Strain-energy function finite strains

Finite Elasticity Theory The VL Representation. While the above description of the finite deformation behavior of elastic materials is very powerful, the limitation on it is that the material parameters W and W2 need to be determined in each geometry of deformation of interest. Hence, the torsional measurements described above only give values of Wi(/i, I2) and W2(/i, I2) for the condition of shear (torsion is anonhomogeneous shear) and that condition is/i = l2 = 3- -y. More measurements need to be made to obtain the parameters in extension, compression, etc. However, in 1967, Valanis and Landel (98) proposed a strain energy function that, rather than being a function of the invariants, is a function of the stretches Xj. The function was assumed to be separable in the stretches as... 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

The stored strain energy can also be determined for the general case of multiaxial stresses [1] and lattices of varying crystal structure and anisotropy. The latter could be important at interfaces where mode mixing can occur, or for fracture of rubber, where f/ is a function of the three stretch rations 1], A2 and A3, for example, via the Mooney-Rivlin equation, or suitable finite deformation strain energy functional. 

The stress relation obtained from an expansion of the internal energy function to fourth order in the finite strain t] takes the following form [79D01] ... 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

Here we begin with a sample of rabber having initial dimensions l, I2, I3. We deform it by an amount A/, A/2, A/3 and define the stretch (ratio) in each direction as A, = (/, -I- A/,)//, = ///,. The purpose of Finite Elasticity Theory has been to relate the deformations of the material to the stresses needed to obtain the deformation. This is done through the strain energy density function, which we will describe using the Valanis-Landel formalism as IT(A, A2, A3). Importantly, as we will see later, this is the mechanical contribution to the Helmholtz free energy. Vala-nis and Landel assumed [60] that the strain energy density function is a separable function of the stretches A, ... 

Finite Elasticity Theory Classical Theory. The finite elasticity theories available today are very powerful and well developed from a phenomenological perspective. Because the K-BKZ (70-72) has the form of a time-dependent finite elasticity (it was developed as a perfect elastic fluid ) it is useful to briefly outline the basics of finite elasticity theory here. In the initial sections of this article, the stress and strain tensors were discussed, and it was noted that the constitutive relationships that arise between the stress and the strain include material parameters called moduli. When a material is classified as hyperelastic then the moduli are related to derivatives of the free energy function (often the Helmholtz free... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

The finite deformation theory developed here is the result of a simple generalisation of linear elasticity. There are much more sophisticated theories available, some of which are motivated by physical arguments. A more secure basis for the development of finite strain theories is the stored energy or strain energy function, which will now be considered. 

The general and detailed constitutive relations of E.H. Lee s elastic-plastic theory at finite strain have been derived by Lubarda and Lee [5]. In this work, let the specid constitutive relations which are employed in the general purpose finite element program be listed as follows. First, the Helmholtz free energy density, E, as a function of the invariants of the elastic Cauchy-Green tensor, c/y, may be expressed as... 

The problem is solved using the calculus of variations. At vanishing strains, the minimum free energy deformation system for the crosslink network is found to be affine. However, at finite strains the free energy criterion of equilibrium shows the deformation system to be non-affine and a functional of the particular function chosen to describe the macromolecule s conformational behavior. Hon-Gaussian and Gaussian chain statistics are examined. 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

It is interesting to note that the diffusion equation contains the transpose of the velocity gradient tensor, but the solution is given in terms of one of the relative finite strain tensors. The tensor a plays an important role in the changes of the thermodynamic functions that occur when a polymer solution goes from a state of equilibrium to a state of flow. The changes in internal energy and entropy are ... 

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