A thermodynamically consistent constitutive equation for describing the response exhibited by several alloys and the study of a meaningful physical problem

• Published in: International journal of solids and structures Vol. 108; pp. 1 - 10
• Main Authors: Devendiran, V.K., Sandeep, R.K., Kannan, K., Rajagopal, K.R.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Ltd 01.03.2017; Elsevier BV
• Subjects: Alloying additive; Alloys; Constitutive equations; Constitutive relationships; Elasticity; Generalization of linearized elastic model; Gibbs potential; Implicit elastic material; Linearization; Materials elasticity; Mathematical analysis; Nonlinear elastic; Nonlinear response; Plate with a hole; Strain; Stress state; Nonlinear elastic; Implicit elastic material; Plate with a hole; Generalization of linearized elastic model; Gibbs potential
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2016.07.036

There are many alloys used in orthopaedic applications that are nonlinear in the elastic regime even when the strains are ‘small’ (see Hao et al., 2005; Saito et al., 2003; Sakaguch et al., 2004). By using conventional theories of elasticity, either Cauchy or Green elasticity, it is impossible to systematically arrive at constitutive equations, which would be applicable in the elastic domain of such metals as such materials exhibit non-linear response for small strains11The approximation is based on the displacement gradient being small which implies both the strain and the rotation are small. where the classical linearized response is supposed to hold in the sense that the norm of the squares of the displacement gradient are much smaller than the displacement gradient. We delineate a new framework for developing constitutive equations for a new class of elastic materials, termed as implicit elastic materials, which can be used to describe the response of such alloys. In addition to a fully implicit constitutive relation, we discuss a non-linear constitutive relation between the linearized strain and the stress that can be properly justified to describe the response of such alloys. By using the example of a rectangular plate with a hole subject to uniform loading, a classical problem, we illustrate the differences in the stress and strain fields when compared to that predicted by the classical linearized relation.