Tensor of deformation

In order to reproduce the temperature variation of the lattice constants, the anisotropy of the lattice expansion has to be taken into account. For this purpose, the tensor of thermal expansion ot is introduced instead of the scalar a , and the tensor of deformation due to the HS <- LS transition is employed instead of the dilation (Fh — Fl)/Fl. Each lattice vector x T) can now be... 

It is convenient to introduce, following to Murnaghan (1954), the tensor of deformation Aik = ASiXsk- The latter is useful to write down a free energy function (1.41) of a deformed network as... 

In the simplest cases it can be reduced to the tensor of deformation of macro-molecular coils... 

Free Energy and Stress Tensor of Deformed Body... 

Let us consider the spatially periodic porous medium consisting of an infinite number of identical unit cells. The spatially periodic medium is subjected to a macroscopic deformation described by the tensor of deformation A, and the local displacement d — A x + d can be decomposed into a macroscopic deformation A x and a microscopic spatially periodic displacement d cf. Poulet et al. (1996). This decomposition introduced into elastostatic Equations (29) and (30) yields... 

Since in this case the only component of the tensor of deformation is Jxz = the general stress-strain relationship is given by Eqs. (5.73) as... 

Here p is the pressure in liquid, v is the kinematic viscosity, p is the liquid density, j is the mass flux on the liquid surface, a is the surface tension, g is gravity acceleration, k is the curvature of the liquid surface, ejj are the components of the tensor of deformation rates, np xp tj are the components of the normal, binormal and tangential vectors on liquid surface. For a constant wall temperature we can write j=A,AT/(ro6), where A, is the thermal conductivity, kv is the latent heat of evaporation, Ts is the vapor temperature and AT=Tw-Ts. 

Then for the free energy we have a dependence F =F(J, 0.p ), where Jj - is the second invariant of the tensor of deformations, then... 

Glassy and hyperelastic states of tightly sewn polymers are determined by values of operator - N,a 0 and 1, respectively, and a transition region between these states by values in interval of 0-1. This make it possible to describe Equations (1)(4) tensor of deformation and phability of dense polymer meshes in all physical states. So, in hyperelastic state cooperative (X -transition is momentary and relaxation operator of... 

The tensor of stresses (Xy is related to the tensor of deformations, to lowest order, by Hooke s law. The relevant elastic constants are the Young modulus Y and the Poisson s coefficient P. 

Theoretically the apparent viscosity of generalized Newtonian fluids can be found using a simple shear flow (i.e. steady state, one-dimensional, constant shear stress). The rate of deformation tensor in a simple shear flow is given as... 

The rate of deformation tensor in a pure elongatioiial flow has the following form... 

The elongation viscosity defined by Equation (1.19) represents a uni-axial extension. Elongational flows based on biaxial extensions can also be considered. In an equi-biaxial extension the rate of deformation tensor is defined as... 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Ec[uation (4.91) is written using the components of the rate of deformation tensor D as ... 

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

C - INITIALIZE TEMPERATURE SECOND IWARIANT OF RATE OF DEFORMATION TENSOR... 

Rate of Deformation Tensor For general three-dimensional flows, where all three velocity components may be important and may vaiy in all three coordinate directions, the concept of deformation previously introduced must be generahzed. The rate of deformation tensor Dy has nine components. In Cartesian coordinates. 

Vorticity The relative motion between two points in a fluid can be decomposed into three components rotation, dilatation, and deformation. The rate of deformation tensor has been defined. Dilatation refers to the volumetric expansion or compression of the fluid, and vanishes for incompressible flow. Rotation is described bv a tensor (Oy = dvj/dxj — dvj/dxi. The vector of vorticity given by one-half the... 

Table 20. Tensor elements referred to the principal axes of the tensor of thermal expansion 0 and the deformation tensor E due to the HS<->LS transition in [Fe(2-pic)3]Cl2 CH3OH according to Ref. [39]...

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