Tensor, stress deviatoric

A strength value associated with a Hugoniot elastic limit can be compared to quasi-static strengths or dynamic strengths observed values at various loading strain rates by the relation of the longitudinal stress component under the shock compression uniaxial strain tensor to the one-dimensional stress tensor. As shown in Sec. 2.3, the longitudinal components of a stress measured in the uniaxial strain condition of shock compression can be expressed in terms of a combination of an isotropic (hydrostatic) component of pressure and its deviatoric or shear stress component. 

The deviatoric stress tensor is related only to fluid motion, since for a fluid at rest the tensor is exactly zero. 

Determine the principal stress components. How does the pressure affect the principal stress components Why is it possible, and advisable, to develop an approach to computing the principal stresses by first subtracting the pressure from the stress tensor, that is, forming the deviatoric stress tensor How is the pressure reintroduced, after having determined the principal stress of the deviatoric stress tensor ... 

Determine the principal axes for the stress tensor. Why are the principal directions the same for the full stress tensor and the deviatoric stress tensor How does this result relate to the Stokes postulates that are used in the derivation of the Navier-Stokes equations ... 

By using the deviatoric stress tensor, Eq. 2.143, we can separate the thermodynamic pressure from the T W term as... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

In fluid mechanics, one common description of the deviatoric stress tensor is the Newtonian model given by,... 

Using the generalized Newtonian constitutive equation, the deviatoric stress tensor is defined as... 

Now that we have discussed the geometric interpretation of the rate of strain tensor, we can proceed with a somewhat more formal mathematical presentation. We noted earlier that the (deviatoric) stress tensor t related to the flow and deformation of the fluid. The kinematic quantity that expresses fluid flow is the velocity gradient. Velocity is a vector and in a general flow field each of its three components can change in any of the three... 

We present first a brief discussion of the stress tensor and the concepts of its mean and deviatoric parts. A rectangular Cartesian coordinate system (x, i 1,2,3) is used throughout. The stress tensor referred to this coordinate system is denoted... 

The resulting physical picture of a rubber-like system as a close-packed collection of mers is radically different from the two-phase image introduced by James and Guth [10]. The latter represents mbber as a network of chains, which act as entropic springs in tension, embedded in a bath of simple liquid. The bath gives rise to an isotropic pressure, whereas the network is responsible for the deviatoric stress. More recent physical pictures consider as well the distribution of network junctions in the liquid and the action of these junctions as constraints on the free motion of a generic chain of the network. The current description is on the mer or atomic level and treats the full stress tensor, both the mean and deviatoric portions, in terms of atomic interactions. 

In Section I, the discussion dealt with the significant role of nonbonded interactions in the development of the full stress tensor, mean plus deviatoric, in rubber elasticity, in the important high reduced density regime p > 1. Here, we present some concepts and formulations that apply to this regime. 

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

With the bulk, deviatoric stress tensor denoted by a, Bossis and Brady define the relative viscosity of the suspension as... 

This difficulty can be overcome by the use of a viscoelastic model limiting the effect of the singularity in the transport equations. In the Modified Upper Convected Maxwell (MUCM) proposed by Apelian et al. (see [1]), the relaxation time X is a function of the trace of the deviatoric part of the extra stress tensor ... 

In solid photoelasticimetry, birefringence is related to local stresses through the stress optical law, which expresses that the principal axes of stress and refractive index tensors are parallel and that the deviatoric parts of the refractive index and stress tensors are proportional ... 

In analogy with the strain, it is possible to express the stress tensor as the sum of a dilatational component, and a deviatoric component, that is,... 

In practice, the Leonard stress is often dominated by the numerical errors inherent in the finite difference (and finite volume) representation and is thus neglected or lumped into the deviatoric stress tensor (e.g., [97] [106] [186], p. 325). Consequently, as the box filter is applied to the Navier-Stokes equation, the residual stresses assume the form of sub-grid scale stresses ... 

An altemative scheme is the von Mises yield condition. In this case, one adopts an approach with a mean-field flavor in which plastic flow is presumed to commence once an averaged version of the shear stresses reaches a critical value. To proceed, we first define the deviatoric stress tensor which is given by,... 

This condition may be rewritten once it is recognized that, in terms of the principal stresses, the deviatoric stress tensor is diagonal and its components are... 

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