Elastic Stress-Strain Equations

To this point the relations between stress and strain (constitutive equations) for viscoelastic materials have been limited to one-dimension. To appreciate the procedure for the extension to three-dimensions recall the generalized Hooke s law for homogeneous and isotropic materials given by Eqs. 2.28, 

Using these stresses and strains, the elastic stress-strain relations given by Eq. 9.1 can be shown to be, 

By rewriting the constitutive equations as in Eq. 9.4, each equation contains only one material property the deviatoric stress and strain are related by the shear modulus, G, while the dilatational stress and strain are related by the bulk modulus, K. 

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Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

If the elastic modulus is obtained from the slope of the elastic stress-strain curve, then we can evaluate the first term on the right-hand side in Equation (8.3) from experimental data elastic stress-strain curves. The second term on the right-hand side in Equation (8.3) can be evaluated from the product of the strain rate, which is set in a constant strain-rate experiment, and the viscosity. As we discussed in Chapter 3, the viscosity of a macromolecule is related to the shape factor v, therefore we can evaluate the second term on the right-hand side of Equation (8.3) from the product of the shape factor and the strain rate. 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

Using identical methods one can write the stress-strain equation for an orthotropic linear elastic solid in terms of the principal values of stress and strain as... 

The four independent constants in the stress/strain equations are Ei, the modulus of elasticity in the fibre direction, E2, the modulus in the transverse direction, v, Poisson s ratio, and Gu, the in-plane shear modulus. The unidirectional lamina plays an essential role in structural engineering, for... 

For the soil with density p, an elastic stress-strain consitutive matrix D (Samuelsson and Wiberg, 1998) and an analogous stress-strain rate visco-elastic matrix standard FE procedures based on form functions N and their spatial derivatives B = V N and a volume integration over the element domain lead to the local matrices as defined in equation (4) ... 

The governing equations for a viscoelastic material are the same as those for an elastic material except all stresses, strains and displacements are time dependent and the stress-strain equations are the integral equations given by Eqs. 9.7 or 9.8. The dependent variables and t are explicitly included to emphasize that in multidimensional problems the stress and strain fields vary spatially in the material and that for viscoelasticity the fields are also time dependent. 

The statistical mechanical theory for rubber elasticity was first qualitatively formulated by Werner Kuhn, Eugene Guth and Herman Mark. The entropy-driven elasticity was explained on the basis of conformational states. The initial theory dealt only with single molecules, but later development by these pioneers and by other scientists formulated the theory also for polymer networks. The first stress—strain equation based on statistical mechanics was formulated by Eugene Guth and Hubert James in 1941. 

Since steel backup and hot copper plates are assumed to exhibit thermo-elastic and -elastic-plastic respectively, isotropic linear elastic stress-strain relation can be solved by constitutive equation (1) using finite element method. 

When a tensile or compressive test is carried out, the applied stress is usually controlled, and the strain is measured. Thus, it is more convenient to use the following equation instead of equation (4) for the elastic stress-strain relationship ... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

J7 In a tensile test on a plastic, the material is subjected to a constant strain rate of 10 s. If this material may have its behaviour modelled by a Maxwell element with the elastic component f = 20 GN/m and the viscous element t) = 1000 GNs/m, then derive an expression for the stress in the material at any instant. Plot the stress-strain curve which would be predicted by this equation for strains up to 0.1% and calculate the initial tangent modulus and 0.1% secant modulus from this graph. 

This equation may be utilised to give elastic properties, strains, curvatures, etc. It is much more general than the approach in the previous section and can accommodate bending as well as plane stresses. Its use is illustrated in the following Examples. 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

Designers of most structures specify material stresses and strains well within the pro-portional/elastic limit. Where required (with no or limited experience on a particular type product materialwise and/or process-wise) this practice builds in a margin of safety to accommodate the effects of improper material processing conditions and/or unforeseen loads and environmental factors. This practice also allows the designer to use design equations based on the assumptions of small deformation and purely elastic material behavior. Other properties derived from stress-strain data that are used include modulus of elasticity and tensile strength. 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Therefore, the rate at which chemical bonds break increases with elastic shear stressing of the material. The rupture of chemical bonds, hence fracture of material, leads to its fragmentation into particles. This reduces the average particle size in powder as fractured particles multiply into even smaller particles. Equation (1.24) points to the importance of elastic shear strains in mechanical activation of chemical bonds for particle size refinement and production of nanoparticles. 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

Figure 10.2. Mean elastic and viscous stress-strain curves for cartilage. Plot of elastic (A) and viscous (B) stress-strain curves for cartilage as a function of visual grade. The visual grade used was 1, shiny and smooth 2, slightly fibrillated 3, mildly fibrillated 4, fibrillated 5, very fibrillated and 6, fissured. The equation for the linear approximation for the stress-strain curve for each group is given, as well as the correlation coefficient. Note the decreased slope with increased severity of osteoarthritis. This data is consistent with down-regulation of mechanochemical transduction and tissue catabolism.

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