Deviatoric stress state

The definition of a solid state reaction implies that the reaction product is a solid. If, for example, one of the reactants is a fluid, no deviatoric stresses are transmitted across the common interface. This situation simplifies the mechanical boundary condition significantly and explains why studies on boundary morphology are often performed with solid/fluid systems. 

Anthony Pearson The deviatoric stress is an important feature. I used the term stress there, but when one does these calculations on multiphase mixtures, suspension, emulsions—one is really looking not at the stresses initially, but one tends to be looking at rates of deformation. Although you get no Brownian motion, you do get very considerable structure development. My question is, is there any way in which thermodynamics can deal with structure development in a nonequilibrium state If you stop shearing the material, the structure disappears. 

Suspended spherical particles, each containing a permanently embedded dipole (e.g., magnetic), are unable to freely rotate (Brenner, 1984 Sellers and Brenner, 1989) in response to the shear and/or vorticity field that they are subjected to whenever a complementary external (e.g., magnetic) field acts on them. This hindered rotation results from the tendency of the dipole to align itself parallel to the external field because of the creation of a couple arising from any orientational misalignment between the directions of the dipole and external field. In accordance with Cauchy s moment-of-momentum equation for continua, these couples in turn give rise to an antisymmetric state of stress in the dipolar suspension, representable as the pseudovector Tx = — e Ta of the antisymmetric portion Ja — (T — Tf) of the deviatoric stress T = P + Ip. 

As expected, the matrix deviatoric stresses will be relaxed away completely. Thereafter, the fiber phase sustains the entire deviatoric stress. As a consequence, in the asymptotic state... 

Also shown on Fig. 1 is the important effect of the state of stress which may be characterized by the negative pressure CTq and the deviatoric stress Sq. These are given by... 

Note that when the state of stress is purely hydrostatic, the deviatoric stress will vanish (as will the shear stresses that arise from the differences in the principal stresses) and hence there will be no plastic flow. 

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. 

Fig. 11,4 (a) The reduction in saturation craze density with time of aging under a deviatoric shear stress = 15.86 MPa at 293 K. (b) The restoration of crazability in specimens that had previously been aged under s = 15.86 MPa for 10 s, and subsequently subjected to a standard stress state after having undergone recovery in an unstressed state for certain times of recovery (from Hannoosh (1975) courtesy of Massachusetts Institute of Technology). 

For a general state of stress and deformation at a material point, how are individual components of plastic strain rate related to stress components in this framework An answer is provided through the work of Rice (1970) on the general structure of stress-strain relations for time-dependent plastic deformation. In the present setting, it is most conveniently expressed in terms of deviatoric stress components Sij defined in terms of stress in... 

Dilatational and Deviatoric Stresses for a General State of Stress For... 

It can be shown that additional invariants exist for both ddatational and deviatoric stresses. For a derivation and description of these see Fung (1965) and Shames, et al. (1992). The invariants for the deviator state will be used briefly in Chapter 11 and are therefore given below. 

A typical set of results of such simulations for the system in uniaxial extension is shown in Fimre 1. In addition to the previously defined quantities, p = no, where n is the atom number density. At the density shown, p = 0.18, it is seen that Tjj is essentially zero for all x. This is in accord with the assumption of the classical molecular theory which states that the noncovalent excluded volume potential contributes only to the mean stress and not to the deviatoric stress. Nevertheless, it is seen that the noncovalent potential does make an indirect contribution to the deviatoric stress, since Tjj is different in the presence of UjjQ (i.e. when o = 0.8a) than when it is absent (i.e. when 0=0). Detailed examination of the molecular dynamics results show that these indirect noncovalent effects comprise changes in the mean bond force, , and the mean bond orientation, <3cos20-1>. 

The composite sphere, with core and shell inseparable at their common surface, is now allowed to swell to equilibrium in a solvent The final situation, as shown in Fig.2, may however be reached by a different route this is, by allowing the core and the shell to swell freely and separately, each achieving its equilibrium radius. The subsequent application of a centrally directed tension / between the surface of the core and the inner surface of the shell to draw them together results in the identical geometry. It is now possible, using the methods described by Eringen , to determine the states of stress in both core and shell. These will result in a (hydrostatic) tension in the core, and a hydrostatic compression, together with a complicated deviatoric stress picture in the shell. 

These criteria describe different states of local excitation and deformation of chain segments. The stress bias criterion [86, 139] refers implicitely to two mechanisms cavitation in a dilatational stress field and stabilization of cavities through a deviatoric stress component. These mechanisms have been more explicitely considered in the mathematical model of cavity expansion in a rigid plastic by Haward et al. [137] and in the molecular models by Argon [152] and Kausch [11]. 

The results indicate that specimens tested at high temperature show higher shear strength (Figure 2a). However, at large strains where (3 /3fj = /dfj = 0) the shear stresses obtained for high temperature tend to the same critical state as samples tested at ambient temperature (Ej is the deviatoric strain). No clear trend can be seen on the volume variation for samples sheared at different temperatures (Figure 2b). 

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