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The present contribution aims to build chiral Cosserat effective models of architectured media prone to such effects, recoursing to dedicated homogenization methods. In order to develop a method in full generality, the effective Cosserat model is obtained from the degeneration of the micromorphic effective continuum, by projecting the micromorphic kinematic degrees of freedom onto the Cosserat variables. The proposed homogenization method relies on variational principles and a decomposition of the microscopic displacement into a so-called homogeneous part and a periodic fluctuation. The homogeneous contribution to the total microscopic displacement accounts for the postulated effective generalized continuum, and it is corrected by a displacement fluctuation. The energy of the fluctuating displacement is shown to define a measure of the accuracy of the homogenized chiral Cosserat model. Applications of the proposed homogenization framework are done for centrosymmetric and non-centrosymmetric chiral architected media. [ABSTRACT FROM AUTHOR]

Unlike inactive systems, living biological systems have the advantage of being able to adapt to their environment through growth. Growth is a phenomenon unique to biological tissues, which is mainly driven by accretion processes, whereby new living tissues are added only on the surface of the growing body, due to the action of generating cells. The aim of this contribution is to develop a model for material accretion in the context of the thermodynamics of continuous media. A relation between the accretion velocity and a conjugated driving force based on Eshelby stress tensor is derived from the optimum condition of the zero thermodynamic potential. Numerical simulations are subsequently carried out in the biomechanical context of bone external remodeling to elucidate the apposition of bone mineral onto the surface of proximal femur samples subjected to physiological loads. [ABSTRACT FROM AUTHOR]

The effective piezoelectric and flexoelectric properties of heterogeneous solid bodies with constituents obeying a piezoelectric behavior are evaluated in full generality, based on the asymptotic expansion method. The successive situations of materials obeying a piezoelectric and flexoelectric behavior at the macroscale is envisaged in the present work. Closed-form expressions for the effective flexoelectric properties are obtained for stratified materials. A general theory for laminated piezoelectric plates is formulated on the basis of the formulated asymptotic models, and the response of the homogeneous substitution plate is evaluated for a loading consisting of a pure bending moment, triggering electric fields and strain and electric fields gradients within the plate thickness. The local mechanical and electric fields at the microscopic level within the initial heterogeneous stratified domain are evaluated by solving unit cell boundary value problems for the localization operators. An effective flexoelectric plate model for a stratified composite is constructed, showing the generation of the gradient of an electric field under application of a pure bending moment. [ABSTRACT FROM AUTHOR]

The present paper aims at developing a homogenization scheme for the identification of strain-gradient elastic moduli, based on the mathematical approach of homogenization introduced by L. Tartar. We expose in the first part of this paper the needed mathematical apparatus in view of the derivation of the effective first and second gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second gradient linear elastic constitutive law. Application of the method of oscillating functions to the homogenization of multilayer materials showing strong strain gradients like composites with a strong contrast of properties of their constituents leads to closed form expressions of the effective first and second gradient moduli, including the coupling coefficients between first and second gradient terms. These results are then applied to the situations of multilayers with interfaces considered as the locus of strong strain gradients, and stratified materials. Analytical expressions of the effective moduli versus the first and second elasticity properties of the plies within the unit cell are obtained in both situations. [ABSTRACT FROM AUTHOR]

The present paper aims at introducing a homogenization scheme for the identification of strain–gradient elastic moduli of composite materials, based on the unfolding mathematical method. We expose in the first part of this paper the necessary mathematical apparatus in view of the derivation of the effective first- and second-gradient mechanical properties of two-phase composite materials, focusing on a one-dimensional situation. Each of the two phases is supposed to obey a second-gradient linear elastic constitutive law. Application of the unfolding method to the homogenization of multi-layer materials provides closed form expressions of all effective first- and second-gradient elastic moduli as well as coupling moduli between first- and second-gradient elasticity. A comparison between the unfolding method and the method of oscillating functions shows that both methods, despite their differences, deliver the same effective second-gradient elastic constitutive law for stratified materials. [ABSTRACT FROM AUTHOR]

As a living tissue, bone is subjected to internal evolutions of its trabecular architecture under normal everyday mechanical loadings leading to damage. The repeating bone remodeling cycle aims at repairing the damaged zones in order to maintain bone structural integrity; this activity of sensing the peak stress at locations where damage or microcracks have occurred, removing old bone and apposing new bone is achieved thanks to a complicated machinery at the cellular level involving specialized cells (osteocytes, osteoclasts, and osteoblasts). This work aims at developing an integrated remodeling-to-fracture model to simulate the bone remodeling process, considering trabecular bone anisotropy. The effective anisotropic continuum mechanical properties of the trabecular bone are derived from an initially discrete planar hexagonal structure representative of femur bone microstructure, relying on the asymptotic homogenization technique. This leads to scaling laws of the effective elastic properties of bone versus effective density at an intermediate mesoscopic scale. An evolution law for the local bone apparent density is formulated in the framework of the thermodynamics of irreversible processes, whereby the driving force for density evolutions is identified as the local strain energy density weighted by the locally accumulated microdamage. We adopt a classical nonlinear damage model for high cycle fatigue under purely elastic strains, where the assumed homogeneous damage is related to the number of cycles bone experiences. Based on this model, we simulate bone remodeling for the chosen initial microstructure, showing the influence of the external mechanical stimuli on the evolution of the density of bone and the incidence of this evolution on trabecular bone effective mechanical properties. [ABSTRACT FROM AUTHOR]

Architected materials have witnessed tremendous research activity in the last two decades, due to their ability to achieve unprecedented effective properties. The up-scaling of the mechanical response towards an effective continuum is challenging, due to the possible emergence of a homogenized macroscopic behavior that may substantially differ from that of the base material at the micro level. The need for enriched continuum models beyond Cauchy's theory arises due to the existence of a new class of artificial materials exhibiting static and dynamic attributes typically not encountered in natural materials, deserving the name meta-materials. In such a situation with a failure of the strict scale separation hypothesis, the enrichment of the Cauchy continuum proceeds by adding new intrinsic parameters and internal length scales representative of the micro-structural impact at the level of the effective medium. Generalized continua including either new kinematic variables or higher gradients of the displacement have proven their ability to provide a faithful description of the response of materials with micro-structural effects; their derivation from a micro-mechanical approach starting directly from the scale of the microstructure raises specific difficulties, in contrast to the phenomenological standpoint, which however lacks a predictive capacity. A survey of the scientific issues raised by higher-order homogenization methods towards generalized continua is provided in this contribution. The main scientific issues and challenges raised by higher-order homogenization methods are reviewed in a critical manner, and solutions to these scientific issues are proposed in light of the more recent contributions. Specific attention is devoted to the elaboration of enriched models for architected materials in both static and dynamic regimes. • The issues raised by higher-order homogenization methods towards enriched continua are reviewed. • A general methodology for deriving micromorphic continua by homogenization method is presented. • The classification of generalized continua issued from the micromorphic medium by degeneration conditions is proposed. • Examples of architected materials obeying a chiral micropolar behavior are proposed. • The large strains and dynamical aspects of elaborating an effective continuum are tackled. [ABSTRACT FROM AUTHOR]

The thermodynamics of open systems exchanging mass, heat, energy, and entropy with their environment is examined as a convenient unifying framework to describe the evolution of growing solid bodies in the context of volumetric growth. Following the theory of non-equilibrium thermodynamics (NET) introduced by De Donder and followers from the Brussels School of Thermodynamics, the formulation of the NET of irreversible processes for multicomponent solid bodies is shortly reviewed. In the second part, extending the framework of NET to open thermodynamic systems, the balance laws for continuum solid bodies undergoing growth phenomena incorporating mass sources and mass fluxes are expressed, leading to a formulation of the second principle highlighting the duality between irreversible fluxes and conjugated driving forces. A connection between NET and the open system thermodynamic formulation for growing continuum solid bodies is obtained by interpreting the balance laws with source terms as contributions from an external reservoir of nutrients. [ABSTRACT FROM AUTHOR]

Models for the internal and external bone remodeling are developed in the framework of the thermodynamics of irreversible processes. Internal remodeling is essentially accounted for by an evolution of the internal bone density versus the trace of the Eshelby stress. The growing surface is endowed with a specific mechanical behavior elaborated from a surface potential, depending upon the elastic surface strain and the external normal vector. The surface remodeling velocity is related to the driving force for growth identified as the surface divergence of Eshelby stress. The developed formalism is applied to simulate bone internal and external remodeling for 2D geometries, showing the influence of external mechanical stimuli on the evolution of the external shape of bone. Simulations of the evolution of the shape and density of bone structures are carried out successively at mesoscopic, microscopic and macroscopic levels, the former representing cellular level simulations of the trabecular unit cell. At the macroscopic level, a linear elastic constitutive law for trabecular bone is derived, relying on the discrete homogenization approach, whereby the continuum parameters such as stiffness evolve with morphology as remodeling occurs within bone. The developed numerical platform is able to simulate combined bone internal and external remodeling at both trabecular and macroscopic scales. [ABSTRACT FROM AUTHOR]

The influence of the loading conditions on the trabecular architecture of a femur is investigated by using topology optimization methods. The response of the bone to physiological loads results in changes of the internal architecture of bone, reflected by a modification of internal effective density and mechanical properties. The homogenization based optimization model is developed for predicting optimal bone density distribution, wherein bone tissue is assumed to be a composite material consisting of a mixture of material and void. The homogenization scheme treats the geometric parameters of the microstructures and their orientation as design variables and homogenizes the properties in that microstructure, which is generally anisotropic. The penalization of the optimal material density then leads to a classical optimal structure which consists of regions with bone material and regions without bone material. The IMD (Isotropic Material Design) approach is next applied to determine the optimal elasticity tensor in terms of the bulk and shear moduli for the present loading applied to the femoral bone sample. IMD is able to provide both the external shape and topology together with the optimal layout of the isotropic moduli. Both topology optimization methods appear to be complementary. Simulations of the internal bone architecture of the human proximal femur results in a density distribution pattern with good consistency with that of the real bone. [ABSTRACT FROM AUTHOR]

This paper aims at developing homogeneous, anisotropic strain-gradient continuum models as substitutes of 3D heterogeneous porous or composite materials and structures. We construct effective first and second order grade continuum models equivalent to such inhomogeneous structures. This in turn leads to the construction of a strain-gradient continuum with effective mechanical properties at the first and second order, accounting for the impact of the underlying microstructure on the overall effective mechanical response of the effective continuum. The effective properties are obtained based on the response of the representative volume element or unit cell of the initial structure under prescribed boundary conditions. Mixed boundary conditions comprising both traction and displacement boundary conditions are applied on the structure boundaries to identify the equivalent 3D strain gradient elasticity. The first and second order mechanical constants of the effective strain-gradient continuum are deduced by an equivalent strain energy method. We perform this study computationally using a finite element approach. The present methodology is exemplified in certain applications, considering sequentially three-dimensional random porous polymer scaffolds, composite reinforced by inclusions, and woven composites, including layered composites and 3D through-thickness orthogonal interlocks. [ABSTRACT FROM AUTHOR]

We presently extend the virial theorem for both discrete and continuous systems of material points with variable mass, relying on developments presented in Ganghoffer (Int J Solids Struct 47:1209-1220, ). The developed framework is applicable to describe physical systems at very different scales, from the evolution of a population of biological cells accounting for growth to mass ejection phenomena occurring within a collection of gravitating objects at the very large astrophysical scales. As a starting basis, the field equations in continuum mechanics are written to account for a mass source and a mass flux, leading to a formulation of the virial theorem accounting for non-constant mass within the considered system. The scalar and tensorial forms of the virial theorem are then written successively in both Lagrangian and Eulerian formats, incorporating the mass flux. As an illustration, the averaged stress tensor in accreting gravitating solid bodies is evaluated based on the generalized virial theorem. [ABSTRACT FROM AUTHOR]

The process of bone remodeling includes a cycle of repair, renewal, and optimization. This adaptation process, in response to variations in external loads and chemical driving factors, involves three main types of bone cells: osteoclasts, which remove the old pre-existing bone; osteoblasts, which form the new bone in a second phase; osteocytes, which are sensing cells embedded into the bone matrix, trigger the aforementioned sequence of events. The remodeling process involves mineralization of the bone in the diffuse interface separating the marrow, which contains all specialized cells, from the newly formed bone. The main objective advocated in this contribution is the setting up of a modeling and simulation framework relying on the phase field method to capture the evolution of the diffuse interface between the new bone and the marrow at the scale of individual trabeculae. The phase field describes the degree of mineralization of this diffuse interface; it varies continuously between the lower value (no mineral) and unity (fully mineralized phase, e.g. new bone), allowing the consideration of a diffuse moving interface. The modeling framework is the theory of continuous media, for which field equations for the mechanical, chemical, and interfacial phenomena are written, based on the thermodynamics of irreversible processes. Additional models for the cellular activity are formulated to describe the coupling of the cell activity responsible for bone production/resorption to the kinetics of the internal variables. Kinetic equations for the internal variables are obtained from a pseudo-potential of dissipation. The combination of the balance equations for the microforce associated to the phase field and the kinetic equations lead to the Ginzburg-Landau equation satisfied by the phase field with a source term accounting for the dissipative microforce. Simulations illustrating the proposed framework are performed in a one-dimensional situation showing the evolution of the diffuse interface separating new bone from marrow. [ABSTRACT FROM AUTHOR]

Abstract: A continuum volumetric growth theory is presently elaborated based on the stress-free configuration. The total deformation gradient is the composition of a growth mapping followed by an elastic mapping ensuring the compatibility of the body. A measure of the growth rate fully lying in the stress-free configuration is elaborated, and the balance of momentum accounting for mass changes due to growth is written in the same configuration, highlighting measures of kinematic incompatibilities. An Eshelby stress is identified as the driving force for growth, in the sense that it appears as the true dissipative thermodynamic force conjugated to the growth rate. Examples of incompatible cylindrical growth with residual stress generation and tumor growth with mechanics coupled to diffusion of nutrients illustrate the proposed theory. [Copyright &y& Elsevier]

Abstract: Composite materials made of auxetic inclusions and giving rise overall to negative Poisson’s ratio are considered, adopting a two-steps micromechanical approach for the calculation of their effective mechanical properties. The inclusions consist of periodic beam lattices, whose equivalent mechanical properties are calculated by a discrete homogenization scheme in a first step. The hexachiral and hexagonal reentrant lattices are considered as representative of the two main deformation mechanisms responsible for auxeticity. In a second step, the equivalent properties of the composite are calculated from numerical homogenization using the finite element method. It is shown that both an auxetic behavior and enhanced moduli can be obtained for not too slender micro-beams. [Copyright &y& Elsevier]

Abstract: Surface growth is presently described as the motion of a moving interface of vanishing thickness, physically representing the generating cells, separating a zone not yet affected by growth from a domain in which growth has occurred. The jump conditions of density, velocity, momentum, energy, and entropy over the moving front are expressed from the general balance laws of open systems in both physical and material format. The writing of the jump of the internal entropy production in material format allows the identification of a driving force for surface growth, thermodynamically conjugated to the material velocity of the moving front. [Copyright &y& Elsevier]

Tissue engineering offers an interesting alternative to current anterior cruciate ligament (ACL) surgeries. Indeed, a tissue-engineered solution could ideally overcome the long-term complications due to actual ACL reconstruction by being gradually replaced by biological tissue. Key requirements concerning the ideal scaffold for ligament tissue engineering are numerous and concern its mechanical properties, biochemical nature, and morphology. This study is aimed at predicting the morphology of a novel scaffold for ligament tissue engineering, based on multilayer braided biodegradable copoly(lactic acid-co-(e-caprolactone)) (PLCL) fibers The process used to create the scaffold is briefly presented, and the degradations of the material before and after the scaffold processing are compared. The process offers varying parameters, such as the number of layers in the scaffold, the pitch length of the braid, and the fibers' diameter. The prediction of the morphology in terms of pore size distribution and pores interconnectivity as a function of these parameters is performed numerically using an original method based on a virtual scaffold. The virtual scaffold geometry and the prediction of pore size distribution are evaluated by comparison with experimental results. The presented process permits creation of a tailorable scaffold for ligament tissue engineering using basic equipment and from minimum amounts of raw material. The virtual scaffold geometry closely mimics the geometry of real scaffolds, and the prediction of the pore size distribution is found to be in good accordance with measurements on real scaffolds. The scaffold offers an interconnected network of pores the sizes of which are adjustable by playing on the process parameters and are able to match the ideal pore size reported for tissue ingrowth. The adjustability of the presented scaffold could permit its application in both classical ACL reconstructions and anatomical double-bundle reconstructions. The precise knowledge of the scaffold morphology using the virtual scaffold will be useful to interpret the activity of cells once it will be seeded into the scaffold. An interesting perspective of the present work is to perform a similar study aiming at predicting the mechanical response of the scaffold according to the same process parameters, by implanting the virtual scaffold into a finite element algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The formal relationships between the scalar and tensorial virials and Eshelby tensors have been presently investigated. The key idea is to evaluate the Eshelby stress from discrete or atomistic simulations for a structured body, conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches. Extending first the writing of the scalar virial to a material format, it is shown that the average of the elaborated scalar material virial is the trace of the (material) Eshelby stress. The spatial and material virials are further related to eachother in the framework of hyperelasticity, and a tensorial extension of the material virial is provided. Interpretation of those results from the microscopic point of view shows that Eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor. Consideration of the material version of the virial theorem further leads to express Eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy. The average scalar virial is further identified to the grand potential in a thermodynamic context. A definition of the material scalar virial for a second order continuum is lastly proposed, based on the identification of a second order Eshelby stress and in line with the second order Cauchy–Born rule. [Copyright &y& Elsevier]

Abstract: Models of the adhesion of a population of cells in a plane flow are developed, considering the dilute regime. Cells considered as rigid punctual entities are virtually injected at regular times within a plane channel limited by two fixed planes. The pressure profile is supposed to be triangular (constant gradient), in accordance with the assumptions of a Poiseuille flow. The cell adherence to the channel wall is governed by the balance of forces, accounting for gravity, non-specific physical interactions, such as electrostatic effects (repulsive) and Van der Waals forces (attractive), specific adhesive forces representing the ligand–receptor interactions, and friction between cells and the fluid in the vicinity of the endothelium wall. The spatial distribution of the adhesion molecules along the wall is supposed to be a random event, accounted for by a stochastic spatial variability of the dipolar moments of those molecules, according to a Gaussian process. Experimental trends reported for the rate of aggregation of L-selectin mediated leukocytes under shear flow are in qualitative accordance with the evolution versus time of adhering cells obtained by the present simulations. The effect of the maximal injection pressure on those kinetics is assessed. [Copyright &y& Elsevier]

Cell adhesion plays an important role in biology: essentially with regard to immunizing defence and the transport of medicinal substances toward specific zones. The focus is here on the mechanical description of adhesion kinetics, in terms of the failure and creation of connections during the rolling phenomenon. Hence, we consider the case of a single cell, which is linked to a rigid substratum. A 2D model is established. We consider that the contact zone cell-wall is rectilinear and composed of vertical fibers and two horizontal rigid beams (complex cell membrane- fibers-vein wall). These connections are modeled by elastic springs having identical elastic properties (e.g stiffness), but different failure strengths. The cell is subjected to the flow of plasma, which can generate the rolling phenomenon; we accordingly consider two distinct zones, one associated with the failure of the old fibers and one with the creation of the new fibers (in the direction opposite to the flow). Several interactions are taken into account in this model: van der Waals (attractive) and electrostatic (repulsive) forces and the effects of fluid pressure, assimilated into a periodic point force applied to the interface zone. We also study the vibration induced failure in the contact zone without mechanical damping using the principle of virtual work and a failure criterion to establish the equation of motion and the time evolution of the failure (dynamical approach). Rupture of a fiber can occur if the stress applied to the fiber is above a certain limit. These limits are determinated with using a probabilistic approach by use of a spectral method to simulate a stochastic and Gaussian field. Modeling of the creation of new fibers is also achieved by the combination of a dynamical and probabilistic method and a kinematical criterion. On the basis of these elements, numerical simulations are developed, that elucidate the rupture and rolling phenomena. [ABSTRACT FROM AUTHOR]

Abstract: The growth of biological tissues is described as the volumetric production of mass within tissue elements, considered as one component potential sites for tissue renewal/resorption at a mesoscopic scale of description. The growth is characterized by a growth transformation gradient, which is accompanied by an additional accommodation transformation, so that the total transformation gradient defines a compatible global displacement field. The continuous change of the domain occupied by each tissue element during growth is considered, hence the mechanical balance laws are written, accounting for the additional terms due to the domain variation. The principle of virtual power is then expressed, considering that the power of internal forces originates from both volumic and surface potentials. The surface potential is thought to express configurational forces tied to the interface motion, extent and orientation. The equilibrium equations of the growing tissue element then follow, associated to surface and line boundary conditions. The writing of the second principle of thermodynamics for a tissue element continuously receiving matter due to transport phenomena allows to identify the driving forces linked to growth, and provides the evolution laws for the growth velocity. The large strains compatibility conditions that characterize the accommodation tensor are further considered as an important aspect of growth. A first insight into the numerical solutions of the compatibility conditions is given, envisaging the situation of radial growth. [Copyright &y& Elsevier]

We present some reflections on the application of the Lagrangian formalism for continuous media locally uniform subjected to internal irreversible evolutions. The Lagrangian density, defined as the time derivative of a non-equilibrium thermodynamic potential, [Thermodynamics of Relaxation Processes using Internal variables within a Lagrange-formalism. P. Germain’s Anniversary Volume 2000. Contiuum Thermomechanics: the Art and Science of Modeling Matter’s Behaviour, 2000], contains all the symmetry properties of the system. The generalised Lagrange co-ordinates correspond to the state and internal variables of the time derivative of the generalised Gibbs potential. The latter being used within the framework of the De Donder’s method, must also account for the memory effect of the physical medium.This first part is devoted to the thermodynamic framework called the distribution of non-linear relaxations approach (DNLR) developed by C. Cunat on the basis of the generalised Gibbs’ relation. [Copyright &y& Elsevier]

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium. [Copyright &y& Elsevier]

A constitutive law for large transformations of soft structures is given, starting from a micropolar constitutive law, where the microrotation takes into account the microstructure effect. In fabrics a wave defines the mesoscopic scale. The hyperelastic law takes into account the extension in the fabric. The mesoscopic wave model is applied in the present paper to predict the uniaxial extension of plain weave fabrics. [Copyright &y& Elsevier]

We propose in this study a two‐dimensional constitutive model for trabecular bone combining continuum damage with embedded strong discontinuity. The model is capable of describing the three failure phases of trabecular bone tissue which is considered herein as a quasi‐brittle material with strains and rotations assumed to be small and without viscous, thermal or other non‐mechanical effects. The finite element implementation of the present model uses constant strain triangle (CST) elements. The displacement jump vector is implicitly solved through a return mapping algorithm at the local (finite element) level, while the global equilibrium equations are dealt with by Newton–Raphson method. The performance, accuracy and applicability of the proposed model for trabecular bone fracture are evaluated and validated against experimental measurements. These comparisons include both global and local aspects through numerical simulations of three‐point bending tests performed on 10 single bovine trabeculae in the quasi‐static regime. [ABSTRACT FROM AUTHOR]

Constitutive models for bone remodeling are established from micromechanical analyses at the scale of individual trabeculae defining the representative unit cell (RUC), accounting for both first- and second-order deformation gradients. On the microscale, trabeculae undergo apposition of new bone modeled by a surface growth velocity field driven by a mechanical stimulus identified to the surface divergence of an Eshelby-like tensor. The static and evolutive first and second gradient effective properties of a periodic network of bone trabeculae are evaluated by numerical simulations with controlled imposed first and second displacement gradient rates over the RUC. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae relates the (average) first and second growth strain rates to the homogenized stress and hyperstress tensors. The constitutive model is identified relying on the framework of TIP (thermodynamics of irreversible processes), adopting a split of the kinematic and static tensors into their deviator and hydrostatic contributions. The obtained results quantify the strength and importance of strain gradient effects on the overall remodeling process. [ABSTRACT FROM AUTHOR]

Highlights • Enriched beam models incorporating section dilatation and in-plane shear for thick beams in a finite strains context are developed; • Macrodilatation, macroshear, and macrostrain beam models are constructed on the basis of generalized beam theories; • Model reduction is performed assuming a uniform hyperstress tensor, leading to new macro-continuums with a reduced set of degrees of freedom and additional microstructural parameters; • A general classification of beam models based on reduction of the micro-continua is proposed; • The proposed macrobeam models better describe the in plane-shear and dilatation of the beam sections along the mean line in comparison to classical beam theories. Abstract The consideration of nonclassical beam theories beyond Timoshenko beam model is necessary in applications involving complex yarns subjected to transverse compaction, for which the section dilatation and in-plane shear require a detailed kinematic and static modeling. In the present work, new enriched beam elements with additional degrees of freedom are derived from the mechanics of generalized continua, allowing a representation of the in plane-shear and dilatation of sections. As a novel aspect, enriched beam model called macrodilatation, macroshear, and macrostrain have been constructed on the basis of generalized beam theories but with a more condensed kinematics, assuming uniform hyperstress tensor within the beam. The choice of the adequate extended continuum for a given beam depends on its macroscopic deformation mechanisms. As a result, the developed beam models include modified material parameters (the tensile, shear and bulk moduli for an isotropic beam) involving additional microstructural parameters. These macrobeam models are expected to better describe the in plane-shear and dilatation of the beam sections along its mean line in comparison to classical beam theories. Numerical illustrative examples show that the microdilatation beam model has the ability to capture the local deformation field around holes, in contrast to Timoshenko beam model. The parameters of the microdilatation beam model provide a very good estimate of the overall porosity. Graphical abstract Cross-section of a beam with the three directors in the deformed configuration Image, graphical abstract [ABSTRACT FROM AUTHOR]

We construct in the present paper constitutive models for bone remodeling based on micromechanical analyses at the scale of a representative unit cell (RUC) including a porous trabecular microstructure. The time evolution of the microstructure is simulated as a surface remodeling process by relating the surface growth remodeling velocity to a surface driving force incorporating a (surface) Eshelby tensor. Adopting the framework of irreversible thermodynamics, a 2D constitutive model based on the setting up of the free energy density and a dissipation potential is identified from FE simulations performed over a unit cell representative of the trabecular architecture obtained from real bone microstructures. The static and evolutive effective properties of bone at the scale of the RUC are obtained by combining a methodology for the evaluation of the average kinematic and static variables over a prototype unit cell and numerical simulations with controlled imposed first gradient rates. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae within the RUC is of viscoplastic type and relates the average growth strain rate to the homogenized stress tensor. The postulated model includes a power law function of an effective stress chosen to depend on the first and second stress invariants. The model coefficients are calibrated from a set of virtual testing performed over the RUC subjected to a sequence of loadings. Numerical simulations show that overall bone growth does not show any growth kinematic hardening. The obtained results quantify the strength and importance of different types of external loads (uniaxial tension, simple shear, and biaxial loading) on the overall remodeling process and the development of elastic deformations within the RUC. [ABSTRACT FROM AUTHOR]

We construct in the present paper the effective growth response of trabecular bone, based on micromechanical analyses at the scale of a representative volume element consisting of individual trabeculae defining the representative unit cell (RUC). From a fundamental perspective, we adopt the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the surface remodeling of the individual trabeculae, the averaging of which leads to a net growth at the mesoscopic scale of the RUC. The effective model for the growing unit cell relies on an average kinematics which has been formulated in terms of the effective growth velocity gradient and effective elastic rate of deformation tensor, both functions of the rate of the effective density and stress applied over the RUC. The evaluation of the relation between the average rate of growth tensor and elastic growth rate versus the external stress applied to the RUC is obtained from a split of the average rate of growth tensor into its spherical and deviatoric parts, each of which being related to the corresponding applied stress quantities. The effective growth model is written for three different loading conditions as a polynomial relation between the components of the rate of growth tensor and similar components of the stress tensor under planar conditions, with a nonlinear function of the effective density as a weighting factor. The developed methodology is quite general and is applicable to any choice of the RUC morphology representative of the internal bone architecture. [ABSTRACT FROM AUTHOR]

The asymptotic homogenization technique is presently developed in the framework of geometrical nonlinearities to derive the large strains effective elastic response of network materials viewed as repetitive beam networks. This works extends the small strains homogenization method developed with special emphasis on textile structures in Goda et al. (J Mech Phys Solids 61(12):2537-2565, 2013). A systematic methodology is established, allowing the prediction of the overall mechanical properties of these structures in the nonlinear regime, reflecting the influence of the geometrical and mechanical micro-parameters of the network structure on the overall response of the chosen equivalent continuum. Internal scale effects of the initially discrete structure are captured by the consideration of a micropolar effective continuum model. Applications to the large strain response of 3D hexagonal lattices and dry textiles exemplify the powerfulness of the proposed method. The effective mechanical responses obtained for different loadings are validated by FE simulations performed over a representative unit cell. [ABSTRACT FROM AUTHOR]

The current paper deals with the wave dispersion characteristics of two-dimensional architected materials, considering both the linear and nonlinear regimes. We study the full non-linear coupling between the shear, extension, and bending deformation modes on the inner material deformation of the square, triangular, hexagonal, and re-entrant hexagonal unit cells. The dynamical analysis is done at high and low frequencies and for two different wave amplitudes. Based on the Linstedt–Poincare perturbation method, the non-linear correction to the dispersion diagram is evaluated. The relative contribution of each non-linear energy mode on the nonlinear wave dispersion attributes is quantified. The obtained results show that both the wavenumber and lattice design play an important role in the nonlinear wave corrections of the dispersion characteristics. The shear mode is shown to have the highest effect in the non-linear correction for all considered architected materials, compared to the flexural mode, which has the lowest energy contribution. The eigenwaves vectors are used to compare the deformation of hexagonal and re-entrant hexagonal unit cells at low and high frequencies over the entire Brillouin zone. Overall, the present work provides some guideline as to the proper selection of the inner design of architected materials prone to strong nonlinear dynamical effects. [ABSTRACT FROM AUTHOR]

Effective elastic properties, and mode I elastic fracture toughness of three isotropic planar lattices – hexagonal, kagome, and triangular – are studied from a micropolar continuum perspective. The hexagonal lattice is bending dominated whereas the kagome and the triangular are stretching dominated for any prescribed macroscopic in-plane loading. Discrete asymptotic homogenization is applied to obtain analytical expressions for the topology governed effective micropolar elastic properties of each lattice and their scaling with relative density: the hexagonal lattice is found to possess the largest micropolar internal length parameter at any given relative density. The homogenized micropolar continuum is then discretized by a four-node finite element model to compute its Cauchy and couple-stress intensities for a central crack under mode I loading. Micropolar scaling exponents for mode I fracture toughness with relative density agree well with the estimates from the literature based on fully discrete beam network models. Hexagonal lattice exhibits an order of magnitude higher couple stress intensity factor when compared to the triangular and the kagome lattices of identical relative density, signifying the importance of micropolar effects in bending dominated architectures. • Micropolar effects are important for topological fracture toughness of lattices. • Cauchy and couple stress intensities are studied. • Micropolar contribution to the strain energy release rate is shown to dominate for a hexagonal lattice. • Discrepancies in scaling laws for toughness dependence on relative densities are resolved by including micropolar effects. • Non-affine bending deformation field and couple-stress intensities are significant for bending dominated lattices. [ABSTRACT FROM AUTHOR]

The notion of strain gradients provides a reliable model for capturing size effects and localization phenomena. The difficulty in identifying corresponding constitutive parameters, on the other hand, restricts the practical applicability of such theory. In this work, we aim at developing homogenization based strain-gradient continuum models to compute the effective Cauchy and strain gradient moduli for a wide class of 3D architectured materials and composites prone to strain gradient effects. The present homogenization method is based on a variational principle in linear elasticity articulated with Hill's lemma, which is extended by considering the generalized kinematics within the framework of periodic homogenization. The microscopic displacement field is decomposed into a polynomial function of the introduced macroscopic kinematic variables and the other is periodic fluctuation. The fluctuating displacement is expressed versus the macroscopic kinematic variables solving sequential classical and higher order unit cell boundary value problems. The computed effective elastic moduli in the current computing framework do not depend on the unit cell size, and are therefore intrinsic to the microstructure within the structure. 3D applications for inclusion based composites, pantographic lattices and thin-walled lattice structures showing pronounced strain gradient effects are carried out in order to exemplify the proposed homogenization strategy. [ABSTRACT FROM AUTHOR]

Abstract: The setting up of predictive models of the pelvic organ motion and deformation may prove an efficient tool in the framework of prostate cancer radiotherapy, in order to deliver doses more accurately and efficiently to the clinical target volume (CTV). A finite element (FE) model of the prostate, rectum and bladder motion has been developed, investigating more specifically the influence of the rectum and bladder repletions on the gland motion. The required organ geometries are obtained after processing the computed tomography (CT) images, using specific softwares. Due to their structural characteristics, a 3D shell discretization is adopted for the rectum and the bladder, whereas a volume discretization is adopted for the prostate. As for the mechanical behavior modelling, first order Ogden hyperelastic constitutive laws for both the rectum and bladder are identified. The prostate is comparatively considered as more rigid and is accordingly modelled as an elastic tissue undergoing small strains. A FE model is then created, accounting for boundary and contact conditions, internal and applied loadings being selected as close as possible to available anatomic data. The order of magnitude of the prostate motion predicted by the FE simulations is similar to the measurements done on a deceased person, accounting for the delineation errors, with a relative error around 8%. Differences are essentially due to uncertainties in the constitutive parameters, pointing towards the need for the setting up of direct measurement of the organs mechanical behavior. [Copyright &y& Elsevier]

The paper addresses the issue of the modelling of the mechanical behaviour of yarns using a mass-spring system of discrete elements. For this purpose, the yarn is discretized into straight, elastic bars modelled by stretching springs which are connected at frictionless hinges by rotational springs. Shear springs are used, in order to model the shearing stiffness of the yarn. An energy study is conducted, taking into account the various strain energy contributions of the yarn and the work of the external forces. From the energy expression, a stability analysis is performed, relying on the Dirichlet-Lagrange stability criterion. The effects of boundary conditions as well as the heterogeneity of the nodes' rigidity are analysed. The equilibrium shapes of the structure, submitted to its own weight, are obtained as shapes associated to the minimum of the total potential energy versus the whole set of kinematic translational and rotational variables. The obtained results show that the effects of boundary conditions and heterogeneity of the yarn's rigidity on the stability of the structure are important. [ABSTRACT FROM AUTHOR]

A discrete model of a woven fabric structure has been established at a macroscopic scale, whereby nodes endowed with a mass and a rotational rigidity are connected by rigid bars to form a two-dimensional truss. The set of four bars that delineate a quadrilateral element – the unit cell of the micromechanical analysis - is further endowed with a torsion deformation mode. The equilibrium shape of the structure is obtained as the minimum of its total potential energy versus the whole set of kinematic translational and rotational variables, accounting for eventual kinematic constraints due to contact with a rigid surface by the Lagrange multipliers method. The critical buckling load resulting from a stability analysis has been evaluated, based on Dirichlet-Lagrange criterion. The effect of structure size and orthotropy, and of the nature of boundary conditions on the stability behavior is further assessed. The potentiality of the model is illustrated by fabric draping simulations. The interaction between yarns is further analyzed at a mesoscopic scale of description. Both the macroscopic and mesoscopic models are formulated in a geometrically nonlinear framework, focusing on large rotations, but with small extensions of the individual beam elements. The initial shape of the yarn, represented by a planar undulated beam supposed to be periodic, is described by a Fourier series. The contact force is expressed vs. the mechanical properties of the transverse yarn and vs. the vertical displacement of the contact point. The potential energy of the interaction is then established. The simulated traction curve reproduces in a satisfactorily manner the observed behavior. The consideration of the coupling enhances the rigidity of the response of the yarn; the effect of the mechanical moduli of the transverse yarn is lastly evidenced. [ABSTRACT FROM AUTHOR]

Abstract: A mesoscopic discrete model of fabric has been developed, accounting for the yarn–yarn interactions occurring at the yarn crossing points. The fabric yarns, described in their initial state by a Fourier series development, are discretized into elastic straight bars represented by stretching springs, and connected at frictionless hinges by rotational springs. In the first part of the paper, the behavior under uniaxial tension of a single yarn has been investigated, and the impact of the interactions of the transverse yarns has been quantitatively assessed. The consideration of the yarn interactions is extended in this second part at the scale of the whole network of interwoven yarns, under uniaxial and biaxial loading conditions. The effect of the transverse yarns properties under uniaxial tension is evidenced, as well as the impact of the biaxial loading ratio. [Copyright &y& Elsevier]

Abstract: A mesoscopic discrete model of dry fabric has been developed, based on the yarn–yarn interactions occurring at the yarns crossing points. The fabric yarns, described initially by a Fourier series development, are discretized into elastic straight bars represented by stretching springs and connected at frictionless hinges by rotational springs. The motion of each node is described by a lateral displacement and a rotation. The expression of the reaction force exerted by the transverse yarns at the contact points is assessed, from which the work of the reaction forces is established. The equilibrium shape of the yarn is obtained as the minimum of its total potential energy, accounting for the work of the reaction forces due to the transverse yarns. Simulations of a traction curve of a single yarn are performed, that evidence the effect of the yarn interactions. The two principal deformation mechanisms, the variation of undulation and the yarn stretching, are separately analysed. [Copyright &y& Elsevier]

Abstract: A discrete model of a fabric has been developed from an analogical description, using a mass-spring system of discrete elements. An element of fabric is modeled by a set of grid nodes endowed with a mass and connected with flexional and stretching springs. This model describes the mechanical behavior of a woven structure at a mesoscopic scale. As a novel contribution, the interactions and the friction between yarns are introduced in the present work. An energy analysis of the discrete system of analogical elements is performed, taking into account the compression strain energy of the yarns and the work of the reaction forces exerted between yarns. A suitable discrete variational principle, accounting for the presence of the nonholonomic forces arising from friction, serves as a basis for a numerical implementation. Simulations of uniaxial tractions are performed, that show the effect of the yarn–yarn friction and yarn compressibility on the fabric response. To cite this article: B. Ben Boubaker et al., C. R. Mecanique 335 (2007). [Copyright &y& Elsevier]

Abstract: Rolling is an important manifestation of cell adhesion [Viallat, A., Akarian, M., Campillo, C., Faivre, M., Pepin-Dona, B., 2005. Dynamique de vésicules géantes et de globules rouges sous écoulement de cisaillement. In: 17e`me Congrès Français de Mécanique, Troyes], especially for the leukocyte cell in the immune process [Bell, G.I., Dembo, M., Bongrand, P., 1984. Cell adhesion: competition between non-specific repulsion and specific bonding. Biophys. J. 45, 1051–1064]. The purpose of this work is the mechanical description of the kinetic of adhesion of a single cell adhering to an extracellular wall, in terms of the failure and creation of connections during the rolling of a single adherent cell subjected to a fluid flow. A 2D model is developed in the present contribution, which describes the local behavior of the contact zone between the cell and the wall. The contact zone is assimilated to two rigid rectilinear beams linked by elastic springs, subjected to the fluid flow and to interaction forces, namely Van der Waals (attractive) and electrostatic (repulsive) forces. The first aspect being investigated concerns the mechanism of failure of the existing connections: the distribution of the failure limits of these fibers is described as a random gaussian field. Moreover, we use a spatial and temporal discretization, thereby assuming that the mass of the cell is concentrated in nodal points, which correspond to the junction points between the cell and the wall. Using the principle of virtual work and a failure criterion, the vibration-failure behavior of the contact interface is described, neglecting mechanical damping in a first step [Mefti, N., Haussy, B., Ganghoffer, J.F., 2005. Mechanical modeling of the rolling phenomenon at the cell scale. Euromech Colloquium 463, Gronnigen, Netherland]. Accounting for the maturation of the fibbers properties results in a pseudo-periodical response due to the competition between the increase of the fibbers stiffness and the increase of the applied pulling effort. The creation of new connections on a part of the cell boundary located opposite to the plasma flow, is integrated into the modeled, and analyzed. A set of free adhesion molecules – ligands and receptors – is distributed on the wall and cell external surfaces. As the ligands are subjected to the Brownian motion of the fluid particles, the orientation and the amplitude of this movement are assume to be described by gaussian random processes. The effects of the chemical affinity are accounted for as molecular Van der Waals interactions. Numerical simulations emphasize the rolling phenomenon, the mechanism of which being the creation and the rupture of the bonds of the contact zone, which involve the rotation of the cell and the relaxation effects of the elastic properties of these connections. [Copyright &y& Elsevier]

The coupling between yarns in a piece of fabric has been analysed at the mesoscopic scale, in terms of its impact on the macroscopic unidirectional behaviour. Starting from a discrete model of a woven structure associated to a variational formulation of the equilibrium of the structure, the coupling between both yarns is introduced, the potential energy of which is calculated. The initial shape of the yarn, represented by a planar undulated beam supposed to be periodic, is described by a Fourier series. The coefficients of the series are expressed vs. the contact force exerted at the top of the undulations, and vs. the mechanical properties of the solicited yarn. The contact force is then expressed vs. the mechanical properties of the transverse yarn and vs. the vertical displacement of the contact point. The potential energy of the coupling is then built, assuming the continuity of the displacement at the contact points. The equilibrium shape of the yarn submitted to unidirectional traction is obtained numerically as the minimum of the total potential energy. The simulated traction curve reproduces in a satisfactorily manner the observed behaviour. The respective contributions of the flexional and extensional effects of the yarn are analysed. The consideration of the coupling enhances the rigidity of the response of the yarn; one demonstrates the effect of the geometrical and mechanical parameters of the transverse yarn. To cite this article: B. Ben Boubaker et al., C. R. Mecanique 331 (2003). [Copyright &y& Elsevier]