Chapter 9

 

Modeling Material Failure

 

 

9.2 Stress and strain based fracture and fatigue criteria

 

Many of the most successful design procedures use simple, experimentally calibrated, functions of stress and strain to assess the likelihood of failure in a component.   Some examples of commonly used failure criteria are summarized in this section.

 

9.2.1 Stress based failure criteria for brittle solids and composites.

 

Experiments show that brittle solids (such as ceramics, glasses, and fiber-reinforced composites) tend to fail when the stress in the solid reaches a critical magnitude.   Materials such as ceramics and glasses can be idealized using an isotropic failure criterion.   Composite materials are stronger when loaded in some directions than others, and must be modeled using an anisotropic failure criterion. 

 

 Failure criteria for isotropic materials

 

The simplest brittle fracture criterion states that fracture is initiated when the greatest tensile principal stress in the solid reaches a critical magnitude,

σ 1max = σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0Jaeq4W dm3aaSbaaSqaaiaadsfacaWGtbaabeaaaaa@412B@

(The subscript TS stands for tensile strength).  To apply the criterion, you must

1.      Measure (or look up) σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaaaaa@3A9D@  for the material.  σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaaaaa@3A9D@  can be measured by conducting tensile tests on specimens MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it is important to test a large number of specimens because the failure stress is likely to show a great deal of statistical scatter.  The tensile strength can also be measured using beam bending tests.  The failure stress measured in a bending test is referred to as the `modulus of rupture’ σ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadkhaaeqaaaaa@39E3@  for the material.  It is nominally equivalent to σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaaaaa@3A9D@  but in practice usually turns out to be somewhat higher.

2.      Calculate the anticipated stress distribution in your component or structure (e.g. using FEM).  Finally, you plot contours of principal stress, and find the maximum value σ 1max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaciGGTbGaaiyyaiaacIhaaeqaaaaa@3C7B@ .  If   σ 1max < σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaciGGTbGaaiyyaiaacIhaaeqaaOGaeyipaWJaeq4W dm3aaSbaaSqaaiaadsfacaWGtbaabeaaaaa@4129@  the design is safe (but be sure to use an appropriate factor of safety!).

 

 

 Failure criteria for anisotropic materials

 

More sophisticated criteria must be used to model anisotropic materials (especially composites).  The criteria must take account for the fact that the material is stronger in some directions than others.  For example, a fiber reinforced composite is usually much stronger when loaded parallel to the fiber direction than when loaded transverse to the fibers.  There are many different ways to account for this anisotropy MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a few approaches are summarized below.

 

Orientation dependent fracture strength.  One approach is to make the tensile strength of the solid orientation dependent.  For example, the tensile strength of a brittle, orthotropic solid (with three distinct, mutually perpendicular characteristic material directions) could be characterized by its tensile strengths σ TS1 , σ TS2 , σ TS3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamivaiaado facaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaamivaiaadofa caaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaamivaiaadofaca aIZaaabeaaaaa@414E@  parallel to the three characteristic directions { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  in the solid.  The tensile strength when loaded  parallel to a general direction n=sinϕcosθ e 1 +sinϕsinθ e 2 +cosϕ e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyypa0Jaci4CaiaacMgaca GGUbGaeqy1dyMaci4yaiaac+gacaGGZbGaeqiUdeNaaCyzamaaBaaa leaacaaIXaaabeaakiabgUcaRiGacohacaGGPbGaaiOBaiabew9aMj GacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqa baGccqGHRaWkciGGJbGaai4BaiaacohacqaHvpGzcaWHLbWaaSbaaS qaaiaaiodaaeqaaaaa@530F@  could be interpolated between these values as

σ TS (n)=( σ TS1 cos 2 θ+ σ TS2 sin 2 θ ) sin 2 ϕ+ σ TS3 cos 2 ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamivaiaado faaeqaaOGaaiikaiaah6gacaGGPaGaeyypa0ZaaeWaaeaacqaHdpWC daWgaaWcbaGaamivaiaadofacaaIXaaabeaakiGacogacaGGVbGaai 4CamaaCaaaleqabaGaaGOmaaaakiabeI7aXjabgUcaRiabeo8aZnaa BaaaleaacaWGubGaam4uaiaaikdaaeqaaOGaci4CaiaacMgacaGGUb WaaWbaaSqabeaacaaIYaaaaOGaeqiUdehacaGLOaGaayzkaaGaci4C aiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqy1dyMaey4kaS Iaeq4Wdm3aaSbaaSqaaiaadsfacaWGtbGaaG4maaqabaGcciGGJbGa ai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaHvpGzaaa@6063@

where (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeqy1dyMaaiilaiabeI7aXj aacMcaaaa@384D@  are illustrated in the figure.  The material fails if the stress acting normal to any plane in the solid exceeds the fracture stress for that plane, i.e.

n i (θ,ϕ) σ ij n j (θ,ϕ)= σ TS (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiabeI7aXjaacYcacqaHvpGzcaGGPaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGcca GGOaGaeqiUdeNaaiilaiabew9aMjaacMcacqGH9aqpcqaHdpWCdaWg aaWcbaGaamivaiaadofaaeqaaOGaaiikaiabew9aMjaacYcacqaH4o qCcaGGPaaaaa@5010@

where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  are the stress components in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@ .  To use this criterion to check for failure at any point in the solid, you must

(i) Find the components of stress in the  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  basis;

(ii) Maximize the function n i (θ,ϕ) σ ij n j (θ,ϕ)/ σ TS (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiabeI7aXjaacYcacqaHvpGzcaGGPaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGcca GGOaGaeqiUdeNaaiilaiabew9aMjaacMcacaGGVaGaeq4Wdm3aaSba aSqaaiaadsfacaWGtbaabeaakiaacIcacqaHvpGzcaGGSaGaeqiUde Naaiykaaaa@4FBD@  with respect to (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeqy1dyMaaiilaiabeI7aXj aacMcaaaa@384D@ ; and

(iii) Check whether the maximum value of   n i (θ,ϕ) σ ij n j (θ,ϕ)/ σ TS (ϕ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiabeI7aXjaacYcacqaHvpGzcaGGPaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGcca GGOaGaeqiUdeNaaiilaiabew9aMjaacMcacaGGVaGaeq4Wdm3aaSba aSqaaiaadsfacaWGtbaabeaakiaacIcacqaHvpGzcaGGSaGaeqiUde Naaiykaaaa@4FBD@  exceeds 1.  If so, the material will fail; if not, it is safe.

 

Goldenblat-Kopnov failure criterion. A very general phenomenological failure criterion can be constructed by simply combining the stress components in a basis oriented with respect to material axes as polynomial function.  The Goldenblat-Kopnov criterion is one example, which states that the critical stresses required to cause failure satisfy the equation

A ij σ ij + B ijkl σ ij σ kl =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHRaWk caWGcbWaaSbaaSqaaiaadMgacaWGQbGaam4AaiaadYgaaeqaaOGaeq 4WdmNaaCjaVpaaBaaaleaacaWGPbGaamOAaaqabaGccqaHdpWCdaWg aaWcbaGaam4AaiaadYgaaeqaaOGaeyypa0JaaGymaaaa@4A0B@

Here A and B are material constants: A is diagonal ( A ij =0ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7caWGPbGaeyiy IKRaamOAaaaa@3FA4@  ) and B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbaaaa@3391@  has the same symmetries as the elasticity tensor, i.e. B ijkl = B klij = B jikl = B ijlk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaOGaeyypa0JaamOqamaaBaaaleaacaWGRbGa amiBaiaadMgacaWGQbaabeaakiabg2da9iaadkeadaWgaaWcbaGaam OAaiaadMgacaWGRbGaamiBaaqabaGccqGH9aqpcaWGcbWaaSbaaSqa aiaadMgacaWGQbGaamiBaiaadUgaaeqaaaaa@48BA@ .  The most general anisotropic material would therefore be characterized by 24 independent material constants, but in practice simplified versions have far fewer parameters.  Most failure criteria for composites are in fact special cases of the Goldenblat-Kopnov criterion, including the Tsai-Hill criterion outlined below.

 

Tsai-Hill criterion:  The Tsai-Hill criterion is used to model damage in brittle laminated fiber-reinforced composites and wood. A specimen of laminated composite subjected to in-plane loading is sketched in the figure.  The Tsai-Hill criterion assumes that a plane stress state exists in the solid.  Let σ 11 , σ 22 , σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGc caGGSaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3E6C@  denote the nonzero components of stress, with basis vectors e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@349B@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa aa@349C@  oriented parallel and perpendicular to the fibers in the sheet, as shown.  The Tsai-Hill failure criterion is

( σ 11 σ TS1 ) 2 + ( σ 22 σ TS2 ) 2 σ 11 σ 22 σ TS1 2 + σ 12 2 σ SS 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaaaOqaaiabeo8aZnaaBaaaleaacaWG ubGaam4uaiaaigdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaakiabgUcaRmaabmaabaWaaSaaaeaacqaHdpWCdaWgaaWc baGaaGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadsfaca WGtbGaaGOmaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaeyOeI0YaaSaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiab eo8aZnaaDaaaleaacaWGubGaam4uaiaaigdaaeaacaaIYaaaaaaaki abgUcaRmaalaaabaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYaaabaGa aGOmaaaaaOqaaiabeo8aZnaaDaaaleaacaWGtbGaam4uaaqaaiaaik daaaaaaOGaeyypa0JaaGymaaaa@60B0@

at failure, where σ TS1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamivaiaado facaaIXaaabeaaaaa@3721@ , σ TS2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamivaiaado facaaIYaaabeaaaaa@3722@  and σ SS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaam4uaiaado faaeqaaaaa@3665@  are material properties.  They are measured as follows:

1.      The laminate is loaded in uniaxial tension parallel to the fibers. The material fails when σ 11 = σ TS1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadsfacaWGtbGaaGym aaqabaaaaa@3B96@

2.      The laminate is loaded in uniaxial tension perpendicular to the fibers.  The material fails when  σ 22 = σ TS2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadsfacaWGtbGaaGOm aaqabaaaaa@3B99@

3.      In principle, the laminate could be loaded in shear MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it would then fail when σ 12 = σ SS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaik daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadofacaWGtbaabeaa aaa@3ADB@ . In practice it is preferable to pull on the laminate in uniaxial tension with stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqaba aaaa@356F@  at 45 degrees to the fibers, which induces stress components σ 11 = σ 22 = σ 12 = σ 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaa kiabg2da9iabeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9a qpcqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaGGVaGaaGOmaaaa@444A@ .  A simple calculation then shows that σ SS = σ TS2 σ 0 / 4 σ TS2 2 σ 0 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaam4uaiaado faaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaadsfacaWGtbGaaGOm aaqabaGccqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaGGVaWaaOaaae aacaaI0aGaeq4Wdm3aa0baaSqaaiaadsfacaWGtbGaaGOmaaqaaiaa ikdaaaGccqGHsislcqaHdpWCdaqhaaWcbaGaaGimaaqaaiaaikdaaa aabeaaaaa@4985@ .

 

 

 

9.2.2 Probabilistic Design Methods for Brittle Fracture  (Weibull Statistics)

 

The fracture criterion σ 1max = σ TS MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0Jaeq4W dm3aaSbaaSqaaiaadsfacaWGtbaabeaaaaa@412B@  is too crude for many applications.  The tensile strength of a brittle solid usually shows considerable statistical scatter, because the likelihood of failure is determined by the probability of finding a large flaw in a highly stressed region of the material.  This makes it difficult to determine an unambiguous value for tensile strength MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  should you use the median value of your experimental data?  Pick the stress level where 95% of specimens survive? It’s better to deal with this problem using a more rigorous statistical approach.

 

Weibull statistics refers to a technique used to predict the probability of failure in a brittle material.  The following approach is used

1.      Test a large number of samples with identical size and shape under uniform tensile stress, and determine their survival probability as a function of stress (survival probability is approximated by the fraction of specimens that survive a given stress level).

2.      Fit the survival probability of these specimens P s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGZbaabeaaaaa@38F6@  is fit by a Weibull distribution

P s ( V 0 )=exp{ ( σ σ 0 ) m } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGZbaabeaakiaacIcacaWGwbWaaSbaaSqaaiaaicdaaeqa aOGaaiykaiabg2da9iGacwgacaGG4bGaaiiCamaacmaabaGaeyOeI0 YaaeWaaeaadaWcaaqaaiabeo8aZbqaaiabeo8aZnaaBaaaleaacaaI WaaabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaad2gaaaaaki aawUhacaGL9baaaaa@4A5B@

where σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@39A6@  and m are material constants.  The index m is typically of the order 5-10 for ceramics, and is independent of specimen volume.  The parameter σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@39A6@  is the stress at which the probability of survival is exp(-1), (about 37%). This critical stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@39A6@  depends on the specimen volume V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaa aa@3487@ , and is smaller for larger specimens.

3.      Given m, σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@39A6@  and the corresponding specimen volume V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaaIWaaabeaaaaa@38BE@ , the survival probability of a volume V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaaaa@36C1@  of material subjected to uniform uniaxial stress σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@38C0@  follows as

P s (V)=exp{ V V 0 ( σ σ 0 ) m } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGZbaabeaakiaacIcacaWGwbGaaiykaiabg2da9iGacwga caGG4bGaaiiCamaacmaabaGaeyOeI0YaaSaaaeaacaWGwbaabaGaam OvamaaBaaaleaacaaIWaaabeaaaaGcdaqadaqaamaalaaabaGaeq4W dmhabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaawM caamaaCaaaleqabaGaamyBaaaaaOGaay5Eaiaaw2haaaaa@4C21@

To see this, note that the volume V can be thought of as containing n=V/ V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabg2 da9iaadAfacaGGVaGaamOvamaaBaaaleaacaaIWaaabeaaaaa@3C45@  specimens.  The probability that they all survive is { P s ( V 0 ) } n = { P s ( V 0 ) } V/ V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaacmaabaGaam iuamaaBaaaleaacaWGZbaabeaakiaacIcacaWGwbWaaSbaaSqaaiaa icdaaeqaaOGaaiykaaGaay5Eaiaaw2haamaaCaaaleqabaGaamOBaa aakiabg2da9maacmaabaGaamiuamaaBaaaleaacaWGZbaabeaakiaa cIcacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaaiykaaGaay5Eaiaaw2 haamaaCaaaleqabaGaamOvaiaac+cacaWGwbWaaSbaaWqaaiaaicda aeqaaaaaaaa@4A96@ .

4.      More generally, the survival probability of a solid subjected to an arbitrary stress distribution with principal values σ 1 , σ 2 , σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaigdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaaa beaakiaacYcacqaHdpWCdaWgaaWcbaGaaG4maaqabaaaaa@4072@  can be computed as

log P s = 1 V 0 σ 0 m V ( σ 1 m + σ 2 m + σ 3 m ) dV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaamiuamaaBaaaleaacaWGZbaabeaakiabg2da9iabgkHi TmaalaaabaGaaGymaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaGccq aHdpWCdaqhaaWcbaGaaGimaaqaaiaad2gaaaaaaOWaa8quaeaadaqa daqaamaaamaabaGaeq4Wdm3aaSbaaSqaaiaaigdaaeqaaaGccaGLPm IaayPkJaWaaWbaaSqabeaacaWGTbaaaOGaey4kaSYaaaWaaeaacqaH dpWCdaWgaaWcbaGaaGOmaaqabaaakiaawMYicaGLQmcadaahaaWcbe qaaiaad2gaaaGccqGHRaWkdaaadaqaaiabeo8aZnaaBaaaleaacaaI ZaaabeaaaOGaayzkJiaawQYiamaaCaaaleqabaGaamyBaaaaaOGaay jkaiaawMcaaaWcbaGaamOvaaqab0Gaey4kIipakiaadsgacaWGwbaa aa@5D49@

where

σ ={ σσ0 0σ0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq aHdpWCaiaawMYicaGLQmcacqGH9aqpdaGabaqaauaabeqaceaaaeaa cqaHdpWCcaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdp WCcqGHLjYScaaIWaaabaGaaGimaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7cqaHdpWCcqGHKjYOcaaIWaaaaaGaay5Eaa aaaa@5AA7@

 

This approach is quite successful in some applications: for example, it explains why brittle materials appear to be stronger in bending than in uniaxial tension.  Like many statistical approaches it has some limitations as a design tool.  The method can predict accurately the stress that gives 30% probability of failure.  But who wants to buy a product that has a 30% probability of failure?  For design applications we need to predict the probability of 1 failure in a million or so.  It is very difficult to measure the tail of a statistical distribution accurately, and a distribution that was fit to predict 63% failure probability may be wildly inaccurate in the region of interest.

 

 

9.2.3 Static Fatigue Criterion for Brittle Materials

 

`Static fatigue’ refers to the progressive reduction in tensile strength of a stressed brittle material with time.  The simplest way to model static fatigue is to make the tensile strength of the material a function of time and applied stress.  The usual approach is to set

σ TS = σ TS0 ( 1α ( σ/ σ TS0 ) m dt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaakiabg2da9iabeo8aZnaaBaaaleaa caWGubGaam4uaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0Iaeq ySde2aa8qaaeaadaqadaqaaiabeo8aZjaac+cacqaHdpWCdaWgaaWc baGaamivaiaadofacaaIWaaabeaaaOGaayjkaiaawMcaamaaCaaale qabaGaamyBaaaakiaadsgacaWG0baaleqabeqdcqGHRiI8aaGccaGL OaGaayzkaaaaaa@5254@

where σ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcaGGOaGaamiDaiaacMcaaa a@36DB@  is the maximum principal stress acting on the solid, which may vary slowly with time t; σ TS0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamivaiaado facaaIWaaabeaaaaa@3720@  is the tensile strength of the solid at time t=0, and α,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGSaGaamyBaaaa@3607@  are two material constants.  Typically m has values between 5 and 10.  For the particular case of a constant stress, we see that

σ TS = σ TS0 ( 1α ( σ/ σ TS0 ) m t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadsfacaWGtbaabeaakiabg2da9iabeo8aZnaaBaaaleaa caWGubGaam4uaiaaicdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0Iaeq ySde2aaeWaaeaacqaHdpWCcaGGVaGaeq4Wdm3aaSbaaSqaaiaadsfa caWGtbGaaGimaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaad2 gaaaGccaWG0baacaGLOaGaayzkaaaaaa@4F66@

Since failure occurs when σ= σ TS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGH9aqpcqaHdpWCdaWgaa WcbaGaamivaiaadofaaeqaaaaa@392F@ , the time to failure follows as

t f = 1 α ( σ TS0 σ ) m ( 1 σ σ TS0 ) 1 α ( σ TS0 σ ) m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWGMbaabeaakiabg2da9maalaaabaGaaGymaaqaaiabeg7a HbaadaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaadsfacaWGtb GaaGimaaqabaaakeaacqaHdpWCaaaacaGLOaGaayzkaaWaaWbaaSqa beaacaWGTbaaaOWaaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacqaHdp WCaeaacqaHdpWCdaWgaaWcbaGaamivaiaadofacaaIWaaabeaaaaaa kiaawIcacaGLPaaacqGHijYUdaWcaaqaaiaaigdaaeaacqaHXoqyaa WaaeWaaeaadaWcaaqaaiabeo8aZnaaBaaaleaacaWGubGaam4uaiaa icdaaeqaaaGcbaGaeq4WdmhaaaGaayjkaiaawMcaamaaCaaaleqaba GaamyBaaaaaaa@5BD2@

so that α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  and m can easily be determined by measuring the time to failure in uniaxial tension as a function of applied stress.

 

Under multi-axial loading, the maximum principal tensile stress should be used for σ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZbaa@322B@ .

 

 

9.2.4 Constitutive laws for crushing failure of brittle materials

 

Brittle materials are generally used in applications where they are subjected primarily to compressive stress. Brittle materials are very strong in compression, but they will fail if subjected to combined hydrostatic compression and shear (e.g. by loading in uniaxial compression).  Failure in compression is a consequence of distributed microcracking in the solid MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  large numbers of small cracks nucleate, propagate for a short while and then arrest.  Failure occurs as a result of coalescence of these cracks.   A typical stress-strain curve during compression of a brittle material, such as concrete, is illustrated in the figure.  Failure in compression is less catastrophic than tension, and in some respects qualitatively resembles metal plasticity.  For plastically deforming metals, however, the stress-strain curve is independent of hydrostatic stress.  In contrast, the crushing resistance of a brittle material increases with hydrostatic compression.

 

This type of crushing is often modeled using constitutive equations based on small-strain metal plasticity.  The governing equations for a simple, small-strain, constitutive model of this form will be summarized briefly here. A more detailed discussion of plasticity theory is given in Section 3.6.

 

The material is characterized by the following properties:

 The Young’s modulus E and Poisson ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@

 The stress-v-plastic strain curve measured from a uniaxial compression test, which is fit by a functional relation of the form σ=Y( ε ¯ p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGH9aqpcqGHsislcaWGzb Gaaiikaiqbew7aLzaaraWaaWbaaSqabeaacaWGWbaaaOGaaiykaaaa @3B9E@ , where ε ¯ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaqeamaaCaaaleqabaGaam iCaaaaaaa@35A7@  is the magnitude of the compressive strain.  Any of the functions listed in Section 3.6.5 could be used for the function Y.

 A material constant c, which controls how rapidly the strength of the material increases with hydrostatic compression.

 

The constitutive equations specify a relationship between an increment in stress d σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaaaaa@377B@  applied to the material and an increment in strain d ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqyTdu2aaSbaaSqaaiaadM gacaWGQbaabeaaaaa@375F@ , as follows

1.      The strain is decomposed into elastic and irreversible (damage) parts  as d ε ij =d ε ij e +d ε ij p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabew 7aLnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGKbGaeqyT du2aa0baaSqaaiaadMgacaWGQbaabaGaamyzaaaakiabgUcaRiaads gacqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaaaa@48A5@ ;

2.      The elastic part of the strain is related to the stress by the linear elastic constitutive equations

d ε ij e = 1+ν E d σ ij ν E d σ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaH1oqzdaqhaaWcbaGaamyAai aadQgaaeaacaWGLbaaaOGaeyypa0ZaaSaaaeaacaaIXaGaey4kaSIa eqyVd4gabaGaamyraaaacaWGKbGaeq4Wdm3aaSbaaSqaaiaadMgaca WGQbaabeaakiabgkHiTmaalaaabaGaeqyVd4gabaGaamyraaaacaWG KbGaeq4Wdm3aaSbaaSqaaiaadUgacaWGRbaabeaakiabes7aKnaaBa aaleaacaWGPbGaamOAaaqabaaaaa@4BD8@

3.      The critical stress that initiates crushing damage is given by a failure criterion (analogous to the yield criterion for a metal) of the form

f( σ ij )= 3 2 S ij S ij +3cp(1c)Y( ε ¯ p )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiabg2da 9maakaaabaWaaSaaaeaacaaIZaaabaGaaGOmaaaacaWGtbWaaSbaaS qaaiaadMgacaWGQbaabeaakiaadofadaWgaaWcbaGaamyAaiaadQga aeqaaaqabaGccqGHRaWkcaaIZaGaam4yaiaadchacqGHsislcaGGOa GaaGymaiabgkHiTiaadogacaGGPaGaamywaiaacIcacuaH1oqzgaqe amaaCaaaleqabaGaamiCaaaakiaacMcacqGH9aqpcaaIWaaaaa@54C0@

where p= σ kk /3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccaGGVaGaaG4m aaaa@3E41@ , S ij = σ ij p δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyOeI0IaamiCaiabes7aKnaaBaaaleaaca WGPbGaamOAaaqabaaaaa@4454@ , and ε ¯ p = 2d ε ij p d ε ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbae badaahaaWcbeqaaiaadchaaaGccqGH9aqpdaWdbaqaamaakaaabaGa aGOmaiaadsgacqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGWb aaaOGaamizaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadcha aaGccaGGVaGaaG4maaWcbeaaaeqabeqdcqGHRiI8aaaa@4A57@  is the accumulated irreversible strain. Notice that the failure criterion depends on the hydrostatic part of the stress: unlike yield in metals, the material becomes more resistant to fracture if p<0.

4.      The plastic strain components are determined using an associated flow rule

d ε ij p = d ε ¯ p 1+2 c 2 df d σ ij = d ε ¯ p 1+2 c 2 { 3 2 S ij 3 S kl S kl /2 +c δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabew 7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadchaaaGccqGH9aqpdaWc aaqaaiaadsgacuaH1oqzgaqeamaaCaaaleqabaGaamiCaaaaaOqaam aakaaabaGaaGymaiabgUcaRiaaikdacaWGJbWaaWbaaSqabeaacaaI YaaaaaqabaaaaOWaaSaaaeaacaWGKbGaamOzaaqaaiaadsgacqaHdp WCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaakiabg2da9maalaaabaGa amizaiqbew7aLzaaraWaaWbaaSqabeaacaWGWbaaaaGcbaWaaOaaae aacaaIXaGaey4kaSIaaGOmaiaadogadaahaaWcbeqaaiaaikdaaaaa beaaaaGcdaGadaqaamaalaaabaGaaG4maaqaaiaaikdaaaWaaSaaae aacaWGtbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqaamaakaaabaGa aG4maiaadofadaWgaaWcbaGaam4AaiaadYgaaeqaaOGaam4uamaaBa aaleaacaWGRbGaamiBaaqabaGccaGGVaGaaGOmaaWcbeaaaaGccqGH RaWkcaWGJbGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGaay 5Eaiaaw2haaaaa@69FB@

 

5.      The magnitude of the plastic strain increment is related to the stress increment by

d ε ¯ p = 1 h(1c) 3 2 S ij d σ ij 3 S kl S kl /2 +cd σ kk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiqbew 7aLzaaraWaaWbaaSqabeaacaWGWbaaaOGaeyypa0ZaaSaaaeaacaaI XaaabaGaamiAaiaacIcacaaIXaGaeyOeI0Iaam4yaiaacMcaaaWaaa WaaeaadaWcaaqaaiaaiodaaeaacaaIYaaaamaalaaabaGaam4uamaa BaaaleaacaWGPbGaamOAaaqabaGccaWGKbGaeq4Wdm3aaSbaaSqaai aadMgacaWGQbaabeaaaOqaamaakaaabaGaaG4maiaadofadaWgaaWc baGaam4AaiaadYgaaeqaaOGaam4uamaaBaaaleaacaWGRbGaamiBaa qabaGccaGGVaGaaGOmaaWcbeaaaaGccqGHRaWkcaWGJbGaamizaiab eo8aZnaaBaaaleaacaWGRbGaam4AaaqabaaakiaawMYicaGLQmcaaa a@5B46@

where h=dY/d ε ¯ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObGaeyypa0JaamizaiaadMfaca GGVaGaamizaiqbew7aLzaaraWaaWbaaSqabeaacaWGWbaaaaaa@3AFD@  is the slope of the uniaxial stress-strain curve, and x =x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadIhaaiaawMYicaGLQm cacqGH9aqpcaWG4baaaa@3796@  for x>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bGaeyOpa4JaaGimaaaa@3585@ , while x =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaadaqaaiaadIhaaiaawMYicaGLQm cacqGH9aqpcaaIWaaaaa@3753@  for x<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bGaeyipaWJaaGimaaaa@3581@ .

 

HEALTH WARNING: These constitutive equations should only be used in regions where the hydrostatic stress is compressive p<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabgY da8iaaicdaaaa@3899@ .  In regions of hydrostatic tension, a tensile brittle fracture criterion should be used MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for example, the material could be assumed to lose all load bearing capacity if the principal tensile stress exceeds a critical magnitude.

 

 

 

9.2.5  Ductile Fracture Criteria

 

Strain to failure approach: Ductile fracture in tension occurs by the nucleation, growth and coalescence of voids in the material.  A crude criterion for ductile failure could be based on the accumulated plastic strain, for example

2 3 d ε ij p d ε ij p = ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada GcaaqaamaalaaabaGaaGOmaaqaaiaaiodaaaGaamizaiabew7aLnaa DaaaleaacaWGPbGaamOAaaqaaiaadchaaaGccaWGKbGaeqyTdu2aa0 baaSqaaiaadMgacaWGQbaabaGaamiCaaaaaeqaaaqaaaqab0Gaey4k Iipakiabg2da9iabew7aLnaaBaaaleaacaWGMbaabeaaaaa@49CB@

at failure, where ε f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadAgaaeqaaaaa@38B4@  is the plastic strain to failure in a uniaxial tensile test.

 

Porous metal plasticity: Experiments show that the strain to cause ductile failure in a material depends on the hydrostatic component of tensile stress acting on the specimen, as shown in the figure.  For example, the strain to failure under torsional loading (which subjects the material to shear with no hydrostatic stress) is much greater than under uniaxial tension. The critical strain is influenced by hydrostatic stress because ductile failure occurs as a result of the nucleation and growth of cavities in the solid.  A hydrostatic stress greatly increases the rate of growth of the cavities. The simple strain-to-failure approach cannot account for this behavior.

 

Porous metal plasticity was developed to address this issue.  The basic idea is simple: the solid is idealized as a plastic matrix which contains a volume fraction V f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaadAgaaeqaaa aa@34B8@  of cavities.  To model the solid, the plastic stress-strain laws outlined in Sections 3.6 and 3.7 are extended to calculate the volume fraction of voids in the material as part of the solution, and also to account for the weakening effect of the voids.  Failure is modeled by constructing the plastic stress-strain law so that the material loses all its strength at a critical void volume fraction.

 

Both rate independent and viscoplastic versions of porous plasticity exist.  The viscoplastic models have some advantages for finite element computations, because the rate dependence can stabilize the effects of strain softening.  A simple small-strain viscoplastic constitutive law with power-law hardening and power-law rate dependence will be outlined here to illustrate the main features of these models.  The constitutive law is known as the `Gurson model.’  

 

 

The material is characterized by the following properties:

 The Young’s modulus E and Poisson ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@ ;

 A characteristic stress Y, a characteristic strain ε 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGimaaqaba aaaa@3553@  and strain hardening exponent n, which govern the strain hardening behavior of the matrix material;

 A characteristic strain rate ε ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGaamaaBaaaleaacaaIWa aabeaaaaa@355C@  and strain rate exponent m, which govern the strain rate sensitivity of the solid;

 A constant N v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGobWaaSbaaSqaaiaadAhaaeqaaa aa@34C0@ , which controls the rate of void nucleation with plastic straining;

 The flow strength of the matrix σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqaba aaaa@356F@ , the void volume fraction, V f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaBa aaleaacaWGMbaabeaaaaa@37E8@ , and the total accumulated effective plastic strain in the matrix material ε ¯ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaqeamaaBaaaleaacaWGTb aabeaaaaa@35A3@  which all evolve with plastic straining.

The constitutive equations specify a relationship between the stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  applied to the material and the resulting strain rate ε ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGaamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@367F@ , as follows

1.      The strain rate is decomposed into elastic and plastic parts  as ε ˙ ij = ε ˙ ij e + ε ˙ ij p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbai aadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JafqyTduMbaiaa daqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaey4kaSIafqyTdu MbaiaadaqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaaaa@4605@ ;

2.      The elastic part of the strain rate is related to the stress rate by the linear elastic constitutive equations

ε ˙ ij e = 1+ν E σ ˙ ij ν E σ ˙ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbew7aLzaacaWaa0baaSqaaiaadMgaca WGQbaabaGaamyzaaaakiabg2da9maalaaabaGaaGymaiabgUcaRiab e27aUbqaaiaadweaaaGafq4WdmNbaiaadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyOeI0YaaSaaaeaacqaH9oGBaeaacaWGfbaaaiqbeo8a ZzaacaWaaSbaaSqaaiaadUgacaWGRbaabeaakiabes7aKnaaBaaale aacaWGPbGaamOAaaqabaaaaa@4938@

3.      The magnitude of the plastic strain rate is determined by the following plastic flow potential

ε ˙ e =g( σ e ,p, σ 0 , V f )= ε ˙ 0 [ ( σ e σ 0 ) 2 +2 V f cosh( 3p 2 σ 0 ) V f 2 ] m/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGaamaaBaaaleaacaWGLb aabeaakiabg2da9iaadEgacaGGOaGaeq4Wdm3aaSbaaSqaaiaadwga aeqaaOGaaiilaiaadchacaGGSaGaeq4Wdm3aaSbaaSqaaiaaicdaae qaaOGaaiilaiaadAfadaWgaaWcbaGaamOzaaqabaGccaGGPaGaeyyp a0JafqyTduMbaiaadaWgaaWcbaGaaGimaaqabaGcdaWadaqaamaabm aabaWaaSaaaeaacqaHdpWCdaWgaaWcbaGaamyzaaqabaaakeaacqaH dpWCdaWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadAfadaWgaaWcbaGaamOz aaqabaGcciGGJbGaai4BaiaacohacaGGObWaaeWaaeaadaWcaaqaai aaiodacaWGWbaabaGaaGOmaiabeo8aZnaaBaaaleaacaaIWaaabeaa aaaakiaawIcacaGLPaaacqGHsislcaWGwbWaa0baaSqaaiaadAgaae aacaaIYaaaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacaWGTbGaai4l aiaaikdaaaaaaa@6579@

where ε ˙ e =2 ε ˙ ij p ε ˙ ij p /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaGaamaaBaaaleaacaWGLb aabeaakiabg2da9iaaikdacuaH1oqzgaGaamaaDaaaleaacaWGPbGa amOAaaqaaiaadchaaaGccuaH1oqzgaGaamaaDaaaleaacaWGPbGaam OAaaqaaiaadchaaaGccaGGVaGaaG4maaaa@423A@   σ e = 3 S ij S ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyzaaqaba GccqGH9aqpdaGcaaqaaiaaiodacaWGtbWaaSbaaSqaaiaadMgacaWG QbaabeaakiaadofadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaai4lai aaikdaaSqabaaaaa@3ECC@ , p= σ kk /3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9iabeo8aZnaaBaaaleaacaWGRbGaam4AaaqabaGccaGGVaGaaG4m aaaa@3E41@  and S ij = σ ij p δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyOeI0IaamiCaiabes7aKnaaBaaaleaaca WGPbGaamOAaaqabaaaaa@4454@ .  Note that for V f >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaSbaaSqaaiaadAgaaeqaaO GaeyOpa4JaaGimaaaa@3684@  the plastic strain rate increases with hydrostatic stress p.

4.      The components of the plastic strain rate tensor are computed from an associated flow law

ε ˙ ij p = 3 2 g( σ e ,p, σ 0 , V f ) [ ( g/ σ kl )( g/ σ kl ) ] 1/2 g σ ij = g( σ e ,p, σ 0 , V f ) σ e 2 / σ 0 2 +( V f 2 /2) sinh 2 (3p/2 σ 0 ) { 3 2 S ij σ 0 + V f 2 sinh( 3p 2 σ 0 ) δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqyTduMbaiaadaqhaaWcbaGaam yAaiaadQgaaeaacaWGWbaaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaa iodaaeaacaaIYaaaaaWcbeaakmaalaaabaGaam4zaiaacIcacqaHdp WCdaWgaaWcbaGaamyzaaqabaGccaGGSaGaamiCaiaacYcacqaHdpWC daWgaaWcbaGaaGimaaqabaGccaGGSaGaamOvamaaBaaaleaacaWGMb aabeaakiaacMcaaeaadaWadaqaamaabmaabaGaeyOaIyRaam4zaiaa c+cacqGHciITcqaHdpWCdaWgaaWcbaGaam4AaiaadYgaaeqaaaGcca GLOaGaayzkaaWaaeWaaeaacqGHciITcaWGNbGaai4laiabgkGi2kab eo8aZnaaBaaaleaacaWGRbGaamiBaaqabaaakiaawIcacaGLPaaaai aawUfacaGLDbaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaGc daWcaaqaaiabgkGi2kaadEgaaeaacqGHciITcqaHdpWCdaWgaaWcba GaamyAaiaadQgaaeqaaaaaaOqaaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabg2da9maalaaabaGaam 4zaiaacIcacqaHdpWCdaWgaaWcbaGaamyzaaqabaGccaGGSaGaamiC aiaacYcacqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaGGSaGaamOvam aaBaaaleaacaWGMbaabeaakiaacMcaaeaadaGcaaqaaiabeo8aZnaa DaaaleaacaWGLbaabaGaaGOmaaaakiaac+cacqaHdpWCdaqhaaWcba GaaGimaaqaaiaaikdaaaGccqGHRaWkcaGGOaGaamOvamaaDaaaleaa caWGMbaabaGaaGOmaaaakiaac+cacaaIYaGaaiykaiGacohacaGGPb GaaiOBaiaacIgadaahaaWcbeqaaiaaikdaaaGccaGGOaGaaG4maiaa dchacaGGVaGaaGOmaiabeo8aZnaaBaaaleaacaaIWaaabeaakiaacM caaSqabaaaaOWaaiWaaeaadaWcaaqaaiaaiodaaeaacaaIYaaaamaa laaabaGaam4uamaaBaaaleaacaWGPbGaamOAaaqabaaakeaacqaHdp WCdaWgaaWcbaGaaGimaaqabaaaaOGaey4kaSYaaSaaaeaacaWGwbWa aSbaaSqaaiaadAgaaeqaaaGcbaGaaGOmaaaaciGGZbGaaiyAaiaac6 gacaGGObWaaeWaaeaadaWcaaqaaiaaiodacaWGWbaabaGaaGOmaiab eo8aZnaaBaaaleaacaaIWaaabeaaaaaakiaawIcacaGLPaaacqaH0o azdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGL7bGaayzFaaaaaaa@B5A1@

5.      Strain hardening in the matrix is modeled by relating its flow stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqaba aaaa@356F@  to the accumulated strain in the matrix ε ¯ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH1oqzgaqeamaaBaaaleaacaWGTb aabeaaaaa@35A3@ . The following power-law hardening model is often used

σ 0 =Y ( 1+ ε ¯ m / ε 0 ) 1/n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGimaaqaba GccqGH9aqpcaWGzbWaaeWaaeaacaaIXaGaey4kaSIafqyTduMbaeba daWgaaWcbaGaamyBaaqabaGccaGGVaGaeqyTdu2aaSbaaSqaaiaaic daaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaa d6gaaaaaaa@4342@

6.      The effective plastic strain in the matrix is calculated from the condition that the plastic dissipation in the matrix must equal the rate of work done by stresses, which requires that

(1 V f ) σ 0 ε ¯ ˙ m = σ ij ε ˙ ij p ε ¯ ˙ m = g( σ e ,p, σ 0 , V f ) (1 V f ) ( σ e 2 σ 0 2 + 3p 2 σ 0 V f sinh( 3p 2 σ 0 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaGymaiabgkHiTiaadAfada WgaaWcbaGaamOzaaqabaGccaGGPaGaeq4Wdm3aaSbaaSqaaiaaicda aeqaaOGafqyTduMbaeHbaiaadaWgaaWcbaGaamyBaaqabaGccqGH9a qpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGafqyTduMbaiaa daqhaaWcbaGaamyAaiaadQgaaeaacaWGWbaaaOGaaGPaVlaaykW7ca aMc8UaeyO0H4TaaGPaVlaaykW7caaMc8UaaGPaVlqbew7aLzaaryaa caWaaSbaaSqaaiaad2gaaeqaaOGaeyypa0ZaaSaaaeaacaWGNbGaai ikaiabeo8aZnaaBaaaleaacaWGLbaabeaakiaacYcacaWGWbGaaiil aiabeo8aZnaaBaaaleaacaaIWaaabeaakiaacYcacaWGwbWaaSbaaS qaaiaadAgaaeqaaOGaaiykaaqaaiaacIcacaaIXaGaeyOeI0IaamOv amaaBaaaleaacaWGMbaabeaakiaacMcaaaWaaeWaaeaadaWcaaqaai abeo8aZnaaDaaaleaacaWGLbaabaGaaGOmaaaaaOqaaiabeo8aZnaa DaaaleaacaaIWaaabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaio dacaWGWbaabaGaaGOmaiabeo8aZnaaBaaaleaacaaIWaaabeaaaaGc caWGwbWaaSbaaSqaaiaadAgaaeqaaOGaci4CaiaacMgacaGGUbGaai iAamaabmaabaWaaSaaaeaacaaIZaGaamiCaaqaaiaaikdacqaHdpWC daWgaaWcbaGaaGimaaqabaaaaaGccaGLOaGaayzkaaaacaGLOaGaay zkaaaaaa@8538@

7.      Finally, the model is completed by specifying the void volume fraction as a function of strain.  The void volume fraction can increase due to growth of existing voids, or nucleation of new ones. To account for both effects, one can set

V ˙ f =(1 V f ) ε ˙ kk p + N v ε ˙ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOvayaaca WaaSbaaSqaaiaadAgaaeqaaOGaeyypa0JaaiikaiaaigdacqGHsisl caWGwbWaaSbaaSqaaiaadAgaaeqaaOGaaiykaiqbew7aLzaacaWaa0 baaSqaaiaadUgacaWGRbaabaGaamiCaaaakiabgUcaRiaad6eadaWg aaWcbaGaamODaaqabaGccuaH1oqzgaGaamaaBaaaleaacaWGLbaabe aaaaa@497D@

where the first term accounts for void growth, and the second accounts for strain controlled void nucleation. 

 
 
 
9.2.6 Ductile failure by strain localization

 

If you test a cylindrical specimen of a very ductile material in uniaxial tension, it will initially deform uniformly, and remain cylindrical. At a critical load (or strain) the specimen will start to neck, as shown in the picture.  Necking, once it starts, is usually unstable MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  there is a concentration in stress near the necked region, increasing the rate of plastic flow near the neck compared with the rest of the specimen, and so increasing the rate of neck formation.   The strains in the necked region rapidly become very large, and quickly lead to failure.

 

Neck formation is a consequence of geometric softening.  A very simple model explains the concept of geometric softening.

1.      Consider a cylindrical specimen with initial cross sectional area A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyqamaaBaaaleaacaaIWaaabeaaaa a@33AE@  and length L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamitamaaBaaaleaacaaIWaaabeaaaa a@33B9@ . The specimen is subjected to a load P, which deforms the material plastically. After straining, the length of the specimen increases to L, and its cross-sectional area decreases to A.

2.      Assume that the material is perfectly plastic and has a true stress-strain curve (Cauchy stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@ v- logarithmic strain) that can be approximated by a power-law σ= σ 0 ε n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeqyTdu2aaWbaaSqa beaacaWGUbaaaaaa@3F40@  with n<1.

3.      The true strain in the specimen is related to its length by ε=log L L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqyTduMaeyypa0JaciiBaiaac+gaca GGNbWaaSaaaeaacaWGmbaabaGaamitamaaBaaaleaacaaIWaaabeaa aaaaaa@3A17@

4.      The force on the specimen is related to the Cauchy stress by P=Aσ=A σ 0 ε n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamiuaiabg2da9iaadgeacqaHdpWCcq GH9aqpcaWGbbGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaeqyTdu2a aWbaaSqabeaacaWGUbaaaaaa@3DAC@

5.      At the point of maximum load dP dL = dA dL σ+A dσ dL =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacaWGKbGaamiuaaqaaiaads gacaWGmbaaaiabg2da9maalaaabaGaamizaiaadgeaaeaacaWGKbGa amitaaaacqaHdpWCcqGHRaWkcaWGbbWaaSaaaeaacaWGKbGaeq4Wdm habaGaamizaiaadYeaaaGaeyypa0JaaGimaaaa@43AA@

6.      We can calculate dA/dL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamizaiaadgeacaGGVaGaamizaiaadY eaaaa@361E@  by noting that the volume of the specimen is constant during plastic straining, which shows that

AL= A 0 L 0 dA dL L+A=0 dA dL = A L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyqaiaadYeacqGH9aqpcaWGbbWaaS baaSqaaiaaicdaaeqaaOGaamitamaaBaaaleaacaaIWaaabeaakiaa ykW7caaMc8UaaGPaVlaaykW7cqGHshI3caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVpaalaaabaGaamizaiaadgeaaeaacaWGKbGaamitaaaa caWGmbGaey4kaSIaamyqaiabg2da9iaaicdacaaMc8UaaGPaVlaayk W7caaMc8UaeyO0H4TaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGa amizaiaadgeaaeaacaWGKbGaamitaaaacqGH9aqpcqGHsisldaWcaa qaaiaadgeaaeaacaWGmbaaaaaa@65D0@

Notice that dA/dL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamizaiaadgeacaGGVaGaamizaiaadY eaaaa@361E@  is negative MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this means that the specimen tends to soften as a result of the change in its cross sectional area.  This is what is meant by geometric softening.

7.      We can calculate dσ/dL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamizaiabeo8aZjaac+cacaWGKbGaam itaaaa@371B@  from (2) and (3) as follows

dσ dL = dσ dε dε dL = n σ 0 ε n1 L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaSaaaeaacaWGKbGaeq4WdmhabaGaam izaiaadYeaaaGaeyypa0ZaaSaaaeaacaWGKbGaeq4WdmhabaGaamiz aiabew7aLbaadaWcaaqaaiaadsgacqaH1oqzaeaacaWGKbGaamitaa aacqGH9aqpdaWcaaqaaiaad6gacqaHdpWCdaWgaaWcbaGaaGimaaqa baGccqaH1oqzdaahaaWcbeqaaiaad6gacqGHsislcaaIXaaaaaGcba Gaamitaaaaaaa@4B2A@

Notice that dσ/dL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamizaiabeo8aZjaac+cacaWGKbGaam itaaaa@371B@  is positive MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  strain hardening in the material tends to compensate for the effects of geometric softening.

8.      Finally, substituting the results of (6) and (7) back into (5) and recalling that  σ= σ 0 ε n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeq4WdmNaeyypa0Jaeq4Wdm3aaSbaaS qaaiaaicdaaeqaaOGaeqyTdu2aaWbaaSqabeaacaWGUbaaaaaa@3A45@  shows that at the point of maximum load, the strain and length of the specimen are

Aσ L +A n σ 0 ε n1 L =0 ε max =n L max = L 0 exp(n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeyOeI0YaaSaaaeaacaWGbbGaeq4Wdm habaGaamitaaaacqGHRaWkcaWGbbWaaSaaaeaacaWGUbGaeq4Wdm3a aSbaaSqaaiaaicdaaeqaaOGaeqyTdu2aaWbaaSqabeaacaWGUbGaey OeI0IaaGymaaaaaOqaaiaadYeaaaGaeyypa0JaaGimaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyO0H4Taeq yTdu2aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqGH9aqpcaWG UbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgk DiElaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadYeadaWg aaWcbaGaciyBaiaacggacaGG4baabeaakiabg2da9iaadYeadaWgaa WcbaGaaGimaaqabaGcciGGLbGaaiiEaiaacchacaGGOaGaamOBaiaa cMcaaaa@7A55@

9.      Finally, note that by volume conservation the cross sectional area is A= A 0 L 0 /L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamyqaiabg2da9iaadgeadaWgaaWcba GaaGimaaqabaGccaWGmbWaaSbaaSqaaiaaicdaaeqaaOGaai4laiaa dYeaaaa@38C9@ , so the maximum load the specimen can withstand follows as

P max =A σ max = A 0 L 0 L σ 0 ( ε max ) n = A 0 σ 0 n n exp(n) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0Jaamyqaiabeo8a ZnaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyypa0ZaaSaaae aacaWGbbWaaSbaaSqaaiaaicdaaeqaaOGaamitamaaBaaaleaacaaI WaaabeaaaOqaaiaadYeaaaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaO Gaaiikaiabew7aLnaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGa aiykamaaCaaaleqabaGaamOBaaaakiabg2da9iaadgeadaWgaaWcba GaaGimaaqabaGccqaHdpWCdaWgaaWcbaGaaGimaaqabaGccaWGUbWa aWbaaSqabeaacaWGUbaaaOGaciyzaiaacIhacaGGWbGaaiikaiabgk HiTiaad6gacaGGPaaaaa@5E5A@

It turns out that the point of maximum load coincides with the condition for unstable neck formation in the bar.  This is plausible MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a falling load displacement curve is always a sign that there might be a possibility of non-unique solutions MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but a rather sophisticated calculation is required to show this rigorously.

 

There are two important points to take away from this discussion.

    Plastic localization, as opposed to material failure, may limit load bearing capacity;

    If you measure the strain to failure of a material in uniaxial tension, it is possible that you have not measured the inherent strength of the material MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  your specimen may have failed due to a geometric effect.  Material behavior does influence the strain to failure, of course: the simple analysis of geometric softening shows that the strain hardening behavior of the material is critical. 

 

Plastic localization can occur for many reasons.  There are two general classes of localization MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it may occur as a consequence of changes in specimen geometry (i.e. geometric softening); or it may occur due to a natural tendency of the material itself to soften at large strains.

 

Examples of geometry induced localization are

1.      Neck formation in a bar under uniaxial tension;

2.      Shear band formation in torsional or shear loading at high strain rate due to thermal softening as a result of plastic heat generation

 

Examples of material induced localization are

1.      Localization in a Gurson solid due to the softening effect of voids at large strains;

2.      Localization in a single crystal due to the softening effect of lattice rotations;

3.      Localization in a brittle microcracking material due to the increase in elastic compliance caused by the cracks.

 

Geometric localization can be modeled quite easily, because it does not rely on any empirical failure criteria.  A straightforward FEM computation, with an appropriate constitutive law and proper consideration of finite strains, will predict localization if it is going to occur MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the only thing you need to worry about is to be sure you understand what triggered the localization.  Localization can start at a geometric imperfection in the model, in which case your prediction is meaningful (but may be sensitive to the nature of the imperfection).  It may also be triggered by numerical errors, in which case the predicted failure load is meaningless.  It is usually exceedingly difficult to compute what happens after localization.  Fortunately it’s rather rare to need to do this for design purposes.

 

 

 

9.2.7 Criteria for failure by high cycle fatigue under constant amplitude cyclic loading

 

Empirical stress or strain based life prediction methods are extensively used in design applications.  The approach is straightforward MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  subject a sample of the material to a cycle of stress (or strain) that resembles service loading, in an environment representative of service conditions, and measure its life as a function of stress (or strain) amplitude, then fit the data with a curve.

 

Here we will review criteria that are used to predict fatigue life under proportional cyclic loading. A typical stress cycle is parameterized by its amplitude ( σ max σ min )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabeo 8aZnaaBaaaleaaciGGTbGaaiyyaiaacIhaaeqaaOGaeyOeI0Iaeq4W dm3aaSbaaSqaaiGac2gacaGGPbGaaiOBaaqabaGccaGGPaGaai4lai aaikdaaaa@444A@  and the mean stress σ m =( σ max + σ min )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaad2gaaeqaaOGaeyypa0Jaaiikaiabeo8aZnaaBaaaleaa ciGGTbGaaiyyaiaacIhaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaai Gac2gacaGGPbGaaiOBaaqabaGccaGGPaGaai4laiaaikdaaaa@4830@

 

For tests run in the high cycle fatigue regime with any fixed value of mean stress, the relationship between stress amplitude σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggaaeqaaaaa@39D2@  and the number of cycles to failure N is fit well by Basquin’s Law

σ a N b =C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggaaeqaaOGaamOtamaaCaaaleqabaGaamOyaaaakiab g2da9iaadoeaaaa@3D9B@

where the exponent b is typically between 0.05 and 0.15.  The constant C is a function of mean stress.

 

There are two ways to account for the effects of mean stress.  Both are based on the same idea: we know that if the mean stress is equal to the tensile strength of the material σ= σ UTS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCcqGH9aqpcqaHdpWCdaWgaa WcbaGaamyvaiaadsfacaWGtbaabeaaaaa@3A09@ , it will fail in 0 cycles of loading.  We also know that for zero mean stress, the fatigue life obeys Basquin’s law.  We can interpolate between these two points.  There are two ways to do this:

 Goodman’s rule uses a linear interpolation, giving

σ a N b = C 0 ( 1 σ m σ UTS ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggaaeqaaOGaamOtamaaCaaaleqabaGaamOyaaaakiab g2da9iaadoeadaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaaigdacq GHsisldaWcaaqaaiabeo8aZnaaBaaaleaacaWGTbaabeaaaOqaaiab eo8aZnaaBaaaleaacaWGvbGaamivaiaadofaaeqaaaaaaOGaayjkai aawMcaaaaa@493B@

where C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIWaaabeaaaaa@37A4@  is the constant in Basquin’s law determined by testing at zero mean stress.

 Gerber’s rule uses a parabolic fit

σ a N b = C 0 { 1 ( σ m σ UTS ) 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadggaaeqaaOGaamOtamaaCaaaleqabaGaamOyaaaakiab g2da9iaadoeadaWgaaWcbaGaaGimaaqabaGcdaGadaqaaiaaigdacq GHsisldaqadaqaamaalaaabaGaeq4Wdm3aaSbaaSqaaiaad2gaaeqa aaGcbaGaeq4Wdm3aaSbaaSqaaiaadwfacaWGubGaam4uaaqabaaaaa GccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGccaGL7bGaayzF aaaaaa@4C5F@

In practice, experimental data seem to lie between these two limits.  Goodman’s rule gives a safe estimate.

 

These criteria are intended to be used for components that are subjected to uniaxial tensile stress.  The criteria can still be used if the loading is proportional (i.e. with fixed directions of principal stress).  In this case, the maximum principal stress should be used to calculate σ a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGHbaabeaaaa a@333D@  and σ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGTbaabeaaaa a@3349@ .  They do not work under non-proportional loading.   A very large number of fatigue models have been developed for more general loading conditions MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  a review can be found in Liu and Mahadevan, Int J. Fatigue, 27 790-800 (2005).

 

 

 

9.2.8 Criteria for failure by low cycle fatigue

 

If a fatigue test is run with a high stress level (sufficient to cause plastic flow in a large section of the solid) the specimen fails very quickly (less than 10 000 cycles).  This regime of behavior is known as `low cycle fatigue’.  The fatigue life correlates best with the plastic strain amplitude rather than stress amplitude, and it is found that the Coffin Manson Law

Δ ε p N b =C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yTdu2aaWbaaSqabeaacaWGWbaaaOGaamOtamaaCaaaleqabaGaamOy aaaakiabg2da9iaadoeaaaa@3EF5@

gives a good fit to empirical data (the constants C and b do not have the same values as for Basquin’s law, of course)

 

 

9.2.9 Criteria for failure under variable amplitude cyclic loading

 

Fatigue tests are usually done at constant stress (or strain) amplitude.  Service loading usually involves cycles with variable (and often random) amplitude.  Fortunately, there’s a remarkably simple way to estimate fatigue life under variable loading using constant stress data.

 

Suppose the load history is comprised of a set of n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaaIXaaabeaaaaa@38D7@  load cycles at a stress amplitude σ a (1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadggaaeaacaGGOaGaaGymaiaacMcaaaaaaa@3BE7@ , followed by a set of n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaaIYaaabeaaaaa@38D8@  cycles at load amplitude σ a (2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadggaaeaacaGGOaGaaGOmaiaacMcaaaaaaa@3BE8@  and so on.  For the ith set of cycles at load amplitude σ a (i) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadggaaeaacaGGOaGaamyAaiaacMcaaaaaaa@3C1A@ , we could compute the number of cycles that would cause the specimen to fail using Basquin’s law

σ a (i) N i b =C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aa0 baaSqaaiaadggaaeaacaGGOaGaamyAaiaacMcaaaGccaWGobWaa0ba aSqaaiaadMgaaeaacaWGIbaaaOGaeyypa0Jaam4qaaaa@40D1@

The Miner-Palmgren failure criterion assumes a linear summation of damage due to each set of load cycles, so that at failure

i n i N i =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada Wcaaqaaiaad6gadaWgaaWcbaGaamyAaaqabaaakeaacaWGobWaaSba aSqaaiaadMgaaeqaaaaakiabg2da9iaaigdaaSqaaiaadMgaaeaaa0 GaeyyeIuoaaaa@400C@

In terms of stress amplitude

i n i ( σ a (i) C ) 1/b =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGUbWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaadaWcaaqaaiabeo8a ZnaaDaaaleaacaWGHbaabaGaaiikaiaadMgacaGGPaaaaaGcbaGaam 4qaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaamOy aaaakiabg2da9iaaigdaaSqaaiaadMgaaeaaa0GaeyyeIuoaaaa@4819@

The same approach works under low cycle fatigue conditions, in which case

i n i ( Δ ε p (i) C ) 1/b =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca WGUbWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaadaWcaaqaaiabfs5a ejabew7aLnaaCaaaleqabaGaamiCaaaakmaaDaaaleaaaeaacaGGOa GaamyAaiaacMcaaaaakeaacaWGdbaaaaGaayjkaiaawMcaamaaCaaa leqabaGaaGymaiaac+cacaWGIbaaaOGaeyypa0JaaGymaaWcbaGaam yAaaqaaaqdcqGHris5aaaa@49A9@

 

The criterion is often used under random loading. To do so, we need to find a way to estimate the number of cycles of load at a given stress level.  There are various ways to do this MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  one approach is to count the peaks in the load history, and compute the probability P(σ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacI cacqaHdpWCcaGGPaaaaa@39D7@  of finding a peak at stress level σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa@37A9@ . (Of course, this only works if the signal has well defined peaks - this is not the case for white noise, for example).

 

Miner’s rule then predicts that the number of cycles to failure satisfies

N 0 P(σ)( σ C ) 1/b dσ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaape habaGaamiuaiaacIcacqaHdpWCcaGGPaWaaeWaaeaadaWcaaqaaiab eo8aZbqaaiaadoeaaaaacaGLOaGaayzkaaaaleaacaaIWaaabaGaey OhIukaniabgUIiYdGcdaahaaWcbeqaaiaaigdacaGGVaGaamOyaaaa kiaadsgacqaHdpWCcqGH9aqpcaaIXaaaaa@4B7C@