Stress tensor definition

The structure of the section is as follows. In Section 2.8.2 we give necessary definitions and construct a Borel measure n which describes the work of the interaction forces, i.e. for a set A c F dr, the value /a(A) characterizes the forces at the set A. The next step is a proof of smoothness of the solution provided the exterior data are regular. In particular, we prove that horizontal displacements W belong to in a neighbourhood of the crack faces. Consequently, the components of the strain and stress tensors belong to the space In this case the measure n is absolutely continuous with respect to the Lebesgue measure. This confirms the existence of a locally integrable function q called a density of the measure n such that... 

For a given deformation or flow, the resulting stress depends on the material. However, the stress tensor does take particular general forms for experimentally used deformations (see section 2). The definitions apply to elastic solids, and viscoelastic liquids and solids. 

Note 1 The stress tensor for a uniaxial deformation is given in Definition 3.1. 

Component stress tensor resulting from a compressive uniaxial deformation. Note See notes 1 and 2 of Definition 3.2. 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

In addition, from the definition of the stress tensor, using the power law model for the viscosity, rrz becomes... 

Therein, D-1 is the positive definite, isotropic, fourth order viscous compliance, where r/ s and (s are the macroscopic viscosity parameters, D, , is the inelastic solid deformation rate and = F, 1 rfEQ F 71 is the corresponding non-equilibrium stress tensor. Furthermore, the superscript ( ) indicates the belonging to the intermediate configuration. 

The methods developed in the theory of liquids (Rice and Gray 1965, Gray 1968) was used by Pokrovskii and Volkov (1978a) to determine the stress tensor for the set of Brownian particles in this case. One can start with the definition of the momentum density, given by (6.3), which is valid for an arbitrary set of Brownian particles. Differentiating (6.3) with respect to time, one finds... 

To calculate the dynamic modulus, we turn to the expression for the stress tensor (6.46) and refer to the definition of equilibrium moments in Section 4.1.2, while memory functions are specified by their transforms as... 

The definition of the density of external angular momentum can be used to express, with the help of equations (8.1) and (8.2), the rate of change of the external angular momentum through the stress tensor crik... 

The set of constitutive equations for the dilute polymer solution consists of the definition of the stress tensor (6.16), which is expressed in terms of the second-order moments of co-ordinates, and the set of relaxation equations (2.39) for the moments. The usage of a special notation for the ratio, namely... 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

As before, let P be the local stress tensor, and denote by an overbar the statistical average of any quantity. The definition of the fluid-velocity field may be analytically extended to the solid-particle interiors and the pressure therein assumed to vanish. As such, taking the statistical average of the... 

The corresponding definitions in the kinetic theory for the properties of the gas mixture will be denoted here by the superscript T these are the mixture stress tensor ajj, where... 

Figure 1.18 The definition of the state-of-stress tensor in terms of force components acting on the faces of a cube. (From Larson 1988, with permission.)...

Having now encountered interfacial tension in the interfacial stress tensor it makes sense to review and compare the various definitions that we have so far encountered. 

Finally we recall [3.6.15] which is a mechanical definition for an infinitesimally thin monolayer. The interfacial stress tensor is a more general quantify than y because it also contains the shear components. When shear stresses are absent reduces to... 

From eqn (6.30) it is clear that the virial of the electronic forces, which is the electronic potential energy, is totally determined by the stress tensor a and hence by the one-electron density matrix. The atomic statement of the virial theorem provides the basis for the definition of the energy of an atom in a molecule, as is discussed in the sections following Section 6.2.2. 

If the cross-sectional area of the macroscopic network is doubled, then twice as large a force is required to obtain the same deformation. This leads naturally to a definition of stress as the ratio of force and cross-sectional area. Both the force and the cross-sectional are have direction and magnitude (the direction of the cross-sectional area being described by the unit vector normal to its surface), making the stress a tensor. The (/-component of the stress tensor is the force applied in the i direction per unit cross-sectional area of a network perpendicular to the j axis. For... 

By definition it is a symmetric second-rank tensor. The stress tensor ffy, i,j= 1,3), is also a symmetric second-rank tensor defined as follows (Landau and Lifchitz ) the element Oy is the i component of the force acting on the unit area normal to the axis x. The symmetry of the stress tensor is imposed by the condition of mechanical equilibrium. 

In different reference systems the strain and stress tensors have different components, the transformation being easily derived starting from the definitions. Let us consider, for example, the sample reference system (y ) and denote by Latin letters etm and stm the components of the strain and stress tensors in this system. If the transformation of the sample reference system (y,) into the crystal reference system (x ) is given by Equation (1) then the transformations of the strain tensors are the following ... 

The total stress tensor, T, is thus interpreted physically as the surface forces per surface unit acting through the infinitesimal surface on the surrounding fluid with normal unit vector n directed out of the CV. This means that the total stress tensor by definition acts on the surrounding fluid. The counteracting force on the fluid element (CV) is therefore expressed in terms of the total stress tensor by introducing a minus sign in (1.65). 

However, without showing all the lengthy details of the method by which the two scalar functions are determined, we briefly sketch the problem definition in which the partial solution (2.247) is used to determine expressions for the viscous-stress tensor o and the heat flux vector q. 

Note that the viscosity parameter p has been introduced as a prefactor in front of the tensor functions by substitution of the kinetic theory transport coefficient expression after comparing the kinetic theory result with the definition of the viscous stress tensor o, (2.69). In other words, this model inter-comparison defines the viscosity parameter in accordance with the Enskog theory. 

A number of papers appeared in the 1980s by Nielsen and Martin [47-49] and one in 2002 by Pendas [50] that employ the classical approach in the definition of pressure as explored by Slater [14] and others and embodied in Equation (17). This approach identifies the pv product with the virial of the external forces acting on the nuclei relating the pressure, in "analogy to classical thinking" [50], to the trace of a stress tensor, Equation (29)... 

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