A micro-macro approach for modeling of rubber elasticity

The micro-structure of elastomeric materials consists of chains of polymers which through cross-linking and entanglements form a network structure. Starting from this micro-structure, a model is developed to describe the striking elastic response of this class of materials. The individual links of the network are modeled as distinct rod-like micro-elements. The deformations of these elements are related to the macro-deformation field through the assumption of affme deformations. Individual micro-elements arek allowed to offer resistance to both axial deformations and reorienting rotations. The elastic energy potential of the macro material region is then calculated through integration over the contribution from each micro element. The resulting equations are specialized to simple deformation configurations of uniaxial extension and pure shear, and the resulting stress response is calculated. The question of what is the most suitable form for the free energy function of the elastomeric materials which is capable of describing the elastic response of the material under varied loading conditions is still not answered satisfactorily. The proposed expressions for the calculation of the free energy fall mainly into two categories: (1) the phenomenological approach and (2) the network approach. In the phenomenological approach, a mathematical form for the strain energy function is hypothesized based on the particular behavior that the material exhibits in simple loading situations. Of course, this mathematical function is so chosen as to not violate the principles of continuum mechanics. The more common theories of MooneyRivlin, Ogden. Valanis-Landel, Peng-Landel, etc., belong to this category. In the second approach, it is acknowledged that the microstructure of a polymeric solid consists of a network of large molecular chains extending from Copywrite O 1994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. points of cross links and entanglements, and in interaction with the filler and reinforcement particles in their immediate environment. It is stipulated that the overall load deflection of the material system is a cumulative contribution from the characteristic force deflection behavior of these individual chains, which can be derived from a presumed model for the chain. In this paper we propose a formulation for the free energy function which is based on a microstructural approach. One of the oldest models of a single polymeric chain is the freely jointed model. The force deflection response of this model can be derived from the statistical treatment of available entropy states based on Gaussian probabilities. At high stretches where limited chain extensibility is encountered Gaussian probabilities are not applicable and non-Gaussian treatment is necessitated. The inverse Langevin probabilities have been used to account for this. The force deflection response that results from the Gaussian statistics is analogous to that of a linear mechanical spring with an unextended length of zero. The extension of Gaussian chain statistics to a network in which each segment of a molecule between successive cross-links is considered to be a Gaussian chain results in a simple form of the elastic energy function which is equivalent to the neo-Hookean form. James & ~ u t h ' proved that the Gaussian network is equivalent to a three chain model where each chain is oriented along one of the three perpendicular directions of a Cartesian system. James & Guth also developed a non-Gaussian network theory in the restricted case of unidirectional loading based on the same three chain cubic topology where the chain deformations were described by inverse Langevin probabilities, first developed by Kuhn & ~ r i i n ~ . The work of Wang & ~ u t h ~ completed the formulation of the three chain inverse Langevin model for the general homogenous strain state. A four chain model where the individual chains extend from the center to the four comers of a tetrahedron elementary cell were proposed by Flory & ~ e h n e r ~ for Gaussian chains, and was extended to non-Gaussian chains by ~ r e l o d . More recently, Arruda & ~ o ~ c e ~ have proposed a model where the elementary cell is a cube and contains eight inverse Langevin chains which extend from the center to the comers of the cube. The simplified geometry of the elementary unit cells of the above models does influence the mechanical response. For example, in the three chain model loading in one direction has no effect on the response in the other directions. The four chain model exhibits anisotropic properties which is inherent to its topology. To alleviate this difficulty ~re loar~ proposed an assembly of chains which in the unstrained state corresponds to a random angular distribution in space. Then the overall behavior of the model can be obtained through an integration over all of the chains. This idea is exploited by Treloar & Riding8 and the response of a total assembly of chains with inverse Langevin distributions is calculated. We note that all of the above network chain models can be regarded as special cases of this total assembly if the distribution function consists of delta functions at particular orientations. Closely related to the model of Treloar & Riding is the theory proposed by ~ s i a o ~ . His model, which is formulated for ply-crystalline materials and in the range of small strains and linear elasticity, consists of a collection of rod-like elements which are only capable of axial deformation. For a given macro deformation field, the strain in each element is obtained through an affine transformation, and the macro-stress field is calculated through integration over the resisting forces which develop in each element. Any changes in the orientation of these elements due to the deformation field is unresisted and is purely a rigid body motion. If the behavior of the rod-like elements is considered to be linear elastic, then a theory of linear elasticity with only one constant parameter is obtained. It can be shown that this theory is equivalent to continuum linear elasticity where the Poisson's ratio is set equal to 0.25. As correctly noted by Treloar & idi in^', in the simple extension situation any fom of the non-Gaussian network theory for rubber elasticity approximates to the Gaussian in the range of approximately l c k 3 . There is abundant evidence that the Gaussian theory does not describe the experimental results in this range well. Therefore, close agreement with experiment cannot be expected for the non-Gaussian theories over the whole range of extension. The utility of non-Gaussian theories lie in their ability to describe the general form of the experimental force-extension curve, at least approximately, over the whole range of strain, including in particular the region of very high strains, where the Gaussian theory becomes quite inadequate. In our present formulation we have strived to avoid the problems which are associated with the elementary cell models, and to achieve a model with as few parameters as possible. We have primarily been concerned with improving the theoretical fit to the experimental data in the lower strain ranges which are most typical of service conditions of actual elastomeric materials. The thermodynamic state of a material region is uniquely determined through the definition of strain, g , temperature T, and a set of i internal or "hidden" variables, cxi, which are not necessarily observable. Let the free energy function for the material region be denoted by Y = Y (5, T, cxi) . Then, through an analysis of the requirements for the second law, it follows that mechanical stress is given by with stress and strain as appropriately defined work conjugate pairs. If the free energy function is exclusively dependent on strain, then we are dealing with an elastic system. In this case the free energy is the same as the strain energy density, i.e., and equation (1) is the hyperelastic formulation of the stress-strain response. We set out to develop an expression for W for an elastic network based on the kinetics of individual chains in the network. Our point of departure consists of an idealization of a network of polymer chains by a collection of rod like micro elements that extend from end to end of each chain. There exist N number of such elements per unit volume. We note that as a result of the incompressibility constraint, which we shall invoke, N retains the same value in all configurations. As a first approximation we shall assume that before any deformation is imposed the micro elements are all of the same length, which we denote by ro. We shall conveniently utilize a spherical coordinate system (p, $, 8) to represent the orientation of each vector in space. Thus, referring to Fig. 1, a single element in the undeformed state is identified with coordinates (ro, $0, 80). Fig. 1: Orientation of the rodlike element representing a single chain in the principal space Since we are going to allow for the possibility of oriented polymeric networks in our model, we shall denote the density of the orientation distribution function for the chains in the undeformed reference configuration by p(@,, go), such that