0 }βslip line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaafa qabeGabaaabaWaaSaaaeaacaWGKbGaamODamaaBaaaleaacaaIXaaa beaaaOqaaiaadsgacaWGZbaaaiabgUcaRiGacshacaGGHbGaaiOBai abew9aMnaalaaabaGaamizaiaadAhadaWgaaWcbaGaaGOmaaqabaaa keaacaWGKbGaam4CaaaacqGH9aqpcaaIWaaabaGaeyOeI0IaaGOmai aadUgadaWcaaqaaiaadsgacqaHvpGzaeaacaWGKbGaam4CaaaacqGH RaWkdaWcaaqaaiaadsgacuaHdpWCgaqeaaqaaiaadsgacaWGZbaaai abg2da9iaaicdaaaaacaGL9baacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlabeg7aHjaaykW7caqGZbGaaeiBai aabMgacaqGWbGaaeiiaiaabYgacaqGPbGaaeOBaiaabwgacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaaciaabaqbaeqabiqaaaqa amaalaaabaGaamizaiaadAhadaWgaaWcbaGaaGymaaqabaaakeaaca WGKbGaam4CaaaacqGHsislciGGJbGaai4BaiaacshacqaHvpGzdaWc aaqaaiaadsgacaWG2bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamizai aadohaaaGaeyypa0JaaGimaaqaaiaaikdacaWGRbWaaSaaaeaacaWG KbGaeqy1dygabaGaamizaiaadohaaaGaey4kaSYaaSaaaeaacaWGKb Gafq4WdmNbaebaaeaacaWGKbGaam4CaaaacqGH9aqpcaaIWaaaaaGa ayzFaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqaHYoGycaaMc8Uaae4CaiaabYgacaqGPbGaaeiCaiaabcca caqGSbGaaeyAaiaab6gacaqGLbaaaa@BBFC@

 

Hencky Equation: Conditions relating σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4WdmNbae baaaa@37C1@  and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37AE@  along slip lines are often expressed as

σ ¯ 2kϕ=constantα slip line σ ¯ +2kϕ=constantβ slip line MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacuaHdp WCgaqeaiabgkHiTiaaikdacaWGRbGaeqy1dyMaeyypa0Jaae4yaiaa b+gacaqGUbGaae4CaiaabshacaqGHbGaaeOBaiaabshacaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeqySdeMaaeiiaiaabohacaqGSbGaaeyAaiaabchacaqGGaGaae iBaiaabMgacaqGUbGaaeyzaaqaaiqbeo8aZzaaraGaey4kaSIaaGOm aiaadUgacqaHvpGzcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaae iDaiaabggacaqGUbGaaeiDaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHYoGycaqGGaGaae 4CaiaabYgacaqGPbGaaeiCaiaabccacaqGSbGaaeyAaiaab6gacaqG Lbaaaaa@8702@

These are known as the Hencky equations

 

Geiringer equations: One can also obtain simpler expressions relating velocity components along slip-lines.  It is convenient to express the velocity vector as components in a basis oriented with the slip-lines

 

The necessary basis-change is

v α = v 1 cosϕ+ v 2 sinϕ v β = v 1 sinϕ+ v 2 cosϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG2b WaaSbaaSqaaiabeg7aHbqabaGccqGH9aqpcaWG2bWaaSbaaSqaaiaa igdaaeqaaOGaci4yaiaac+gacaGGZbGaeqy1dyMaey4kaSIaamODam aaBaaaleaacaaIYaaabeaakiGacohacaGGPbGaaiOBaiabew9aMbqa aiaadAhadaWgaaWcbaGaeqOSdigabeaakiabg2da9iabgkHiTiaadA hadaWgaaWcbaGaaGymaaqabaGcciGGZbGaaiyAaiaac6gacqaHvpGz cqGHRaWkcaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaci4yaiaac+gaca GGZbGaeqy1dygaaaa@5B8B@

A straightforward algebraic exercise then yields

d v α ds = v β dϕ ds d v β ds = v α dϕ ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaamODamaaBaaaleaacqaHXoqyaeqaaaGcbaGaamizaiaadoha aaGaeyypa0JaamODamaaBaaaleaacqaHYoGyaeqaaOWaaSaaaeaaca WGKbGaeqy1dygabaGaamizaiaadohaaaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaSaaaeaacaWGKb GaamODamaaBaaaleaacqaHYoGyaeqaaaGcbaGaamizaiaadohaaaGa eyypa0JaeyOeI0IaamODamaaBaaaleaacqaHXoqyaeqaaOWaaSaaae aacaWGKbGaeqy1dygabaGaamizaiaadohaaaaaaa@6215@

These are known as the Geiringer equations.

 

 

 

6.1.3 Examples of slip-line field solutions to boundary value problems

 

When using slip-line field theory, the first step is always to find the characteristics (known as the slip line field).  This is usually done by trial and error, and can be exceedingly difficult.  These days, we usually hope that some smart person has already been able to find the slip-line field, and if we can’t find the solution in some ancient book we give up and clobber the problem with an FEM package.  If the slip-line field is known, the stress and velocity everywhere in the solid can be determined using the Hencky and Geiringer equations.

 

In this section we give several examples of slip-line field solutions to boundary value problems.

 

 

Plane Strain Extrusion (Hill)

 

A slip-line field solution to plane strain extrusion through a tapered die is shown in the picture on the right. Friction between the die and workpiece is neglected.

 

It is of particular interest to calculate the force P required to extrude the bar. The easiest way to do this is to consider the forces acting on the region ABCDEF.  Note that

(i) The resultant force on EF is P e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWGqbGaaCyzamaaBaaale aacaaIXaaabeaaaaa@366D@

(ii) The resultant force on CB is zero (you can see this by noting that no external forces act on the material to the left of CB)

 (iii) The stress state at a point b on the line CD can be calculated by tracing a slip-line from a to b. The Mohr’s circle construction for this purpose is shown on the right. At point a, the slip-lines intersect CB at 45 degrees, so that ϕ a = 45 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaWGHbaabeaaki abg2da9iabgkHiTiaaisdacaaI1aWaaWbaaSqabeaacaaIWaaaaaaa @37C3@ ; we also know that σ 11 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaaGimaaaa@37F5@  on CB (because the solid to the left of CB has no forces acting on it). These conditions can be satisfied by choosing σ ¯ =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeaiabg2da9iabgkHiTi aadUgaaaa@3784@ , so that the stress state at a is σ 11 =0 σ 12 =0 σ 22 =2k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlabeo8aZnaa BaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaaGOmaiaaikda aeqaaOGaeyypa0JaeyOeI0IaaGOmaiaadUgaaaa@508D@ .  Tracing a β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGyaaa@3467@  slip-line from a to b, we see that σ ¯ b = σ ¯ a +2k( ϕ a ϕ b )=πk/3k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGIb aabeaakiabg2da9iqbeo8aZzaaraWaaSbaaSqaaiaadggaaeqaaOGa ey4kaSIaaGOmaiaadUgacaGGOaGaeqy1dy2aaSbaaSqaaiaadggaae qaaOGaeyOeI0Iaeqy1dy2aaSbaaSqaaiaadkgaaeqaaOGaaiykaiab g2da9iabgkHiTiabec8aWjaadUgacaGGVaGaaG4maiabgkHiTiaadU gaaaa@4C45@ .  Finally, the slip lines intersect CD at 45 degrees, so CD is subjected to a pressure σ nn = σ ¯ b k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOBaiaad6 gaaeqaaOGaeyypa0Jafq4WdmNbaebadaWgaaWcbaGaamOyaaqabaGc cqGHsislcaWGRbaaaa@3C80@  acting normal to CD, while the component of traction tangent to CD is zero.

(iv) CD has length H, so the resultant force acting on CD is H σ nn sin30 e 1 H σ nn cos30 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGibGaeq4Wdm3aaSbaaSqaaiaad6 gacaWGUbaabeaakiGacohacaGGPbGaaiOBaiaaiodacaaIWaGaaCyz amaaBaaaleaacaaIXaaabeaakiabgkHiTiaadIeacqaHdpWCdaWgaa WcbaGaamOBaiaad6gaaeqaaOGaci4yaiaac+gacaGGZbGaaG4maiaa icdacaWHLbWaaSbaaSqaaiaaikdaaeqaaaaa@4959@

(v) By symmetry, the resultant force acting on AB is H σ nn sin30 e 1 +H σ nn cos30 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGibGaeq4Wdm3aaSbaaSqaaiaad6 gacaWGUbaabeaakiGacohacaGGPbGaaiOBaiaaiodacaaIWaGaaCyz amaaBaaaleaacaaIXaaabeaakiabgUcaRiaadIeacqaHdpWCdaWgaa WcbaGaamOBaiaad6gaaeqaaOGaci4yaiaac+gacaGGZbGaaG4maiaa icdacaWHLbWaaSbaaSqaaiaaikdaaeqaaaaa@494E@

(vi) Equilibrium then gives

P=kH( π 3 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiabg2 da9iaadUgacaWGibWaaeWaaeaadaWcaaqaaiabec8aWbqaaiaaioda aaGaey4kaSIaaGOmaaGaayjkaiaawMcaaaaa@3F2F@

 

 

Double-notched plate in tension

 

A slip-line field solution for a double-notched plate under tensile loading is shown in the picture.  The stress state in the neck, and the load P are of particular interest.

 

Both can be found by tracing a slip-line from either boundary into the constant stress region at the center of the solid.

 

Consider the slip-line starting at A and ending at B, for example.  At A the slip-lines meet the free surface at 45 degrees.  With α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGSaGaeqOSdigaaa@36B6@  designated as shown, ϕ A =απ/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamyqaaqaba GccqGH9aqpcqaHXoqycqGHsislcqaHapaCcaGGVaGaaGinaaaa@3C4A@  and σ ¯ A =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGbb aabeaakiabg2da9iaadUgaaaa@3793@ .  Following the slip-line to b, we see that ϕ B =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamOqaaqaba GccqGH9aqpcqaHapaCcaGGVaGaaGinaaaa@39BF@ , so the Hencky equation gives σ ¯ B =k( π2α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGcb aabeaakiabg2da9iaadUgadaqadaqaaiabec8aWjabgkHiTiaaikda cqaHXoqyaiaawIcacaGLPaaaaaa@3E22@ .  The state of stress at b follows as

σ 11 =(π2α)k σ 22 =(π2α+2)k σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaakiabg2da9iaacIcacqaHapaCcqGH sislcaaIYaGaeqySdeMaaiykaiaadUgacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaI Yaaabeaakiabg2da9iaacIcacqaHapaCcqGHsislcaaIYaGaeqySde Maey4kaSIaaGOmaiaacMcacaWGRbGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaaGymai aaikdaaeqaaOGaeyypa0JaaGimaaaa@6C78@

The state of stress is clearly constant in the region ABCD, (and so is constant along the line connecting the two notches).  The force required to deform the solid is therefore P=ak(π2α+2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbGaeyypa0JaamyyaiaadUgaca GGOaGaeqiWdaNaeyOeI0IaaGOmaiabeg7aHjabgUcaRiaaikdacaGG Paaaaa@3E73@ .

 

 

 

Pressurized cylindrical cavity

 

The slip-line field solution to an internally pressurized rigid-plastic cylinder is shown on the right. The goal is to determine the stress state everywhere in the cylinder, and to calculate the internal pressure necessary to drive the deformation.

 

Consider the α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@  slip-line, which starts at point A (with cylindrical-polar coordinates r=a,θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamyyaiaacYcaca aMc8UaaGPaVlaaykW7caaMc8UaeqiUdeNaeyypa0JaaGimaaaa@3FFB@  ), and ends at B (with cylindrical-polar coordinates  r=b,θ= θ b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamOyaiaacYcaca aMc8UaaGPaVlaaykW7caaMc8UaeqiUdeNaeyypa0JaeqiUde3aaSba aSqaaiaadkgaaeqaaaaa@420B@

1.      At point B, the surface is traction free, which requires σ rr = σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadkhacqaH4oqCaeqa aOGaeyypa0JaaGimaaaa@3E19@ .  To satisfy σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiabeI 7aXbqabaGccqGH9aqpcaaIWaaaaa@392C@ , the slip-line must meet the surface at 45 degrees ( ϕ B = θ b +π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamOqaaqaba GccqGH9aqpcqaH4oqCdaWgaaWcbaGaamOyaaqabaGccqGHRaWkcqaH apaCcaGGVaGaaGinaaaa@3D74@  ).  In addition, to satisfy σ rr =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaaGimaaaa@386D@  the hydrostatic stress σ ¯ B =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGcb aabeaakiabg2da9iaadUgaaaa@3794@ .

2.      Note that the shear stress component σ rθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiabeI 7aXbqabaGccqGH9aqpcaaIWaaaaa@392C@  throughout the cylinder.  This means that the slip-line must cross every radial line at 45 degrees (or, if you prefer, it must cross every circumferential line at 45 degrees).

3.      Consider a small segment ds of the slip-line.  Since the slip-line is at 45 degrees to the radial direction, dr=rdθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamOCaiabg2da9iaadkhaca WGKbGaeqiUdehaaa@3942@ .

4.      Integrating this result from r=a,θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamyyaiaacYcaca aMc8UaaGPaVlaaykW7caaMc8UaeqiUdeNaeyypa0JaaGimaaaa@3FFB@  to (r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGPaaaaa@377C@  gives r=a e θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0Jaamyyaiaadwgada ahaaWcbeqaaiabeI7aXbaaaaa@3876@  - i.e. the slip-lines are logarithmic spirals.

5.      At B, this gives b=a e θ B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOyaiabg2da9iaadggacaWGLbWaaW baaSqabeaacqaH4oqCdaWgaaadbaGaamOqaaqabaaaaaaa@38A6@  or θ B =log(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaamOqaaqaba GccqGH9aqpciGGSbGaai4BaiaacEgacaGGOaGaamOyaiaac+cacaWG HbGaaiykaaaa@3D28@

6.      Note that ϕ A =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamyqaaqaba GccqGH9aqpcqaHapaCcaGGVaGaaGinaaaa@39BE@  and apply the Hencky equation from B to A to see that σ ¯ A =k2k θ B =k2klog(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGbb aabeaakiabg2da9iaadUgacqGHsislcaaIYaGaam4AaiabeI7aXnaa BaaaleaacaWGcbaabeaakiabg2da9iaadUgacqGHsislcaaIYaGaam 4AaiGacYgacaGGVbGaai4zaiaacIcacaWGIbGaai4laiaadggacaGG Paaaaa@4817@

7.      Finally, the basis change equation shows that σ rr = p A = σ ¯ A k=2klog(b/a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaeyOeI0IaamiCamaaBaaaleaacaWGbbaabeaa kiabg2da9iqbeo8aZzaaraWaaSbaaSqaaiaadgeaaeqaaOGaeyOeI0 Iaam4Aaiabg2da9iabgkHiTiaaikdacaWGRbGaciiBaiaac+gacaGG NbGaaiikaiaadkgacaGGVaGaamyyaiaacMcaaaa@4A93@

8.      At a generic point (r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGPaaaaa@377C@ , the same procedure gives σ rr =2klog(b/a) σ θθ =2klog(b/a)+2k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaeyOeI0IaaGOmaiaadUgaciGGSbGaai4Baiaa cEgacaGGOaGaamOyaiaac+cacaWGHbGaaiykaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpcqGHsislcaaIYaGaam4AaiGacYgacaGGVbGaai4zai aacIcacaWGIbGaai4laiaadggacaGGPaGaey4kaSIaaGOmaiaadUga aaa@5AE7@

This result can be compared with the axisymmetric elastic-plastic solution in Section 4.2.

 

 

 

Notched Bar in Bending

 

The figure on the right shows a slip-line field solution for a notched bar subjected to a pure bending moment.  The solution is valid for ω1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcqGHLjYScaaIXaaaaa@3714@  (radian).

 

The slip-line field can be used to determine the moment M required to deform the bar as a function of the notch angle ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDaaa@3493@ . To do so, note that

  1. The stress acting on the line NO is constant, since slip-lines are straight.
  2. You can determine the stress at a point D between O and N by following the slip-line CD. The stress must satisfy σ 22 =0 σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVlabeo8aZnaa BaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@41C8@  at C, so the slip-lines must meet the surface at 45 degrees ( ϕ C =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaam4qaaqaba GccqGH9aqpcqaHapaCcaGGVaGaaGinaaaa@39C0@  ) and we must choose σ ¯ C =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGdb aabeaakiabg2da9iabgkHiTiaadUgaaaa@3882@ .  This gives σ 11 =2k σ 22 = σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaeyOeI0IaaGOmaiaadUgacaaMc8UaaGPaVlaa ykW7cqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0Jaeq 4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaaicdaaaa@481C@  at D.
  3. Similarly, the stress acting on the line OP is constant, since slip-lines are straight.  You can calculate the stress at some point B between P and O by following the slip-line AB.  At point A, the surface is free of traction, so the slip-line must meet the surface at 45 degrees ( ϕ A =π/4ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamyqaaqaba GccqGH9aqpcqaHapaCcaGGVaGaaGinaiabgkHiTiabeM8a3baa@3C78@  ), and the hydrostatic stress must satisfy σ ¯ A =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGbb aabeaakiabg2da9iaadUgaaaa@3793@ .  At B, we see that ϕ B =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzdaWgaaWcbaGaamOqaaqaba GccqGH9aqpcqGHsislcqaHapaCcaGGVaGaaGinaaaa@3AAC@ .  Using the Hencky equation along the β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGyaaa@3467@  slip-line AB, we find that σ ¯ B =k(1+π2ω) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqeamaaBaaaleaacaWGcb aabeaakiabg2da9iaadUgacaGGOaGaaGymaiabgUcaRiabec8aWjab gkHiTiaaikdacqaHjpWDcaGGPaaaaa@3FBD@ .  Finally σ 11 =k(2+π2ω) σ 22 =k(π2ω) σ 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0Jaam4AaiaacIcacaaIYaGaey4kaSIaeqiWdaNa eyOeI0IaaGOmaiabeM8a3jaacMcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9iaadUgacaGGOaGaeqiWdaNaeyOeI0IaaGOmaiabeM 8a3jaacMcacaaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqa aiaaigdacaaIYaaabeaakiabg2da9iaaicdaaaa@6271@  from the basis change formulas.
  4. The height d of point O can be found from the condition that the axial force applied to the bar must vanish.  Integrating σ 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaaaa@362B@  along the line NOP and setting the result to zero shows that

d= 2+π2ω 4+π2ω (ha) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeyypa0ZaaSaaaeaacaaIYa Gaey4kaSIaeqiWdaNaeyOeI0IaaGOmaiabeM8a3bqaaiaaisdacqGH RaWkcqaHapaCcqGHsislcaaIYaGaeqyYdChaaiaacIcacaWGObGaey OeI0IaamyyaiaacMcaaaa@4682@

  1. Finally, taking moments for the region of the bar to the right of NOP about O shows that

d 2 2 (2k)+ ( had ) 2 2 (2+π2ω)kM=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgadaahaaWcbeqaai aaikdaaaaakeaacaaIYaaaaiaacIcacaaIYaGaam4AaiaacMcacqGH RaWkdaWcaaqaamaabmaabaGaamiAaiabgkHiTiaadggacqGHsislca WGKbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOm aaaacaGGOaGaaGOmaiabgUcaRiabec8aWjabgkHiTiaaikdacqaHjp WDcaGGPaGaam4AaiabgkHiTiaad2eacqGH9aqpcaaIWaaaaa@4DCC@

Substituting for d and simplifying shows that

M=k (ha) 2 2+π2ω 4+π2ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGnbGaeyypa0Jaam4AaiaacIcaca WGObGaeyOeI0IaamyyaiaacMcadaahaaWcbeqaaiaaikdaaaGcdaWc aaqaaiaaikdacqGHRaWkcqaHapaCcqGHsislcaaIYaGaeqyYdChaba GaaGinaiabgUcaRiabec8aWjabgkHiTiaaikdacqaHjpWDaaaaaa@484E@

 

Overstressing: At first sight, this solution is valid for any notch angle ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDaaa@34A3@ , but in fact this is not the case.  A slip-line field is valid only if the rigid regions in the field do not exceed yield.  This means that it must be possible to find a static equilibrium distribution of stress which does not violate the yield criterion anywhere in the rigid part of the solid.  If this cannot be done, the solid is said to be over-stressed.

 

The slip-line field for a notched bar has a peculiar state of stress at point O MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  there is a stress discontinuity (and singularity) at the corner, and it turns out that the region that was assumed rigid in this solution is over-stressed (the maximum principal shear stress exceeds k) if the notch is too sharp. 

To see this, consider the rigid region of the solid just to the left of O, as shown in the picture.  The lines OE and OF are adjacent to α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  slip lines, and so are subjected to a combined shear stress k and normal stresses σ ¯ B , σ ¯ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaaiilaiqbeo8aZzaaraWaaSbaaSqaaiaadseaaeqaaaaa@36E0@  as shown.  When the value of σ ¯ B σ ¯ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaeyOeI0Iafq4WdmNbaebadaWgaaWcbaGaamiraaqabaaaaa@371D@  gets too large, the rigid region OEFO collapses plastically MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a possible slip-line field at collapse is shown in the figure.  The slip-line field consists of a 90 degree fan, centered at O.  Applying the Hencky relation along a generic β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3229@  slip-line shows that, at collapse σ ¯ B σ ¯ D =kπ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaeyOeI0Iafq4WdmNbaebadaWgaaWcbaGaamiraaqabaGccqGH 9aqpcaWGRbGaeqiWdahaaa@3ADA@ , and so for the rigid region to remain below yield σ ¯ B σ ¯ D kπ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaeyOeI0Iafq4WdmNbaebadaWgaaWcbaGaamiraaqabaGccqGH KjYOcaWGRbGaeqiWdahaaa@3B89@ .  Substituting the values of σ ¯ B , σ ¯ D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaaiilaiqbeo8aZzaaraWaaSbaaSqaaiaadseaaeqaaaaa@36E0@  from parts (2) and (3) then gives ω1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcqGHLjYScaaIXaaaaa@3714@ .

 

A solution for a sharp notch is shown in the figure to the right. In the modified field, the region PBNFG is rigid.   The left hand part of the bar rotates about point O, shearing along a pair slip lines formed by the circular arcs AB and GF.  To calculate the moment, we need first to calculate the angles θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@323E@  and ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5baa@3256@ , the radius R of the arc BC, the length b of the constant stress regions adjacent to the notch, and the height d of point O above the base of the beam.  To this end, note that

  1. At point A, the surface of the wedge is traction free. The slip-lines must intersect the surface at 45 degrees, which shows that ϕ A =ω3π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaWGbbaabeaaki abg2da9iabeM8a3jabgkHiTiaaiodacqaHapaCcaGGVaGaaGinaaaa @3AF7@  and that σ ¯ A =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadgeaae qaaOGaeyypa0Jaam4Aaaaa@3555@ .
  2. Tracing the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3227@  slip-line from A to B and noting ϕ B =ω+θ3π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaWGcbaabeaaki abg2da9iabeM8a3jabgUcaRiabeI7aXjabgkHiTiaaiodacqaHapaC caGGVaGaaGinaaaa@3D90@  gives σ ¯ B =k+2kθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaeyypa0Jaam4AaiabgUcaRiaaikdacaWGRbGaeqiUdehaaa@399A@ .
  3. At point D at the base of the beam, the surface is traction free, so the slip-lines must meet the surface at 45 degrees.  This gives ϕ D =π/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMnaaBaaaleaacaWGebaabeaaki abg2da9iabec8aWjaac+cacaaI0aaaaa@3783@  and σ ¯ D =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadseaae qaaOGaeyypa0JaeyOeI0Iaam4Aaaaa@3645@ .
  4. The stress is uniform in the region CDEF, so that σ ¯ C =k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadoeaae qaaOGaeyypa0JaeyOeI0Iaam4Aaaaa@3644@ .
  5. The hydrostatic stresses at B and C must be related by the Hencky equation for a β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIbaa@3229@  slip-line, which gives σ ¯ B = σ ¯ C +2kψψθ=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaae qaaOGaeyypa0Jafq4WdmNbaebadaWgaaWcbaGaam4qaaqabaGccqGH RaWkcaaIYaGaam4AaiabeI8a5jabgkDiElabeI8a5jabgkHiTiabeI 7aXjabg2da9iaaigdaaaa@442A@ .
  6. Finally, elementary geometry shows that ω+θ+ψ=π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabgUcaRiabeI7aXjabgUcaRi abeI8a5jabg2da9iabec8aWbaa@3A60@ .
  7. Hence, solving (5) and (6) gives θ=(πω1)/2ψ=(πω+1)/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaacIcacqaHapaCcq GHsislcqaHjpWDcqGHsislcaaIXaGaaiykaiaac+cacaaIYaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7cqaHipqEcqGH9aqpcaGGOaGaeq iWdaNaeyOeI0IaeqyYdCNaey4kaSIaaGymaiaacMcacaGGVaGaaGOm aaaa@4F92@ .
  8. Geometry gives d+Rsin(ψπ/4)+bcos(ψπ/4)=ha MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqGHRaWkcaWGsbGaci4CaiaacM gacaGGUbGaaiikaiabeI8a5jabgkHiTiabec8aWjaac+cacaaI0aGa aiykaiabgUcaRiaadkgaciGGJbGaai4BaiaacohacaGGOaGaeqiYdK NaeyOeI0IaeqiWdaNaai4laiaaisdacaGGPaGaeyypa0JaamiAaiab gkHiTiaadggaaaa@4CE8@ .
  9. We obtain two more equations relating the unknown variables from the condition that the resultant force acting on any surface that extends from the top of the beam to the bottom must vanish.  The resultant force acting on the surface to the right of PBCD can be calculated as

σ ¯ B b( cos(ψπ/4) e 1 +sin(ψπ/4) e 2 )kb( sin(ψπ/4) e 1 +cos(ψπ/4) e 2 ) + π/4ψ π/4 [ k(sinλ e 1 cosλ e 2 ) σ ¯ BC (λ)(cosλ e 1 +sinλ e 2 ) ]Rdλ +(d 2 R)[ k( e 1 + e 2 )/ 2 σ ¯ C ( e 1 e 2 )/ 2 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafq4WdmNbaebadaWgaaWcbaGaam OqaaqabaGccaWGIbWaaeWaaeaaciGGJbGaai4BaiaacohacaGGOaGa eqiYdKNaeyOeI0IaeqiWdaNaai4laiaaisdacaGGPaGaaCyzamaaBa aaleaacaaIXaaabeaakiabgUcaRiGacohacaGGPbGaaiOBaiaacIca cqaHipqEcqGHsislcqaHapaCcaGGVaGaaGinaiaacMcacaWHLbWaaS baaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0Iaam4Aaiaa dkgadaqadaqaaiGacohacaGGPbGaaiOBaiaacIcacqaHipqEcqGHsi slcqaHapaCcaGGVaGaaGinaiaacMcacaWHLbWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaci4yaiaac+gacaGGZbGaaiikaiabeI8a5jabgk HiTiabec8aWjaac+cacaaI0aGaaiykaiaahwgadaWgaaWcbaGaaGOm aaqabaaakiaawIcacaGLPaaaaeaacqGHRaWkdaWdXbqaamaadmaaba Gaam4AaiaacIcaciGGZbGaaiyAaiaac6gacqaH7oaBcaWHLbWaaSba aSqaaiaaigdaaeqaaOGaeyOeI0Iaci4yaiaac+gacaGGZbGaeq4UdW MaaCyzamaaBaaaleaacaaIYaaabeaakiaacMcacqGHsislcuaHdpWC gaqeamaaBaaaleaacaWGcbGaam4qaaqabaGccaGGOaGaeq4UdWMaai ykaiaacIcaciGGJbGaai4BaiaacohacqaH7oaBcaWHLbWaaSbaaSqa aiaaigdaaeqaaOGaey4kaSIaci4CaiaacMgacaGGUbGaeq4UdWMaaC yzamaaBaaaleaacaaIYaaabeaakiaacMcaaiaawUfacaGLDbaacaWG sbGaamizaiabeU7aSbWcbaGaeqiWdaNaai4laiaaisdacqGHsislcq aHipqEaeaacqaHapaCcaGGVaGaaGinaaqdcqGHRiI8aaGcbaGaey4k aSIaaiikaiaadsgadaGcaaqaaiaaikdaaSqabaGccqGHsislcaWGsb GaaiykamaadmaabaGaam4AaiaacIcacaWHLbWaaSbaaSqaaiaaigda aeqaaOGaey4kaSIaaCyzamaaBaaaleaacaaIYaaabeaakiaacMcaca GGVaWaaOaaaeaacaaIYaaaleqaaOGaeyOeI0Iafq4WdmNbaebadaWg aaWcbaGaam4qaaqabaGccaGGOaGaaCyzamaaBaaaleaacaaIXaaabe aakiabgkHiTiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaai4l amaakaaabaGaaGOmaaWcbeaaaOGaay5waiaaw2faaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqpcaWHWaaaaaa@CC69@

where σ ¯ BC = σ ¯ C +2k(π/4λ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadkeaca WGdbaabeaakiabg2da9iqbeo8aZzaaraWaaSbaaSqaaiaadoeaaeqa aOGaey4kaSIaaGOmaiaadUgacaGGOaGaeqiWdaNaai4laiaaisdacq GHsislcqaH7oaBcaGGPaaaaa@41BD@  is the hydrostatic stress along the slip-line BC.  The results of (7), (8) and (9) can be solved for d, R and b

  1. Finally, taking moments about O gives M= b 2 σ ¯ B /2+kRb+k R 2 ψ σ ¯ C (2 d 2 R 2 )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eacqGH9aqpcaWGIbWaaWbaaSqabe aacaaIYaaaaOGafq4WdmNbaebadaWgaaWcbaGaamOqaaqabaGccaGG VaGaaGOmaiabgUcaRiaadUgacaWGsbGaamOyaiabgUcaRiaadUgaca WGsbWaaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaeyOeI0Iafq4WdmNb aebadaWgaaWcbaGaam4qaaqabaGccaGGOaGaaGOmaiaadsgadaahaa WcbeqaaiaaikdaaaGccqGHsislcaWGsbWaaWbaaSqabeaacaaIYaaa aOGaaiykaiaac+cacaaIYaaaaa@4D58@ . Thus,

M=k (ha) 2 { (ωπ) 2 1 }/{ (ωπ1) 2 4 cos 2 [ (ω1)/2 ] } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eacqGH9aqpcaWGRbGaaiikaiaadI gacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaaGOmaaaakmaacmaa baGaaiikaiabeM8a3jabgkHiTiabec8aWjaacMcadaahaaWcbeqaai aaikdaaaGccqGHsislcaaIXaaacaGL7bGaayzFaaGaai4lamaacmaa baGaaiikaiabeM8a3jabgkHiTiabec8aWjabgkHiTiaaigdacaGGPa WaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiGacogacaGGVbGa ai4CamaaCaaaleqabaGaaGOmaaaakmaadmaabaGaaiikaiabeM8a3j abgkHiTiaaigdacaGGPaGaai4laiaaikdaaiaawUfacaGLDbaaaiaa wUhacaGL9baaaaa@5BE7@

This result is valid only if b0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgacqGHLjYScaaIWaaaaa@33EF@ , which requires ω>0.056 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabg6da+iaaicdacaGGUaGaaG imaiaaiwdacaaI2aaaaa@3702@ .  In addition, the notch angle must satisfy ω1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3jabgsMiJkaaigdaaaa@34C5@  to avoid overstressing the rigid corner at P.