J. Merodio

Azimuthal shear of a transversely isotropic elastic solid

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.

The role of malignant tissue on the thermal distribution of cancerous breast

The present work focuses on the integration of analytical and numerical strategies to investigate the thermal distribution of cancerous breasts. Coupled stationary bioheat transfer equations are considered for the glandular and heterogeneous tumor regions, which are characterized by different thermophysical properties. The cross-section of the cancerous breast is identified by a homogeneous glandular tissue that surrounds the heterogeneous tumor tissue, which is assumed to be a two-phase periodic composite with non-overlapping circular inclusions and a square lattice distribution, wherein the constituents exhibit isotropic thermal conductivity behavior. Asymptotic periodic homogenization method is used to find the effective properties in the heterogeneous region. The tissue effective thermal conductivities are computed analytically and then used in the homogenized model, which is solved numerically. Results are compared with appropriate experimental data reported in the literature. In particular, the tissue scale temperature profile agrees with experimental observations. Moreover, as a novelty result we find that the tumor volume fraction in the heterogeneous zone influences the breast surface temperature.

MATHEMATICAL MODELING OF THE INTERPLAY BETWEEN STRESS AND ANISOTROPIC GROWTH OF AVASCULAR TUMORS

In this work, we propose a new mathematical framework for the study of the mutual interplay between anisotropic growth and stresses of an avascular tumor surrounded by an external medium. The mechanical response of the tumor is dictated by anisotropic growth, and reduces to that of an elastic, isotropic, and incompressible material when the latter is not taking place. Both proliferation and death of tumor cells are in turn assumed to depend on the stresses. We perform a parametric analysis in terms of key parameters representing growth anisotropy and the influence of stresses on tumor growth in order to determine how these effects affect tumor progression. We observe that tumor progression is enhanced when anisotropic growth is considered, and that mechanical stresses play a major role in limiting tumor growth.

A semi-analytical heterogeneous model for thermal analysis of cancerous breasts

In the present work coupled stationary bioheat transfer equations are considered. The cancerous breast is characterized by two areas of dissimilar thermal properties: the glandular and tumor tissues. The tumorous region is modeled as a two-phase composite where parallel periodic isotropic circular fibers are embedded in the glandular isotropic matrix. The periodic cell is assumed square. The local problem on the periodic cell and the homogenized equation are stated and solved. The temperature distribution of the cancerous breast is found through a numerical computation. A mathematical and computational model is integrated by FreeFem++.