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 quantities C and d are the second-order tensors that are an array of nine Cij(iJ = 1,2,3) and dy terms C is known as the Green deformation tensor and d is the rate of deformation tensor. It is always possible to find a principal axis coordinate system where C and d have diagonal forms, that is, where Cy and dy are zero when i is not equal toj. In this principal axis coordinate system, all deformations are representable as multi-axial elongational and may be expressed through extension ratios li, In, and Ini corresponding to these principal axes. 

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... 

The Green deformation tensor C can now be expressed in terms of the rotated deformation tensor F, such that ... 

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... 

---