Strain deviator tensor

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. 

Eij is the infinitesimal strain tensor is the strain deviator Cij and the volumetric strain e = ea Ui denotes the material displacement in j-th direction g = T - and 0 = 2 - X / are the increments of temperature and liquid content with respect to reference values... 

An arbitrary symmetric strain tensor has been thus subdivided into two additive components a spherical hydrostatic expansion (or contraction) tensor and a deviator tensor, characterizing shear deformations without any change in volume. 

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 ... 

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). 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

The most important coupling to deformations of the network is the one that is linear in both the strain of the network and the nematic order parameter. As has been discussed earlier in this section this leads to the consequence that the strain tensor can be used as an order parameter for the nematic-isotropic transition in nematic sidechain elastomers, just as the dielectric or the diamagnetic tensor are used as macroscopic order parameters to characterize this phase transition in low molecular weight materials. But it has also been stressed that nonlinear elastic effects as well as nonlinear coupling terms between the nematic order parameter and the strain tensor must be taken into account as soon as effects that are nonlinear in the nematic order parameter are studied [4, 25]. So far, no deviation from classical mean field behavior concerning the critical exponents has been detected in the static properties of this transition and correspondingly there are no reports as yet discussing static critical fluctuations. 

Obviously the plastic strains are parallel to the derivative of the yield function w.r.t. the deviator of the stress tensor. The coefficient dA characterizes the absolute value of deP and must be iteratively determined by means of the consistency condition, cf. (Ansys theoly ref., 2007). 

For small deformations, the terms eu are small and can be neglected. In this case, Green s strain tensor G approaches the strain tensor e. The deviation increases with increasing deformation. 

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... 

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. 

Stress is a concept similar to pressure. It measures how the total energy of a surface changes with the strain e, which is the relative deviation of the lattice from its equilibrium value. Each of the two lattice dimensions can be distorted in two independent lateral directions with the surface strain as a tensor e = ,j. The surface stress becomes n = nj- = l/A dE/dsij, where A is the area of the surface, E = Ay is the total surface energy, and y is the... 

---