This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

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The Poisson function is introduced to study in a simple tension test the lateral contractive response of compressible and incompressible, isotropic elastic materials in finite strain. The relation of the Poisson function to the classical Poisson’s ratio and its behavior for certain constrained materials are discussed. Some experimental results for several elastomers, including two natural rubber compounds of the same kind studied in earlier basic experiments by Rivlin and Saunders, are compared with the derived relations. A special class of compressible materials is also considered. It is proved that the only class of compressible hyperelastic materials whose response functions depend on only the third principal invariant of the deformation tensor is the class first introduced in experiments by Blatz and Ko. Poisson functions for the Blatz-Ko polyurethane elastomers are derived; and our experimental data are reviewed in relation to a volume constraint equation used in their experiments.

Dedicated to Professor Cornelius O. Horgan in esteem and friendship, with highest regard. Abstract:The response function for a general class of elastic molecular based materials characterized by their limiting molecular chain extensibility and depending on only the first principal invariant of the Cauchy-Green deformation tensor together with a certain molecular based limiting extensibility parameter is introduced. The constitutive response function for the Gent material is then derived inversely as the [0/1] Padé approximant of this class, a result that leads naturally to an infinite geometric series representation of its response function. Truncation of this series function characterizes a familiar class of quadratic materials now having physically relevant material constants. It is shown that the [0/2] approximant of the response function for the general class of restricted elastic materials leads inversely to a new constitutive model and its series representation. Of course, many familiar limited elastic material models are members of the general class. The Padé approximants for some response functions are not, and empirical modifications that admit these as members of the general class are described. Examples of two limited elastic models in the class that are not Padé approximants are noted. The strain energy functions for a few of the restricted elastic models described are presented.