A model is examined for thermoelastic materials, such as those that display the shape memory and pseudoelastic effect. As is common with models for these materials, an internal variable is utilized which gives the phase fraction of austenite at the microstructural level within the continua. Evolution equations are developed that govern the time history of the system based on changes in stress, strain and temperature. Hysteresis is inherent in the system due to the Duhem- Madelung form of one of these equations. Attention is focused on the connection between complete transformation phenomena and arrested loading/unloading (outer loops vs. subloops); the transition from isothermal to adiabatic loading via loading in a heat convective environment; the identification of attracting states associated with both temperature cycling and periodic stressing; and the deter mination of mathematical restrictions on otherwise rather general constitutive entities entering the model so as to ensure well-posedness, proper qualitative behavior and admissible thermodynamic behavior.
We study the effect of swelling on the mechanical response of fiber reinforced tubes within the context of finite elastic deformation. The fibers themselves do not swell, setting up a competition between the matrix, for which swelling tends to open the tube, and the fibers, for which swelling tends to constrict the tube. Balancing these tendencies in the constitutive response can lead to an internal channel opening that remains relatively constant over a wide range of swelling. Further, the hoop stress on the inner wall in such a situation may be compressive, rather than tensile. Both effects may be advantageous in certain settings, including biological organ systems.
In this paper 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 at an angle with the radial direction that depends only on the radius. 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 (anticlockwise 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 stress-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 reinforced neo-Hookean material, 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. Absolutely 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 increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.
Swelling, generally referring to volumetric change and typically due to mass addition from some diffusive or transport mechanism, is central to a variety of physical phenomena. Here we consider the role of swelling as it relates to the inflation of hollow spheres and to cavity formation at the center of solid spheres. The swelling is modeled in terms of a prescribed scalar field that gives the local free volume. The finite deformation theory of incompressible hyperelasticity is generalized so as to include the effect of this swelling field directly in the stored energy density. The general framework is based on global energy minimization wherein the stored energy density is minimized at the locally prescribed swollen state. On this basis it is found that both inflation and cavitation can be caused solely by swelling. This result is intuitive with respect to inflation where it follows from a simple uniform swelling field. In contrast, to obtain swelling-induced cavitation we consider a non-uniform swelling field and study how this field can cause a cavity to nucleate, grow, shrink and disappear.
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