Left Cauchy-Green tensor

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... 

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... 

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 ... 

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 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... 

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]... 

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... 

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