On the modeling of the non-linear response of soft elastic bodies

• A new class of models is proposed to describe the response of elastic bodies.
• Models are developed wherein one has a non-linear relation between linearized strain and stress.
• One does not have to introduce a Lagrange multiplier to enforce the constraint for the models.
• The model complies with the requirements of causality that underlies Newtonian Mechanics.

In this short note we articulate the need for a new approach to develop constitutive models for the non-linear response of materials wherein one is interested in describing the Cauchy–Green stretch as a non-linear function of the Cauchy stress, with the relationship not in general being invertible. Such a material is neither Cauchy nor Green elastic. The new class of materials has several advantages over classical elastic bodies. When linearized under the assumption that the displacement gradient be small, the classical theory leads unerringly to the classical linearized model for elastic response, while the current theory would allow for the possibility that the linearized strain be a non-linear function of the stress. Such bodies also exhibit a very desirable property when viewed within the context of constraints. One does not need to introduce a Lagrange multiplier as is usually done in the classical approach to incompressibility and the models are also more suitable when considering nearly incompressible materials. The class of materials considered in this paper belongs to a new class of implicit elastic bodies introduced by Rajagopal [19] and [20]. We show how such a model can be used to interpret the data for an experiment on rubber by Penn [18].