Cauchy-Green strain tensor

The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as... 

The pressure P0 represents the arbitrary additive contribution to the normal components of stress in an incompressible system, 8i is the Kronecker delta, C[ j 1(t t) is the inverse of the Cauchy-Green strain tensor for the configuration of material at t with respect to the configuration at the current time t [a description of the motion (221)], and M(t) is the junction age distribution or memory function of the fluid. 

Components of the Cauchy-Green strain tensor and its inverse, with the current configuration as the reference configuration. 

A typical choice to model compressible materials is to decompose the left Cauchy-Green strain tensor into a pure isochoric and a pure volumetric part [110, 111]... 

Hence, the right-Cauchy-Green strain tensor reads as C(t)... 

It can be easily shown that the Cauchy-Green strain tensor also transforms like this. 

In the above two equations, as well as in the rest of the equations in this section, subscripts 1, 2, and 3 indicate x, y, and z directions, respectively. The deformation tensor and its transpose can be combined to yield the right relative Cauchy-Green strain tensor, C, with components... 

The components of the Cauchy-Green strain tensor, C, are calculated from Eq. 6.85. The only nonzero components are... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

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) ... 

In the large strain situation, we can split the deviatoric and volumetric terms 9] by redefining the deformation gradient tensor as F = Then, the right Cauchy-Green deformation tensor invariants become... 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

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 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... 

It is also important to note that the 2nd Piola-Kirchhoff stress tensor is energetically conjugate to the Green-Lagrange strain tensor and the Cauchy stress is energetically conjugate to the infinitesimal strain tensor. In other words, we have... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

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... 

We have shown here that the Cauchy-Green and Finger tensors are not equivalent measures of finite strain, which is a very important fact to remember in the formulation of constitutive equations, as is discussed in Chapter 3. 

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