Macroscopic Elasticity Tensor Research Articles

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting.

On the identification of material parameters in the relaxed micromorphic continuum

AbstractFor modeling materials with size‐effects, the classical Cauchy linear elasticity theory and first‐order homogenization procedures are insufficient. Enriched continua are able to capture such phenomena in principle. In this work, we model the size‐effects of metamaterial beams via the relaxed micromorphic continuum. The material parameters of the relaxed micromorphic model are evaluated. We present a simple procedure for the determination of the material parameters based on the fact that the model operates between two well‐defined scales that are the classical continuum scales of macroscopic linear elasticity, whose stiffness is given by a known macroscopic elasticity tensor, and a microscopic scale with a microscopic elasticity tensor yet to be identified. The size‐effects of metamaterial cantilevers under traction load are captured successfully.

Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures

In this paper, the Proportional Topology Optimization (PTO) algorithm is extended for the two-scale concurrent topology optimization, in which both the structure and material cellular micro-structure are subject to design. PTO was originally developed on the concept that the amount of material being distributed to an element would be proportional to the contribution of that element in the objective function. Sensitivity analysis is not required. In a two-scale concurrent topology optimization problem, two sets of design variables are defined, one for macro-structure and one for micro-structure. Here, the objective function is reformulated such that the contribution of each micro-scale design variable can be determined, facilitating the employment of PTO. The macroscopic effective elastic tensor is evaluated by the energy-based homogenization method (EBHM), providing a link between micro-structure and macro-structure. Feasibility and efficiency of the proposed PTO approach are demonstrated via several benchmark examples of both two and three dimensional structures.

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.

Linking snow microstructure to its macroscopic elastic stiffness tensor: A numerical homogenization method and its application to 3‐D images from X‐ray tomography

AbstractThe full 3‐D macroscopic mechanical behavior of snow is investigated by solving kinematically uniform boundary condition problems derived from homogenization theories over 3‐D images obtained by X‐ray tomography. Snow is modeled as a porous cohesive material, and its mechanical stiffness tensor is computed within the framework of the elastic behavior of ice. The size of the optimal representative elementary volume, expressed in terms of correlation lengths, is determined through a convergence analysis of the computed effective properties. A wide range of snow densities is explored, and power laws with high regression coefficients are proposed to link the Young's and shear moduli of snow to its density. The degree of anisotropy of these properties is quantified, and Poisson's ratios are also provided. Finally, the influence of the main types of metamorphism (isothermal, temperature gradient, and wet snow metamorphism) on the elastic properties of snow and on their anisotropy is reported.

Microstructure and crystallographic texture of Ranella Olearia Shell

Molluscs are soft-bodied animals, that why many of them invented complex strategies for protecting themselves [1]. One of these strategies consists in creating a rigid biomineral called mollusc shell. Mollusc shells are complex biocomposites of mineral and organic material (5% in volume) with high mechanical performances, compared to the geological mineral. Mollusc shells are mainly made of calcite and aragonite crystalline polymorphs of calcium carbonate. This organic part behaves as nanometer growth-control of the inorganic crystals. This has stimulated chemists and materials scientists to design materials with a microstructure similar to that of nacre. To achieve this, understanding the microstructure of the nacre at various scales is certainly the key [2]. In this work we made use of the Combined Analysis [3] to determine the structure and preferred orientations of constituting aragonite crystallites of the Ranella olerea shell (fig. 1a) layers using scanning electron microscopy and X-rays diffraction. SEM analyses (fig. 1b) show the presence of an inner layer composed of Radial Lamellar, an intermediate comarginal crossed lamellar layer and an outer crossed lamellar layer. The refinement of X-ray diffraction diagrams gives the textures of the three layers, their respective aragonite unit-cell distortions, and the macroscopic elastic tensor of their mineral parts. The textures of the three layers were found to be of high level, especially for the inner layer. Both intermediate and outer layers exhibit regular texture patterns for crossed lamellae with a split of the c-axis component around the normal to the shell. An anisotropic unit-cell distortion is quantified for the three layers which is attributed to the combined effects of inter- and intra-crystalline macromolecules. The simulation of the macroscopic elastic tensor shows that the strong orientations present in the successive layers give an optimisation in terms of rigidity and shear resistance.

COMPUTATIONAL HOMOGENIZATION METHOD AND REDUCED DATABASE MODEL FOR HYPERELASTIC HETEROGENEOUS STRUCTURES

A nonconcurrent multiscale homogenization method is proposed to compute the response of structures made of heterogeneous hyperelastic materials. The method uses a database describing the effective strain energy density function (potential) in the macroscopic right Cauchy-Green strain tensor space. Each value of the database is computed numerically by means of the finite element method on a representative volume element, the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is then introduced to provide a continuous representation of the potential, from which the macroscopic stress and elastic tangent tensors can be derived during macroscopic structures calculations. To efficiently compute the interpolations at the macroscopic scale, the full database is reduced by a tensor product approximation. Several extensions are provided to handle issues related to finite strains. The accuracy of the method is tested through different numerical tests involving composites at finite strains with isotropic or anisotropic microstructures. Second-order accuracy is achieved during the macroscopic Newton-Raphson iterations.

Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis

SUMMARYThis paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a nonconcurrent multiscale approach. The geometry of the microstructure is random. The nonconcurrent multiscale approach for micro‐macro nonlinear mechanics is extended to the stochastic case. Because the nonconcurrent multiscale approach is based on the use of a tensorial decomposition, which is then submitted to the curse of dimensionality, we perform an analysis with respect to the stochastic dimension. The technique uses a database describing the strain energy density function (potential) in both the macroscopic Cauchy green strain space and the geometrical random parameters domain. Each value of the potential is numerically computed by means of the FEM on an elementary cell whose geometry is given by the random parameters and the corresponding macroscopic strains being prescribed as boundary conditions. An interpolation scheme is finally introduced to obtain a continuous explicit form of the potential, which, by derivation, allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations. Two numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.

A two-parameter model of the effective elastic tensor for cortical bone

Multiscale models of cortical bone elasticity require a large number of parameters to describe the organization and composition of the tissue. We hypothesize that the macro-scale anisotropic elastic properties of different bones can be modeled retaining only two variable parameters, and setting the others to universal values identical for all bones. Cortical bone is regarded as a two-phase composite material: a dense mineralized matrix (ultrastructure) and a soft phase (pores). The ultrastructure is assumed to be a homogeneous and transversely isotropic tissue whose elastic properties in different directions are mutually dependent and can be scaled with a single parameter driving the overall rigidity. This parameter is taken to be the volume fraction of mineral fha. The pore network is modeled as an ensemble of water-filled cylinders and described only by the porosity p. The effective macroscopic elasticity tensor Cij(fha,p) is calculated with a multiscale micromechanics approach starting from existing models. The modeled stiffness coefficients compare favorably to four literature datasets which were chosen because they provide the full stiffness tensors of groups of human samples. Since the physical counterparts of fha and p were unknown for the datasets, their values which allow the best fit of experimental tensors by the modeled ones were determined by optimization. Optimum values of fha and p are found to be unique and realistic. These results suggest that a two-parameter model may be sufficient to model the elasticity of different samples of human femora and tibiae. Such a model would in particular be useful in large-scale parametric studies of bone mechanical response.

Topological derivative for multi‐scale linear elasticity models applied to the synthesis of microstructures

AbstractThis paper proposes an algorithm for the synthesis/optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation. The macroscopic elasticity tensor is estimated by a standard multi‐scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element. The algorithm is of simple computational implementation. In particular, it does not require artificial algorithmic parameters or strategies. This is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept. This concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems. The effectiveness of the proposed methodology is illustrated in a set of finite element‐based numerical examples.Copyright © 2010 John Wiley & Sons, Ltd.

Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions

This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula—a symmetric fourth-order tensor field over the RVE domain—is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

Topological sensitivity analysis of a multi-scale constitutive model considering a cracked microstructure

This paper deals with the sensitivity analysis of the macroscopic elasticity tensor to topological microstructural changes of the underlying material. In particular, the microstucture is topologicaly perturbed by the nucleation of a small circular inclusion. The derivation of the proposed sensitivity relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are defined as volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. We consider that the RVE can contain a number of voids, inclusions and/or cracks. It is assumed that non-penetration conditions are imposed at the crack faces, which do not allow the opposite crack faces to penetrate each other. The derived sensitivity leads to a symmetric fourth-order tensor field over the unperturbed RVE domain, which measures how the macroscopic elasticity parameters estimated within the multi-scale framework changes when a small circular inclusion is introduced at the micro-scale level. Copyright © 2009 John Wiley & Sons, Ltd.

A homogenisation-based continuum damage mechanics model for cyclic damage in 3D composites

AbstractThis paper develops a 3D homogenisation based continuum damage mechanics (HCDM) model for fibre-reinforced composites undergoing micromechanical damage under cyclic loading. Micromechanical damage in a representative volume element (RVE) of the material occurs by fibre-matrix interfacial debonding, which is incorporated in the model through a hysteretic bilinear cohesive zone model. The proposed model expresses a damage evolution surface in the strain space in the principal damage co-ordinate system or PDCS. PDCS enables the model to account for the effect of non-proportional load history. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterised as a macroscopic internal variable. Cyclic damage parameters are introduced in the monotonic HCDM model to describe the material degradation due to fatigue. Three dimensional damage in composites is accounted for through functional forms of the fourth order damage tensor in terms of components of macroscopic strain and elastic stiffness tensor. The HCDM model parameters are calibrated from homogenisation of micromechanical solutions of the RVE for a few representative cyclic strain histories. The proposed model is validated by comparing results of the HCDM model with pure micromechanical analysis results followed by homogenisation. Finally, the potential of cyclic HCDM model as a design tool is demonstrated through macro-micro analysis of cyclic damage progression in composite structures.

Sensitivity of the macroscopic elasticity tensor to topological microstructural changes

A remarkably simple analytical expression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topological microstructural changes of the underlying material is proposed. The derivation of the proposed formula relies on the concept of topological derivative, applied within a variational multi-scale constitutive framework where the macroscopic strain and stress at each point of the macroscopic continuum are volume averages of their microscopic counterparts over a representative volume element (RVE) of material associated with that point. The derived sensitivity—a symmetric fourth order tensor field over the RVE domain—measures how the estimated two-dimensional macroscopic elasticity tensor changes when a small circular hole is introduced at the microscale level. This information has potential use in the design and optimisation of microstructures.

Assessment of Mixed Uniform Boundary Conditions for Predicting the Mechanical Behavior of Elastic and Inelastic Discontinuously Reinforced Composites

The combination of heterogeneous volume elements and numerical analysis schemes such as the Finite Element method provides a powerful and well proven tool for studying the mechanical behavior of composite materials. Periodicity boundary conditions (PBC), homogeneous displacement boundary conditions (KUBC) and homogeneous traction boundary conditions (SUBC) have been widely used in such studies. Recently Pahr and Zysset (2008) proposed a special set of mixed uniform boundary conditions for evaluating the macroscopic elasticity tensor of human trabecular bone. These boundary conditions are not restricted to periodic phase geometries, but were found to give the same predictions as PBC for the effective elastic properties of periodic open cell microstructures of orthotropic symmetry. Accordingly, they have been referred to as “periodicity compatible MUBC” (PMUBC). The present study uses periodic volume elements that contain randomly positioned spherical particles or randomly oriented short fibers at moderate volume fractions for assessing the applicability of PMUBC to modeling composite materials via volume elements that deviate from orthotropic symmetry. Macroscopic elasticity tensors are evaluated with PBC, PMUBC and KUBC for elastic contrasts in the range 2 ≤ sr ≤ 30. For one configuration the isotropic contributions to the macroscopic elastic tensors obtained with PBC and PMUBC are extracted and compared. In addition, macroscopic elasticplastic responses for different hardening behaviors are studied with PBC and PMUBC. Only small differences between the predictions obtained with PBC and PMUBC are found, validating the PMUBC for studying volume elements the overall behavior of which shows minor contributions of lower than orthotropic symmetry. keyword: Mixed Uniform Boundary Conditions, Periodicity Boundary Conditions, Finite Element Simulation, Composites 1 Institute of Lightweight Design and Structural Biomechanics, Vienna University of Technology, A-1040 Vienna, Austria

Homogenization based continuum damage mechanics model for monotonic and cyclic damage evolution in 3D composites

This paper develops a 3D homogenization based continuum damage mechanics (HCDM) model for fiber reinforced composites undergoing micromechanical damage under monotonic and cyclic loading. Micromechanical damage in a representative volume element (RVE) of the material occurs by fiber-matrix interfacial debonding, which is incorporated in the model through a hysteretic bilinear cohesive zone model. The proposed model expresses a damage evolution surface in the strain space in the principal damage coordinate system or PDCS. PDCS enables the model to account for the effect of non-proportional load history. The loading/unloading criterion during cyclic loading is based on the scalar product of the strain increment and the normal to the damage surface in strain space. The material constitutive law involves a fourth order orthotropic tensor with stiffness characterized as a macroscopic internal variable. Three dimensional damage in composites is accounted for through functional forms of the fourth order damage tensor in terms of components of macroscopic strain and elastic stiffness tensors. The HCDM model parameters are calibrated from homogenization of micromechanical solutions of the RVE for a few representative strain histories. The proposed model is validated by comparing results of the HCDM model with pure micromechanical analysis results followed by homogenization. Finally, the potential of HCDM model as a design tool is demonstrated through macro-micro analysis of monotonic and cyclic damage progression in composite structures.

Microstructure and crystallographic texture of Charonia lampas lampas shell

Charonia lampas lampas shell is studied using scanning electron microscopy and X-ray diffraction combined analysis of the preferred orientations and cell parameters. The Charonia shell is composed of three crossed lamellar layers of biogenic aragonite. The outer layer exhibits a 001 fibre texture, the intermediate crossed lamellar layer is radial with a split of its c-axis and single twin pattern of its a-axis, and the inner layer is comarginal with split c-axis and double twinning. A lost of texture strength is quantified from the inner layer outward. Unit-cell refinements evidence the intercrystalline organic influence on the aragonite unit-cell parameters anisotropic distortion and volume changes in the three layers. The simulation of the macroscopic elastic tensors of the mineral part of the three layers, from texture data, reveals an optimisation of the elastic coefficient to compression and shear in all directions of the shell as an overall.

Effects of the Orientational Distribution of Cracks in Solids

We derive a theory for the elastic characterization of multicracked solids based on a homogenization technique. We consider a material containing a two-dimensional arbitrary distribution of parallel slit cracks which is elastically equivalent to a crystal with orthorhombic symmetry. We obtain explicit expressions for the macroscopic elastic stiffness tensor which is found to depend upon both the density of cracks and their angular distribution, here described by a suitable order parameter. For the isotropic case, we find that the degradation depends exponentially on the crack density. In addition, we show an unusual elastic behavior of a multicracked medium in the plane strain condition: for a negative Poisson ratio, we obtain an effective Young modulus greater than the actual value of the host matrix.

Optimality criteria method for topology optimization under multiple constraints

The existing framework of optimality criteria method is limited to the optimization of a simple energy functional with a single constraint on material resource. The present work extends the optimality criteria method to the case of multiple constraints. The difficulty in updating the Lagrangian multipliers is treated by gradient-split Taylor series expansion. Applications of the method are illustrated by computing the optimal structures under multiple displacement constraints, and by designing the material cells under given macroscopic elastic tensors that correspond to both positive and negative Poisson's ratios.

X-Ray Diffraction Determination of Residual Stresses in Narrow Copper Interconnects with Submicronic Widths

In relation with the formation of voids or cracks in Cu interconnections for integrated circuits, we are interested to measure the stresses resulting from the thermal cycles during the process. X ray diffraction measurements have been performed on periodic arrays of Cu lines fabricated with the damascene process. The parameter investigated was the linewidth ranging from 0.34 μm to 2.59 μm. Textures analysis has showed that the grains grow from the bottom of the lines when the linewidth is greater than the thickness and from the sidewall in the opposite case. The knowledge of these grains orientations has allowed an evaluation of the macroscopic elastic tensors of the lines taking into account the strong anisotropy of Cu. Starting from crystallographic planes strain measurements, we have then determine the Cu state of stress. We deal with a triaxial tensile state which is quasi hydrostatic when linewidth and thickness are the same. Its level increases with decreasing linewidth.