Tensor deformation gradient

Deformation Gradient Tensor, Strain Tensor, Veiocity Gradient Tensor and Rate-of-Strain Tensor 

For the description of motion given by Eq. (2.3), consider two particles in the reference configuration (at t = 0) that are a distance dX apart. Then in the configuration x 

One may also interpret F as a linear operator, which maps the neighborhood of the particles X in the reference configuration k into the configuration x- 

Using the relative configuration x, defined by Eq. (2.5), we can also define 

For many purposes it is convenient to describe the history of the velocity gradient by another quantity. Consider the motion of a point r f) fixed on the material. In the homogeneous flow in which the velocity field is given by 

Since this is a linear equation, its solution is written as 

This is analogous to eqn (7.75). The tensor E t, t ) denotes the deformation gradient at time t referred to the state at time t. From eqn (7.152) it follows that 

This equation, together with the initial condition 

3 Constitutive equation derived from Rome model 

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

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

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 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. 

In codeformational equations, the basic kinematic quantities are the displacement functions. This generally means using the respective Cauchy and Finger tensors [Pg.303]

Here we have introduced a tensor F known as the deformation gradient tensor whose components reflect the various gradients in the deformation mapping and are given by... 

This corresponds in turn to a deformation gradient tensor of the form... 

In Chapter 5, we defined the deformation gradient tensor E, Cauchy tensor C, and Finger tensor B, respectively, as... 

In matrix form, the second-rank deformation gradient tensor can be expressed by... 

Now we have two entities that link the undeformed body to the deformed body, that is, the displacement vector u and the deformation gradient tensor F. Next, we would like to establish connections between these two entities, which will help us establish the link between the displacement and strain and the deformation gradient and strain for the case with finite strain. Let us differentiate both sides of Equation (4.2) with respect to X ... 

Obviously, the left-hand side is the deformation gradient tensor F. The first term on the right-hand side is the Kronecker delta second-order tensor and the second term on the right-hand side is the displacement gradient, which is also the strain tensor in the undeformed body. Therefore, Equation (4.6) can be rewritten as... 

The deformation compatibility at the interface requires that the deformation gradient tensors of each phase be used in Equation (5.18). To take into account the molecular alignment in the amorphous phase, the deformation gradient of the amorphous phase is incorporated as follows ... 

Note that both N and n are unit tangent vectors to the area elements dA and da, respectively. In Eqs. (8.5) and (8.6), F denotes the deformation gradient tensor to be calculated from the fluid element trajectories through integration of Eq. (8.3). The ratio of intermaterial areas, also called area stretch, r) can be defined as ... 

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