Mario M. Attard

The authors provided a novel explicit expression for the dot derivative of an isotropic tensor function in the work by Kellermann et al. An alternate proof is provided in this addendum without the power series assumption.

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

A strain energy density function for rubber under finite strain presented in Attard & Hunt (2004) is described here. The compressibility component of the strain energy is a generalization of the Simo and Pister (1984) proposal while the incompressibility component is a generalized Mooney expression. The constitutive relationship for the second Piola Kirchhoff and Eulerian stress tensors are presented, as well as the associated constitutive relationships for the principal Lagrangian and Eulerian physical stresses. The equi-biaxial tension problem of Treloar (1944) is solved assuming various levels of compressibility.

A numerical strategy is developed and used to investigate the localized lateral buckling of circular pipelines under thermal loading and friction. The constitutive relations for circular pipelines are derived for thermal stresses and finite strain based on a hyperelastic constitutive model. The prebuckling lateral expansion and localized postbuckling deformation are investigated. A critical included angle for circular profiles is studied. Beyond the critical included angle, increasing the included angle of the pipeline or changing the boundary conditions does not influence the localized buckling behavior. Parametric studies are performed and the results are validated with ansys.