The Strain Invariants

Any second-order tensor has a number of invariants associated with it. One such is the trace of the tensor, equal to the sum of its diagonal terms, applicable to any strain tensor. We define the first invariant h as the trace of the Cauchy-Green strain measure tr(C)  

Adopting principal directions, this can be expressed in terms of the extension ratios via Equation (3.17)  

The second invariant/2 can also defined in terms of the trace function and C  

The third invariant is related to the volume of the material, which, like energy, is the same for all axis sets. A volume of material that is initially a unit cube is deformed to a 

These three invariants are all that is necessary for our purposes in defining strain energy functions. 

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For viscoelastic fluids, both strain energy and stress can be assumed to depend on the strain history through the strain invariants ... 

From experiments in our laboratory on biaxial deformations of thin sheets, it is found that in some materials cracks are formed without any evidence of necking, while at the same levels of strain in uniaxial extension necking had already occurred. This is not surprising since the potential function w depends on the strain Invariants and for biaxial experiments, the solution given in section III has to be modified because the strain potential now has to be differentiated with respect to the first and second strain invariants. More work in biaxial deformations will lead to a better description of the failure mechanism In general. 

A blood vessel generally exhibits anisotropic behavior when subjected to large variations in internal pressure and distending force. When the degree of anisotropy is small, the blood vessel may be treated as isotropic. For isotropic materials it is convenient to introduce the strain invariants ... 

An alternative isotropic strain energy density function which can predict the appropriate type of stress stiffening for blood vessels is an exponential where the arguments is a polynomial of the strain invariants. 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

The symbol I represents the strain invariants analogous to the stress invariants given as J in Eqs. (1.22e) and (1.23). The coefficients in Eq. (1.98c) are the results of the engineering shear strain being ... 

In addition to these theoretical considerations, which suggest that we need not be restricted to squares of extension ratios in formulating the strain energy function, it has been found by experimentalists that there is high sensitivity to experimental error when small values of the strain invariants I and f are involved. It is therefore natural to postulate that the only constraint on the form U is that imposed by the invariance of U with respect to the axis lables, which implies that U i, X-i, A3) should be a symmetric function of the extension ratios, i.e. invariant to any permutation of the indices 1,2, 3. 

From Eq. (23), the relation between the Cauchy stress and the strain invariants... 

Since in a simple tension test, the lateral surfaces of the specimen are supposed to be unloaded, the principal stresses corresponding to the directions 2 and 3 vanish. For deformation (59) the strain invariants become ... 

We add to these set of equations the constitutive equations that relate stresses to strains. One form of constitutive equations for isotropic highly deformable materials is of the generalized Mooney-Rivlin type [2] in which the strain energy density W is expressed in terms of the strain invariants ... 

A natural extension of linear elasticity is hyperelasticity [17]. Hyperelasticity is a collective term for a family of models that have a strain energy density that only depends on the currently apphed deformation state (and not on the history of deformations). This class of material models is characterized by a nonhnear elastic response and does not capture yielding, viscoplasticity, or time dependence. Strain energy density is the energy that is stored in the material as it is deformed and is typically represented either in terms of the strain invariants h,h, and/, where... 

The nonlinear behavior shown in Fig. 44.1 is not the result of plastic deformation akin to that for ductile metals. It can best be understood in the context of the strain invariants that eventually resulted in the development of the SIFT (Strain-Invariant Failure Theory) failure... 

The strain-invariant polymer failure model (SIFT) permits the superposition of separate analyses for shear and induced peel loads. There is no interaction between the two failure mechanisms. While the need for this has been minimized through sound design practice (gentle tapering of the ends of the adherends to minimize the induced peel stresses, as explained later), the new model puts this technique on a secure scientific foundation and also accommodates any applied transverse shear loads. 

Hart-Smith LJ (2010) Application of the strain invariant failure theory (SIFT) to metals and fibre-polymer composites. Philos Mag 90(31) 4263-4331... 

For this deformation, the strain invariants for an incompressible material are ... 

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