Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

Abstract

In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.

Extension, inflation and torsion of a residually stressed circular cylindrical tube

“…For the specific forms of τ RR and τ ΘΘ given in (46) and (47) with R ∈ [A, B], the strong ellipticity inequalities…”

“…Again using the specific forms of τ RR and τ ΘΘ given in (46) and (47), restrictions on the parameters α * and η that guarantee strong ellipticity are obtained as…”

“…Here we have considered a specialization of the list of invariants given by Hoger [34] for a transversely isotropic material with residual stress, and also used in [46]. We emphasize that the above set of nine invariants, or an equivalent set of alternative invariants, forms a complete set of invariants of C and τ in three dimensions.…”

“…Residual stress is regarded as a stress that is in equilibrium in the absence of external loads (tractions or body forces), as in the definition found in [33]. A formulation of the constitutive law for a residually stressed transversely isotropic hyperelastic material in terms of invariants has been provided by Hoger [34], and this formulation (and its specialization to the case without a preferred direction associated with transverse isotropy) has been used as a basis for analyzing various wave propagation problems in initially stressed (as distinct from residually stressed) elastic materials in [35,36,37,38].…”

Classical plate buckling theory as the small‐thickness limit of three‐dimensional incremental elasticity

“…In contrast, here the pre-stress is induced by the underlying o(1) strain; the symmetry of the material in κ manifests itself only through the moduli via (100), whereas the pre-stress is delivered by the linearly elastic boundary-value problem. The importance of the distinction between initial stress and pre-stress is discussed further in [17].…”