Shear deviatoric stress

The last terms in each of Eqn. (5.2-9) and Eqn. (5.2-10) represent the divergence of the deviatoric stress including viscosity and pseudo-turbulence. The quantities /n L and uG are effective viscosities of each phase including bulk and shear viscosities. Fl and Fg represent the volume-averaged forces exerted on the liquid-and gas-phase (respectively) by the other phases across the common interfaces. 

Normal Stresses in Shear Flow. The tendency of polymer molecules to curl-up while they are being stretched in shear flow results in normal stresses in the fluid. For example, shear flows exhibit a deviatoric stress defined by... 

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. 

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,... 

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. 

This section incorporates the unpublished work of Palmer and Weaver subsequently the fatigue analysis was included as an integral part of the FMP Shaft Design Guide which Palmer and Weaver compiled. Results are quoted, for brevity the reader is referred to references dealing with the Distortion Energy Theory of Failure (also called deviatoric stress, octahedral, von Mises, or shear strain) for a complete analysis. 

The deviatoric stresses are thus related to the maximum shear stresses, a fact which is in accord with experimentally observed relatirMiships between shearing and plastic flow, thus lending support to this intuitive approach. On these grounds, the following type of yield criterion should be acceptable ... 

In the physically separate and distinct representation of shear and volumetric response we introduce the concepts of the deviatoric stress and the deviatoric strain which are free of volumetric response and represent only the shear response. These are defined as ... 

Figure 7.20(a) gives the simulation results of the plastic response of this particular structure in a tensile-flow experiment at a given constant strain rate at 0 K and 300 K. The response is given as a deviatoric shear resistance (stress) shear resistance r at 0 K, plotted as a function of the total deviatoric shear strain y, where, in a formal application of a Tresca connection between tensile and shear response, a in shear is taken as half of the tensile deviatoric plastic resistance and y is twice the total uniaxial deviatoric strain (McClintock and Argon 1966). The initial quenched-in level of tp for this alloy is... 

In cases for which a system-wide simple shear representation was desired for mechanistic interpretation this information was obtained by the well-known von Mises operations to give the system deviatoric stress and the deviatoric shear-... 

The computed trial elastic stress is relaxed by the plastic-corrector method, as described in Table 5.1. In Table 5.1, Qg is the hardening tensor, 2 is the inelastic multiplier, which shows the magnitude of inelastic deformation, and fi is the shear modulus. The second-order tensor hy defines the direction of the inelastic flow for example, in the case of the associated formulation it is hy = dif//dffy, where the yield surface is = 0. The deviatoric stress and back stress tensors are, respectively, identified by Sy = [Pg.195]

The result of a simple tension experiment for a metal is schematically shown in Fig. 2.17 with axes of axial stress ai and axial sdain 1 or deviatoric stress s and deviatoric strain e. In metals the volumetric plastic strain can generally be ignored (sP = 0) therefore we can treat the behavior as a uniaxial response. On the other hand, the shearing behavior of geomaterials is inevitably accompanied by volume changes that are plastic, therefore we have to modify the original flow theory developed for metallic materials (Kachanov 2005 Lubliner 1990). Note that in small strain plasticity we assume that the plastic increment de can be decomposed into incremental elastic and plastic components ... 

By rewriting the constitutive equations as in Eq. 9.4, each equation contains only one material property the deviatoric stress and strain are related by the shear modulus, G, while the dilatational stress and strain are related by the bulk modulus, K. 

A very valuable technique, well known in plasticity theory, is to split the stress into its hydrostatic and deviatoric components, associated respectively with volumetric and shear strain. On the basis that creep is caused by shearing of molecules past one another, we would expect creep to be only associated with deviatoric stress. This is constructed by subtracting the hydrostatic component from the stress tensor E to give the deviatoric stress E ... 

Solution, For this unconfined flow, we can unambiguously let P = 1 atm. We neglect such things as surface-tension forces on the sides of the rod and find that the deviatoric stress tensor has only one component, axial direction. This is the familiar tensile stress. It will obviously produce a tensile elongation, so there will be a rate of tensile elongation component of the rate-of-strain tensor kn. For an incompressible material (Poisson s ratio = 0.5), the lateral dimensions of the rod will contract to maintain the volume constant as the rod is extended, and the lateral contractile strains will be one-half the axial extension strain. Thus, — 2b22 — — 2 33. There is no shearing, so all the shear strains are zero. Therefore, the deviatoric stress and rate-of-strain tensors are... 

The net force due to shear stresses per unit volume, t is the deviatoric stress tensor, meaning that the pressure has been subtracted from the total stress tensor, so that the sum of the three diagonal elements is zero. This essentially leaves us with the shear stresses. 

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. 

Cd2 Cd3 coefficients of pressure drag, drag due to deviatoric normal stress, and drag due to shear stress... 

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