A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Appli
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O R I G I NA L PA P E R
M. H. B. M. Shariff · R. Bustamante
A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Applications to rock mechanics
Received: 30 April 2020 / Revised: 11 July 2020 / Accepted: 9 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract A constitutive equation is provided for nonlinear transversely isotropic elastic solids, wherein the linearized strain tensor is assumed to be a function of the Cauchy stress tensor; this elastic constitutive equation belongs to a subclass of a more general set of implicit constitutive relations proposed in the recent years. The proposed constitutive equation is valid for both compressible and incompressible bodies and can be simply modified, to exclude the mechanical influence of compressed fibres and to model inextensible fibres. A crude specific constitutive model is proposed to compare with a uniaxial experimental data on Marcellus shale. Some simple boundary value problems are analyzed.
1 Introduction In recent years, some new constitutive equations and relations have been proposed for elastic [1–5] and inelastic bodies [6,7], where in general it is not assumed that the stresses are functions of the strains. One of such relatively new types of constitutive equation corresponds to the class wherein we have that the linearized strain tensor ε (infinitesimal strain tensor) is given as a function of the Cauchy stress tensor T , i.e. ε = h(T ) (see, for example, [8–12]). A subclass of the above equation is when we assume that there exists a scalar potential = (T ) ∂ such that (see [10]) h = ∂ ∂ T . The above constitutive equation ε = ∂ T has many potential applications, where we can observe a nonlinear behaviour for a solid when the strains and rotations are very small. Applications can be found, for example, in the mathematical modelling of the mechanical behaviour of concrete [13], some metal alloys [14,15], rock [16], fracture mechanics [17,18], and in the study of fibre-reinforced bodies [19]. With the exception of [19], in most of the works published on this type of constitutive equation so far, only isotropic bodies have been studied (see, for example, [20]). But there are many potential applications for ε = ∂ ∂ T , where is a potential function for a transversely isotropic solid. For example, in [19] a model of inextensible bodies in a preferred direction was studied, for a matrix filled with a family of fibres, where it was very easy to model inextensibility in the direction of the fibres. In the case of the modelling of the mechanical behaviour of rocks, from the different literature available (see the short review at the beginning of Sect. 5), it is well known that many types of rocks are anisotropic, such as some metamorphic rocks like shale, schist, slates and gneiss, and sedimentary rocks such as sandstone and shales. Anisotropy of rocks can also appear as a product of the presence of micro-cracks, cracks, joints, bedding, and stratification1 . 1
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