Isotropic Constituents Research Articles

Analysis of micropolar elastic multi-laminated composite and its application to bioceramic materials for bone reconstruction.

The asymptotic homogenization method is applied to characterize the effective behaviour of periodic multi-laminated micropolar elastic heterogeneous composites under perfect contact conditions. The local problem formulations and the analytical expressions for the effective stiffness and torque coefficients are derived for the centrosymmetric case. One of the main findings in this work is the analysis of the rotations effect of the layers' constitutive properties on the mechanical response of bi-laminated composites. The effects of microstructure and interfacial interactions on the composite's mechanical behaviour are captured through the independent effective moduli. Comparisons with the classical elastic case show the approach validation. Some numerical examples are shown. Furthermore, considering the micropolar media's prevalence in bio-inspired systems, the model's applicability is evaluated for reconstructing bone fractures using multi-laminated biocomposites. An important finding in this bio-inspired simulation is related to the analysis of a periodic bi-laminated micropolar composite whose isotropic constituents are a bioceramic material and a compact bone. This artificial bio-inspired material should integrate with host tissue to support cell growth and be stable and compatible. These characteristics are crucial in the enhancement of the fractured bone.

Effective properties of multi-laminated micropolar composites with Fibonacci and random structures

In this work, the two-scale asymptotic homogenization method (AHM) is developed to describe the effective behavior of multi-laminated elastic micropolar composites with Fibonacci and random structure under perfect contact conditions at the interfaces. The local problem statements over the periodic cell [Formula: see text] are presented, and the corresponding effective stiffness and torque properties are reported. The transversal cross-section of the periodic cell [Formula: see text] is characterized by a laminated structure where the pattern for the layers follows two distinct configurations: (a) a Fibonacci arrangement, and (b) a random sequence focused on the probabilistic binomial function. The non-null effective properties of multi-laminated Cosserat elastic composites with isotropic centro-symmetric constituents are listed. Numerical results for multi-laminated elastic micropolar composites with both types of structures and centro-symmetric isotropic constituents are illustrated and discussed. The overall effective behavior for both cases converges to specific effective values of periodic structures as the number of layers increases.

A semi-analytic method for the computation of the effective properties of composites of two isotropic constituent materials

We develop a computational framework for the semi-analytic representation of the effective transport properties of two-component composite materials, in the spirit of Bergman’s spectral representation. The components of the effective permittivity (or conductivity) tensor are expanded as power series in a contrast parameter between the two phases. The coefficients (the moments of a spectral measure) of these expansions are determined recursively through the numerical evaluations of integrals defined solely on the boundary between the constituent phases, and discretized by high-order (possibly exponentially converging) quadrature schemes. The high precision reached by these coefficients allows the series representation to be recast into Padé approximants (or continued fractions) to provide analytical expressions valid over large (possibly infinite) regions of the parameter complex plane. Examples, thus far limited to two-dimensional rhombic unit cells, demonstrate that the method is reliable for microgeometries and material parameters for which other methods are far less dependable. The analytical representations are particularly useful to study the optical or electrostatic resonant behaviour of some composite materials.

Effective properties of centro-symmetric micropolar composites with non-uniform imperfect contact conditions

In this work, the homogenization theory is addressed within the framework of three-dimensional linear micropolar composite materials with centro-symmetric constituents and non-uniform imperfect interface conditions. The imperfect contact conditions are modeled like a generalization of the spring model, where tractions and coupled stresses are continuous, but displacements and microrotations are discontinuous across the interface. The jumps in displacement and microrotation components are proportional to the interface traction and coupled stress components in terms of a partition of different spring-factor-type interface parameters, respectively. The two-scale asymptotic homogenization method (AHM) is developed, through series expansions for displacements and micro-rotations, to find the analytical statement of the local problems on the periodic cell and the corresponding effective coefficients. In particular, centro-symmetric multi-laminated micropolar composites with non-uniform imperfect contact conditions are studied, and their corresponding effective properties are explicitly declared. Numerical results show the effects of the interface partition lengths, the non-uniform imperfection values, and the constituent’s fraction volumes on the effective properties of centro-symmetric bi-laminated composite with isotropic constituent materials. We also analyze and discuss the effective behaviors illustrated in the results. In general, the effective properties are always affected by a non-uniform imperfect interface, and they are bounded between those achieved when the contact conditions are perfect and imperfect uniform. The reported formulas and data may be helpful as benchmarks for checking other experimental and numerical results.

Effect of imperfect interface on the effective properties of elastic micropolar multilaminated nanostructures

AbstractIn this paper, the problem of a heterogeneous elastic micropolar nanostructure with a periodic structure subject to imperfect contact conditions is analyzed through the two‐scale asymptotic homogenization method (AHM). The imperfect interface is modeled as a generalization of the well‐known spring model; that is, the homogeneous imperfect interface is described by the following conditions: tractions and coupled stress are continuous, but displacements and microrotations are discontinuous across the imperfect interface. The jumps in displacements and microrotations are proportional to the interface traction and coupled stress components, respectively. In particular, micropolar multilaminated nanocomposites with centro‐symmetric isotropic constituents and imperfect contact conditions are studied. From AHM, the solutions for the displacement and microrotation fields are found by means of two‐scale series expansions depending on a local (microscopic) variable and a global (macroscopic) variable. The local problem statements and the corresponding effective properties are explicitly described. The formulation depends on the constituent physical properties, the imperfection parameters, the cell length in the y3‐direction, and the phase's volume fractions. Numerical results are illustrated and discussed. We concluded that the effective moduli are affected by the imperfections and the cell length in the y3‐direction.

Surface instability of sheared active skeletal muscle tissue with loss of muscle mass

In the present work, we study the surface instability of the wrinkling kind for the active-elastic skeletal muscle tissue subject to shear when the tissues are modeled by the mixture active strain approach. We consider two different models for the strain energy density functions, one as a sum of an incompressible neo-Hookean for the isotropic tissue constituents with the shear modulus μ and the anisotropic standard reinforcing model with the fiber stiffness E, and the other as a sum of the Gent material as an isotropic part and a function of the anisotropic variant I4 to model the anisotropic part. In both models, we introduce the activation parameter γ which describes the microstructural degree of contraction of the muscle. With the help of incremental deformation over the equations of equilibrium and incremental boundary conditions in the form of sinusoidal perturbations, we investigate the solutions for the surface instabilities of the wrinkling kind. First, we show that the critical amount of shear for instability, Kcr, decreases with an increasing E/μ ratio for the first model and the surface loses its stability with an increase in γ for a given E/μ ratio up to that angle Φ between the fibers and the direction of shear at which Kcr abruptly drops to a very low value for γ=0. For the second model, we find that Kcr increases with an increase in limiting chain extensibility of the Gent model and it either increases or almost remains constant with an increase in γ. Then we also introduce a decay deformation tensor involving a dimensionless parameter g into the multiplicative decomposition of the deformation gradient tensor, which helps model the loss of muscle mass during sarcopenia. We find that Kcr is almost independent with an increase in g for the first model but decreases with the increase in g for the second model for most Φ’s. Based on our results, we suggest that the second model is more compatible with the intuitive reasoning that the stability of the skeletal muscle surface should decrease with an increase in the loss of muscle mass.

The multiscale meso-mechanics model of viscoelastic cortical bone.

Cortical bone is a complex hierarchical structure consisting of biological fiber composites with transversely isotropic constituents, whose microstructures deserve extensive study to understand the mechanism of living organisms and explore development of biomimetic materials. Based on this, we establish a three-level hierarchical structure from microscale to macroscale and propose a multiscale micromechanics model of cortical bone, which considers Haversian canal, osteonal lamellae, cement line and interstitial lamellae. In order to study the microstructural effect on the elastic behavior of hierarchical structures, the Mori-Tanaka model and locally exact homogenization theory are introduced for the homogenization of heterogeneous materials of microstructure at each level. Within sub-microscale, Haversian canal and Osteonal lamella are treated as fiber and matrix, whose homogenization is surrounded with cement line matrix in microstructure (or what we called "osteon") for the second homogenization; finally, osteon and interstitial lamella establish the meso-structure for the third homogenization, predicting the effective moduli of cortical bone. The correctness of the model in this paper is verified against the data in literature with good agreement. Finally, the dynamic viscoelastic response of cortical bones is investigated from a multiscale perspective, where the measured data are substituted into the present models to study the hydration and aging effect on bones' stiffness and viscoelasticity. It is demonstrated that the hydration is much more influential in affecting the storage and loss moduli of cortical bone than the aging effect. We also present a few numerical investigations on microstructural material and geometric parameters on the overall mechanical properties of cortical bone.

Closed-form expressions for the effective properties of piezoelastic composites reinforced with cylindrical fibers by Maxwell scheme

In this work, the Maxwell scheme is implemented to characterize the effective properties of piezoelastic fiber composites reinforced by unidirectional cylindrical fibers distributed along the x3-axis. The piezoelastic composites are assumed to have transversely isotropic constituent materials. Analytical expressions for the effective properties (elastic, piezoelectric, and dielectric) are reported. The Mori–Tanaka approach is also developed for piezoelastic composites and it is shown that their analytical results coincide with the Maxwell scheme and the effective field method. Numerical analysis and comparisons with the two-scale asymptotic homogenization method and the semi-analytical finite element method are shown and discussed. As a summary, the novelty value of this work is that the Maxwell and Mori–Tanaka schemes have been extended to heterogeneous piezoelectric structures. Closed formulas are derived for the effective elastic, piezoelectric, and dielectric moduli. A numerical implementation of the effective moduli is carried out and the overall properties are validated with other methods.

Macro-modelling of orthotropic damage in masonry: Combining micro-mechanics and continuum FE analysis

• An efficient homogenisation scheme is proposed and validated. • Orthotropic properties of masonry are calculated from its isotropic constituents. • Stresses and strains in joints and units are calculated from macroscopic strain. • Damage due to mechanical action on the joints and units can be simply quantified.

Micro–macro asymptotic approach applied to heterogeneous elastic micropolar media. Analysis of some examples

In this work, the derivation of the effective properties for heterogeneous micropolar media with periodic structure using the two-scale asymptotic homogenization method (AHM) is reported. Analytical expressions for the local problems and the effective coefficients are explicitly described. As a particular case, periodic laminated composites are also analyzed, focusing on centro-symmetric Cosserat composites with isotropic and cubic constituents. Also, closed-form formulae of the effective properties are obtained for both constituent symmetries, and numerical values are reported and discussed. The resulting composite belongs to the orthotropic symmetry under rotations of 90°about the unitary vector e3, i.e., it has eighteen effective independent properties: nine stiffness and nine torques. As a limit case, a comparison between classical and Cosserat effective elastic properties is shown for a laminated composite with isotropic constituents. Finally, the engineering moduli of centro-symmetric laminated Cosserat materials with isotropic and cubic constituents are reported, and the numerical values are analyzed.

Viscoelastic-viscoplastic polymer composites: Development and evaluation of two very dissimilar mean-field homogenization models

This paper deals with the micromechanical modeling of polymer composites with viscoelastic-viscoplastic (VE-VP) constituents. Two mean-field homogenization (MFH) models based on completely dissimilar theoretical approaches are extended from elasto-viscoplasticity (EVP) to VE-VP and assessed. The first approach is the incremental-secant method. It relies on a fictitious unloading of the composite at the beginning of each time step. Then, a thermoelastic-like Linear Comparison Composite (LCC) is constructed from the computed residual state directly in the time domain. The method provides naturally isotropic per-phase incremental-secant operators for isotropic VE-VP constituents. It takes into account both the first and the second statistical moment estimates of the equivalent stress micro-field. The second approach is the integral affine method. It starts by linearizing the rates of viscoplastic strain and internal variables. The linearized constitutive equations are then recast in a hereditary integral format to which the Laplace-Carson (L-C) transform is applied. A thermoelastic-like LCC is built in the L-C domain, where MFH is carried out. Finally, the composite’s response in the time domain is recovered by numerical inversions of L-C transforms. The method is able to overcome the issue of heterogeneous viscous stresses encountered by time domain MFH models. The two proposed MFH formulations are able to handle non-monotonic, non-proportional and multi-axial loading histories. Their accuracy was assessed against full-field finite element (FE) results for different microstructures and loadings. The computational cost of both methods is negligible compared to FE analyses. Overall, the incremental-secant approach is much simpler mathematically and numerically than the integral affine formulation, its accuracy ranges from acceptable to excellent, and important improvements can be expected in the future by controlling the virtual unloading time increment.

Ellipsoidal inhomogeneity in elliptically orthotropic elastic solid

An unbounded elliptically orthotropic elastic solid containing a single anisotropic ellipsoidal inhomogeneity is considered. The problem statement assumes arbitrary orientation of ellipsoid with respect to the orthotropy axes and non-uniform far loading. By applying the appropriate affine transformations, the boundary value problem is reduced to that one with the isotropic constituents. The complete displacement solution to the transformed problem is obtained using the Papkovich-Neuber representation with the scalar potentials written in terms of ellipsoidal solid harmonics. By accurate fulfilling the interface conditions, the boundary value problem is reduced to the system of the linear algebraic equations for the constants entering the potentials. For the Eshelby problem (uniform remotely applied loading and perfect interface), an explicit analytical solution is written in terms of ellipsoidal harmonics. This result yields solution for the elliptical crack problem in the remarkably simple way. The numerical examples illustrate an effect of the matrix anisotropy on the stress concentration and stress intensity factors. An application of the obtained solution to the micromechanics is discussed. The effective elastic stiffness tensor of a particulate composite with elliptically orthotropic matrix is evaluated using Maxwell homogenization scheme.

Numerical algorithm of multipole expansion method for conductivity of ellipsoidal particle composite

The paper addresses the numerical implementation of the multipole expansion method in application to the conductivity problem for the ellipsoidal particle composite with isotropic constituents and imperfect interfaces. The main steps of the numerical algorithm are considered in detail including generation of the geometry model, evaluating the re-expansion coefficients, fulfilling the periodicity and interface conditions and determination of the series expansion coefficients by iterative solving a set of linear equations. For each step, the numerical tests are performed which illustrate an accuracy and efficiency of the method. The rational computational strategies and the ways of boosting the computer code performance are discussed. The algorithm-based code provides a fast and accurate analysis of the local potential fields and the effective conductivity of composite with an adequate account for the arrangement and orientation of inhomogeneities. The imperfect thermal contact between the matrix and inhomogeneities is taken into account accurately. The application area covers the polydisperse and multiphase composites of matrix type. Its extension to the ellipsoidal particle composites with anisotropic constituents is straightforward. Incorporation of this feature yields possibly the most general model of composite that may be considered in the framework of analytical approach, and the proposed numerical algorithm is by far the most efficient method for studying these models.

Simultaneous focusing and rotation of a bifunctional thermal metamaterial with constant anisotropic conductivity

We show that a constant anisotropic conductivity tensor can lead to two different functionalities at the same time, thermal focusing or a shielding effect together with a rotating effect. This design is of practical applications, as in energy harvesting and dissipation devices, and it is desirable to concentrate or to shield the thermal energy from a selected direction. We demonstrate that it is possible to control the way that heat flows within a medium to achieve the bifunctional effect solely through a suitable balance among the three material parameters of a homogeneous medium. It is seen that the diagonal entries of the conductivity tensor will govern the thermal focusing or shielding effect, while the off diagonal terms control the rotational effect. To fabricate the designed anisotropic material, we show that the material can be realized via the construction of a layered medium with two alternating isotropic constituents. Theoretical analysis and numerical calculations based on finite element calculations are presented to validate the performance of the design. The use of uniform conductivity to manage heat conduction brings exciting new advances for practical applications, with which one can manipulate thermal energy and offer unprecedented control of heat conduction at the same time.

A modulus gradient model for inhomogeneous materials with isotropic linear elastic constituents

The one-dimensional modulus gradient (E-grad) model proposed in Gülaşık et al. (2018) is extended to more general three-dimensional inhomogeneous materials with isotropic linear elastic constituents. In addition to the constitutive equations and balance relations, length scale dependent differential relations for the material parameters of isotropic linear elasticity are provided. The finite element formulation for axisymmetric problems is derived and a model problem of a soft cylindrical rod with a stiff spherical inclusion is solved. It is seen that discontinuities and/or very sharp changes in the modulus, displacement, strain and stress fields that exist in local formulations are smoothed with the proposed gradient model. Furthermore, a graded interphase region around the stiff inclusion is obtained by the model as reported in the literature of polymer nanocomposites. The thickness of the graded interphase region is interpreted as a characteristic length scale of the polymer nanocomposite. It is also shown that increasing the internal length scale parameter increases the stiffness of the model. The proposed model is compared with a micromechanical model from literature and experiments conducted with polyimide/silica nanocomposites. The results obtained by the proposed approach capture the experimentally measured values of the nanocomposite modulus. Finally, the model is extended to obtain anisotropic macroscopic response by choosing different length scale parameters in different directions.

Materials knowledge system for nonlinear composites

In this contribution, we present a new Materials Knowledge System framework for microstructure-sensitive predictions of effective stress–strain responses in composite materials. The model is developed for composites with a wide range of combinations of strain hardening laws and topologies of the constituents. The theoretical foundation of the model is inspired by statistical continuum theories, leveraged by mean-field approximation of self-consistent models, and calibrated to data obtained from micromechanical finite element simulations. The model also relies on newly formulated data-driven linkages between micromechanical responses (phase-average strain rates and effective strength) and microstructure as well as strength contrast of the constituents. The paper describes in detail the theoretical development of the model, its implementation into an efficient computational plasticity framework, calibration of the linkages, and demonstration of the model predictions on two-phase composites with isotropic constituents exhibiting linear and power-law strain hardening laws. It is shown that the model reproduces finite element results reasonably well with significant savings of the computational cost.

Interfacial characteristics of hybrid nanocomposite under thermomechanical loading

AbstractIn this work, an improved shear lag model was developed to investigate the interfacial characteristics of three-phase hybrid nanocomposite which is reinforced with microscale fibers augmented with carbon nanotubes on their circumferential surfaces. The shear lag model accounts for (i) radial and axial deformations of different transversely isotropic constituents, (ii) thermomechanical loads on the representative volume element (RVE), and (iii) staggering effect of adjacent RVEs. The results from the current newly developed shear lag model are validated with the finite element simulations and found to be in good agreement. This study reveals that the reduction in the maximum value of the axial stress in the fiber and the interfacial shear stress along its length become more pronounced in the presence of applied thermomechanical loads on the staggered RVEs. The existence of shear tractions along the RVE length plays a significant role in the interfacial characteristics and cannot be ignored.

Multi-scale modal analysis for axisymmetric and spherical symmetric structures with periodic configurations

A new modal analysis method with second-order two-scale (SOTS) asymptotic expansion is presented for axisymmetric and spherical symmetric structures. The symmetric structures considered are periodically distributed with homogeneous and isotropic constituent materials. By the asymptotic expansion of the eigenfunctions, the homogenized modal equations, the effective materials coefficients, the first- and second-order correctors are obtained. The derived homogenized constitutive relationships are the same as the ones which serve to homogenize the corresponding static problems. The eigenvalues are also expanded to the second-order terms and using the so called “corrector equation”, the correctors of the eigenvalues are expressed in terms of the first- and second-order correctors of the eigenfunctions. The anisotropic materials are obtained by homogenization with different properties in the circumferential direction. Especially for the two-dimensional axisymmetric layered structure, the one-dimensional plane axisymmetric and spherical symmetric structures, the homogenized eigenfunctions and eigenvalues, as well as their corresponding correctors are all solved analytically. The finite element algorithm is established, three typical numerical experiments are carried out and the necessity of the second-order correctors is discussed. Based on the numerical results, it is validated that the SOTS asymptotic expansion homogenization method is effective to identify the eigenvalues of the axisymmetric and spherical symmetric structures with periodic configurations and the original eigenfunctions with periodic oscillation can be reproduced by adding the correctors to the homogenized eigenfunctions.

Static Analysis of Al - ZrO2FG Thick Plate Using Graded FEM

Functionally graded materials(FGM) have attracted a lot of attention these days on account of their diversified and potential applications in aerospace and other industries. Hence much focus is on Functionally Graded material structural forms. The Mindlin plate theory which includes the first order shear deformation effects is very much appropriate for thick plates, was taken up to analyze the behaviour of Al-ZrO2FG plates subjected to static loads. The FG plates are made up of two isotropic constituents and variation in properties is in thickness direction, in accordance with power law. Graded Finite element method which accommodates for continuous variation in material properties of the elements was resorted to for the numerical analysis. Tailored MATLAB code was used for obtaining solutions. A parametric study with different power-law indices, thickness ratios, aspect ratios, support conditions and load parameters on non-dimensional centroidal deflection had been performed. The obtained results are in good agreement with published results.

Computation of effective properties in elastic composites under imperfect contact with different inclusion shapes

A fibrous elastic composite is considered with transversely isotropic constituents. Three types of fibers are studied: circular, square, and rhombic. Fibers are distributed with the same periodicity along the two perpendicular directions to the fiber orientation, that is, the periodic cell of the composite is square. The composite exhibits imperfect contact at the interface between the fiber and matrix. Effective properties of this composite are calculated by means of a semi‐analytic method, that is, the differential equations that described the local problems obtained by asymptotic homogenization method are solved using the finite element method. The finite element formulation can be applied to any type of element; particularly, three approaches are used: quadrilateral element of 4 nodes, quadrilateral element of 8 nodes, and quadrilateral element of 12 nodes. Numerical computations are implemented, and different comparisons are presented. Copyright © 2016 John Wiley & Sons, Ltd.