An approach for modeling non-ageing linear viscoelastic composites with general periodicity

International audienceThe present work deals with the modeling of non-ageing linear viscoelastic composite materials and quasi-periodic microstructure. The stratified functions and the curvilinear coordinates play an important role in the design of different geometrical shapes. The main objective focuses on the application of two-scales Asymptotic Homogenization Method (AHM) to obtain the overall behavior of the viscoelastic composite materials. Although the whole process is based on the analysis of laminated configurations, a multi-step homogenization scheme to estimate the effective properties of a structure reinforced with long rectangular fibers and wavy effects is used. The associated local problems, the homogenized problem and the analytical expressions for the effective coefficients are obtained by using the correspondence principle and the Laplace-Carson transform. Also, the inter-connection between the effective relaxation modulus and the effective creep compliance is performed. Finally, the inversion to the original temporal space is calculated. Some comparisons between the proposed approach and Finite Elements Method (FEM) results are displayed

Asymptotic Homogenization Method Applied to Linear Viscoelastic Composites. Examples

In this paper, the Asymptotic Homogenization Method (AHM) is applied to anisotropic viscoelastic composites. The local problems are considered and the effective viscoelastic moduli are explicitly determined. A layer viscoelastic composite with periodic structure is studied. Each layer is isotropic and homogeneous. Analytic expressions for the effective coefficients are derived. Numerical results for predicting the viscoelastic properties of layer composite with periodic structure, in particular, two-layer medium is presented. Some comparisons with other theories are done

Analysis Of Effective Elastic Properties For Shell With Complex Geometrical Shapes

The manuscript offers a methodology to solve the local problem derived from the homogenization technique, considering composite materials with generalized periodicity and imperfect spring contact at the interface. The general expressions of the local problem for an anisotropic composite with perfect and imperfect contact at the interface are derived. The analytical solutions of the local problems are obtained by solving a system of partial differential equations. In order to validate the model, the effective properties of the structure presented in the literature are obtained as particular cases. The solution of the local problem is used to extend the study to more complex structures, such as, wavy laminates shell composites with imperfect spring type contact at the interface. Also, the results are compared with the results for perfect and imperfect contact models available in the literature

Soft and hard anisotropic interface in composite materials

For a large class of composites, the adhesion at the fiber-matrix interface is imperfect i.e. the continuity conditions for displacements and often for stresses is not satisfied. In the present contribution, effective elastic moduli for this kind of composites are obtained by means of the Asymptotic Homogenization Method (AHM). Interaction between fiber and matrix is considered for linear elastic fibrous composites with parallelogram periodic cell. In this case, the contrast or jump in the displacements on the boundary of each phase is proportional to the corresponding component of the tension on the interface. A general anisotropic behavior of the interphase is assumed and the interface stiffnesses are explicitly given in terms of the elastic constants of the interphase. The constituents of the composites exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions is considered. Comparisons with theoretical and experimental results verified that the present model is efficient for the analysis of composites with presence of imperfect interface and parallelogram cell. The present method can provide benchmark results for other numerical and approximate methods

Effective predictions of heterogeneous flexoelectric multilayered composite with generalized periodicity

International audienceIn this work, the general mathematical statements for flexoelectric heterogeneous equilibrium boundary value problems are reported. A methodology to find the local problems and the effective properties of flexoelectric composites with generalized periodicity is presented, using the two-scales asymptotic homogenization method. The statement of the homogenized boundary values problem is given. A procedure to solve the local problems of stratified multilayered composites with complex geometry and perfect contact at the interface is proposed. Consequently, the analytical expressions of the effective coefficients are obtained. The piezoelectric limit case for rectangular bi-laminated composites is validated. Finally, numerical analysis to illustrate the behavior of the effective properties for rectangular and wavy flexoelectric bi-layered structures are shown

Effective transport properties for periodic multiphase fiber-reinforced composites with complex constituents and parallelogram unit cells

[EN] The two-scale asymptotic homogenization method is used to find closed-form formulas for effective properties of periodic multi-phase fiber-reinforced composites where constituents have complexvalued transport properties and parallelogram unit cells. An antiplane problem relevant to linear elasticity is formulated in the frame of transport properties. The application of the method leads to the need of solving some local problems whose solution is found using potential theory and shear effective coefficients are explicitly obtained for nphase fiber-reinforced composites. Simple formulae are explicitly given for three- and four-phase fiber-reinforced composites. The broad applicability, accuracy and generality of this model is determined through comparison with other methods reported in the literature in relation to shear elastic moduli and several transport problems of multi-phase fiber-reinforced composites in their realm, such as conductivity in a biological context and permittivity leading to gain and loss enhancement of dielectrics. Also, the example of gain enhancement of inertial mass density is looked into. Good agreement with other theoretical approaches is obtained. The formulas may be useful as benchmarks for checking experimental and numerical results.YE gratefully acknowledges the Program of Postdoctoral Scholarships of DGAPA from UNAM, Mexico. RG and RR would like to thank to COIC/STIA/9042 and COIC/STIA/9045/2019. RR thanks the International Research Training Group GRK 2078 "Integrated engineering of continuous-discontinuous long fiber reinforced polymer structures"(CoDiCoFRP) funded by German Research Foundation (DFG) for inviting him as a guest scientist, parts of the manuscript were written during this stay. JB and FJS acknowledge the funding of PAPIIT-DGAPA-UNAM IA100919. TB acknowledges the support by DFG under the grant GRK 2078/2. Thanks to the Department of Mathematics and Mechanics, IIMAS-UNAM, for its support and Ramiro Chavez Tovar and Ana Perez Arteaga for computational assistance.Sabina, FJ.; Guinovart-Díaz, R.; Espinosa-Almeyda, Y.; Rodríguez-Ramos, R.; Bravo-Castillero, J.; López-Realpozo, JC.; Guinovart-Sanjuán, D.... (2020). Effective transport properties for periodic multiphase fiber-reinforced composites with complex constituents and parallelogram unit cells. International Journal of Solids and Structures. 204:96-113. https://doi.org/10.1016/j.ijsolstr.2020.08.001S9611320

Mathematical modeling of anisotropic avascular tumor growth

Cancer represents one the most challenging problems in medicine and biology nowadays, and is being actively addressed by many researchers from different areas of knowledge. The increasing development of sophisticated mathematical models and computer-based procedures has had a positive impact on our understanding of cancer-related mechanisms and the design of anticancer treatment strategies. However, further investigation and experimentation are still required to completely elucidate the tumor-associated mechanical responses, as well as the effect of mechanical forces on the net tumor growth. In this work we develop a theoretical framework in the context of continuum mechanics to investigate the anisotropic growth of avascular tumor spheroids. To that end, a specific anisotropic growth deformation tensor is considered, which also describes an isotropic growth law as a particular case. Mixtures theory and the notion of multiple natural configurations are then used to formulate a mathematical model of avascular tumor growth. More precisely, mass, momentum balance and nutrients diffusion equations are derived, where solid tumors are assumed as hyperelastic and compressible materials. Moreover, mechanical interactions with a rigid extracellular matrix (ECM) are considered, and the mechanical modulation of growing tumors in a rigid surrounding tissue is investigated by means of numerical simulations. Finally, the model results are compared with experimental data previously reported in the literature

Action of body forces in tumor growth

In the present work two mathematical models are proposed to investigate tumor growth within the framework of continuum mechanics. In particular, the tumor is modeled as an ideal saturated mixture, where the mechanical description is based on both the mixtures theory and the notion of multiple natural configurations. The mixture is considered as a porous material composed by a hyperelastic compressible solid and an incompressible viscous fluid. In addition, the growth of a tumor considered as a single hyperelastic solid material is also studied as a particular case. Then, a general mathematical model is formulated using particular constitutive laws for each component involved. The resulting constitutive equation is used to describe the isotropic inhomogeneous growth of an encapsulated spherical solid tumor. During that process, the mixture is assumed to be isothermal. Furthermore, growth is understood as a change in the body mass of the constituents supplemented with diffusion of nutrients. The mechanical modulation of growth by body forces is then illustrated and analyzed by means of computer numerical simulations. To that end, the material parameter values considered were taken from experimental data, and model results describe realistic tumor growth dynamics

Asymptotic and numerical homogenization methods applied to fibrous viscoelastic composites using Prony’s series

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