Stress deviator tensor

In the new coordinate system, one essential feature that was not visible in the principal coordinate system became revealed. Namely, an arbitrary stressed state can be viewed as the sum of two tensors the spherical tensor characterizing the equilateral extension or compression and the shear stress deviator tensor. Either of these tensors (or both of than) can be zero. 

In mathematical terms, the system of equations shows the divergence of the stress tensor. For a continuum, the complete dynamic formulation of the mechanical problem requires that the stress tensor be known. Rheology is the discipline of mechanics which deals with the determination of the stress tensor for a given material, whether fluid or solid. In Chapter 7, we introduce some concepts of rheology, or rather rheometiy. This discipline makes use of certain techniques (e.g. the use of rheometers) to determine the relationship that links the stress deviator tensor to the strain tensor or to the strain rate tensor, for a given material. This relationship is called constitutive equation. ... 

For Newtonian fluids the constitutive law is a linear relationship between the stress deviator tensor and the strain rate tensor ... 

Now, specific attention is given to the forrrth term, i.e. the rate of energy dissipation. For a Newtonian fluid, bringing the expression of the stress deviator tensor (Chapter 1, Table 1.1) in, the rate of energy dissipation, integrated over the volnme, can be written as ... 

The left-hand side of OEq. 23.134 is related to the second invariant, of the stress deviator tensor by the relation ... 

Here sy are the components of the stress deviator such that sy = ay — Syaukl, Sy are the components of the deviator of the back stress tensor Oy, ao is the initial switching (yield) strength of the material in tension or compression, and A is the as yet undetermined plastic multiplier. 

The viscous behaviour of bulk solids during rapid shear flow is known since Bagnold [ 15] 1954 and Savage [16] 1984. Hutter and Hwang have shown in [17] 1994, that the velocity-dependent behaviour of the general constitutive stress deviator can be derived from a rate-dependent functional, where the dynamic extension is represented by an additional term which includes the deformation tensor coupled with a viscous parameter. So, one can write both Eq.(3) and (4) in the following form by seperating T in a static part T, and a dynamic part T,. ... 

The stress and strain tensors are separated into their deviators fey, e,y) and volume e[I hydrostatic) tensors (Section 16). The stress deviator is written as... 

Concerning the deformability, especially because of its sensitive dependence on the state of stress, there is, typically, insufficient information available. This indicates the importance of the stress state while only the deviator-tensor is responsible for plastic deformation, fracture is determined by the whole stress state. The clearly ductile fracture is a consequence of the joining of concentrated dislocation masses to a continuos deficiency at the grain boundaries. This process is retarded by the component of hydrostatic compression of the stress state. Ductile fracture is determined by the crack propagation, which is a consequence of an... 

Maier [10] represented first the plastic strain to fracture against the ratio of the average of the three principal stresses and of the Misesian yield stress (see Equation 8). The importance of Ae Maier s stress parameter is that it is the ratio of two principal components of the stress state, i.e. it is the quotient of the hydrostatic sphere-tensor and the flow stress, which is connected to the second invariant of the deviator-tensor (see Equation 15). 

As it is well known, the stress tensor can be decomposed into two components, a deviator-tensor which describes the deviation from the hydrostatic sphere tensor and a hydrostatic sphere tensor ... 

The stress deviator, which expresses the difference from the hydrostatic stress state (see Equation 16), causes plastic deformation, while the sphere tensor causes only elastic deformation. 

The stress deviator is responsible for the plastic deformation, but the deformability is governed by the hydrostatic sphere tensor. In other words, the plastic deformation of the near surface layer of the first body is caused by the deviator of the local stress state, but the particle detachment, which is a consequence of the ductile fracture of the near surface layer, is determined by the whole stress state. For the analysis of the behaviour of the near surface layer, we have to take the dependence of the deformability on the actual stress state into consideration too, beside the dependence of the plasticity on the temperature and the strain rate. 

In the case K > fi, the usual diffusion determines the kinetics for any gel shapes. Here the deviation of the stress tensor is nearly equal to — K(V u)8ij since the shear stress is small, so that V u should be held at a constant at the boundary from the zero osmotic pressure condition. Because -u obeys the diffusion equation (4.18), the problem is trivially reduced to that of heat conduction under a constant boundary temperature. The slowest relaxation rate fi0 is hence n2D/R2 for spheres with radius R, 6D/R2 for cylinders with radius R (see the sentences below Eq. (6.49)), and n2D/L2 for disks with thickness L. However, in the case K < [i, the process is more intriguing, where the macroscopic critical mode slows down as exp(- Q0t) with Q0 oc K. 

In this equation is the deviator and a is the spherical part of the stress tensor <7, eij is the strain deviator and e the volumetric part of the strain tensor ij, K = (2M + 3A) /3 is bulk modulus with M and A corresponding to the familiar Lame coefficients in the theory of elasticity, while r) and n can be termed the viscous shear and bulk moduli. 

Scattering or form birefringence contributions will cause a deviation in the stress optical rule. As seen in equation (7.36), these effects do not depend on the second-moment tensor, but increase linearly with chain extension. 

In this situation, which is also discussed in Section 7.5, we refer to experimental evidence according to which components of the relative permittivity tensor are strongly related to components of the stress tensor. It is usually stated (Doi and Edwards 1986) that the stress-optical law, that is proportionality between the tensor of relative permittivity and the stress tensor, is valid for an entangled polymer system, though one can see (for example, in some plots of the paper by Kannon and Kornfield (1994)) deviations from the stress-optical law in the region of very low frequencies for some samples. In linear approximation for the region of low frequencies, one can write the following relation... 

The elements present in the deviator of the stress tensor cause the stretching... 

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