Thermo-mechanics of rubberlike materials

Abstract

A theoretical study is made in this paper of the formulation of constitutive equations describing the thermo-mechanical response of solid polymers in the temperature range in which they exhibit rubberlike behaviour. An expression for the Helmholtz free energy of such a material is first constructed on the basis of two assumptions which are motivated by physical arguments concerning the relation of the molecular structure of a cross-linked polymer to its bulk response. The constitutive equations for the stress and the entropy generated by the proposed form of the free energy function are then employed m a detailed investigation of the extension of a cylindrical specimen under prescribed conditions of temperature and pressure, a situation which serves as a model of the experimental arrangement most frequently used in laboratory studies of the mechanical and thermo-mechanical properties of rubberlike solids. Qualitative consistency of the theoretical predictions with observed behaviour is shown to be assured, over the full range of circumstances for which measurements have been reported, by two simple inequalities affecting one of the three response functions appearing in the stress-deformation-temperature relations. The function concerned is closely related to the strain energy governing isothermal deformations of the material at a selected reference temperature and it is associated, by the physical considerations referred to above, with the contribution to the stress of the polymer network. This conclusion shows that a rational macroscopic theory of rubberlike thermoelasticity can be developed in rather general terms. In particular, the requirement that the constitutive equations shall reproduce the anomalous thermo-mechanical effects which are characteristic of solid elastomers imposes restrictions on the response functions no more severe than those which ensure that the purely mechanical behaviour of the material is physically realistic. In the remainder of the paper the capability of the basic theory for furnishing results quantitatively agreeing with experiment is examined. Empirical forms of the three response functions are presented which accurately represent measurements made in tests involving compression at different fixed temperatures and stretching at the reference temperature. Numerical calculations relating to the analysis of the extension of a cylinder, given earlier, are then described and compared with the results of experiments in which thermoelastic inversion phenomena occur. Satisfactory agreement is secured, but it is noted that insufficiency of material data for the rubbers used in the tests precludes an exact correlation of theory and experiment. The final section of the paper is concerned with isothermal deformations of rubberlike materials which are mechanically incompressible (in the sense that volume changes can be brought about by thermal expansion but not by loading at fixed temperature). This property closely approximates the typical behaviour of natural and synthetic rubbers, but its incorporation into a general treatment of rubberlike thermoelasticity presents difficulties and places an undesirable limitation on the scope of the theory. An analogue is shown to exist between the constitutive equations for deformations at the reference temperature and their counterparts in respect of isothermal deformations at other temperatures, and with its aid the problem of the combined extension, torsion and uniform heating of a circular cylinder is solved. Again, a numerical evaluation of the solution is compared with available experimental data.