×

zbMATH — the first resource for mathematics

Waves in constrained linear elastic materials. (English) Zbl 1171.74365
Summary: This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by A. Hoger and B. E. Johnson in [J. Elasticity 38, No. 1, 69–93 (1995; Zbl 0824.73007); ibid. 38, No. 1, 95–120 (1995; Zbl 0829.73002)]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant.

MSC:
74J30 Nonlinear waves in solid mechanics
74B99 Elastic materials
74J10 Bulk waves in solid mechanics
74B20 Nonlinear elasticity
PDF BibTeX XML Cite
Full Text: DOI