Displacement gradient tensor

Another combination of the displacement gradient tensors which are often used are the Cauchy strain tensor and the Finger strain tensor defined by B —1 = Afc A and B = EEt, respectively. 

E displacement gradient tensor N. numberofprimitivepathstepsoc-ctqpied by the matrix chains in a... 

The material morphology is specified by a set of nodal points in the continuum description. The inclusion boundary is defined by a mesh of vertices xf b for boundary). The exterior of the inclusion contains the vertices xt (c for continuum). Inside the atomistic system, the (affine) transformations obtained by altering the scaling matrix from ho to h can be expressed by the overall displacement gradient tensor matrix M(h) = hho, The Lagrange strain tensor [40] of the atomistic system is then... 

Let a network be stretched along the x-direction. Let denote the component of the displacement gradient tensor in the direction of stretch. may be decomposed into a volumetric and a distortional part as... 

To describe deformation, we will examine the relative displacement of two neighboring fluid particles. At time tg, these particles are separated by the vector dx(fo) and at time by the vector dx(ti). A quantity that provides complete information about the relative displacement of any two such particles in a very small volume of the fluid is the displacement gradient tensor F whose components are given by ... 

Another tensor that relates the two vectors is the inverse of the displacement gradient tensor, which is often represented as E. Its components are ... 

And the inverse displacement gradient tensor can be used to calculate dx t ) given dx(t2) ... 

The displacement functions can be used to define the components of two displacement gradient tensors ... 

In this section we consider the simplest approach to the thermodynamics of a deformed network, for which a tensor of displacement gradients is given by... 

As an example, we shall consider simple shear when z/12 0, and find components of the tensor of the recoverable displacement gradients A12, An, A22, A33 the components of the tensor are calculated from the relaxation equations (9.49) or (9.58). In this case the matrix of the deformation tensor is determined as follows... 

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i" displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1],... 

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... 

In the case of a simple shear deformation, schematically indicated in Figure 4.6b, the only nonzero components of the displacement gradient and strain tensors are given by... 

Here we have used the fact that Fij = 5 + Uij. In addition, we have invoked the summation convention in which all repeated indices (in this case the index k) are summed over. For the case in which all the displacement gradient components satisfy Uij 1, the final term in the expression above may be neglected, resulting in the identification of the small strain (or infinitesimal strain) tensor,... 

Corresponding system plastic-strain increments are also obtained at the atomic level from the displacement gradients between the four relevant neighboring corner atoms of Delaunay tetrahedra for each external distortion increment and are allocated subsequently as an atomic site average to each Voronoi polyhedral atom environment by a special procedure of double space tessellation developed by Mott et al. (1992) for this purpose, leading eventually to volume averages of strain-increment tensors of all Voronoi atom environments to attain the system-wide strain-inerement tensor. 

The simulated dilatations involved increasing steps of imposed dilatation on the simulation cell. To permit a detailed understanding of the dilatational response of the polymer at the atomic level the entire volume of the simulation cell was tessellated into Voronoi polyhedra at each atomic site, permitting determination of strain-increment tensor elements dcy for each site from local displacement gradients by a technique described by Mott et al. (1992). Such increments of imposed dilatation at a level of 3 x 10 were applied 100 times to obtain total system dilatations of 0.3 (Mott et al. 1993b). For eaeh dilatation increment the atomic site strain-tensor increments de were obtained for each site n. The two invariants, de", the atomic site dilatation increment, and the work-equivalent shear-strain increment, dy", were obtained from the individual increments as... 

In the continuum limit the local deformation of a crystal can be described by the gradient of the displacement field (/, t) = Ri(t) - Ri where Ji, and JJ, are the temporary and equilibrium lattice positions, respectively. The gradient tensor is given by = dUi (i)ldR with k, l = x, y, z. The general deformation can be described as a pure strain deformation tj followed by a rotation D... 

For a small strain, the following relationship holds between the tensor strain and the displacement gradient ... 

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... 

It is noted that the above strain can also be expressed by the displacement gradient. For instance, the Green-Lagrange strain in Equation (4.20), after operation in terms of the tensor index, can be rewritten as... 

Partial differentiation of Eq. (3.9) with respect to the material coordinates yields the tensor coordinates of the displacement gradient... 

Frequently it is useful to express the coordinates of the Lagrange strain tensor by means of the displacement gradients (3.10) ... 

When the displacement gradient is split into its symmetric and skew symmetric portions, the infinitesimal strain tensor of Eq. (3.18) is identified to be the former, while the latter represents infinitesimal rotations that do not... 

In Chapter I (recall Figure 1.4.1) we were also concerned with describing how points separate. There we used the displacement of (mints to find F here we need rate of displacement Clearly F, the deformation gradient tensor, and L, the velocity gradient tensor, are related. Recall eq. 1.4.3... 

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