Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

 

The equations listed in Chapter 2 are universal MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they apply to all deformable solids.  They can’t be solved, however, unless the deformation measure can be related to the internal forces.

 

The constitutive model for a material is a set of equations relating stress to strain (and possibly strain history, strain rate, and other field quantities).  Unlike the governing equations in the previous chapter, these equations cannot generally be calculated using fundamental physical laws (although people are trying to do this).  Instead, constitutive models are fit to experimental measurements.

 

Before discussing specific constitutive models, it is helpful to review the basic assumptions that we take for granted in developing stress-strain laws.  They are listed below.

 A very small sample that is extracted from the solid has uniform properties;

 When the solid is deformed, initially straight lines in the solid are deformed into smooth curves (with continuous slope);

 This means that very short line segments (much shorter than the radius of curvature of the curves) are just stretched and rotated by the deformation.   Consequently, the deformation of a sufficiently small volume element can be characterized by the deformation gradient;

 The stress at a point in the solid depends only on the change in shape of a vanishingly small volume element surrounding the point.  It must therefore be a function of the deformation gradient or a strain measure that is derived from it.

If we accept the preceding assumptions, it means that we can measure the relationship between stress and strain by doing an experiment that induces a uniform strain in a suitable sample of the material.  According to our assumptions, the stress should also be uniform, and can be calculated from the forces acting on the specimen.

 

These are clearly approximations.  Materials are not really uniform at small scales, whether you choose to look at the atomic scale, or the microstructural scale.  However, these features are usually much smaller than the size of the solid part or component, and the material can be regarded as statistically uniform, in the sense that if you cut two specimens with similar size out of the material they will behave in the same way.  A continuum model then describes the average stress and deformation in a region of the material that is larger than microstructural features, but small compared with the dimensions of the part.

 

 

3.1 General requirements for constitutive equations

 

You may be called upon to develop a stress-strain law for a new material at some point of your career.  If so, it is essential to make sure that the stress-strain law satisfies two conditions:

(i) It must obeys the laws of thermodynamics.

(ii) It must satisfy the condition of objectivity, or material frame indifference

In addition, it is a good idea to ensure that the material satisfies the Drucker stability criterion discussed in more detail below.  Of course, your proposed law must conform to experimental measurements, and if possible should be based on some understanding of the physical processes that govern the response of the solid.

 

Thermodynamic restrictions: The laws of thermodynamics impose two restrictions on stress-strain laws:

1.      The first law requires that the work done by stresses must either be stored as recoverable internal energy in the solid, or be dissipated as heat (or a combination of both).

2.      The second law requires that if a sample of the material is subjected to a cycle of deformation that starts and ends with an identical strain and internal energy (at constant temperature, or without heat exchange with the surroundings) the total work done must be positive or zero.

 

Objectivity: Strictly speaking, the term “objectivity” or “material frame independence” is the condition that the tensor-valued functions that relate stress to deformation measure must transform correctly under a change of basis and change of origin for the coordinate system.   A detailed mathematical derivation of the consequences of objectivity will not be given here (see for example Malvern “Introduction to the Mechanics of a Continuous Medium,” Prentice Hall).  However, you can check whether a constitutive law is objective using the following simple (analytical) test:

1.      Load the solid (quasi-statically) by subjecting its boundary to prescribed forces, to induce a Cauchy stress σ ij (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaacIcacaaIWaGaaiykaaaaaaa@3648@  in the solid.

2.      Subject both the solid, and the loads acting on the solid, to a quasi-static rigid rotation, characterized by a rotation tensor R ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3348@ .

3.      The constitutive law must predict that after the rotation, the stress components change to new values given by σ ij (1) = R ik σ kl (0) R jl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaacIcacaaIXaGaaiykaaaakiabg2da9iaadkfadaWgaaWcbaGa amyAaiaadUgaaeqaaOGaeq4Wdm3aa0baaSqaaiaadUgacaWGSbaaba GaaiikaiaaicdacaGGPaaaaOGaamOuamaaBaaaleaacaWGQbGaamiB aaqabaaaaa@4315@

 

To see this, note that since the loads rotate together with the solid, the components of traction acting normal and tangent to any interior material plane in the solid must remain constant.  With this in mind:

1.      Suppose that, just before the rotation is applied, T i (0) = n k (0) σ ki (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaqhaaWcbaGaamyAaaqaaiaacI cacaaIWaGaaiykaaaakiabg2da9iaad6gadaqhaaWcbaGaam4Aaaqa aiaacIcacaaIWaGaaiykaaaakiabeo8aZnaaDaaaleaacaWGRbGaam yAaaqaaiaacIcacaaIWaGaaiykaaaaaaa@3F8D@  denotes the traction acting on an interior material plane with normal n i (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaqhaaWcbaGaamyAaaqaaiaacI cacaaIWaGaaiykaaaaaaa@3489@ .

2.      The traction vector and the normal to the interior plane rotate with the solid, and therefore have components T i (1) = R ij T j (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaqhaaWcbaGaamyAaaqaaiaacI cacaaIXaGaaiykaaaakiabg2da9iaadkfadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaamivamaaDaaaleaacaWGQbaabaGaaiikaiaaicdaca GGPaaaaaaa@3C72@  and n i (1) = R ij n j (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaqhaaWcbaGaamyAaaqaaiaacI cacaaIXaGaaiykaaaakiabg2da9iaadkfadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaamOBamaaDaaaleaacaWGQbaabaGaaiikaiaaicdaca GGPaaaaaaa@3CA6@  after rotation.

3.      By definition, T i (1) = n j (1) σ ji (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaqhaaWcbaGaamyAaaqaaiaacI cacaaIXaGaaiykaaaakiabg2da9iaad6gadaqhaaWcbaGaamOAaaqa aiaacIcacaaIXaGaaiykaaaakiabeo8aZnaaDaaaleaacaWGQbGaam yAaaqaaiaacIcacaaIXaGaaiykaaaaaaa@3F8E@ .  From (2), we see therefore that R ik T k (0) = R kl n l (0) σ ki (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamivamaaDaaaleaacaWGRbaabaGaaiikaiaaicdacaGGPaaa aOGaeyypa0JaamOuamaaBaaaleaacaWGRbGaamiBaaqabaGccaWGUb Waa0baaSqaaiaadYgaaeaacaGGOaGaaGimaiaacMcaaaGccqaHdpWC daqhaaWcbaGaam4AaiaadMgaaeaacaGGOaGaaGymaiaacMcaaaaaaa@456A@

4.      Multiply both sides of this equation by R ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyAaiaadchaae qaaaaa@334E@  and recall that R ip R ik = δ kp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyAaiaadchaae qaaOGaamOuamaaBaaaleaacaWGPbGaam4AaaqabaGccqGH9aqpcqaH 0oazdaWgaaWcbaGaam4Aaiaadchaaeqaaaaa@3AFF@  to see that T p (0) = n l (0) R kl σ ki (1) R ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaqhaaWcbaGaamiCaaqaaiaacI cacaaIWaGaaiykaaaakiabg2da9iaad6gadaqhaaWcbaGaamiBaaqa aiaacIcacaaIWaGaaiykaaaakiaadkfadaWgaaWcbaGaam4AaiaadY gaaeqaaOGaeq4Wdm3aa0baaSqaaiaadUgacaWGPbaabaGaaiikaiaa igdacaGGPaaaaOGaamOuamaaBaaaleaacaWGPbGaamiCaaqabaaaaa@4574@

5.      Comparing this with (1), we conclude that R kl σ ki (1) R ip = σ lp (0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaam4AaiaadYgaae qaaOGaeq4Wdm3aa0baaSqaaiaadUgacaWGPbaabaGaaiikaiaaigda caGGPaaaaOGaamOuamaaBaaaleaacaWGPbGaamiCaaqabaGccqGH9a qpcqaHdpWCdaqhaaWcbaGaamiBaiaadchaaeaacaGGOaGaaGimaiaa cMcaaaaaaa@4321@ .  Finally, multiplying both sides of this equation by R jl R np MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamOAaiaadYgaae qaaOGaamOuamaaBaaaleaacaWGUbGaamiCaaqabaaaaa@3640@  we conclude that σ jn (1) = R jl σ lp (0) R np MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGQbGaamOBaa qaaiaacIcacaaIXaGaaiykaaaakiabg2da9iaadkfadaWgaaWcbaGa amOAaiaadYgaaeqaaOGaeq4Wdm3aa0baaSqaaiaadYgacaWGWbaaba GaaiikaiaaicdacaGGPaaaaOGaamOuamaaBaaaleaacaWGUbGaamiC aaqabaaaaa@4329@ , giving the required result.

 

 

Drucker Stability:  For most practical applications, the constitutive equation must satisfy a condition known as the Drucker stability criterion, which can be expressed as follows.  Consider a deformable solid, subjected to boundary tractions t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaamyAaaqabaaaaa@327A@ , which induce some displacement field u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaaaaa@327C@ .  Suppose that the tractions are increased to t i +Δ t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaamyAaaqabaGccq GHRaWkcqqHuoarcaWG0bWaaSbaaSqaaiaadMgaaeqaaaaa@36DF@ , resulting in an additional displacement Δ u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa qabaaaaa@33E2@ . The material is said to be stable in the sense of Drucker if the work done by the tractions Δ t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadshadaWgaaWcbaGaamyAaa qabaaaaa@33F0@  through the displacements Δ u i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa qabaaaaa@33F1@  is positive or zero for all Δ t i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadshadaWgaaWcbaGaamyAaa qabaaaaa@33F0@ :

ΔW= { A Δ t i dΔ u i dt } dt0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadEfacqGH9aqpdaWdbaqaam aacmaabaWaa8quaeaacqqHuoarcaWG0bWaaSbaaSqaaiaadMgaaeqa aOWaaSaaaeaacaWGKbGaeuiLdqKaamyDamaaBaaaleaacaWGPbaabe aaaOqaaiaadsgacaWG0baaaaWcbaGaamyqaaqab0Gaey4kIipaaOGa ay5Eaiaaw2haaaWcbeqab0Gaey4kIipakiaadsgacaWG0bGaeyyzIm RaaGimaaaa@4953@

You can show that this condition is satisfied as long as the stress-strain relation obeys

Δ τ ij Δ ε ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabes8a0naaBaaaleaacaWGPb GaamOAaaqabaGccqqHuoarcqaH1oqzdaWgaaWcbaGaamyAaiaadQga aeqaaOGaeyyzImRaaGimaaaa@3D46@

where Δ τ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabes8a0naaBaaaleaacaWGPb GaamOAaaqabaaaaa@359C@  is the change in Kirchhoff stress, and Δ ε ij =(Δ u i / x j +Δ u j / x i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcaGGOaGaeyOaIyRaeuiLdqKaamyDamaa BaaaleaacaWGPbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaai aadQgaaeqaaOGaey4kaSIaeyOaIyRaeuiLdqKaamyDamaaBaaaleaa caWGQbaabeaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadMgaae qaaOGaaiykaaaa@4B13@  is an increment in strain resulting from an infinitesimal change in displacement Δ u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadwhadaWgaaWcbaGaamyAaa qabaaaaa@33E2@ .

 

This is not a thermodynamic law (the work done by the change in tractions is not a physically meaningful quantity), and there is nothing to say that real materials have to satisfy Drucker stability. In fact, many materials show clear signs that they are not stable in the sense of Drucker.   However, if you try to solve a boundary value problem for a material that violates the Drucker stability criterion, you are likely to run into trouble.  The problem will probably not have a unique solution, and in addition you are likely to find that smooth curves on the undeformed solid develop kinks (and may not even be continuous) after the solid is deformed.  This kind of deformation violates one of the fundamental assumptions underlying continuum constitutive equations. 

 

A simple example of a stress-strain curve for material that is not stable in the sense of Drucker is shown in the picture.  The stability criterion is violated wherever the stress decreases with strain in tension, or increases with strain in compression. For the former we see that Δ σ 11 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH8aapcaaIWaaaaa@36FB@ , while Δ ε 11 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH+aGpcaaIWaaaaa@36E3@ ; for the latter Δ σ 11 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH+aGpcaaIWaaaaa@36FF@ , while Δ ε 11 <0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabew7aLnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH8aapcaaIWaaaaa@36DF@ .

 

In the following chapter, we outline constitutive laws that were developed to approximate the behavior of a wide range of materials, including polycrystalline metals and non-metals; elastomers; polymers; biological tissue; soils; and metal single crystals.  A few additional material models, which account for material failure, are also described in Chapter 9.