In this thesis we examine three boundary-value problems combined with the presence of dead-load tractions in respect of transersely isotropic elastic materials.In particular, Chapter 1 mainly consists of existing preliminary remarks on the continuum (phenomenological) approach used here to study the mechanical response of elastic materials under large strains. More specifically, we discuss, always within the continuous framework, basic kinematical concepts, fundamental stress principles as well as balance laws; those also being appropriately specialized for material bodies under the state of equilibrium, i.e. for static problems. Description of the governing constitutive theory for Cauchy elastic isotropic and transversely isotropic solids follows, with reference to which, the notion of a hyperelastic solid is then prescribed. Further, the necessary connections with the classical linear theory of transversely isotropic solids are generated and finally some typical constitutive inequalities are summaraized.Then, in Chapter 2, we examine the classical problem of finite bending of a rectangular block of elastic material into a sector of a circular cylindrical tube in respect of compressible transversely isotropic elastic materials. More specifically, we consider the possible existence of isochoric solutions. In contrast to the corresponding problem for isotropic materials, for which such solutions do not exist for a compressible material [38], we determine conditions on the form of the strain-energy function for which isochoric solutions are possible. Based on those conditions, some general forms of strain-energy functions that admit isochoric bending are derived. We also, for the considered geometry and deformation, examine aspects of stability predicated on the notion of strong ellipticity. Expressly, for plane strain, we provide necessary and sufficient conditions for strong ellipticity to hold. The material incorporated in the chapter has been accepted for publication in [42]. In Chapter 3 we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is disposed so as to preserve the cylindrical symmetry. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either 'with' or 'against' the preferred direction (anti-clockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear-strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a widely used reinforced neo-Hookean material (see e.g., [77, 63, 62, 47, 48]), we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absoulutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function. The work of this chapter has been accepted for publication and will appear in [41]. Also, parts of this work have already been presented in SES-Penn State (2006) by the third author.In Chapter 4 we are concerned with circular cylindrical tubes composed of incompressible transversely isotropic elastic material subject to simultaneous finite axial extension, inflation and torsion. Here, a great deal of attention is given to the actual kinematics of the problem. Due to the incompressibility constraint, three independent deformaions quantities associated with each one of the processes comprising the combined deformation are identified. These serve, in essence, to measure stretch in the axial and azimuthal direction of the body as well as the amount of shear in the planes normal to its radial direction and hence they suffice to fully characterize the resulting strain. Analogously to the azimuthal shear problem examined in the previous chapter, the preferred direction associated with the transverse isotropy is distributed in the planes normal to the tube axis and is disposed so as, in any case, to preserve the cylindrical symmetry. For the considered geometry, the material line elements in the preferred direction always contract when axial extension of the tube is applied. Assuming that the body is held fixed in that extended state, inflation of the tube may be responsible for either further contraction (at least in early stages of the process) or relaxation of the preferred direction. In this situation, the sense of shear is of no importance since the torsional aspect of the deformation has no actual impact on the length of line aliments in that direction. The cylindrical polar components of the Cauchy stress tensor are written down by means of a general form of strain-energy function and then a new universal relation applying for the considered geometry and deformation is generated. In the special situation where the preferred direction lies along, in the undeformed configuration, the radial direction of the body, coaxiality between the Cauchy stress and the left stretch tensors is accomplished and the latter constitutive relation, under appropriate specialization, recovers a well known result holding in the corresponding isotropic theory (see, e.g., [32]). Finally, based on the governing equilibrium equations and in conjunction with the kinematics of the problem, we provide general formulas for the applied loads necessary to support the combined deformation. These are found to apply for a wide range of transversely isotropic materials as well as for isotropic materials. Analogous remarks are briefly made with respect to a specific class of cylindrically orthotropic tubes.
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Boundary-value problems for transversely isotropic hyperelastic solids
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