Deformation gradient

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

Consider a body undergoing a smooth homogeneous admissible motion. In the closed time interval [fj, fj] with < fj, let the motion be such that the material particle velocity v(t) and deformation gradient /"(t), and hence (r), and p(r), have the same values at times tj and tj. Such a finite smooth closed cycle of homogeneous deformation will be denoted by tj). Consider an arbitrary region in the body of volume which has a smooth closed boundary of surface area with outward unit normal vector n. The work W done by the stress s on and by the body force A in during... 

The partial derivatives of x are the velocity vector y and the deformation gradient tensor f, respectively. 

Consequently, E has components relative to the reference configuration, and is a referential strain tensor. A complementary strain tensor may be defined from the inverse deformation gradient F ... 

The relative motion of materials points in a solid body in finite strain is best represented by a deformation gradient having components... 

The deformation gradient tensor A is related to the strain tensor n by the equation ... 

Tensor whose components are deformation gradients in an elastic solid. 

Symmetric tensor that results when a deformation gradient tensor is factorized into a rotation tensor followed or preceded by a symmetric tensor. 

Derivative, for a viscoelastic liquid or solid in homogeneous deformation, of the rotational part of the deformation-gradient tensor at reference time, t. 

Note 2 The Xi are effectively deformation gradients, or, for finite deformations, the deformation ratios characterizing the deformation. 

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

Note 3 The deformation gradient tensor for the simple shear of an elastic solid is... 

In such cases W will be a function of position as well as of temperature and the coordinates of the deformation gradient tensor. Finally, most materials, in particular polymers, are anelastic. Energy is dissipated in them during a deformation and the stored energy function W cannot be defined. It is still of value, however, to consider ideal materials in which W does exist and to seek its form since such ideal materials may approximate quite closely to the real ones. 

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]

In codeformational equations, the basic kinematic quantities are the displacement functions. This generally means using the respective Cauchy and Finger tensors deformation gradient tensor X /9 Xfi ] by the following equations... 

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