Elastic strain tensor

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

Here, V denotes specific volume, K denotes bulk modulus, subscripts P,V, S and T denote isobaric, isochoric, isentopic and isothermal conditions, respectively s is the second-rank strain tensor, and C is the fourth-rank elastic tensor. 

Elastic Tensor, Volume Fluctuations, and Isotropic Moduli. Rahman and Parrinello [83] showed that the fourth-rank elastic tensor for an anisotropic crystalline solid can be calculated using fluctuations of the microscopic strain tensor ... 

Surface stress — The surface area A of a solid electrode can be varied in two ways In a plastic deformation, such as cleavage, the number of surface atoms is changed, while in an elastic deformation, such as stretching, the number of surface atoms is constant. Therefore, the differential dUs of the internal surface energy, at constant entropy and composition, is given by dUs = ydAp + A m g m denm, where y is the interfacial tension, dAp is the change in area due to a plastic deformation, gnm is the surface stress, and enm the surface strain caused by an elastic deformation. Surface stress and strain are tensors, and the indices denote the directions of space. From this follows the generalized Lippmann equation for a solid electrode ... 

It is important to use the exact strain tensor definition, Eq. (6), to achieve rotational invariance with respect to lattice rotation the conventional linear strain tensor only provides differential rotational invariance of u in Eq. (7).hierarchy of approximations may be used for the elastic tensor 7. The most rigorous approach is to transform the bulk elastic tensor c according to... 

In formal rheology, relations between these three tensors are formulated and analyzed. Only for the two extremes of viscoelastic behaviour are such relations simple. For purely elastic materials there is a relation between the stress tensor and the strain tensor it contains the elasticity modulus and the Poisson ratio, accounting for the extent to which extension in one direction is accompamied by concomitant compression in the other two. For purely viscous fluids there is a relation between the stress tensor and the strain rate tensor. As extension in one direction is concomitant with (viscous) compression in the other two, in this case only one viscosity is required. For incompressible Newton fluids eventually an expression with only one viscosity results, see (1.6.1.131. 

The coupling of the mechanical field with the chemical field is realized as follows As there are bound charges present in the gel, a jump in the concentrations of the mobile ions at the interface between the gel and the solution is obtained. This difference in the concentrations leads to an osmotic pressure difference Ati between gel and solution. As a consequence of this pressure difference, the gel takes up solvent, which leads to a change of the swelling of the gel. This deformation is described by the prescribed strain e. This means that the mechanical stress is obtained by the product of the elasticity tensor C and the difference of the total (geometrical) strain e and the prescribed strain ... 

Marsh and Casabella induced elastic strain upon single crystals of NaCl and NaBr by plying static pressure (up to 6.9 MPa) and noted, as expected, that the C1 and Br NMR lineshapes broadened and became less intense. The authors determined that the purely ionic model of vK was inadequate to describe the changing field gradients at the chlorine and bromine nuclei with respect to changing pressure. It was concluded that ion orbital overlap between nearest and NNN atoms was a satisfactory model to rationalize their observations and that pure covalent effects did not need to be included. The effects of static elastic strain on chlorine SSNMR spectra were observed to determine the gradient-elastic tensors for LiCl and RbCl by Flackeloer and Kanert. ... 

Newton s law of motion for liquids describes a linear relationship between the deformation of a fluid and the corresponding stress, as indicated in Equation 22.16, where the constant of proportionality is the Newtonian viscosity of the fluid. The generalized Newtonian fluid (GNF) refers to a family of equations having the structure of Equation 22.16 but written in tensorial form, in which the term corresponding to viscosity can be written as a function of scalar invariants of the stress tensor (x) or the strain rate tensor (y). For the GNF, no elastic effects are taken into account [12, 33] ... 

Tensors for the small-strain elastic properties of materials... 

For both magnesite and calcite, the elastic bulk modulus Bq was computed straightforwardly by the Murnaghan interpolation formula, while of the elasticity tensor only the C33 component and the C + C 2 linear combination could be calculated in a simple way. The relations used are C = (l/Vo)c (d L /crystal structure. To derive other elastic constants, the symmetry must be lowered with a consequent need of complex calculations for structural relaxation. A detailed account of how to compute the Ml tensor of crystal elasticity by use of simple lattice strains and structure relaxation was given previously[10, 11]. For the present deformations only the c-o ( ) relaxation need be considered. The results are reported in Table 6, together with the corresponding values extrapolated to 0 K from experimental data (Table 2). For calcite, the mea-... 

The return mapping techniques in inelastic solutions are a natural consequence of splitting the total strain into elastic and inelastic strains. Let tensor uy, an incremental field to describe the deformation, and its gradient, Vt/,y, show the deformation rate. The solution is implemented by the following steps. Step 1 introduces a loading condition such as F = (/, 4- V ,t)Fj." where ly is the unity second-rank tensor and the superscripts n and n 4-1 represent, respectively, the previous and current load steps. In step 2 the material is elastically stretched... 

Deviatoric stress tensor Norm of deviatoric stress Lode s angle for stress Second invariant of deviatoric stress Third invariant of deviatoric stress Strain increment tensor Elastic strain increment tensor Plastic strain increment tensor Volumetric plastic strain increment tensor... 

Because the stress and the strain tensor contain only 6 independent components each, due to their symmetry, the elasticity tensor C needs only 6 = 36 independent parameters. 

The elasticity tensor is thus the derivative of the stress with respect to the strain. 

The considerations of the previous sections dealt with the two most extreme load cases If loaded in fibre direction, the stiffening effect of the fibres is maximal, if loaded perpendicularly, it is minimal. Under arbitrary loads, it is necessary to calculate the components of the elasticity tensor (see section 2.4.2). Depending on the fibre arrangement, couplings between normal and shear components can occur. For example, this can be exploited to construct aerofoils that twist on bending, with normal stresses causing shear strains. 

In elasticity theory (section 2.4.2), Hooke s law uses a double contraction between the elasticity tensor of order four and the strain tensor of order two ... 

Cartesian components of the 2nd Piola-Kirchhoff stress tensor Cartesian components of the Green-Lagrange strain tensor Components of the linear elasticity tensor Increment in the /th displacement component... 

The Cauchy stress tensor cr and Green Lagrange strain tensor Cgl are of second order and may be connected for a general anisotropic linear elastic material via a fourth-order tensor. The originally 81 constants of such an elasticity tensor reduce to 36 due to the symmetry of the stress and strain tensor, and may be represented by a square matrix of dimension six. Because of the potential property of elastic materials, such a matrix is symmetric and thus the number of independent components is further reduced to 21. For small displacements, the mechanical constitutive relation with the stiffness matrix C or with the compliance matrix S reads... 

The stress tensor has been introduced in Chapter 2. In small strain elasticity theory, the components of stress are defined by considering the equilibrium of an elemental cube within the body. When the strains are small, the dimensions of the body, and therefore the areas of the cube faces, are to a first approximation unaffected by the strain. It is then of no consequence whether the components of stress are defined with respect to the cube before deformation or the cube after deformation. For finite strains, however, this is not true and there are alternative definitions of stress depending on whether the deformed or undeformed state is chosen as a reference. We will choose to adopt the stress associated with the deformed state - the true stress or Cauchy stress - throughout this work. In our present axis notation, we can express this stress tensor X as... 

The condensed notation for the elements of the torsional elasticity tensor is normally used, and the torsional strain elements are written as a column vector with the compo-... 

For the moment, we shall not go into great detail about the compatibility conditions which the vectorial quantities must satisfy. In the above equations, the interface is considered to be a two-dimensional fluid medium. This consideration is by no means obligatory, though. We could also envisage interfaces with the behavior of an elastic surface, for example. This exploits the product pressure tensor by the strain rate tensor (see Chapter 3 of [PRU 12]). We retain the option of bringing these tensors into play in applications when it becomes necessary to do so. However, in the demonstrations given below, so as not to complicate the discussion, we shall suppose that the interface behaves like a fluid. 

Curly brackets represent an average over all possible orientations of term ( ). In Eq. (1), the terms Cj and C2 are the elastic stiffness tensors of the matrix material and the particle, respectively, is the particle volume fraction and A2 is the strain concentration tensor defined as ... 

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