Components 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. 

Each cry is known as a deviatoric stress component it is the amount by which the stress component deviates from the static pressure. From equation A. 18, the stress components are related to the deviatoric stress components as follows ... 

It is the deviatoric stress components that are related to the rate of deformation, ie to the flow. 

This result may be contrasted with potential flow past a sphere, where the streamlines again have fore-and-aft symmetry but p is an even function of 9 so that there is no net pressure force (see Chapter 1). Additional drag components arise from the deviatoric normal stress ... 

Two thirds of this drag arises from skin friction, one third from form drag, and the component due to deviatoric normal stress is zero. The corresponding terminal velocity follows from Eq. (3-15) as ... 

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 ... 

A comparison of eqs. (2.27) and (2.28) clearly shows that the deviatoric components of these tensors are proportional to each other. As a consequence, the tensors are coaxial. The proportionality factor reads ... 

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. 

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... 

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)... 

The stress bias criterion [24,25] refers implicitly to two mechanisms of microvoid formation in a dilatational stress field and stabilization of the microvoids through a deviatoric stress component and local plasticity. Its definition is... 

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 order to describe the viscoelastic behavior of a system subjected to multiaxial tensions, it is convenient to separate the shear (deviatoric) effects from the purely dilatational components. This is due to the fact that in... 

If the dilational and deviatoric components are separated, reasoning similar to that used in the case of integral representation leads to the equations (5, p. 78)... 

The decomposition in deviatoric and dilatational components of both the stress and strain tensors are... 

The normal component of the collisional deviatoric pressure tensor resembles the bulk viscosity term and is thus denoted by pB,d-... 

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... 

In a sense we treat the material as a gas dissolved in a continuum. The continuum part is chemically inert but responds in the classical way to the total stress field, both the mean stress and the deviatoric stress and their gradients it supports the whole of the nonhydrostatic stress components. The gas part has a completely different mobility coefficient, and the idea of its being affected by the deviation of the stress state from hydrostatic is rejected. 

To obtain more information, stress-whitened zones were prepared at Section B. For all the rubber-modified specimens, the size of the stress-whitened zone increased from zero at the outer surface to a maximum at the midsection (Section A). Figure 5 shows the whitened zone at Section B of specimen RF5. The whitened zone of RF series specimens can be three-dimensionally visualized, as shown in Figure 6. The shape of the whitened zone is in opposition to that of the plastic zone ahead of a crack tip in dense materials, in which the size of the plastic zone decreases to a minimum at the midsection because of the state of plane strain. At the outer surface there will always be plane stress, and hence the stress in the thickness direction, a, is zero at the surface. Concurrently, plane strain prevails in the interior, thus increasing the a in the interior. It can accordingly be seen that the maximum hydrostatic stress is found at the midsection (Section A). Thus, stress-whitening appears to be due to the hydrostatic stress components rather than the deviatoric stress components. 

Shock wave compression cannot only induce deformation in the form of high density of defects such as dislocations and twins but can also result in phase transition, structural changes and chemical reaction. These changes in the material are controlled by different components of stress, the mean stress and the deviatoric stress. The mean stress causes pressure-induced changes such as phase transformations while the deviators control the generation and motion of dislocations. 

Where eJ is the elastic volumetric strain and efj components of the elastic deviatoric strain. Kq and Po are respectively the initial bulk modulus and shear modulus of undamaged material. The scalar variable d characterises the isotropic damage. The... 

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

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