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

 

 

 

3.5 Hyperelasticity MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  time independent behavior of rubbers and foams subjected to large strains

 

Hyperelastic constitutive laws are used to model materials that respond elastically when subjected to very large strains. They account both for nonlinear material behavior and large shape changes.  The main applications of the theory are (i) to model the rubbery behavior of a polymeric material, and (ii) to model polymeric foams that can be subjected to large reversible shape changes (e.g. a sponge). 

 

In general, the response of a typical polymer is strongly dependent on temperature, strain history and loading rate.  The behavior will be described in more detail in the next section, where we present the theory of viscoelasticity.  For now, we note that polymers have various regimes of mechanical behavior, referred to as `glassy,’ `viscoelastic’ and `rubbery.’   The various regimes can be identified for a particular polymer by applying a sinusoidal variation of shear stress to the solid and measuring the resulting shear strain amplitude.  A typical result is illustrated in the figure, which shows the apparent shear modulus (ratio of stress amplitude to strain amplitude) as a function of temperature.

 

At a critical temperature known as the glass transition temperature, a polymeric material undergoes a dramatic change in mechanical response.  Below this temperature, it behaves like a glass, with a stiff response. Near the glass transition temperature, the stress depends strongly on the strain rate.  At the glass transition temperature, there is a dramatic drop in modulus.  Above this temperature, there is a regime where the polymer shows `rubbery’ behavior MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the response is elastic; the stress does not depend strongly on strain rate or strain history, and the modulus increases with temperature.  All polymers show these general trends, but the extent of each regime, and the detailed behavior within each regime, depend on the solid’s molecular structure.  Heavily cross-linked polymers (elastomers) are the most likely to show ideal rubbery behavior.   Hyperelastic constitutive laws are intended to approximate this `rubbery’ behavior.

 

Features of the behavior of a solid rubber:

1.      The material is close to ideally elastic. i.e. (i) when deformed at constant temperature or adiabatically, stress is a function only of current strain and independent of history or rate of loading, (ii) the behavior is reversible: no net work is done on the solid when subjected to a closed cycle of strain under adiabatic or isothermal conditions.

2.      The material strongly resists volume changes.  The bulk modulus (the ratio of volume change to hydrostatic component of stress) is comparable to that of metals or covalently bonded solids;

3.      The material is very compliant in shear MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  shear modulus is of the order of 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaGymaiaaicdadaahaaWcbeqaaiabgk HiTiaaiwdaaaaaaa@342B@  times that of most metals;

4.      The material is isotropic MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  its stress-strain response is independent of material orientation.

5.      The shear modulus is temperature dependent: the material becomes stiffer as it is heated, in sharp contrast to metals;

6.      When stretched, the material gives off heat.

 

 

Polymeric foams (e.g. a sponge) share some of these properties:

1.      They are close to reversible, and show little rate or history dependence.

2.      In contrast to rubbers, most foams are highly compressible MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  bulk and shear moduli are comparable.

3.      Foams have a complicated true stress-true strain response, generally resembling the figure to the right.  The finite strain response of the foam in compression is quite different to that in tension, because of buckling in the cell walls.

4.      Foams can be anisotropic, depending on their cell structure.   Foams with a random cell structure are isotropic. 

 

The literature on stress-strain relations for finite elasticity can be hard to follow, partly because nearly every paper uses a different notation, and partly because there are many different ways to write down the same stress-strain law.   You should find that most of the published literature is consistent with the framework given below MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but it may take some work to show the equivalence.

 

All hyperelastic models are constructed as follows:

1.      Define the stress-strain relation for the solid by specifying its strain energy density W  as a function of deformation gradient tensor: W=W(F).    This ensures that the material is perfectly elastic, and also means that we only need to work with a scalar function.  The general form of the strain energy density is guided by experiment; and the formula for strain energy density always contains material properties that can be adjusted to describe a particular material.

2.      The undeformed material is usually assumed to be isotropic MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqacKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E8@  i.e the behavior of the material is independent of the initial orientation of the material with respect to the loading.  If the strain energy density is a function of the Left Cauchy-Green deformation tensor B=F F T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaaCOraiabgwSixl aahAeadaahaaWcbeqaaiaadsfaaaaaaa@3984@  the constitutive equation is automatically isotropic.   If B is used as the deformation measure, then the strain energy must be a function of the invariants of B to ensure that the constitutive equation is objective (recall that the invariants of a tensor remain constant under a change of basis)

3.      Formulas for stress in terms of strain are calculated by differentiating the strain energy density as outlined below.

 

 

 

3.5.1 Deformation Measures used in finite elasticity

 

Suppose that a solid is subjected to a displacement field u i ( x k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@3860@ . Define

 The deformation gradient and its Jacobian

F ij = δ ij + u i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqaba aakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaaa@4256@        J=det(F) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bGaaiikaiaahAeacaGGPaaaaa@398E@

 The Left Cauchy-Green deformation tensor

B=F F T B ij = F ik F jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaaCOraiabgwSixl aahAeadaahaaWcbeqaaiaadsfaaaGccaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamOqamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaa dAeadaWgaaWcbaGaamOAaiaadUgaaeqaaaaa@4DF0@

 Invariants of B (these are the conventional definitions)

I 1 =trace(B)= B kk I 2 = 1 2 ( I 1 2 BB )= 1 2 ( I 1 2 B ik B ki ) I 3 =detB= J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMeadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaqG0bGaaeOCaiaabggacaqGJbGaaeyzaiaacIca caWHcbGaaiykaiabg2da9iaadkeadaWgaaWcbaGaam4AaiaadUgaae qaaaGcbaGaamysamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaaaWaaeWaaeaacaWGjbWaa0baaSqaaiaaig daaeaacaaIYaaaaOGaeyOeI0IaaCOqaiabgwSixlabgwSixlaahkea aiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaGaamysamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHi TiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOqamaaBaaale aacaWGRbGaamyAaaqabaaakiaawIcacaGLPaaaaeaacaWGjbWaaSba aSqaaiaaiodaaeqaaOGaeyypa0JaciizaiaacwgacaGG0bGaaCOqai abg2da9iaadQeadaahaaWcbeqaaiaaikdaaaaaaaa@65F1@

 An alternative set of invariants of B (more convenient for models of nearly incompressible materials MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  note that I ¯ 1 , I ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGjbGbaebadaWgaaWcbaGaaGymaa qabaGccaGGSaGabmysayaaraWaaSbaaSqaaiaaikdaaeqaaaaa@371B@  remain constant under a pure volume change)

I ¯ 1 = I 1 J 2/3 = B kk J 2/3 I ¯ 2 = I 2 J 4/3 = 1 2 ( I ¯ 1 2 BB J 4/3 )= 1 2 ( I ¯ 1 2 B ik B ki J 4/3 ) J= detB MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiqadMeagaqeamaaBaaaleaaca aIXaaabeaakiabg2da9maalaaabaGaamysamaaBaaaleaacaaIXaaa beaaaOqaaiaadQeadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaa GccaqG9aWaaSaaaeaacaWGcbWaaSbaaSqaaiaadUgacaWGRbaabeaa aOqaaiaadQeadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaaake aaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqa aiaadMeadaWgaaWcbaGaaGOmaaqabaaakeaacaWGkbWaaWbaaSqabe aacaaI0aGaai4laiaaiodaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaa baGaaGOmaaaadaqadaqaaiqadMeagaqeamaaDaaaleaacaaIXaaaba GaaGOmaaaakiabgkHiTmaalaaabaGaaCOqaiabgwSixlabgwSixlaa hkeaaeaacaWGkbWaaWbaaSqabeaacaaI0aGaai4laiaaiodaaaaaaa GccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaa daqadaqaaiqadMeagaqeamaaDaaaleaacaaIXaaabaGaaGOmaaaaki abgkHiTmaalaaabaGaamOqamaaBaaaleaacaWGPbGaam4AaaqabaGc caWGcbWaaSbaaSqaaiaadUgacaWGPbaabeaaaOqaaiaadQeadaahaa WcbeqaaiaaisdacaGGVaGaaG4maaaaaaaakiaawIcacaGLPaaaaeaa caWGkbGaeyypa0ZaaOaaaeaaciGGKbGaaiyzaiaacshacaWHcbaale qaaaaaaa@7088@

 

 Principal stretches and principal stretch directions

1.      Let e 1 , e 2 , e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamyzamaaBaaaleaacaaIYaaabeaakiaacYcacaWGLbWaaSba aSqaaiaaiodaaeqaaaaa@3752@  denote the three eigenvalues of B.  The principal stretches are

λ 1 = e 1 , λ 2 = e 2 , λ 3 = e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaBaaaleaacaaIXaaabeaaki abg2da9maakaaabaGaamyzamaaBaaaleaacaaIXaaabeaaaeqaaOGa aiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH7oaBdaWgaaWc baGaaGOmaaqabaGccqGH9aqpdaGcaaqaaiaadwgadaWgaaWcbaGaaG OmaaqabaaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabeU7aSnaaBaaaleaacaaIZaaabeaakiabg2da9maakaaa baGaamyzamaaBaaaleaacaaIZaaabeaaaeqaaaaa@6C2F@

2.      Let b 1 , b 2 , b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCOyamaaBaaaleaacaaIYaaabeaakiaacYcacaWHIbWaaSba aSqaaiaaiodaaeqaaaaa@3755@  denote three, mutually perpendicular unit eigenvectors of B. These define the principal stretch directions.  (Note: since B is symmetric its eigenvectors are automatically mutually perpendicular as long as no two eigenvalues are the same.  If two, or all three eigenvalues are the same, the eignevectors are not uniquely defined MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in this case any convenient mutually perpendicular set of eigenvectors can be used).

3.      Recall that B can be expressed in terms of its eigenvectors and eigenvalues as B= λ 1 2 b (1) b (1) + λ 2 2 b (2) b (2) + λ 3 2 b (3) b (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOqaiabg2da9iabeU7aSnaaDaaale aacaaIXaaabaGaaGOmaaaakiaahkgadaahaaWcbeqaaiaacIcacaaI XaGaaiykaaaakiabgEPielaahkgadaahaaWcbeqaaiaacIcacaaIXa GaaiykaaaakiabgUcaRiabeU7aSnaaDaaaleaacaaIYaaabaGaaGOm aaaakiaahkgadaahaaWcbeqaaiaacIcacaaIYaGaaiykaaaakiabgE PielaahkgadaahaaWcbeqaaiaacIcacaaIYaGaaiykaaaakiabgUca RiabeU7aSnaaDaaaleaacaaIZaaabaGaaGOmaaaakiaahkgadaahaa WcbeqaaiaacIcacaaIZaGaaiykaaaakiabgEPielaahkgadaahaaWc beqaaiaacIcacaaIZaGaaiykaaaaaaa@57F5@

 

 

 

3.5.2 Stress Measures used in finite elasticity

 

Usually stress-strain laws are given as equations relating Cauchy stress (`true’ stress) σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  to left Cauchy-Green deformation tensor.  For some computations it may be more convenient to use other stress measures.  They are defined below, for convenience.

 The Cauchy (“true”) stress represents the force per unit deformed area in the solid and is defined by

n i σ ij = Lim dA0 d P j (n) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maaxaba baGaamitaiaadMgacaWGTbaaleaacaWGKbGaamyqaiabgkziUkaaic daaeqaaOWaaSaaaeaacaWGKbGaamiuamaaDaaaleaacaWGQbaabaGa aiikaiaah6gacaGGPaaaaaGcbaGaamizaiaadgeaaaaaaa@47F6@

 Kirchhoff stress  τ=Jσ τ ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaCiXdOGaeyypa0JaamOsaKaaal aaho8acaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabes8a0PWaaSba aSqaaiaadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaa WcbaGaamyAaiaadQgaaeqaaaaa@4819@

 Nominal (First Piola-Kirchhoff) stress   S=J F 1 σ S ij =J F ik 1 σ kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4uaiabg2da9iaadQeacaWHgbWaaW baaSqabeaacqGHsislcaaIXaaaaOGaeyyXICDcaaSaaC4WdiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4uaO WaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadQeacaWGgbWa a0baaSqaaiaadMgacaWGRbaabaGaeyOeI0IaaGymaaaakiabeo8aZn aaBaaaleaacaWGRbGaamOAaaqabaaaaa@5407@

 Material (Second Piola-Kirchhoff) stress   Σ=J F 1 σ F T Σ ij =J F ik 1 σ kl F jl 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaC4Odiabg2da9iaadQeacaWHgbWaaW baaSqabeaacqGHsislcaaIXaaaaOGaeyyXICDcaaSaaC4WdiabgwSi xRGaaCOramaaCaaaleqabaGaeyOeI0IaamivaaaakiaaykW7caaMc8 UaaGPaVlaaykW7cqqHJoWudaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyypa0JaamOsaiaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacqGHsi slcaaIXaaaaOGaeq4Wdm3aaSbaaSqaaiaadUgacaWGSbaabeaakiaa dAeadaqhaaWcbaGaamOAaiaadYgaaeaacqGHsislcaaIXaaaaaaa@587C@

 

 

 

3.5.3 Calculating stress-strain relations from the strain energy density

 

The constitutive law for an isotropic hyperelastic material is defined by an equation relating the strain energy density of the material to the deformation gradient, or, for an isotropic solid, to the three invariants of the strain tensor

W(F)=U( I 1 , I 2 , I 3 )= U ¯ ( I ¯ 1 , I ¯ 2 ,J)= U ˜ ( λ 1 , λ 2 , λ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGxbGaaiikaiaahAeacaGGPaGaey ypa0JaamyvaiaacIcacaWGjbWaaSbaaSqaaiaaigdaaeqaaOGaaiil aiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamysamaaBaaale aacaaIZaaabeaakiaacMcacqGH9aqpceWGvbGbaebacaGGOaGabmys ayaaraWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiqadMeagaqeamaaBa aaleaacaaIYaaabeaakiaacYcacaWGkbGaaiykaiabg2da9maaGaaa baGaamyvaaGaay5adaGaaiikaiabeU7aSnaaBaaaleaacaaIXaaabe aakiaacYcacqaH7oaBdaWgaaWcbaGaaGOmaaqabaGccaGGSaGaeq4U dW2aaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@561F@

The stress-strain law must then be deduced by differentiating the strain energy density.   This can involve some tedious algebra.  Formulas are listed below for the stress-strain relations for each choice of strain invariant.  The results are derived below

 

Strain energy density in terms of F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@359A@

σ ij = 1 J F ik W F kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadQeaaaGaamOramaa BaaaleaacaWGPbGaam4AaaqabaGcdaWcaaqaaiabgkGi2kaadEfaae aacqGHciITcaWGgbWaaSbaaSqaaiaadUgacaWGQbaabeaaaaaaaa@40D1@

Strain energy density in terms of I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamysamaa BaaaleaacaaIZaaabeaaaaa@395C@

σ ij = 2 I 3 [ ( U I 1 + I 1 U I 2 ) B ij U I 2 B ik B kj ]+2 I 3 U I 3 δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGOmaaqaamaakaaabaGaamysamaa BaaaleaacaaIZaaabeaaaeqaaaaakmaadmaabaWaaeWaaeaadaWcaa qaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaigda aeqaaaaakiabgUcaRiaadMeadaWgaaWcbaGaaGymaaqabaGcdaWcaa qaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikda aeqaaaaaaOGaayjkaiaawMcaaiaadkeadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyOeI0YaaSaaaeaacqGHciITcaWGvbaabaGaeyOaIyRa amysamaaBaaaleaacaaIYaaabeaaaaGccaWGcbWaaSbaaSqaaiaadM gacaWGRbaabeaakiaadkeadaWgaaWcbaGaam4AaiaadQgaaeqaaaGc caGLBbGaayzxaaGaey4kaSIaaGOmamaakaaabaGaamysamaaBaaale aacaaIZaaabeaaaeqaaOWaaSaaaeaacqGHciITcaWGvbaabaGaeyOa IyRaamysamaaBaaaleaacaaIZaaabeaaaaGccqaH0oazdaWgaaWcba GaamyAaiaadQgaaeqaaaaa@64D7@

 

Strain energy density in terms of I ¯ 1 , I ¯ 2 ,J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGjbGbaebadaWgaaWcbaGaaGymaa qabaGccaGGSaGabmysayaaraWaaSbaaSqaaiaaikdaaeqaaOGaaiil aiaadQeaaaa@38A4@

σ ij = 2 J [ 1 J 2/3 ( U ¯ I ¯ 1 + I ¯ 1 U ¯ I ¯ 2 ) B ij ( I ¯ 1 U ¯ I ¯ 1 +2 I ¯ 2 U ¯ I ¯ 2 ) δ ij 3 1 J 4/3 U ¯ I ¯ 2 B ik B kj ]+ U ¯ J δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGOmaaqaaiaadQeaaaWaamWaaeaa daWcaaqaaiaaigdaaeaacaWGkbWaaWbaaSqabeaacaaIYaGaai4lai aaiodaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi2kqadwfagaqeaaqa aiabgkGi2kqadMeagaqeamaaBaaaleaacaaIXaaabeaaaaGccqGHRa WkceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiabgkGi 2kqadwfagaqeaaqaaiabgkGi2kqadMeagaqeamaaBaaaleaacaaIYa aabeaaaaaakiaawIcacaGLPaaacaWGcbWaaSbaaSqaaiaadMgacaWG QbaabeaakiabgkHiTmaabmaabaGabmysayaaraWaaSbaaSqaaiaaig daaeqaaOWaaSaaaeaacqGHciITceWGvbGbaebaaeaacqGHciITceWG jbGbaebadaWgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaaGOmaiqadM eagaqeamaaBaaaleaacaaIYaaabeaakmaalaaabaGaeyOaIyRabmyv ayaaraaabaGaeyOaIyRabmysayaaraWaaSbaaSqaaiaaikdaaeqaaa aaaOGaayjkaiaawMcaamaalaaabaGaeqiTdq2aaSbaaSqaaiaadMga caWGQbaabeaaaOqaaiaaiodaaaGaeyOeI0YaaSaaaeaacaaIXaaaba GaamOsamaaCaaaleqabaGaaGinaiaac+cacaaIZaaaaaaakmaalaaa baGaeyOaIyRabmyvayaaraaabaGaeyOaIyRabmysayaaraWaaSbaaS qaaiaaikdaaeqaaaaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqa aOGaamOqamaaBaaaleaacaWGRbGaamOAaaqabaaakiaawUfacaGLDb aacqGHRaWkdaWcaaqaaiabgkGi2kqadwfagaqeaaqaaiabgkGi2kaa dQeaaaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaaa@80A5@

 

Strain energy density in terms of λ 1 , λ 2 , λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaWgaaWcbaGaaGymaaqaba GccaGGSaGaeq4UdW2aaSbaaSqaaiaaikdaaeqaaOGaaiilaiabeU7a SnaaBaaaleaacaaIZaaabeaaaaa@3C0E@

σ ij = λ 1 λ 1 λ 2 λ 3 U ˜ λ 1 b i (1) b j (1) + λ 2 λ 1 λ 2 λ 3 U ˜ λ 2 b i (2) b j (2) + λ 3 λ 1 λ 2 λ 3 U ˜ λ 3 b i (3) b j (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeq4UdW2aaSbaaSqaaiaaigdaaeqa aaGcbaGaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeq4UdW2aaSbaaS qaaiaaikdaaeqaaOGaeq4UdW2aaSbaaSqaaiaaiodaaeqaaaaakmaa laaabaGaeyOaIy7aaacaaeaacaWGvbaacaGLdmaaaeaacqGHciITcq aH7oaBdaWgaaWcbaGaaGymaaqabaaaaOGaamOyamaaDaaaleaacaWG PbaabaGaaiikaiaaigdacaGGPaaaaOGaamOyamaaDaaaleaacaWGQb aabaGaaiikaiaaigdacaGGPaaaaOGaey4kaSYaaSaaaeaacqaH7oaB daWgaaWcbaGaaGOmaaqabaaakeaacqaH7oaBdaWgaaWcbaGaaGymaa qabaGccqaH7oaBdaWgaaWcbaGaaGOmaaqabaGccqaH7oaBdaWgaaWc baGaaG4maaqabaaaaOWaaSaaaeaacqGHciITdaaiaaqaaiaadwfaai aawoWaaaqaaiabgkGi2kabeU7aSnaaBaaaleaacaaIYaaabeaaaaGc caWGIbWaa0baaSqaaiaadMgaaeaacaGGOaGaaGOmaiaacMcaaaGcca WGIbWaa0baaSqaaiaadQgaaeaacaGGOaGaaGOmaiaacMcaaaGccqGH RaWkdaWcaaqaaiabeU7aSnaaBaaaleaacaaIZaaabeaaaOqaaiabeU 7aSnaaBaaaleaacaaIXaaabeaakiabeU7aSnaaBaaaleaacaaIYaaa beaakiabeU7aSnaaBaaaleaacaaIZaaabeaaaaGcdaWcaaqaaiabgk Gi2oaaGaaabaGaamyvaaGaay5adaaabaGaeyOaIyRaeq4UdW2aaSba aSqaaiaaiodaaeqaaaaakiaadkgadaqhaaWcbaGaamyAaaqaaiaacI cacaaIZaGaaiykaaaakiaadkgadaqhaaWcbaGaamOAaaqaaiaacIca caaIZaGaaiykaaaaaaa@859F@

 

Derivations:   We start by deriving the general formula for stress in terms of W(F) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEfacaGGOaGaaCOraiaacMcaaaa@336C@ :

1.      Note that, by definition, if the solid is subjected to some history of strain, the rate of change of the strain energy density W (F)  must equal the rate of mechanical work done on the material per unit reference volume.

2.      Recall that the rate of work done per unit undeformed volume by body forces and surface tractions is expressed in terms of the nominal stress S ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3349@  as S ji F ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamOAaiaadMgaae qaaOGabmOrayaacaWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@3630@ .

3.      Therefore, for any deformation gradient Fij,

dW dt = W F ij F ij t = S ji F ij t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadEfaaeaacaWGKb GaamiDaaaacqGH9aqpdaWcaaqaaiabgkGi2kaadEfaaeaacqGHciIT caWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaaaaGcdaWcaaqaaiabgk Gi2kaadAeadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeyOaIyRa amiDaaaacqGH9aqpcaWGtbWaaSbaaSqaaiaadQgacaWGPbaabeaakm aalaaabaGaeyOaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaa keaacqGHciITcaWG0baaaaaa@4D12@

This must hold for all possible F ˙ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadAeagaGaamaaBaaaleaacaWGPbGaam OAaaqabaaaaa@3345@ ,so that

W F ij = S ji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaSaaaeaacqGHciITcaWGxbaabaGaey OaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0Ja am4uamaaBaaaleaacaWGQbGaamyAaaqabaaaaa@3B4B@

4.      Finally, the formula for Cauchy stress follows from the equation relating σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  to S ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3349@

σ ij = 1 J F ik S kj = 1 J F ik W F kj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadQeaaaGaamOramaa BaaaleaacaWGPbGaam4AaaqabaGccaWGtbWaaSbaaSqaaiaadUgaca WGQbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadQeaaaGaamOr amaaBaaaleaacaWGPbGaam4AaaqabaGcdaWcaaqaaiabgkGi2kaadE faaeaacqGHciITcaWGgbWaaSbaaSqaaiaadUgacaWGQbaabeaaaaaa aa@491D@

 

For an isotropic material, it is necessary to find derivatives of the invariants with respect to the components of F in order to compute the stress-strain function for a given strain energy density.  It is straightforward, but somewhat tedious to show that:

                                  I 1 F ij =2 F ij , I 2 F ij =2( I 1 F ij B ik F kj ), I 3 F ij =2 I 3 F ji 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamOr amaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0JaaGOmaiaadA eadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiilaiaaykW7caaMc8+a aSaaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaey OaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0Ja aGOmamaabmaabaGaamysamaaBaaaleaacaaIXaaabeaakiaadAeada WgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0IaamOqamaaBaaaleaa caWGPbGaam4AaaqabaGccaWGgbWaaSbaaSqaaiaadUgacaWGQbaabe aaaOGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVpaalaaabaGaeyOa IyRaamysamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadAeada WgaaWcbaGaamyAaiaadQgaaeqaaaaakiabg2da9iaaikdacaWGjbWa aSbaaSqaaiaaiodaaeqaaOGaamOramaaDaaaleaacaWGQbGaamyAaa qaaiabgkHiTiaaigdaaaaaaa@7027@

Then,

W F ij = U I 1 I 1 F ij + U I ¯ 2 I 2 F ij + U I 3 I 3 F ij =2( U I 1 + I 1 U I 2 ) F ij 2 U I 2 B ik F kj 2 I 3 U I 3 F ji 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaam4vaaqaaiabgkGi2kaadAeadaWgaaWcbaGaamyAaiaadQga aeqaaaaakiabg2da9maalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2k aadMeadaWgaaWcbaGaaGymaaqabaaaaOWaaSaaaeaacqGHciITcaWG jbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamOramaaBaaale aacaWGPbGaamOAaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITcaWG vbaabaGaeyOaIyRabmysayaaraWaaSbaaSqaaiaaikdaaeqaaaaakm aalaaabaGaeyOaIyRaamysamaaBaaaleaacaaIYaaabeaaaOqaaiab gkGi2kaadAeadaWgaaWcbaGaamyAaiaadQgaaeqaaaaakiabgUcaRm aalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2kaadMeadaWgaaWcbaGa aG4maaqabaaaaOWaaSaaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaio daaeqaaaGcbaGaeyOaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqa baaaaOGaeyypa0JaaGOmamaabmaabaWaaSaaaeaacqGHciITcaWGvb aabaGaeyOaIyRaamysamaaBaaaleaacaaIXaaabeaaaaGccqGHRaWk caWGjbWaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacqGHciITcaWGvb aabaGaeyOaIyRaamysamaaBaaaleaacaaIYaaabeaaaaaakiaawIca caGLPaaacaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTi aaikdadaWcaaqaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSba aSqaaiaaikdaaeqaaaaakiaadkeadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamOramaaBaaaleaacaWGRbGaamOAaaqabaGccaaIYaGaamys amaaBaaaleaacaaIZaaabeaakmaalaaabaGaeyOaIyRaamyvaaqaai abgkGi2kaadMeadaWgaaWcbaGaaG4maaqabaaaaOGaamOramaaDaaa leaacaWGQbGaamyAaaqaaiabgkHiTiaaigdaaaaaaa@9381@

and

σ ij = 1 I 3 F ik W F kj = 2 I 3 [ ( U I 1 + I 1 U I 2 ) B ij U I 2 B ik B kj ]+2 I 3 U I 3 δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaabaGaamysamaa BaaaleaacaaIZaaabeaaaeqaaaaakiaadAeadaWgaaWcbaGaamyAai aadUgaaeqaaOWaaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaamOr amaaBaaaleaacaWGRbGaamOAaaqabaaaaOGaeyypa0ZaaSaaaeaaca aIYaaabaWaaOaaaeaacaWGjbWaaSbaaSqaaiaaiodaaeqaaaqabaaa aOWaamWaaeaadaqadaqaamaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaamysamaa BaaaleaacaaIXaaabeaakmaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGa amOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaai abgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaaeqa aaaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOqamaaBa aaleaacaWGRbGaamOAaaqabaaakiaawUfacaGLDbaacqGHRaWkcaaI YaWaaOaaaeaacaWGjbWaaSbaaSqaaiaaiodaaeqaaaqabaGcdaWcaa qaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaioda aeqaaaaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@71F0@

 

When using a strain energy density of the form U ¯ ( I ¯ 1 , I ¯ 2 ,J) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaraGaaiikaiqadMeagaqeam aaBaaaleaacaaIXaaabeaakiaacYcaceWGjbGbaebadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaamOsaiaacMcaaaa@3906@ ,  we will have to compute the derivatives of the invariants I ¯ 1 , I ¯ 2  and J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmysayaaraWaaSbaaSqaaiaaigdaae qaaOGaaiilaiqadMeagaqeamaaBaaaleaacaaIYaaabeaakiaabcca caqGHbGaaeOBaiaabsgacaqGGaGaamOsaaaa@3A0D@  with respect to the components of F in order to find

                                              W F ij = U ¯ I ¯ 1 I ¯ 1 F ij + U ¯ I ¯ 2 I ¯ 2 F ij + U ¯ J J F ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaey OaIyRaam4vaaqaaiabgkGi2kaadAeadaWgaaWcbaGaamyAaiaadQga aeqaaaaakiabg2da9maalaaabaGaeyOaIyRabmyvayaaraaabaGaey OaIyRabmysayaaraWaaSbaaSqaaiaaigdaaeqaaaaakmaalaaabaGa eyOaIyRabmysayaaraWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIy RaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaey4kaSYaaSaa aeaacqGHciITceWGvbGbaebaaeaacqGHciITceWGjbGbaebadaWgaa WcbaGaaGOmaaqabaaaaOWaaSaaaeaacqGHciITceWGjbGbaebadaWg aaWcbaGaaGOmaaqabaaakeaacqGHciITcaWGgbWaaSbaaSqaaiaadM gacaWGQbaabeaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kqadwfagaqe aaqaaiabgkGi2kaadQeaaaWaaSaaaeaacqGHciITcaWGkbaabaGaey OaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaaaa@6523@

We find that

J F ij =J F ji 1 I ¯ 1 F ij = 1 J 2/3 I 1 F ij 2 I 1 3 J 5/3 J F ij = 2 J 2/3 ( F ij I 1 3 F ji 1 )= 2 J 2/3 F ij 2 3 I ¯ 1 F ji 1 I ¯ 2 F ij = 1 J 4/3 I 2 F ij 4 I 2 3 J 7/3 J F ij = 2 J 4/3 ( I 1 F ij B ik F kj 2 I 2 3 F ji 1 )= 2 J 2/3 I ¯ 1 F ij 2 J 4/3 B ik F kj 4 I ¯ 2 3 F ji 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaGabeaadaWcaaqaaiabgkGi2kaadQeaae aacqGHciITcaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaaaaGccqGH 9aqpcaWGkbGaamOramaaDaaaleaacaWGQbGaamyAaaqaaiabgkHiTi aaigdaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7daWcaaqaaiabgkGi2kqadMeagaqeamaa BaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadAeadaWgaaWcbaGaam yAaiaadQgaaeqaaaaakiabg2da9maalaaabaGaaGymaaqaaiaadQea daahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaGcdaWcaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWGgbWa aSbaaSqaaiaadMgacaWGQbaabeaaaaGccqGHsisldaWcaaqaaiaaik dacaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaG4maiaadQeadaah aaWcbeqaaiaaiwdacaGGVaGaaG4maaaaaaGcdaWcaaqaaiabgkGi2k aadQeaaeaacqGHciITcaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaa aaGccaaMc8Uaeyypa0ZaaSaaaeaacaaIYaaabaGaamOsamaaCaaale qabaGaaGOmaiaac+cacaaIZaaaaaaakmaabmaabaGaamOramaaBaaa leaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaaiaadMeadaWgaa WcbaGaaGymaaqabaaakeaacaaIZaaaaiaaykW7caWGgbWaa0baaSqa aiaadQgacaWGPbaabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaai abg2da9maalaaabaGaaGOmaaqaaiaadQeadaahaaWcbeqaaiaaikda caGGVaGaaG4maaaaaaGccaWGgbWaaSbaaSqaaiaadMgacaWGQbaabe aakiabgkHiTmaalaaabaGaaGOmaaqaaiaaiodaaaGabmysayaaraWa aSbaaSqaaiaaigdaaeqaaOGaaGPaVlaadAeadaqhaaWcbaGaamOAai aadMgaaeaacqGHsislcaaIXaaaaaGcbaWaaSaaaeaacqGHciITceWG jbGbaebadaWgaaWcbaGaaGOmaaqabaaakeaacqGHciITcaWGgbWaaS baaSqaaiaadMgacaWGQbaabeaaaaGccqGH9aqpdaWcaaqaaiaaigda aeaacaWGkbWaaWbaaSqabeaacaaI0aGaai4laiaaiodaaaaaaOWaaS aaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOa IyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyOeI0YaaS aaaeaacaaI0aGaamysamaaBaaaleaacaaIYaaabeaaaOqaaiaaioda caWGkbWaaWbaaSqabeaacaaI3aGaai4laiaaiodaaaaaaOWaaSaaae aacqGHciITcaWGkbaabaGaeyOaIyRaamOramaaBaaaleaacaWGPbGa amOAaaqabaaaaOGaeyypa0ZaaSaaaeaacaaIYaaabaGaamOsamaaCa aaleqabaGaaGinaiaac+cacaaIZaaaaaaakmaabmaabaGaamysamaa BaaaleaacaaIXaaabeaakiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyOeI0IaamOqamaaBaaaleaacaWGPbGaam4AaaqabaGccaWG gbWaaSbaaSqaaiaadUgacaWGQbaabeaakiabgkHiTmaalaaabaGaaG OmaiaadMeadaWgaaWcbaGaaGOmaaqabaaakeaacaaIZaaaaiaadAea daqhaaWcbaGaamOAaiaadMgaaeaacqGHsislcaaIXaaaaaGccaGLOa GaayzkaaGaeyypa0ZaaSaaaeaacaaIYaaabaGaamOsamaaCaaaleqa baGaaGOmaiaac+cacaaIZaaaaaaakiqadMeagaqeamaaBaaaleaaca aIXaaabeaakiaadAeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOe I0YaaSaaaeaacaaIYaaabaGaamOsamaaCaaaleqabaGaaGinaiaac+ cacaaIZaaaaaaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amOramaaBaaaleaacaWGRbGaamOAaaqabaGccqGHsisldaWcaaqaai aaisdaceWGjbGbaebadaWgaaWcbaGaaGOmaaqabaaakeaacaaIZaaa aiaadAeadaqhaaWcbaGaamOAaiaadMgaaeaacqGHsislcaaIXaaaaa aaaa@FC0B@

 

Thus,

σ ij = 1 J F ik W F kj = 2 J 5/3 ( U ¯ I ¯ 1 + I ¯ 1 U ¯ I ¯ 2 ) B ij 2 3J ( I ¯ 1 U ¯ I ¯ 1 +2 I ¯ 2 U ¯ I ¯ 2 ) δ ij 2 J 7/3 U ¯ I ¯ 2 B ik B kj + U ¯ J δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaaiaadQeaaaGaamOramaa BaaaleaacaWGPbGaam4AaaqabaGcdaWcaaqaaiabgkGi2kaadEfaae aacqGHciITcaWGgbWaaSbaaSqaaiaadUgacaWGQbaabeaaaaGccqGH 9aqpdaWcaaqaaiaaikdaaeaacaWGkbWaaWbaaSqabeaacaaI1aGaai 4laiaaiodaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi2kqadwfagaqe aaqaaiabgkGi2kqadMeagaqeamaaBaaaleaacaaIXaaabeaaaaGccq GHRaWkceWGjbGbaebadaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiab gkGi2kqadwfagaqeaaqaaiabgkGi2kqadMeagaqeamaaBaaaleaaca aIYaaabeaaaaaakiaawIcacaGLPaaacaWGcbWaaSbaaSqaaiaadMga caWGQbaabeaakiabgkHiTmaalaaabaGaaGOmaaqaaiaaiodacaWGkb aaamaabmaabaGabmysayaaraWaaSbaaSqaaiaaigdaaeqaaOWaaSaa aeaacqGHciITceWGvbGbaebaaeaacqGHciITceWGjbGbaebadaWgaa WcbaGaaGymaaqabaaaaOGaey4kaSIaaGOmaiqadMeagaqeamaaBaaa leaacaaIYaaabeaakmaalaaabaGaeyOaIyRabmyvayaaraaabaGaey OaIyRabmysayaaraWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaa wMcaaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislda WcaaqaaiaaikdaaeaacaWGkbWaaWbaaSqabeaacaaI3aGaai4laiaa iodaaaaaaOWaaSaaaeaacqGHciITceWGvbGbaebaaeaacqGHciITce WGjbGbaebadaWgaaWcbaGaaGOmaaqabaaaaOGaamOqamaaBaaaleaa caWGPbGaam4AaaqabaGccaWGcbWaaSbaaSqaaiaadUgacaWGQbaabe aakiabgUcaRmaalaaabaGaeyOaIyRabmyvayaaraaabaGaeyOaIyRa amOsaaaacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa@8AC2@

 

 

 

Next, we derive the stress-strain relation in terms of a strain energy density U ˜ ( λ 1 , λ 2 , λ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaiaaqaaiaadwfaaiaawoWaaiaacI cacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2aaSba aSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZaaabe aakiaacMcaaaa@3F0D@  that is expressed as a function of the principal  strains.  Note first that

        U ˜ ( λ 1 , λ 2 , λ 3 )=U( I 1 , I 2 , I 3 ), I 1 = λ 1 2 + λ 2 2 + λ 3 2 , I 2 = λ 1 2 λ 3 2 + λ 2 2 λ 3 2 + λ 1 2 λ 3 2 , I 3 = λ 1 2 λ 2 2 λ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaaiaaqaaiaadwfaaiaawoWaaiaacI cacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2aaSba aSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZaaabe aakiaacMcacqGH9aqpcaWGvbGaaiikaiaadMeadaWgaaWcbaGaaGym aaqabaGccaGGSaGaamysamaaBaaaleaacaaIYaaabeaakiaacYcaca WGjbWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaacYcacaaMc8UaaGPa VlaadMeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBdaqhaa WcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkcqaH7oaBdaqhaaWcbaGa aGOmaaqaaiaaikdaaaGccqGHRaWkcqaH7oaBdaqhaaWcbaGaaG4maa qaaiaaikdaaaGccaGGSaGaaGPaVlaaykW7caaMc8UaamysamaaBaaa leaacaaIYaaabeaakiabg2da9iabeU7aSnaaDaaaleaacaaIXaaaba GaaGOmaaaakiabeU7aSnaaDaaaleaacaaIZaaabaGaaGOmaaaakiab gUcaRiabeU7aSnaaDaaaleaacaaIYaaabaGaaGOmaaaakiabeU7aSn aaDaaaleaacaaIZaaabaGaaGOmaaaakiabgUcaRiabeU7aSnaaDaaa leaacaaIXaaabaGaaGOmaaaakiabeU7aSnaaDaaaleaacaaIZaaaba GaaGOmaaaakiaacYcacaaMc8UaaGPaVlaadMeadaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcqaH7oaBdaqhaaWcbaGaaGymaaqaaiaaikdaaa GccqaH7oaBdaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccqaH7oaBdaqh aaWcbaGaaG4maaqaaiaaikdaaaaaaa@8A38@

so that the chain rule gives

U ˜ λ i =2 λ i ( U I 1 +( I 1 λ i 2 ) U I 2 + I 3 λ i 2 U I 3 ),(i=1,2,3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaGaaabaGaam yvaaGaay5adaaabaGaeyOaIyRaeq4UdW2aaSbaaSqaaiaadMgaaeqa aaaakiabg2da9iaaikdacqaH7oaBdaWgaaWcbaGaamyAaaqabaGcda qadaqaamaalaaabaGaeyOaIyRaamyvaaqaaiabgkGi2kaadMeadaWg aaWcbaGaaGymaaqabaaaaOGaey4kaSYaaeWaaeaacaWGjbWaaSbaaS qaaiaaigdaaeqaaOGaeyOeI0Iaeq4UdW2aa0baaSqaaiaadMgaaeaa caaIYaaaaaGccaGLOaGaayzkaaWaaSaaaeaacqGHciITcaWGvbaaba GaeyOaIyRaamysamaaBaaaleaacaaIYaaabeaaaaGccqGHRaWkdaWc aaqaaiaadMeadaWgaaWcbaGaaG4maaqabaaakeaacqaH7oaBdaqhaa WcbaGaamyAaaqaaiaaikdaaaaaaOWaaSaaaeaacqGHciITcaWGvbaa baGaeyOaIyRaamysamaaBaaaleaacaaIZaaabeaaaaaakiaawIcaca GLPaaacaGGSaGaaGPaVlaaykW7caGGOaGaamyAaiabg2da9iaaigda caGGSaGaaGOmaiaacYcacaaIZaGaaiykaaaa@6A35@

Using this and the expression that relates the stress components to the derivatives of U,

σ ij = 1 I 3 F ik W F kj = 2 I 3 [ ( U I 1 + I 1 U I 2 ) B ij U I 2 B ik B kj ]+2 I 3 U I 3 δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaaGymaaqaamaakaaabaGaamysamaa BaaaleaacaaIZaaabeaaaeqaaaaakiaadAeadaWgaaWcbaGaamyAai aadUgaaeqaaOWaaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaamOr amaaBaaaleaacaWGRbGaamOAaaqabaaaaOGaeyypa0ZaaSaaaeaaca aIYaaabaWaaOaaaeaacaWGjbWaaSbaaSqaaiaaiodaaeqaaaqabaaa aOWaamWaaeaadaqadaqaamaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaamysamaa BaaaleaacaaIXaaabeaakmaalaaabaGaeyOaIyRaamyvaaqaaiabgk Gi2kaadMeadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGa amOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaai abgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaaeqa aaaakiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOqamaaBa aaleaacaWGRbGaamOAaaqabaaakiaawUfacaGLDbaacqGHRaWkcaaI YaWaaOaaaeaacaWGjbWaaSbaaSqaaiaaiodaaeqaaaqabaGcdaWcaa qaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaioda aeqaaaaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@71F0@

we find that the principal stresses σ 1 , σ 2 , σ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGaaGOmaaqa baGccaGGSaGaeq4Wdm3aaSbaaSqaaiaaiodaaeqaaaaa@3FAF@  are related to the corresponding principal stretches λ 1 , λ 2 , λ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSnaaBa aaleaacaaIXaaabeaakiaacYcacqaH7oaBdaWgaaWcbaGaaGOmaaqa baGccaGGSaGaeq4UdW2aaSbaaSqaaiaaiodaaeqaaaaa@3F82@  (square-roots of the eigenvalues of B) through

σ i = λ i λ 1 λ 2 λ 3 U ˜ λ i ,(i=1,2,3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaaqaba GccqGH9aqpdaWcaaqaaiabeU7aSnaaBaaaleaacaWGPbaabeaaaOqa aiabeU7aSnaaBaaaleaacaaIXaaabeaakiabeU7aSnaaBaaaleaaca aIYaaabeaakiabeU7aSnaaBaaaleaacaaIZaaabeaaaaGcdaWcaaqa aiabgkGi2oaaGaaabaGaamyvaaGaay5adaaabaGaeyOaIyRaeq4UdW 2aaSbaaSqaaiaadMgaaeqaaaaakiaacYcacaaMc8UaaGPaVlaacIca caWGPbGaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGPa aaaa@5383@

The spectral decomposition for B in terms of its eigenvalues λ 1 2 , λ 2 2 , λ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSnaaDa aaleaacaaIXaaabaGaaGOmaaaakiaacYcacqaH7oaBdaqhaaWcbaGa aGOmaaqaaiaaikdaaaGccaGGSaGaeq4UdW2aa0baaSqaaiaaiodaae aacaaIYaaaaaaa@41B9@  and eigenvectors b (1) , b (2) , b (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOyamaaCaaaleqabaGaaiikaiaaig dacaGGPaaaaOGaaiilaiaahkgadaahaaWcbeqaaiaacIcacaaIYaGa aiykaaaakiaacYcacaWHIbWaaWbaaSqabeaacaGGOaGaaG4maiaacM caaaaaaa@3BD7@ :

  B ij = λ 1 2 b i (1) b j (1) + λ 2 2 b i (2) b j (2) + λ 3 2 b i (3) b j (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOqamaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH7oaBdaqhaaWcbaGaaGymaaqaaiaaikdaaaGc caWGIbWaa0baaSqaaiaadMgaaeaacaGGOaGaaGymaiaacMcaaaGcca WGIbWaa0baaSqaaiaadQgaaeaacaGGOaGaaGymaiaacMcaaaGccqGH RaWkcqaH7oaBdaqhaaWcbaGaaGOmaaqaaiaaikdaaaGccaWGIbWaa0 baaSqaaiaadMgaaeaacaGGOaGaaGOmaiaacMcaaaGccaWGIbWaa0ba aSqaaiaadQgaaeaacaGGOaGaaGOmaiaacMcaaaGccqGHRaWkcqaH7o aBdaqhaaWcbaGaaG4maaqaaiaaikdaaaGccaWGIbWaa0baaSqaaiaa dMgaaeaacaGGOaGaaG4maiaacMcaaaGccaWGIbWaa0baaSqaaiaadQ gaaeaacaGGOaGaaG4maiaacMcaaaaaaa@595A@  now allows the stress tensor to be written as

σ ij = λ 1 λ 1 λ 2 λ 3 U ˜ λ 1 b i (1) b j (1) + λ 2 λ 1 λ 2 λ 3 U ˜ λ 2 b i (2) b j (2) + λ 3 λ 1 λ 2 λ 3 U ˜ λ 3 b i (3) b j (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeq4UdW2aaSbaaSqaaiaaigdaaeqa aaGcbaGaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeq4UdW2aaSbaaS qaaiaaikdaaeqaaOGaeq4UdW2aaSbaaSqaaiaaiodaaeqaaaaakmaa laaabaGaeyOaIy7aaacaaeaacaWGvbaacaGLdmaaaeaacqGHciITcq aH7oaBdaWgaaWcbaGaaGymaaqabaaaaOGaamOyamaaDaaaleaacaWG PbaabaGaaiikaiaaigdacaGGPaaaaOGaamOyamaaDaaaleaacaWGQb aabaGaaiikaiaaigdacaGGPaaaaOGaey4kaSYaaSaaaeaacqaH7oaB daWgaaWcbaGaaGOmaaqabaaakeaacqaH7oaBdaWgaaWcbaGaaGymaa qabaGccqaH7oaBdaWgaaWcbaGaaGOmaaqabaGccqaH7oaBdaWgaaWc baGaaG4maaqabaaaaOWaaSaaaeaacqGHciITdaaiaaqaaiaadwfaai aawoWaaaqaaiabgkGi2kabeU7aSnaaBaaaleaacaaIYaaabeaaaaGc caWGIbWaa0baaSqaaiaadMgaaeaacaGGOaGaaGOmaiaacMcaaaGcca WGIbWaa0baaSqaaiaadQgaaeaacaGGOaGaaGOmaiaacMcaaaGccqGH RaWkdaWcaaqaaiabeU7aSnaaBaaaleaacaaIZaaabeaaaOqaaiabeU 7aSnaaBaaaleaacaaIXaaabeaakiabeU7aSnaaBaaaleaacaaIYaaa beaakiabeU7aSnaaBaaaleaacaaIZaaabeaaaaGcdaWcaaqaaiabgk Gi2oaaGaaabaGaamyvaaGaay5adaaabaGaeyOaIyRaeq4UdW2aaSba aSqaaiaaiodaaeqaaaaakiaadkgadaqhaaWcbaGaamyAaaqaaiaacI cacaaIZaGaaiykaaaakiaadkgadaqhaaWcbaGaamOAaaqaaiaacIca caaIZaGaaiykaaaaaaa@857F@

 

 

 

3.5.4 A note on perfectly incompressible materials

 

The preceding formulas assume that the material has some (perhaps small) compressibility MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  that is to say, if you load it with hydrostatic pressure, its volume will change by a measurable amount.   Most rubbers strongly resist volume changes, and in hand calculations it is sometimes convenient to approximate them as perfectly incompressible.   The material model for incompressible materials is specified as follows:

 The deformation must satisfy J=1 to preserve volume.

 The strain energy density is therefore only a function of two invariants; furthermore, both sets of invariants defined above are identical.  We can use a strain energy density of the form U( I 1 , I 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacaGGOaGaamysamaaBaaaleaaca aIXaaabeaakiaacYcacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@36CA@ .

 Because you can apply any pressure to an incompressible solid without changing its shape, the stress cannot be uniquely determined from the strains.   Consequently, the stress-strain law only specifies the deviatoric stress σ ¯ ij = σ ij σ kk δ ij /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaaraWaaSbaaSqaaiaadMgaca WGQbaabeaakiabg2da9iabeo8aZnaaBaaaleaacaWGPbGaamOAaaqa baGccqGHsislcqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeq iTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiaac+cacaaIZaaaaa@4320@ .  In problems involving quasi-static loading, the hydrostatic stress p= σ kk /3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcqaHdpWCdaWgaaWcba Gaam4AaiaadUgaaeqaaOGaai4laiaaiodaaaa@37AC@  can usually be calculated, by solving the equilibrium equations (together with appropriate boundary conditions).   Incompressible materials should not be used in a dynamic analysis, because the speed of elastic pressure waves is infinite.

 The formula for stress in terms of U( I 1 , I 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacaGGOaGaamysamaaBaaaleaaca aIXaaabeaakiaacYcacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@36CA@  has the form

σ ij =2[ ( U I 1 + I 1 U I 2 ) B ij ( I 1 U I 1 +2 I 2 U I 2 ) δ ij 3 U I 2 B ik B kj ]+p δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaaikdadaWadaqaamaabmaabaWaaSaaaeaacqGH ciITcaWGvbaabaGaeyOaIyRaamysamaaBaaaleaacaaIXaaabeaaaa GccqGHRaWkcaWGjbWaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacqGH ciITcaWGvbaabaGaeyOaIyRaamysamaaBaaaleaacaaIYaaabeaaaa aakiaawIcacaGLPaaacaWGcbWaaSbaaSqaaiaadMgacaWGQbaabeaa kiabgkHiTmaabmaabaGaamysamaaBaaaleaacaaIXaaabeaakmaala aabaGaeyOaIyRaamyvaaqaaiabgkGi2kaadMeadaWgaaWcbaGaaGym aaqabaaaaOGaey4kaSIaaGOmaiaadMeadaWgaaWcbaGaaGOmaaqaba GcdaWcaaqaaiabgkGi2kaadwfaaeaacqGHciITcaWGjbWaaSbaaSqa aiaaikdaaeqaaaaaaOGaayjkaiaawMcaamaalaaabaGaeqiTdq2aaS baaSqaaiaadMgacaWGQbaabeaaaOqaaiaaiodaaaGaeyOeI0YaaSaa aeaacqGHciITcaWGvbaabaGaeyOaIyRaamysamaaBaaaleaacaaIYa aabeaaaaGccaWGcbWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadkea daWgaaWcbaGaam4AaiaadQgaaeqaaaGccaGLBbGaayzxaaGaey4kaS IaamiCaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaaaa@72EA@

The hydrostatic stress p is an unknown variable, which must be calculated by solving the boundary value problem.

 

 

 

 

3.5.5 Specific forms of the strain energy density

 

 Generalized Neo-Hookean solid  (Adapted from Treloar, Proc Phys Soc 60 135-44 1948)

U ¯ = μ 1 2 ( I ¯ 1 3)+ K 1 2 (J1) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaraGaeyypa0ZaaSaaaeaacq aH8oqBdaWgaaWcbaGaaGymaaqabaaakeaacaaIYaaaaiaacIcaceWG jbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaGaaiykai abgUcaRmaalaaabaGaam4samaaBaaaleaacaaIXaaabeaaaOqaaiaa ikdaaaGaaiikaiaadQeacqGHsislcaaIXaGaaiykamaaCaaaleqaba GaaGOmaaaaaaa@433C@

where μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaa aa@337A@  and K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaaaaa@321F@  are material properties (for small deformations, μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaa aa@337A@  and K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaaaaa@321F@  are the shear modulus and bulk modulus of the solid, respectively). Elementary statistical mechanics treatments predict that μ 1 =NkT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamOtaiaadUgacaWGubaaaa@3718@ , where N is the number of polymer chains per unit volume, k is the Boltzmann constant, and T is temperature.  This is a rubber elasticity model, for rubbers with very limited compressibility, and should be used with K 1 >> μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaGccq GH+aGpcqGH+aGpcqaH8oqBdaWgaaWcbaGaaGymaaqabaaaaa@36D6@ .  The stress-strain relation follows as

σ ij = μ 1 J 5/3 ( B ij 1 3 B kk δ ij )+ K 1 ( J1 ) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqa aaGcbaGaamOsamaaCaaaleqabaGaaGynaiaac+cacaaIZaaaaaaakm aabmaabaGaamOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl daWcaaqaaiaaigdaaeaacaaIZaaaaiaadkeadaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGa ayjkaiaawMcaaiabgUcaRiaadUeadaWgaaWcbaGaaGymaaqabaGcda qadaqaaiaadQeacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiTdq2a aSbaaSqaaiaadMgacaWGQbaabeaaaaa@5355@

The fully incompressible limit can be obtained by setting K 1 (J1)=p/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamOsaiabgkHiTiaaigdacaGGPaGaeyypa0JaamiCaiaac+ca caaIZaaaaa@3964@  in the stress-strain law.

 

 Generalized Mooney-Rivlin solid (Adapted from Mooney, J Appl Phys 11 582 1940)

U ¯ = μ 1 2 ( I ¯ 1 3)+ μ 2 2 ( I ¯ 2 3)+ K 1 2 (J1) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaraGaeyypa0ZaaSaaaeaacq aH8oqBdaWgaaWcbaGaaGymaaqabaaakeaacaaIYaaaaiaacIcaceWG jbGbaebadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaGaaiykai abgUcaRmaalaaabaGaeqiVd02aaSbaaSqaaiaaikdaaeqaaaGcbaGa aGOmaaaacaGGOaGabmysayaaraWaaSbaaSqaaiaaikdaaeqaaOGaey OeI0IaaG4maiaacMcacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGa aGymaaqabaaakeaacaaIYaaaaiaacIcacaWGkbGaeyOeI0IaaGymai aacMcadaahaaWcbeqaaiaaikdaaaaaaa@4C6D@

where μ 1 , μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeY7aTnaaBaaaleaacaaIYaaabeaaaaa@36C4@  and K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4samaaBaaaleaacaaIXaaabeaaaa a@3286@  are material properties.  For small deformations, the shear modulus and bulk modulus of the solid are μ= μ 1 + μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9iabeY7aTnaaBaaale aacaaIXaaabeaakiabgUcaRiabeY7aTnaaBaaaleaacaaIYaaabeaa aaa@394B@  and K= K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacqGH9aqpcaWGlbWaaSbaaSqaai aaigdaaeqaaaaa@33F5@ .  This is a rubber elasticity model, and should be used with K 1 >> μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaGccq GH+aGpcqGH+aGpcqaH8oqBdaWgaaWcbaGaaGymaaqabaaaaa@36D6@ . The stress-strain relation follows as

σ ij = μ 1 J 5/3 ( B ij 1 3 B kk δ ij )+ μ 2 J 7/3 ( B kk B ij 1 3 [ B kk ] 2 δ ij B ik B kj + 1 3 B kn B nk δ ij )+ K 1 ( J1 ) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqa aaGcbaGaamOsamaaCaaaleqabaGaaGynaiaac+cacaaIZaaaaaaakm aabmaabaGaamOqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsisl daWcaaqaaiaaigdaaeaacaaIZaaaaiaadkeadaWgaaWcbaGaam4Aai aadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaOGa ayjkaiaawMcaaiabgUcaRmaalaaabaGaeqiVd02aaSbaaSqaaiaaik daaeqaaaGcbaGaamOsamaaCaaaleqabaGaaG4naiaac+cacaaIZaaa aaaakmaabmaabaGaamOqamaaBaaaleaacaWGRbGaam4AaaqabaGcca WGcbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaalaaabaGa aGymaaqaaiaaiodaaaGaai4waiaadkeadaqhaaWcbaGaam4AaiaadU gaaeaaaaGccaGGDbWaaWbaaSqabeaacaaIYaaaaOGaeqiTdq2aaSba aSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadkeadaWgaaWcbaGaam yAaiaadUgaaeqaaOGaamOqamaaBaaaleaacaWGRbGaamOAaaqabaGc cqGHRaWkdaWcaaqaaiaaigdaaeaacaaIZaaaaiaadkeadaWgaaWcba Gaam4Aaiaad6gaaeqaaOGaamOqamaaBaaaleaacaWGUbGaam4Aaaqa baGccqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLOaGaay zkaaGaey4kaSIaam4samaaBaaaleaacaaIXaaabeaakmaabmaabaGa amOsaiabgkHiTiaaigdaaiaawIcacaGLPaaacqaH0oazdaWgaaWcba GaamyAaiaadQgaaeqaaaaa@7FAA@

 

 Generalized polynomial rubber elasticity potential

U ¯ = i+j=1 N C ij ( I ¯ 1 3) i ( I ¯ 2 3) j + i=1 N K i 2 (J1) 2i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaraGaeyypa0ZaaabCaeaaca WGdbWaaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGPbGaey4kaSIa amOAaiabg2da9iaaigdaaeaacaWGobaaniabggHiLdGccaGGOaGabm ysayaaraWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaG4maiaacMca daahaaWcbeqaaiaadMgaaaGccaGGOaGabmysayaaraWaaSbaaSqaai aaikdaaeqaaOGaeyOeI0IaaG4maiaacMcadaahaaWcbeqaaiaadQga aaGccqGHRaWkdaaeWbqaamaalaaabaGaam4samaaBaaaleaacaWGPb aabeaaaOqaaiaaikdaaaGaaiikaiaadQeacqGHsislcaaIXaGaaiyk amaaCaaaleqabaGaaGOmaiaadMgaaaaabaGaamyAaiabg2da9iaaig daaeaacaWGobaaniabggHiLdaaaa@582E@

where C ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3339@  and K i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaamyAaaqabaaaaa@3252@  are material properties.  For small strains the shear modulus and bulk modulus follow as μ=2( C 01 + C 10 ),K=2 K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9iaaikdacaGGOaGaam 4qamaaBaaaleaacaaIWaGaaGymaaqabaGccqGHRaWkcaWGdbWaaSba aSqaaiaaigdacaaIWaaabeaakiaacMcacaGGSaGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadUeacqGH9aqpcaaIYaGa am4samaaBaaaleaacaaIXaaabeaaaaa@4AC7@ . This model is implemented in many finite element codes.  Both the neo-Hookean solid and the Mooney-Rivlin solid are special cases of the law (with N=1 and appropriate choices of C ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@3339@  ).  Values of N>2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacqGH+aGpcaaIYaaaaa@32FF@  are rarely used, because it is difficult to fit such a large number of material properties to experimental data. 

 

 Ogden model (Ogden, Proc R Soc Lond A326, 565-84 (1972), ibid A328 567-83 (1972))

U ˜ = i=1 N 2 μ i α i 2 ( λ ¯ 1 α i + λ ¯ 2 α i + λ ¯ 3 α i 3)+ K 1 2 (J1) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaiaGaeyypa0ZaaabCaeaada WcaaqaaiaaikdacqaH8oqBdaWgaaWcbaGaamyAaaqabaaakeaacqaH XoqydaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaOGaaiikaiqbeU7aSz aaraWaa0baaSqaaiaaigdaaeaacqaHXoqydaWgaaadbaGaamyAaaqa baaaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGobaaniabggHiLd GccqGHRaWkcuaH7oaBgaqeamaaDaaaleaacaaIYaaabaGaeqySde2a aSbaaWqaaiaadMgaaeqaaaaakiabgUcaRiqbeU7aSzaaraWaa0baaS qaaiaaiodaaeaacqaHXoqydaWgaaadbaGaamyAaaqabaaaaOGaeyOe I0IaaG4maiaacMcacqGHRaWkdaWcaaqaaiaadUeadaWgaaWcbaGaaG ymaaqabaaakeaacaaIYaaaaiaacIcacaWGkbGaeyOeI0IaaGymaiaa cMcadaahaaWcbeqaaiaaikdaaaaaaa@5D02@

where λ ¯ i = λ i / J 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeU7aSzaaraWaaSbaaSqaaiaadMgaae qaaOGaeyypa0Jaeq4UdW2aaSbaaSqaaiaadMgaaeqaaOGaai4laiaa dQeadaahaaWcbeqaaiaaigdacaGGVaGaaG4maaaaaaa@3B10@  , and μ i , α i ,K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaWGPbaabeaaki aacYcacqaHXoqydaWgaaWcbaGaamyAaaqabaGccaGGSaGaam4saaaa @3835@  are material properties.  For small strains the shear modulus and bulk modulus follow as μ= i=1 N μ i ,K= K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9maaqadabaGaeqiVd0 2aaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGa amOtaaqdcqGHris5aOGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGlbGaeyypa0Jaam4samaaBaaaleaacaaI Xaaabeaaaaa@4A81@ . This is a rubber elasticity model, and is intended to be used with K 1 >> μ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaGccq GH+aGpcqGH+aGpcqaH8oqBdaWgaaWcbaGaamyAaaqabaaaaa@3709@ .  The stress can be computed using the formulas in 3.4.3, but are too lengthy to write out in full here.

 

 Arruda-Boyce 8 chain model (J. Mech. Phys. Solids, 41, (2) 389-412, 1992)

U ¯ =μ{ 1 2 ( I ¯ 1 3)+ 1 20 β 2 ( I ¯ 1 2 9)+ 11 1050 β 4 ( I ¯ 1 3 27)+... }+ K 2 ( J1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaraGaeyypa0JaeqiVd02aai WaaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiaacIcaceWGjbGbaeba daqhaaWcbaGaaGymaaqaaaaakiabgkHiTiaaiodacaGGPaGaey4kaS YaaSaaaeaacaaIXaaabaGaaGOmaiaaicdacqaHYoGydaahaaWcbeqa aiaaikdaaaaaaOGaaiikaiqadMeagaqeamaaDaaaleaacaaIXaaaba GaaGOmaaaakiabgkHiTiaaiMdacaGGPaGaey4kaSYaaSaaaeaacaaI XaGaaGymaaqaaiaaigdacaaIWaGaaGynaiaaicdacqaHYoGydaahaa WcbeqaaiaaisdaaaaaaOGaaiikaiqadMeagaqeamaaDaaaleaacaaI XaaabaGaaG4maaaakiabgkHiTiaaikdacaaI3aGaaiykaiabgUcaRi aac6cacaGGUaGaaiOlaaGaay5Eaiaaw2haaiabgUcaRmaalaaabaGa am4saaqaaiaaikdaaaWaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@6107@

where μ,β,K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjaacYcacqaHYoGycaGGSaGaam 4saaaa@35EF@  are material properties.  For small deformations μ,K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjaacYcacaWGlbaaaa@339E@  are the shear and bulk modulus, respectively. This is a rubber elasticity model, so K>>μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeacqGH+aGpcqGH+aGpcqaH8oqBaa a@34FE@ .    The potential was derived by calculating the entropy of a simple network of long-chain molecules, and the series is the result of a Taylor expansion of an inverse Langevin function.  The reference provided lists more terms if you need them.  The stress-strain law is

σ ij = μ J 5/3 ( 1+ B kk 5 J 2/3 β 2 + 33 ( B kk ) 2 525 β 4 J 4/3 +... )( B ij B kk 3 δ ij )+K(J1) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiabeY7aTbqaaiaadQeadaahaaWcbeqa aiaaiwdacaGGVaGaaG4maaaaaaGcdaqadaqaaiaaigdacqGHRaWkda WcaaqaaiaadkeadaWgaaWcbaGaam4AaiaadUgaaeqaaaGcbaGaaGyn aiaadQeadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaakiabek7aIn aaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaiodacaaI ZaGaaiikaiaadkeadaWgaaWcbaGaam4AaiaadUgaaeqaaOGaaiykam aaCaaaleqabaGaaGOmaaaaaOqaaiaaiwdacaaIYaGaaGynaiabek7a InaaCaaaleqabaGaaGinaaaakiaadQeadaahaaWcbeqaaiaaisdaca GGVaGaaG4maaaaaaGccqGHRaWkcaGGUaGaaiOlaiaac6caaiaawIca caGLPaaadaqadaqaaiaadkeadaWgaaWcbaGaamyAaiaadQgaaeqaaO GaeyOeI0YaaSaaaeaacaWGcbWaaSbaaSqaaiaadUgacaWGRbaabeaa aOqaaiaaiodaaaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaO GaayjkaiaawMcaaiabgUcaRiaadUeacaGGOaGaamOsaiabgkHiTiaa igdacaGGPaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaaa@6F4A@

 

 Ogden-Storakers hyperelastic foam

U ˜ = i=1 N 2 μ i α i 2 ( λ 1 α i + λ 2 α i + λ 3 α i 3+ 1 β i ( J α i β i 1) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGabmyvayaaiaGaeyypa0ZaaabCaeaada WcaaqaaiaaikdacqaH8oqBdaWgaaWcbaGaamyAaaqabaaakeaacqaH XoqydaqhaaWcbaGaamyAaaqaaiaaikdaaaaaaOWaaeWaaeaacqaH7o aBdaqhaaWcbaGaaGymaaqaaiabeg7aHnaaBaaameaacaWGPbaabeaa aaGccqGHRaWkcqaH7oaBdaqhaaWcbaGaaGOmaaqaaiabeg7aHnaaBa aameaacaWGPbaabeaaaaGccqGHRaWkcqaH7oaBdaqhaaWcbaGaaG4m aaqaaiabeg7aHnaaBaaameaacaWGPbaabeaaaaGccqGHsislcaaIZa Gaey4kaSYaaSaaaeaacaaIXaaabaGaeqOSdi2aaSbaaSqaaiaadMga aeqaaaaakiaacIcacaWGkbWaaWbaaSqabeaacqGHsislcqaHXoqyda WgaaadbaGaamyAaaqabaWccqaHYoGydaWgaaadbaGaamyAaaqabaaa aOGaeyOeI0IaaGymaiaacMcaaiaawIcacaGLPaaaaSqaaiaadMgacq GH9aqpcaaIXaaabaGaamOtaaqdcqGHris5aaaa@63C8@

where μ i , α i , β i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaWGPbaabeaaki aacYcacqaHXoqydaWgaaWcbaGaamyAaaqabaGccaGGSaGaeqOSdi2a aSbaaSqaaiaadMgaaeqaaaaa@3A20@  are material properties.   For small strains the shear modulus and bulk modulus follow as μ= i=1 N μ i ,K= i=1 N 2 μ i ( β i +1/3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTjabg2da9maaqadabaGaeqiVd0 2aaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaabaGa amOtaaqdcqGHris5aOGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGlbGaeyypa0ZaaabmaeaacaaIYaGaeqiV d02aaSbaaSqaaiaadMgaaeqaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOtaaqdcqGHris5aOWaaeWaaeaacqaHYoGydaWgaaWcbaGaamyA aaqabaGccqGHRaWkcaaIXaGaai4laiaaiodaaiaawIcacaGLPaaaaa a@5934@ .   This is a foam model, and can model highly compressible materials.  The shear and compression responses are coupled.

 

 Blatz-Ko foam rubber

                                                  U( I 1 , I 2 , I 3 )= μ 2 ( I 1 I 2 +2 I 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwfacaGGOa GaamysamaaBaaaleaacaaIXaaabeaakiaacYcacaWGjbWaaSbaaSqa aiaaikdaaeqaaOGaaiilaiaadMeadaWgaaWcbaGaaG4maaqabaGcca GGPaGaeyypa0ZaaSaaaeaacqaH8oqBaeaacaaIYaaaamaabmaabaWa aSaaaeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamysamaaBa aaleaacaaIYaaabeaaaaGccqGHRaWkcaaIYaWaaOaaaeaacaWGjbWa aSbaaSqaaiaaiodaaeqaaaqabaaakiaawIcacaGLPaaaaaa@4B1B@

where μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@321E@  is a material parameter corresponding to the shear modulus at infinitesimal strains. Poisson’s ratio for such a material is 0.25.                                              

 

 

 

3.5.6 Calibrating nonlinear elasticity models

 

To use any of these constitutive relations, you will need to determine values for the material constants.  In some cases this is quite simple (the incompressible neo-Hookean material only has 1 constant!); for models like the generalized polynomial or Ogden’s it is considerably more involved. 

 

Conceptually, however, the procedure is straightforward.  You can perform various types of test on a sample of the material, including simple tension, pure shear, equibiaxial tension, or volumetric compression. It is straightforward to calculate the predicted stress-strain behavior for the specimen for each constitutive law.  The parameters can then be chosen to give the best fit to experimental behavior. 

 

Here are some guidelines on how best to do this:

1.      When modeling the behavior of rubber under ambient pressure, you can usually assume that the material is nearly incompressible, and don’t need to characterize response to volumetric compression in detail.  For the rubber elasticity models listed above, you can take K 1 10 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaWgaaWcbaGaaGymaaqabaGccq GHijYUcaaIXaGaaGimamaaCaaaleqabaGaaGynaaaaaaa@362B@  MPa. To fit the remaining parameters, you can assume the material is perfectly incompressible.

2.      If rubber is subjected to large hydrostatic stress (>100 MPa) its volumetric and shear responses are strongly coupled. Compression increases the shear modulus, and high enough pressure can even induce a glass transition (see, e.g. D.L. Quested, K.D. Pae, J.L. Sheinbein and B.A. Newman, J. Appl. Phys, 52, (10) 5977 (1981)).  To account for this, you would have to use one of the foam models: in the rubber models the volumetric and shear responses are decoupled. You would also have to determine the material constants by testing the material under combined hydrostatic and shear loading. 

3.      For the simpler material models, (e.g. the neo-Hookean solid, the Mooney-Rivlin material, or the Arruda-Boyce model, which contain only two material parameters in addition to the bulk modulus) you can estimate material parameters by fitting to the results of a uniaxial tension test.  There are various ways to actually do the fit MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you could match the small-strain shear modulus to experiment, and then select the remaining parameter to fit the stress-strain curve at a larger stretch.  Least-squared fits are also often used.  However, models calibrated in this way do not always predict material behavior under multiaxial loading accurately.

4.      A more accurate description of material response to multiaxial loading can be obtained by fitting the material parameters to multiaxial tests.  To help in this exercise, the nominal stress (i.e. force/unit undeformed area) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@ v- extension predicted by several constitutive laws are listed in the table below (assuming perfectly incompressible behavior, as suggested in 1.)

 

 

 

 

Uniaxial Tension

Biaxial Tension

Pure Shear

Invariants

l 1 / L 1 =λ l 2 / L 2 = l 3 / L 3 = λ 1/2 I 1 = λ 2 +2 λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamiBamaaBaaaleaacaaIXaaabe aakiaac+cacaWGmbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaeq4U dWgabaGaamiBamaaBaaaleaacaaIYaaabeaakiaac+cacaWGmbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaamiBamaaBaaaleaacaaIZaaa beaakiaac+cacaWGmbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaeq 4UdW2aaWbaaSqabeaacqGHsislcaaIXaGaai4laiaaikdaaaaakeaa caWGjbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaeq4UdW2aaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGOmaiabeU7aSnaaCaaaleqabaGa eyOeI0IaaGymaaaaaaaa@51C6@

l 1 / L 1 = l 2 / L 2 =λ l 3 / L 3 = λ 2 I 1 =2 λ 2 + λ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamiBamaaBaaaleaacaaIXaaabe aakiaac+cacaWGmbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaamiB amaaBaaaleaacaaIYaaabeaakiaac+cacaWGmbWaaSbaaSqaaiaaik daaeqaaOGaeyypa0Jaeq4UdWgabaGaamiBamaaBaaaleaacaaIZaaa beaakiaac+cacaWGmbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaeq 4UdW2aaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamysamaaBaaa leaacaaIXaaabeaakiabg2da9iaaikdacqaH7oaBdaahaaWcbeqaai aaikdaaaGccqGHRaWkcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaisda aaaaaaa@505B@

l 1 / L 1 =λ l 2 / L 2 =1 l 3 / L 3 = λ 1 I 1 =1+ λ 2 + λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8qkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamiBamaaBaaaleaacaaIXaaabe aakiaac+cacaWGmbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaeq4U dWMaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaamiBamaaBaaaleaacaaIYaaabeaakiaac+cacaWGmbWa aSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGymaaqaaiaadYgadaWgaa WcbaGaaG4maaqabaGccaGGVaGaamitamaaBaaaleaacaaIZaaabeaa kiabg2da9iabeU7aSnaaCaaaleqabaGaeyOeI0IaaGymaaaaaOqaai aadMeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIXaGaey4kaSIa eq4UdW2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeq4UdW2aaWbaaS qabeaacqGHsislcaaIYaaaaaaaaa@5FD7@

Neo-Hookean

S 1 = μ 1 ( λ λ 2 ) S 2 = S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iabeY7aTnaaBaaaleaacaaIXaaabeaakmaabmaabaGa eq4UdWMaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqGHsislcaaIYaaaaa GccaGLOaGaayzkaaaabaGaam4uamaaBaaaleaacaaIYaaabeaakiab g2da9iaadofadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIWaaaaa a@43ED@

S 1 = S 2 = μ 1 ( λ λ 5 ) S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadofadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabeU7aSjabgkHiTi abeU7aSnaaCaaaleqabaGaeyOeI0IaaGynaaaaaOGaayjkaiaawMca aaqaaiaadofadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIWaaaaa a@43F0@

S 1 = μ 1 ( λ λ 3 ) S 2 = μ 1 ( λ 1 λ 3 ) S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iabeY7aTnaaBaaaleaacaaIXaaabeaakmaabmaabaGa eq4UdWMaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqGHsislcaaIZaaaaa GccaGLOaGaayzkaaaabaGaam4uamaaBaaaleaacaaIYaaabeaakiab g2da9iabeY7aTnaaBaaaleaacaaIXaaabeaakmaabmaabaGaeq4UdW 2aaWbaaSqabeaacqGHsislcaaIXaaaaOGaeyOeI0Iaeq4UdW2aaWba aSqabeaacqGHsislcaaIZaaaaaGccaGLOaGaayzkaaaabaGaam4uam aaBaaaleaacaaIZaaabeaakiabg2da9iaaicdaaaaa@5034@

Mooney-Rivlin

S 1 = μ 1 ( λ λ 2 ) + μ 2 (1 λ 3 ) S 2 = S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iabeY7aTnaaBaaaleaacaaIXaaabeaakmaabmaabaGa eq4UdWMaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqGHsislcaaIYaaaaa GccaGLOaGaayzkaaGaaGPaVdqaaiaaykW7caaMc8UaaGPaVlaaykW7 cqGHRaWkcqaH8oqBdaWgaaWcbaGaaGOmaaqabaGccaGGOaGaaGymai abgkHiTiabeU7aSnaaCaaaleqabaGaeyOeI0IaaG4maaaakiaacMca aeaacaWGtbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaam4uamaaBa aaleaacaaIZaaabeaakiabg2da9iaaicdaaaaa@55C5@

S 1 = S 2 = μ 1 ( λ λ 5 ) + μ 2 ( λ 3 λ 3 ) S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadofadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabeU7aSjabgkHiTi abeU7aSnaaCaaaleqabaGaeyOeI0IaaGynaaaaaOGaayjkaiaawMca aaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7cqGHRaWkcqaH8oqBdaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeq 4UdW2aaWbaaSqabeaacaaIZaaaaOGaeyOeI0Iaeq4UdW2aaWbaaSqa beaacqGHsislcaaIZaaaaOGaaiykaaqaaiaadofadaWgaaWcbaGaaG 4maaqabaGccqGH9aqpcaaIWaaaaaa@68AE@

S 1 =( μ 1 + μ 2 )( λ λ 3 ) S 2 = μ 1 ( 1 λ 1 )+ μ 2 ( λ 2 1 ) S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9maabmaabaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaeqiVd02aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaa WaaeWaaeaacqaH7oaBcqGHsislcqaH7oaBdaahaaWcbeqaaiabgkHi TiaaiodaaaaakiaawIcacaGLPaaaaeaacaWGtbWaaSbaaSqaaiaaik daaeqaaOGaeyypa0JaeqiVd02aaSbaaSqaaiaaigdaaeqaaOWaaeWa aeaacaaIXaGaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqGHsislcaaIXa aaaaGccaGLOaGaayzkaaGaey4kaSIaeqiVd02aaSbaaSqaaiaaikda aeqaaOWaaeWaaeaacqaH7oaBdaahaaWcbeqaaiaaikdaaaGccqGHsi slcaaIXaaacaGLOaGaayzkaaaabaGaam4uamaaBaaaleaacaaIZaaa beaakiabg2da9iaaicdaaaaa@5BCF@

Arruda-Boyce

S 1 =C( λ λ 2 ) S 2 = S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadoeadaqadaqaaiabeU7aSjabgkHiTiabeU7aSnaa CaaaleqabaGaeyOeI0IaaGOmaaaaaOGaayjkaiaawMcaaaqaaiaado fadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaa iodaaeqaaOGaeyypa0JaaGimaaaaaa@420E@

S 1 = S 2 =C( λ λ 5 ) S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadofadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWG dbWaaeWaaeaacqaH7oaBcqGHsislcqaH7oaBdaahaaWcbeqaaiabgk HiTiaaiwdaaaaakiaawIcacaGLPaaaaeaacaWGtbWaaSbaaSqaaiaa iodaaeqaaOGaeyypa0JaaGimaaaaaa@4211@

S 1 =C( λ λ 3 ) S 2 =C( 1 λ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadoeadaqadaqaaiabeU7aSjabgkHiTiabeU7aSnaa CaaaleqabaGaeyOeI0IaaG4maaaaaOGaayjkaiaawMcaaaqaaiaado fadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWGdbWaaeWaaeaacaaI XaGaeyOeI0Iaeq4UdW2aaWbaaSqabeaacqGHsislcaaIYaaaaaGcca GLOaGaayzkaaaaaaa@4611@

C=μ( 1+ I 1 5 β 2 + 33 I 1 2 525 β 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeacqGH9aqpcqaH8oqBdaqadaqaai aaigdacqGHRaWkdaWcaaqaaiaadMeadaWgaaWcbaGaaGymaaqabaaa keaacaaI1aGaeqOSdi2aaWbaaSqabeaacaaIYaaaaaaakiabgUcaRm aalaaabaGaaG4maiaaiodacaWGjbWaa0baaSqaaiaaigdaaeaacaaI YaaaaaGcbaGaaGynaiaaikdacaaI1aGaeqOSdi2aaWbaaSqabeaaca aI0aaaaaaaaOGaayjkaiaawMcaaaaa@45DB@

Ogden

S 1 = n μ n ( λ α n λ α n /2 )/λ S 2 = S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9maaqafabaGaeqiVd02aaSbaaSqaaiaad6gaaeqaaOGa aiikaiabeU7aSnaaCaaaleqabaGaeqySde2aaSbaaWqaaiaad6gaae qaaaaakiabgkHiTiabeU7aSnaaCaaaleqabaGaeyOeI0IaeqySde2a aSbaaWqaaiaad6gaaeqaaSGaai4laiaaikdaaaGccaGGPaGaai4lai abeU7aSbWcbaGaamOBaaqab0GaeyyeIuoaaOqaaiaadofadaWgaaWc baGaaGOmaaqabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaaiodaaeqaaO Gaeyypa0JaaGimaaaaaa@4FEF@

S 1 = S 2 = n μ n ( λ α n λ 2 α n )/λ S 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9iaadofadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaae qbqaaiabeY7aTnaaBaaaleaacaWGUbaabeaakiaacIcacqaH7oaBda ahaaWcbeqaaiabeg7aHnaaBaaameaacaWGUbaabeaaaaGccqGHsisl cqaH7oaBdaahaaWcbeqaaiabgkHiTiaaikdacqaHXoqydaWgaaadba GaamOBaaqabaaaaOGaaiykaiaac+cacqaH7oaBaSqaaiaad6gaaeqa niabggHiLdaakeaacaWGtbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0 JaaGimaaaaaa@4F31@

S 1 = n μ n ( λ α n λ α n )/λ S 2 = n μ n (1 λ α n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4uamaaBaaaleaacaaIXaaabe aakiabg2da9maaqafabaGaeqiVd02aaSbaaSqaaiaad6gaaeqaaOGa aiikaiabeU7aSnaaCaaaleqabaGaeqySde2aaSbaaWqaaiaad6gaae qaaaaakiabgkHiTiabeU7aSnaaCaaaleqabaGaeyOeI0IaeqySde2a aSbaaWqaaiaad6gaaeqaaaaakiaacMcacaGGVaGaeq4UdWgaleaaca WGUbaabeqdcqGHris5aaGcbaGaam4uamaaBaaaleaacaaIYaaabeaa kiabg2da9maaqafabaGaeqiVd02aaSbaaSqaaiaad6gaaeqaaOGaai ikaiaaigdacqGHsislcqaH7oaBdaahaaWcbeqaaiabgkHiTiabeg7a HnaaBaaameaacaWGUbaabeaaaaGccaGGPaaaleaacaWGUbaabeqdcq GHris5aaaaaa@5986@

 

 

 

 

3.5.7 Representative values of material properties for rubbers

 

The properties of rubber are strongly sensitive to its molecular structure, and for accurate predictions you will need to obtain experimental data for the particular material you plan to use.    As a rough guide, the experimental data of Treloar  (Trans. Faraday Soc. 40, 59.1944) for the behavior of vulcanized rubber under uniaxial tension, biaxial tension, and pure shear is shown in the picture.  The solid lines in the figure show the predictions of the Ogden model (which gives the best fit to the data).

 

Material parameters fit to this data for several constitutive laws are listed below.

 

 

 

 

 

 

Neo-Hookean

μ 1 =0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGUaGaaGinaaaa@363F@  MPa

Mooney-Rivlin

μ 1 =0.39 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGUaGaaG4maiaaiMdaaaa@3701@  MPa,  μ 2 =0.015 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIYaaabeaaki abg2da9iaaicdacaGGUaGaaGimaiaaigdacaaI1aaaaa@37B6@  MPa

Arruda-Boyce

μ 1 =0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGUaGaaGinaaaa@363F@  MPa, β=10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjabg2da9iaaigdacaaIWaaaaa@3484@

Ogden

μ 1 =0.62 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGUaGaaGOnaiaaikdaaaa@36FD@  MPa, α 1 =1.3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHnaaBaaaleaacaaIXaaabeaaki abg2da9iaaigdacaGGUaGaaG4maaaa@3628@

μ 2 =0.00118 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIYaaabeaaki abg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIXaGaaGymaiaaiIda aaa@392E@  MPa, α 2 =5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHnaaBaaaleaacaaIYaaabeaaki abg2da9iaaiwdaaaa@34BE@

μ 3 =0.00981 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTnaaBaaaleaacaaIZaaabeaaki abg2da9iabgkHiTiaaicdacaGGUaGaaGimaiaaicdacaaI5aGaaGio aiaaigdaaaa@3A24@  MPa, α 3 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHnaaBaaaleaacaaIZaaabeaaki abg2da9iabgkHiTiaaikdaaaa@35A9@