Cauchy-Green deformation tensor

Customarily, the following three invariants of Cauchy-Green deformation tensor,... 

Therein, Cs = FsFj and Jl,g = detFL, g are the right Cauchy-Green deformation tensor of the solid and the Jacobian of the gas and liquid phases respectively, where Fa denotes the deformation gradient of free energies and the specific entropies of the constituents [Pg.333]

Corresponding to U and V, two new tensors can be defined, which are used to calculate U and V. We have the right Cauchy-Green deformation tensor C and the left Cauchy-Green deformation tensor B ... 

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... 

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... 

The deviatoric part of the left Cauchy-Green deformation tensor, or Finger tensor, is given as... 

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

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]... 

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... 

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... 

To transform between different axis sets, we use the same method as for deformation gradients as given by Equation (3.21). Note that, unlike the deformation gradient, the stress tensor is always symmetric. This is the consequence of the equilibrium of torques applied to a material element, as pointed out in Chapter 2. As a second-order tensor, the stress is subject to the same axis transformation operations as the deformation gradient and Cauchy-Green measure (Equations (3.21) and (3.22)). The principal stresses are the eigenvalues... 

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

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... 

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

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