Microscopic stress tensor

By way of concluding remarks, let us reiterate the limitations of the theory presented here. The ensemble averages of the microscopic stress tensors appearing in the equations of motion Eqs. (12) and (13) were... 

The total pressure is the average of the diagonal part of the microscopic stress tensor,... 

Explicit forms for the stress tensors d1 are deduced from the microscopic expressions for the component stress tensors and from the scheme of the total stress devision between the components [164]. Within this model almost all essential features of the viscoelastic phase separation observable experimentally can be reproduced [165] (see Fig. 20) existence of a frozen period after the quench nucleation of the less viscous phase in a droplet pattern the volume shrinking of the more viscous phase transient formation of the bicontinuous network structure phase inversion in the final stage. 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

Having constructed the microscopic mesh, we specify the microscopic problem based on the macroscopic nodal displacements. The displacements of the elemental boundaries are given by the macroscopic solution (although the internal microscopic scale displacements are not necessarily affine). The microscopic problem is to find node positions and segment lengths such that the boundary nodes are as specified by the macroscopic displacements and the internal nodes experience no net force. The boundary nodes have displacement specified and are subjected to a non-zero net force. The next step in the solution process is to convert those forces into the macroscopic stress tensor. 

The continuum mechanics of solids and fluids serves as fhe prototypical example of the strategy of turning a blind eye to some subset of the full set of microscopic degrees of freedom. From a continuum perspective, the deformation of the material is captured kinematically through the existence of displacement or velocity fields, while fhe forces exerted on one part of the continuum by the rest are described via a stress tensor field. For many problems of interest to the mechanical behavior of materials, it suffices to build a description purely in terms of deformation fields and their attendant forces. A review of the key elements of such theories is the subject of this chapter. However, we should also note that the purview of continuum models is wider than that described here, and includes generalizations to liquid crystals, magnetic materials, superconductors and a variety of other contexts. 

This is a statement of the product rule for the divergence of the vector dot product of a tensor with a vector, which is valid when the tensor is symmetric. In other words, r = r, where is the transpose of the viscous stress tensor. Synunetry of the viscous stress tensor is a controversial topic in fluid dynamics, bnt one that is invariably assumed. is short-hand notation for the scalar double-dot product of two tensors. If the viscous stress tensor is not symmetric, then r must be replaced by in the second term on the right side of the (25-29). The left side of (25-29), with a negative sign, corresponds to the rate of work done on the fluid by viscous forces. The microscopic equation of change for total energy is written in the following form ... 

To solve the equations for the pressure and velocity of the fluid, one must specify boundary conditions. Usually one assumes that the fluid sticks to a solid wall, so that v(r, t) = 0 when r is on the solid surface. (This no-slip boundary condition may not be completely accurate at microscopic length scales.) The other boundary condition that is often important is that flows at infinity are unperturbed by the boundaries. Finally, at surfaces or interfaces, there is continuity of the normal and tangential forces. The force on a surface is related to the normal component of the stress tensor, Eq. (1.143) the ith component of the force per unit area, fs = dFs/dS, obeys... 

Fig. 3.8. Microscopic definition of the stress tensor. The stress component = X, y, z) is the a component of the force (per area) that the material above the plane (denoted by the dashed line) exerts on the material below the plane. The force S consists of two parts and is the force acting...

Microscopic expression for die stress tensor Let us now study the viscoelastic properties using molecular models. As was discussed in Chapter 3, the macroscofnc stress of the polymer solutions is written as (see eqn (3.133))... 

Expression for the stress tensor. The microscopic expression for the stress tensor can be obtained by taking the average of eqn (7.4) for a given conformation of the primitive chain. Alternatively, it can derived by an elementary argument explained in Fig. 7.10. In both cases the result is... 

General Extensions. - Bader applied ideas of AIM to the atomic force microscope (AFM). In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is an IAS. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach is contrasted with literature results that equate it to the Hellmann-Feynman forces exerted on the nuclei of the atoms in the tip. 

One application of the stress theorem is the study of elastic properties of solids, which becomes straightforward when a suitable finite macroscopic strain is applied to the solid. When the wavefunctions of the distorted solid are known, the stress tensor is evaluated with the stress theorem. In the harmonic approximation elastic constants are defined as the ratio of stress to strain, and it is furthermore possible to go to large strains to obtain all nonlinear elastic properties. In general it is necessary to be concerned with internal strains that may appear microscopically owing to the lower symmetry of the strained solid. In section 6 we show in detail how this problem is solved by combining the stress and force theorems. 

The macroscopic equations are solved by a time-marching scheme in which the stress tensor is obtained from a micro simulation and treated as a constant body force. The microscopic simulations reflect the dynamics of the specific fluid at hand and yield the stress tensor for a given velocity field. 

The usage of the term micro/macro is justified by the two-level description of the fluid that lies at its core a microscopic approach permits evaluation of the stress tensor to be used in closing the macroscopic conservation equations. The macro part can be thought of as corresponding to the Eulerian description of the fluid, while the micro part has Lagrangian character. 

The results from recent decades allow us to describe a picture of thermal motion of long macromolecules in a system of entangled macromolecules. The basic picture is, of coarse, a picture of thermal rotational movement of the interacting rigid segments connected in chains - Kuhn - Kramers chains. One can refer to this model as to a microscopic model. In the simplest case (linear macromolecules, see Sect. 3.3), the tensor of the mean orientation (e,ej) of all (independently of the position in the chain) segments can be introduced, so that the stress tensor and the relative optical permittivity tensor can be expressed through mean orientation as... 

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