Homogenized out-of-plane shear response three-scale fiber-reinforced composites

In the present work we embrace a three scales asymptotic homogenization approach to investigate the effective behavior of hierarchical linear elastic composites reinforced by cylindrical, uniaxially aligned fibers and possessing a periodic structure at each hierarchical level of organization. We present our novel results assuming isotropy of the constituents and focusing on the effective out-of-plane shear modulus, which is computed exploiting the solution of the arising anti-plane problems. The latter are solved semi-analytically by means of complex variables and successfully benchmarked against the results obtained by finite elements. Our findings can pave the way for multiscale modeling of complex hierarchical materials (such as bone and tendons) at a negligible computational cost

Three scales asymptotic homogenization and its application to layered hierarchical hard tissues

In the present work a novel multiple scales asymptotic homogenization approach is proposed to study the effective properties of hierarchical composites with periodic structure at different length scales. The method is exemplified by solving a linear elastic problem for a composite material with layered hierarchical structure. We recover classical results of two-scale and reiterated homogenization as particular cases of our formulation. The analytical effective coefficients for two phase layered composites with two structural levels of hierarchy are also derived. The method is finally applied to investigate the effective mechanical properties of a single osteon, revealing its practical applicability in the context of biomechanical and engineering applications

The influence of anisotropic growth and geometry on the stress of solid tumors

Solid stresses can affect tumor patho-physiology in at least two ways: directly, by compressing cancer and stromal cells, and indirectly, by deforming blood and lymphatic vessels. In this work, we model the tumor mass as a growing hyperelastic material. We enforce a multiplicative decomposition of the deformation gradient to study the role of anisotropic tumor growth on the evolution and spatial distribution of stresses. Specifically, we exploit radial symmetry and analyze the response of circumferential and radial stresses to (a) degree of anisotropy, (b) geometry of the tumor mass (cylindrical versus spherical shape), and (c) different tumor types (in terms of mechanical properties). According to our results, both radial and circumferential stresses are compressive in the tumor inner regions, whereas circumferential stresses are tensile at the periphery. Furthermore, we show that the growth rate is inversely correlated with the stresses’ magnitudes. These qualitative trends are consistent with experimental results. Our findings therefore elucidate the role of anisotropic growth on the tumor stress state. The potential of stress-alleviation strategies working together with anticancer therapies can result in better treatments

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

Effective coefficients of isotropic complex dielectric composites in a hexagonal array

Based on the asymptotic homogenization method, the local problems related to two-phase periodic fibrous dielectric composites with isotropic and complex constituents are solved. A hexagonal periodicity distribution of the fibers is considered. Explicit formulas for the real and imaginary parts of the effective dielectric properties are derived. Such formulas can be computed for any desired precision related to a truncation order of an infinite system of algebraic linear equations. Two simple analytical expressions are specified for the first two truncation orders. Comparisons with results via other approachess how a good concordance. Hexagonal periodic lattices of acoustic scatterers are useful structures for acoustic applications

Computation of effective properties in two-phase piezocomposites with a rectangular periodic array

Based on the Asymptotic Homogenization Method, the electromechanical global behavior of a two-phase piezoelectric unidirectional periodic fibrous composite is investigated. The composite is made of homogeneous and linear transversely isotropic piezoelectric materials that belong to the symmetry crystal class 622. The cross-sections of the fibers are circular and are centered in a periodic array of rectangular cells. The composite state is anti-plane shear piezoelectric. Local problems that arise from the two-scale analysis using the Asymptotic Homogenization Method are solved by means of a complex variable, leading to an infinite system of algebraic linear equations. This infinite system is solved here using different truncation orders, allowing a numerical study of the effective properties. Some numerical examples are shown

Homogeneização assintótica da equação do calor para meios unidimensionais periódicos continuamente heterogêneos

O método de homogeneização assintótica consiste na transformação do problema com coeficientes rapidamente oscilantes de um meio heterogêneo (chamado problema original), em outro sobre um meio homogêneo, assintoticamente equivalente ao heterogêneo (chamado problema homogeneizado), mediante uma sequência recorrente de problemas que se inicia com o problema homogeneizado e cujas soluções são os coeficientes de uma série assintótica que aproxima a solução do problema original. Este método tem se mostrado uma importante ferramenta na modelagem e simulação de fenômenos físicos em meios heterogêneos. O presente trabalho descreve o processo formal da homogeneização na equação do calor, em meios continuamente heterogêneos e periódicos. Além disso, estabelece-se uma relação de proximidade entre a solução do problema original e a do problema homogeneizado, a qual é ilustrada mediante um exemplo implementado computacionalmente

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

Exactness of formal asymptotic solutions of a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods

In their usual form, homogenization methods produce first-order approximations of the exact solutions of problems for differential equations with rapidly oscillating coefficients which model the physical behavior of microstructured media. However, there is need of approximations containing higher-order terms when the usual first-order approximations, which are formed by superposing a macroscopic trend and a local perturbation, are not capable of reproducing the local details of the exact solutions. Here, two-scale asymptotic solutions with second-order terms are provided for a Dirichlet problem modeling the steady state of functionally-graded microperiodic nonlinear rods. The need of considering higherorder terms is illustrated through numerical examples for various power-law nonlinearities