Green tensor

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

The inverse of the Cauchy-Green tensor, Cf, is called the Finger strain tensor. Physically the single-integral constitutive models define the viscoelastic extra stress Tv for a fluid particle as a time integral of the defonnation history, i.e. 

Note 4 The X, are elements of the deformation gradient tensor F and the resulting Cauchy and Green tensors C and B are... 

It can be shown [8] that Hadamard instabilities are possible for admissible motions if a is in the interval (—1,1), e.g., in extensional flows. On the other hand, restrictions on the eigenvalues of r prevent Hadamard instabilities for a = 1. This is immediately seen from the integral forms qf (4)-(5) for the upper- and lower-convected Maxwell models, which imply constraints on the eigenvalues of the Cauchy-Green tensors. (See, for instance, [12].)... 

In this equation, 0 is a material parameter related to the first and second normal stress differences of the polymer N is the number of relaxation modes a, and k. are relaxation modulus and times, respectively and C stands for the Cauchy-Green tensor. 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

To hold the incompressibihty assumption the volume preserving part of the apphed deformation gradient needs to be utilized in the trial elastic part, with the left and right Cauchy-Green tensors given by [14]... 

The consideration of the assumed right Cauchy-Green tensor C +i = leads... 

C is referred to as the right Cauchy-Green tensor and B the left Cauchy-Green tensor. Then the deformation measure can be written as... 

Note 2.4 (Generalized strain measure (Hill 1978)9t). Since the right Cauchy-Green tensor C = is symmetric and the components are real numbers, there are three real eigenvalues that are set as A ( = 1,2,3) and the corresponding eigenvectors are given by Ni then we have... 

Note that the right Cauchy-Green tensor C and the Green strain E are not frame... 

Affinity at reaction stage r Acceleration Activity of species a Left Cauchy-Green tensor Body force per unit mass Diffusive body force Right Cauchy-Green tensor Mass-molar concentration Mass concentration of species a Mass-energy flux of the system... 

Exactly the same technique can be used to analyse the Cauchy-Green tensor C. When a deformation gradient F includes rigid-body rotation, it is necessary to first form the Cauchy-Green measure C and then find its principal components and directions using the methods outlined above for V. The principal directions of C are the same as those of the pure deformation V that underlies F (F = VR). Writing the analogue of Equation (3.12) forC... 

We can also express deformation in terms of length change. This comes from the Green or Cauchy-Green tensor... 

If we had used the Green tensor C instead of B in writing the constitutive equation, we would have obtained different results for the normal stresses in shear, (note eq. 1.4.25). Using B gives results that agree with observations for rubber. 

Section 1.4 mentioned that physically the Green tensor C is an operator that gives length changes that is, prove eq. 1.4.22. 

The general and detailed constitutive relations of E.H. Lee s elastic-plastic theory at finite strain have been derived by Lubarda and Lee [5]. In this work, let the specid constitutive relations which are employed in the general purpose finite element program be listed as follows. First, the Helmholtz free energy density, E, as a function of the invariants of the elastic Cauchy-Green tensor, c/y, may be expressed as... 

Using the relative Cauchy-Green tensor C (t, t) one can define other rate tensors, such as (Rivlin and Ericksen 1955)... 

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