Stress tensor Cartesian expression

In equation [1.1] and [1.2], the stress tensor is expressed in a Cartesian coordinate system (Q, x, y, z). In this coordinate system, the single-column matrices define the normal vector h and force dF. We only have to multiply matrix [Z] by h to calculate the force. 

In this section, we use the Cartesian force of Section VI to derive several equivalent expressions for the stress tensor of a constrained system of pointlike particles in a flow field with a macroscopic velocity gradient Vv. The excess stress of any system of interacting beads (i.e., point centers of hydrodynamic resistance) in a Newtonian solvent, beyond the Newtonian contribution that would be present at the applied deformation rate in the absence of the beads, is given by the Kramers-Kirkwood expression [1,4,18]... 

A fully Cartesian form for the stress tensor may be obtained by using Eq. (2.182) to expand terms arising from the generalized divergence of in Eq. (2.384). To begin, we move the derivative of In within In v /gq into the second term in Eq. (2.384), to obtain the equivalent expression... 

Consider a plane-strain problem in the x-z plane, as shown in Fig. 8.2. The stress tensor, expressed in Cartesian coordinates, takes the form... 

For a three-dimensional body, discussions of elastic responses in the framework of Hooke s law become more complicated. One defines a 3 x 3 stress tensor P [12], which is the force (with emits of newtons) expressed in a Cartesian coordinate system ... 

The remaining six quantities are called shear stresses. They have two subscripts associated with the coordinates, and are referred to as the components of the molecular momentum flow tensor, or the components of the molecular stress tensor, as they are associated with molecular motion. Usually, the viscous stress tensor, t, and the molecular stress tensor, it, are simply referred to as stress tensors. For a Newtonian fluid, we may express the stresses in terms of velocity gradients and viscosities in Cartesian coordinates as follows ... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

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