IJPAM: Volume 91, No. 4 (2014)

ON THE THREE STRESS TENSORS FOR
LINEARLY ELASTIC CONSTRAINED MATERIALS

Maria Luisa Tonon
Department of Mathematics ``Giuseppe Peano''
University of Turin
Via Carlo Alberto 10, Turin, 10123, ITALY


Abstract. In this paper we obtain the constitutive equation for the second Piola-Kirchhoff stress tensor according to the linearized finite theory of elasticity for hyperelastic constrained materials. We show that in such a theory the three stress tensors (Cauchy stress tensor, first and second Piola-Kirchhoff stress tensor) differ by terms that are first order in the strain, while in classical linear theory of elasticity they are indistinguishable to first order of approximation both for unconstrained and constrained materials. Moreover we show that the constitutive equations for the three stress tensors usually adopted in classical linear elasticity are not correct to first order in the strain. Finally we provide an example for a particular material symmetry and for a particular constraint in which the three stress tensors coincide, while in general they are different.

Received: October 30, 2013

AMS Subject Classification: 74B99, 74A10

Key Words and Phrases: hyperelastic constrained materials, linearized finite elasticity, stress tensors

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DOI: 10.12732/ijpam.v91i4.3 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 91
Issue: 4