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In the present work, we apply the asymptotic homogenization technique to the equations describing the dynamics of a heterogeneous material with evolving micro-structure, thereby obtaining a set of upscaled, effective equations. We consider the case in which the heterogeneous body comprises two hyperelastic materials and we assume that the evolution of their micro-structure occurs through the development of plastic-like distortions, the latter ones being accounted for by means of the Bilby–Kröner–Lee (BKL) decomposition. The asymptotic homogenization approach is applied simultaneously to the linear momentum balance law of the body and to the evolution law for the plastic-like distortions. Such evolution law models a stress-driven production of inelastic distortions, and stems from phenomenological observations done on cellular aggregates. The whole study is also framed within the limit of small elastic distortions, and provides a robust framework that can be readily generalized to growth and remodeling of nonlinear composites. Finally, we complete our theoretical model by performing numerical simulations

Effective properties of hierarchical fiber-reinforced composites via a three-scale asymptotic homogenization approach

The study of the properties of multiscale composites is of great interest in engineering and biology. Particularly, hierarchical composite structures can be found in nature and in engineering. During the past decades, the multiscale asymptotic homogenization technique has shown its potential in the description of such composites by taking advantage of their characteristics at the smaller scales, ciphered in the so-called effective coefficients. Here, we extend previous works by studying the in-plane and out-of-plane effective properties of hierarchical linear elastic solid composites via a three-scale asymptotic homogenization technique. In particular, the approach is adjusted for a multiscale composite with a square-symmetric arrangement of uniaxially aligned cylindrical fibers, and the formulae for computing its effective properties are provided. Finally, we show the potential of the proposed asymptotic homogenization procedure by modeling the effective properties of musculoskeletal mineralized tissues, and we compare the results with theoretical and experimental data for bone and tendon tissues

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

Effective properties of fractional viscoelastic composites via two-scale asymptotic homogenization

Driven by the growing interest in fractional constitutive modeling, we adopt a two-scale asymptotic homogenization approach to study the effective properties of fractional viscoelastic composites. We focus on a purely mechanical setting and derive the cell and homogenized problems corresponding to the balance of linear momentum equation in the absence of body forces and inertial terms. In doing this, we reformulate the original framework in the Laplace–Carson domain and discuss how to obtain the effective coefficients in the time domain. We particularize the general setting of our work by considering memory functions that describe special types of fractional linear viscoelastic behaviors, and after presenting the limit cases of our selections, we framed the homogenization results to account for benchmark problems with different combinations of constitutive models. Specifically, these latter involve elastic, fractional Kelvin–Voigt, fractional Zener and fractional Maxwell constituents. Our results permit us to reinforce the interpretation of the theoretical findings and to elucidate the role of the fractional constitutive models on the effective properties of the composites under investigation

Two-scale, non-local diffusion in homogenised heterogeneous media

We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods

Influence of non-local diffusion in avascular tumour growth

The availability and evolution of chemical agents play an important role in the growth of a tumour and, therefore, the mathematical description of their consumption is of special interest. Usually, Fick’s law of diffusion is adopted for describing the local character of the evolution of chemicals. However, in a highly complex, heterogeneous medium, as is a tumour, the progression of chemical species could be influenced by non-local interactions. In this respect, our goal is to investigate the influence of such types of diffusion on the growth of a tumour in the avascular stage. For our purposes, we consider a diffusion equation for the evolution of the chemical agents that accounts for the existence of non-local interactions in a non-Fickean manner, and that involves notions of fractional calculus. In particular, the introduction of derivatives or integrals of fractional type of order α ∈ ℝ has proven to be an effective mathematical tool in the description of various non-local phenomena. To achieve our goals, we adopt part of the modelling assumptions outlined in previous works, in which the growth of a tumour is described in terms of mass transfer among the tumour’s constituents and structural changes that occur in the tumour itself in response to growth. The latter ones are characterised by means of the Bilby–Kröner–Lee decomposition of the deformation gradient tensor. We perform numerical simulations, whose results indicate the relevance of embracing a fractional framework in modelling tumour growth. Specifically, the real parameter α ‘dominates’ the way in which the tumour grows, since it permits the modelling of a variety of growth patterns ranging from the standard growth to no growth at all

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