Linear elasticity for constrained materials: General theory for hyperelasticity
The authors' derivation of the linear constitutive equations for linear hyperelastic constrained materials is obtained by formal linearization of the appropriate finite theory of hyperelasticity. In this theory the Cauchy stress tensor is the sum of the reaction stress, representing the stress generated by the forces that act against the constraints, and the determinate stress, which depends on the gradient of strain energy function for constrained material. This derivation is afterwards compared in Sect. 5 with the so-called ``classical approach'' which results in the ``classical linear constitutive equations'' for constrained hyperelastic materials [see for example, \textit{A. J. M. Spencer}, Continuum theory of the mechanics of fibre-reinforced composites, CISM Courses Lect. 282, 1- 32 (1984; Zbl 0588.73117)]. A general survey of standard results regarding the constitutive equations for hyperelastic materials, material symmetry, and constitutive equations of a constrained hyperelastic material is made in Sect. 2 of the paper. Sect. 3 is devoted to the derivation of the linear constitutive equations for a constrained material. The general constitutive equations derived in the previous sections are made specific for the constraints of incompressibility and inextensibility in Sect. 4. A summary of the classical procedure and a comparison of resulting constitutive equations with those derived in previous sections are presented in the final Sect. 6. The authors point out that the classical constitutive laws omit terms that must be included in order to have accuracy to the first order in the strain, as is usually expected for a linear elastic material. Such a problem was discussed in an earlier authors' paper [J. Elasticity 38, 69- 93 (1995)].
scientific article; zbMATH DE number 753727