Solutions of some boundary value problems for a new class of elastic bodies undergoing small strains. Comparison with the predictions of the classical theory of linearized elasticity: Part I. Problems with cylindrical symmetry

Abstract

There is considerable evidence that shows that for a large class of materials the relationship between the stress and the strain is nonlinear even in the range of strain that is considered small enough for the classical linearized theory of elasticity to be applicable (see Saito et al. in Science 300:464-467, 2003; Li et al. in Phys Rev Lett 98:105503, 2007; Talling et al. in Scr Mater 59:669-672, 2008; Withey et al. in Mater Sci Eng A 493:26-32, 2008; Zhang et al. in Scr Mater 60:733-736, 2009). A proper description of the experiments requires an alternative theory which when linearized would allow the possibility of such a nonlinear relationship between the stress and the strain. Recently, such a theory of elastic bodies has been put into place (see Rajagopal in Appl Math 48:279-319, 2003; Bustamante in Proc R Soc A 465:1377-1392, 2009; Rajagopal in Math Mech Solids 16:536-562, 2011). In this paper, we consider a special class of bodies that belong to the new generalization of response relations for elastic bodies that have a nonlinear relationship between the linearized strain and the stress. We use the special class of bodies that exhibit limited small strain to study two boundary value problems, the first concerning the telescopic shearing and inflation of a tube and the second being the extension, inflation and circumferential shearing of a tube. The results that we obtain for the models under consideration are markedly different from the predictions of the classical linearized elastic model with regard to the same boundary value problems.