Stress deviatoric

From (A.81), /3T, = k, and this equation implies that the yield surface in stress space is a circular cylinder of radius k, shown in a FI plane projection in Fig. 5.7(a). The corresponding yield surface in strain space may be obtained by inserting the deviatoric stress relation (5.86) into the yield function (5.92)... 

Following usual conventions, repeated indices indicate summation and fy denotes df/dXj. The permutation S5mibol is used to present the vector cross product in indicial notation. Due to the anisotropic nature, traction and body couples can exist, and thus the angular momentum equation must be considered. For purely viscous fluids this equation says simply that the deviatoric stresses are symmetric. 

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. 

If we assume that the body force is only due to gravity and use the definition r = —prL +1 where pr is the isotropic resin pressure, / is the unit tensor, and r is the deviatoric stress as well as assuming a constant density and define a new pressure Pr —pr + prgh, Equation 5.21 simplifies to,... 

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... 

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. 

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. 

When all the SE s of a solid with non-hydrostatic (deviatoric) stresses are immobile, no chemical potential of the solid exists, although transport between differently stressed surfaces takes place provided external transport paths are available. Attention should be given to crystals with immobile SE s which contain an (equilibrium) network of mobile dislocations. In these crystals, no bulk diffusion takes place although there may be gradients of the chemical free energy density and, in multicomponent systems, composition gradients (e.g., Cottrell atmospheres [A.H. Cottrell (1953)]). 

Non-Newtonian Viscosity. A Newtonian fluid is one where the deviatoric stresses that occur during deformation, t, are directly proportional to the rate of deformation tensor, 7,... 

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... 

Figure 5.5 Effect of deviatoric stresses as the fluid element travels along its streamline.

In fluid flow, however, it is necessary to split the total stress, hydrostatic stress, aH. The deviatoric stress is the one that leads to deformation (Fig. 5.5) and the hydrostatic stress is the one that is described by pressure (Fig. 5.6). 

Table 5.2 presents the momentum balance in terms of deviatoric stress in the Cartesian, cylindrical and spherical coordinate systems. 

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... 

Because of the large ratio K/G, most studies of rubber elastic behavior are restricted to constant volume deformations, i.e., to Dey, and to the resulting deviatoric stress Dty. Although this restriction is widely made, it is often tacit. Because of its importance, we make it explicit here because it has important consequences for the current discussion. 

In the usual development of the theory, the important assumption was made that the nonbonded interactions, although certainly present, contributed only to the mean stress p and made zero contribution to the deviatoric stress Dty. Because as noted, the earlier restricted theories of rubber elasticity were... 

The initial series of simulations was performed for 0 < p < 1. Because the purpose of these simulations was to test the common assumption that the nonbonded interactions did not contribute to the deviatoric stress, the sum in Eq. (6) was separated as in Eq. (9) to show the bonded and nonbonded interactions explicitly. The first sum is over all atom pairs that interact through u/, the second is over all pairs that interact through unh. 

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. 

The significant nonbonded contribution to deviatoric stress in the melt is made by the strong, short-range, repulsive excluded volume interaction. 

The deviatoric stress contribution made by a generic atom is caused by the anisotropy of the atomic distribution in its neighborhood. As the anisotropy is localized, the attractive long-range interaction has only a minor effect. 

We next consider the contributions to the deviatoric stresses Dty, which are made by the chain bonds by the spring potential Ub(r), in the MD simulation of our atomic model corresponding to a network (not a melt) under extension X = 2 in the x direction. The average force / in a covalent bond, modeled by Eq. (4), is... 

In Sections IIB through IID, we presented important characteristics of the nonbonded interactions that contribute to the deviatoric stress. In Section II.C, it was observed that the short-range repulsive EV portion of unb made the dominant contribution, whereas the attractive portion of unb was of negligible importance. Furthermore, the contribution cry (/ ) to the deviatoric stress that is made by nonbonded interactions with atom [> depends on the steric shielding of that atom by its neighbors. 

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