Hill Mandel Condition Research Articles

Analytical formulae for the higher-order effective coefficients of periodic laminated elastic composites with imperfect contact

Analytical formulae are derived for the effective coefficients of linear elastic second-order laminate composite materials made of any finite number of linear elastic first-order layers. Imperfect contact conditions are considered at the interfaces, and small strains are assumed at the micro and macro scales. The “micro-macro” homogenization procedure used here is reported in Continuum Mech. Thermodyn. (2020) 32:1251-1270 wherein only numerical studies are presented. Asymptotic homogenization method results at the micro-scale level are combined with the macro-scale parameters based on the equivalence of the stored energies on the periodic cell (i.e., the energy-averaging theorem known as the Hill-Mandel condition). Some comparisons are considered for validation. To the best of our knowledge, the fully analytical application of this methodology to the case of laminate media has not been reported previously, that is, with the analytical solution of the local problems. The formulas obtained could be useful to control numerical codes in more complex periodic cells. Projects PAPIIT DGAPA UNAM IN101822, Mexico, and CNPq Universal Nº 402857/2021-6, Brazil, are gratefully acknowledged. CAPES Brazil Program (PRAPG) – Edital No 14/2023 is also recognized.

Onset of dynamic void coalescence in porous ductile solids

This paper investigates void growth and coalescence in porous ductile solids under dynamic loading condition. A physical definition for the onset of void coalescence in porous ductile solids under dynamic loading is proposed. The onset is deemed to occur when the third invariant of the tensorial form of the Hill–Mandel condition attains a zero value. The definition allows for systematic investigations on the effects of dimensionless stress rate κ and stress state, defined by the stress triaxiality T and Lode parameter L, on the onset of void coalescence via micromechanical analyses. The analyses reveal that the critical macroscopic effective strain for the onset of void coalescence displays an increasing–decreasing transition trend as the dimensionless stress rate increases, for all levels of T and L considered. The macroscopic effective strain at the transition is identified as the “ductile–brittle” transition strain. The dimensionless stress rate at which the transition strain occurs is found to be relatively constant. A mapping in the κ−T space for L=−1, representative of a generalized uniaxial tension typical in spall fracture experiments, is established which depicts regions where coalescence and non-coalescence can take place, as well as the ductile–brittle regions demarcated by a ductile–brittle transition curve. The results also show that the critical void volume fraction and macroscopic effective strain at the onset of void coalescence are insensitive to inertia at high stress triaxialities at L=−1.

Homogenisation method based on energy conservation and independent of boundary conditions

The foundation of homogenisation methods rests on the postulate of Hill–Mandel, describing energy consistency throughout the transition of scales. The consideration of this principle is therefore crucial in the discipline of Digital Rock Physics which focuses on the upscaling of rock properties. For this reason, numerous studies have developed numerical schemes for porous media to enforce the Hill–Mandel condition to be respected. The most common method is to impose specific boundary conditions, such as periodic ones. However, these boundary conditions influence both the effective property and the size of the REV. The recent study of Thovert and Mourzenko (2020) has shown that most boundary conditions still result in the same intrinsic effective physical property if the averaging is applied outside the range of the boundary layer. From this discovery, it becomes logical to question the status of Hill–Mandel postulate in porous media when homogenising away from the boundary. In this contribution, we simulate Stokes flow through random packings of spheres and a range of rock microstructures. For each, we plot the evolution of the ratio micro- vs macro-scale of the energy of the fluid transport outside the boundary layer, for a growing subsample size of porous media. Here, we prove that we naturally find energy consistency across scales when reaching the size of the Representative Elementary Volume (REV), which is a known condition for rigorous upscaling. Furthermore, we show that this index for the energy consistency is a more accurate indicator of REV convergence since the mean value is already known to be unitary.

Homogenization assumptions for the two-scale analysis of first-order shear deformable shells

This contribution presents a multiscale approach for the analysis of shell structures using Reissner–Mindlin kinematics. A distinctive feature is that the thickness of the representative volume element (RVE) corresponds to the shell thickness. The main focus of this paper is on the choice of correct boundary conditions for the RVE. Three different types of boundary conditions, which fulfil the Hill–Mandel condition, are presented to bridge the two scales. A common feature is the application of zero-traction boundary conditions at the top and bottom surfaces of the RVE. Furthermore, an internal constraint is used to reduce the dependency of the stiffness components on the RVE size. The introduced boundary conditions differ mainly in the application of shear strains and their symmetry requirements on the RVE. The characteristic features are compared by means of linear-elastic benchmark tests. It is shown that the stress resultants and tangent stiffness components are obtained correctly. Moreover, the presented approach is verified using different macroscopic shell structures and different mesostructures. Both, linear and nonlinear small strain examples are compared to analytical values or full-scale solutions and demonstrate a wide applicability of the present formulation.

Second-order computational homogenisation enhanced with non-uniform body forces for non-linear cellular materials and metamaterials

Although “classical” multi-scale methods can capture the behaviour of cellular, including lattice, materials, when considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale cannot capture localisation bands inherent to cell buckling propagation. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size, which is problematic when the RVE has to consider several cells to recover periodicity during local instability. In this paper we reformulate in a finite-strain setting the second-order computational homogenisation using the idea of equivalent homogenised volume. From this equivalence, arises at the micro-scale a non-uniform body force that acts as a supplementary volume term over the RVE. In the presented method, this non-uniform body-force term arises from the equivalence of energy, i.e. the Hill–Mandel condition, between the micro- and macroscopic volumes and depends mainly on the relation between the micro-scale and macro-scale deformation gradient. We show by considering elastic and elasto-plastic metamaterials and cellular materials that this approach reduces the RVE size dependency on the homogenised response.

An extended Hill’s lemma for non-Cauchy continua based on the modified couple stress and surface elasticity theories

An extended Hill’s lemma is provided for non-Cauchy continua satisfying the modified couple stress theory in the bulk and the surface elasticity theory in the surface layer. Based on the Hill–Mandel condition, four sets of boundary conditions (BCs) are identified. It is shown that each of these four sets of BCs satisfies the admissibility and average field requirements. In addition, the set of kinetic BCs meets the force and moment balance conditions, and the set of modified kinematic BCs satisfies the displacement and micro-rotation compatibility constraint. By applying the newly extended Hill’s lemma, a homogenization analysis of a two-phase composite is performed, in which the average strain energy of the composite is computed using a finite element model constructed using COMSOL. The effective elastic constants obtained in this analysis are found to satisfy the Voigt and Reuss bounds.

Unified mean-field modeling of viscous short-fiber suspensions and solid short-fiber reinforced composites

Mean-field homogenization is an established and computationally efficient method estimating the effective linear elastic behavior of composites. In view of short-fiber reinforced materials, it is important to homogenize consistently during process simulation. This paper aims to comprehensively reflect and expand the basics of mean-field homogenization of anisotropic linear viscous properties and to show the parallelism to the anisotropic linear elastic properties. In particular, the Hill–Mandel condition, which is generally independent of a specific material behavior, is revisited in the context of boundary conditions for viscous suspensions. This study is limited to isothermal conditions, linear viscous and incompressible fiber suspensions and to linear elastic solid composites, both of which consisting of isotropic phases with phase-wise constant properties. In the context of homogenization of viscous properties, the fibers are considered as rigid bodies. Based on a chosen fiber orientation state, different mean-field models are compared with each other, all of which are formulated with respect to orientation averaging. Within a consistent mean-field modeling for both fluid suspensions and solid composites, all considered methods can be recommended to be applied for fiber volume fractions up to 10%. With respect to larger, industrial-relevant, fiber volume fractions up to 20%, the (two-step) Mori–Tanaka model and the lower Hashin–Shtrikman bound are well suited.

Exact second moments of strain for composites with isotropic phases

In micromechanical approaches to failure prediction of inhomogeneous microstructures or homogenization of inelastic materials, local field fluctuations beyond the mean field are relevant, but not easily accessible except for resorting to full-field simulations. However, any analytical estimate of the effective strain energy density combined with the Hill–Mandel condition implies some information about the probability distributions of local fields such as stresses and strains. In this paper, analytical equations for the second moments of local fields are derived for particulate composites with isotropic phases. Explicit expressions are provided using the Mori–Tanaka approach, the Hashin–Shtrikman bounds and the singular approximation. The results are compared to full-field simulations of a random sphere assemblage. As the Hashin–Shtrikman covariance is isotropic, it is not well-suited for predicting the behavior under deviatoric loadings. The singular approximation and Mori–Tanaka methods yield sensible approximations for the second moment of matrix strain.

A kinematically consistent second-order computational homogenisation framework for thick shell models

This paper presents a kinematically consistent second-order computational homogenisation scheme for shear deformable shells. The proposed framework can accurately evaluate the membrane, bending, and transverse shear components of the shell resultants and tangent operators, whilst showing no size dependency on the fine scale model. To date, a proper extension of second-order homogenisation to a thick shell model, such as the 5-parameter formulation, remains non-trivial due to the difficulties in projecting the macroscopic transverse shear strains to the fine scale whilst satisfying the stress boundary conditions on the top and bottom faces. To overcome this, the paper proposes a novel volumetric constraint on the fluctuation moment field, that is used in conjunction with a set of constraints obtained through an orthogonality condition. The result is a consistent downscaling procedure that can properly downscale all the macroscopic shell strains. More notably, a pure transverse shear deformation can be achieved, thus producing a uniform parabolic shear stress distribution in homogeneous materials. The key equations for upscaling are obtained through the modified Hill–Mandel condition. In particular, the shell tangent operators are derived in closed form as a function of the Taylor upper bound and softening terms coming from the fluctuation matrices. The proposed framework is general and can incorporate full geometric and material nonlinearities across all the length scales of interest. Through a series of benchmarks, it is demonstrated that the constitutive tangents computed through the present framework correspond well with analytical solutions. Finally, excellent agreements in model responses, including through-thickness stress distributions are found between the multi-scale (FE2) and the full-scale models in the numerical benchmarks, featuring the nonlinear loading of thin and thick heterogeneous panels.

NTFA-enabled goal-oriented adaptive space–time finite elements for micro-heterogeneous elastoplasticity problems

In this work, we establish a goal-oriented space–time finite element method for a class of dissipative heterogeneous materials. Those materials are modeled on both micro- and macroscale, with a scale transition of volume averaging type satisfying the Hill–Mandel condition. A nonuniform transformation field analysis is performed on the microscopic inelastic strain fields for a model reduction. Reduced variables are deduced from a space–time decomposition of those inelastic strain fields. Closed-form constitutive relations are derived from some dissipative considerations, thus resulting into a reduced order homogenization problem. The resulting model error is sufficiently small for the considered class of materials, thus leaving the discretization error of the finite element method as a main error source. For ease of error estimate, we rewrite the reduced order problem in a multifield formulation. Based on duality techniques, a backward-in-time dual problem is derived from a Lagrange method, rendering error representations of a user-defined quantity of interest. Combining a patch recovery technique, a computable error estimator is developed to quantify both spatial and temporal discretization errors. By means of a localization technique, local error estimators are used to drive a greedy adaptive refinement algorithm in space and time. The effectiveness of the resulting algorithm is confirmed by several numerical examples w.r.t. a prototype model.

Scaling in Anti-Plane Elasticity on Random Shear Modulus Fields with Fractal and Hurst Effects

The scale dependence of the effective anti-plane shear modulus response in microstructures with statistical ergodicity and spatial wide-sense stationarity is investigated. In particular, Cauchy and Dagum autocorrelation functions which can decouple the fractal and the Hurst effects are used to describe the random shear modulus fields. The resulting stochastic boundary value problems (BVPs) are set up in line with the Hill–Mandel condition of elastostatics for different sizes of statistical volume elements (SVEs). These BVPs are solved using a physics-based cellular automaton (CA) method that is applicable for anti-plane elasticity to study the scaling of SVEs towards a representative volume element (RVE). This progression from SVE to RVE is described through a scaling function, which is best approximated by the same form as the Cauchy and Dagum autocorrelation functions. The scaling function is obtained by fitting the scaling data from simulations conducted over a large number of random field realizations. The numerical simulation results show that the scaling function is strongly dependent on the fractal dimension D, the Hurst parameter H, and the mesoscale δ, and is weakly dependent on the autocorrelation function. Specifically, it is found that a larger D and a smaller H results in a higher rate of convergence towards an RVE with respect to δ.

Efficient prediction of the effective nonlinear properties of porous material by FEM-Cluster based Analysis (FCA)

The present paper extends the FEM-Cluster based Analysis (FCA) method to the efficient prediction of effective properties of porous materials in the nonlinear range. Different from the traditional methods of homogenization or mean field theory, this new approach does not fill the pore with low modulus materials which causes numerical inaccuracy and convergence difficulties as reported in several literature. With no material filling in pores, the average strain theorem and the Hill–Mandel condition which is the basis of the Representative Volume Element (RVE) method for the prediction of effective properties of heterogeneous materials is almost impossible to implement. Furthermore, because of the lack of pore strain information, the compatibility condition of cluster strains cannot be applied in the incremental analysis of reduced order model in the nonlinear stage. In the present study we have overcome these difficulties by several approaches. First, it is found that the energy equivalence principle provides a more suitable basis for effective properties prediction of porous materials. Second, the self-equilibrium cluster stress field obtained from the interaction matrix and the cluster-based minimum complementary energy in FCA can be successfully applied to the incremental analysis of reduced order model in the nonlinear stage instead of the strain compatibility condition. However, due to the errors introduced by averaging stress and strain over each cluster, the interaction matrix becomes non-singular and the self-equilibrium properties of the cluster stress field generated from the interaction matrix become very poor. Fortunately, the most exciting discovery of the present study is that we have found a way to recover the singularity of the interaction matrix by minimizing its rank on the premise that the elastic energy norm of RVE remains unchanged within the given tolerance. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed approach.

Numerical homogenization of second gradient, linear elastic constitutive models for cubic 3D beam-lattice metamaterials

Generalized continuum mechanical theories such as second gradient elasticity can consider size and localization effects, which motivates their use for multiscale modeling of periodic lattice structures and metamaterials. For this purpose, a numerical homogenization method for computing the effective second gradient constitutive models of cubic lattice metamaterials in the infinitesimal deformation regime is introduced here. Based on the modeling of lattice unit cells as shear-deformable 3D beam structures, the relationship between effective macroscopic strain and stress measures and the microscopic boundary deformations and rotations is derived. From this Hill–Mandel condition, admissible kinematic boundary conditions for the homogenization are concluded. The method is numerically verified and applied to various lattice unit cell types, where the influence of cell type, cell size and aspect ratio on the effective constitutive parameters of the linear elastic second gradient model is investigated and discussed. To facilitate their use in multiscale simulations with second gradient linear elasticity, these effective constitutive coefficients are parameterized in terms of the aspect ratio of the lattices structures.

A strain energy-based homogenization method for 2-D and 3-D cellular materials using the micropolar elasticity theory

A new strain energy-based method for homogenization of 2-D and 3-D cellular materials is developed using an extended version of Hill’s Lemma for the non-Cauchy continuum satisfying the micropolar elasticity theory. Both kinematic and mixed boundary conditions (BCs) that obey the Hill-Mandel condition are identified. The newly proposed method requires that the nodal equilibrium equations be explicitly satisfied at all boundary and interior nodes, unlike the existing approach which does not enforce the equilibrium conditions at interior nodes. For 2-D cellular materials, it is shown that purely kinematic BCs (with prescribed displacement and micro-rotation vectors) and mixed BCs (with prescribed couple stress traction and displacement vectors) can both be used for homogenization, and periodic constraints can be readily accommodated in the former. For 3-D cellular materials, it is demonstrated that mixed BCs (with prescribed couple stress traction and displacement vectors) enable the homogenization satisfying the Hill-Mandel condition. Two sample cases of homogenization of cellular materials are studied by applying the new method – one for a 2-D lattice structure and the other for a 3-D pentamode material. For the 2-D lattice material, the effective classical and micropolar stiffness tensors are obtained using purely kinematic BCs (with and without periodic constraints) and mixed BCs. The closed-form expressions of the effective stiffness tensor components derived here are compared to those given by the existing strain energy-based homogenization method and are seen to be more accurate. For the 3-D pentamode material, mixed BCs are employed in the homogenization, and the effective stiffness tensor components are obtained in closed-form expressions for the first time. The numerical results for the pentamode material given by these analytical formulas are found to agree well with those provided by a finite element model constructed using COMSOL.

A study on equivalence of nonlinear energy dissipation between first-order computational homogenization (FOCH) and reduced-order homogenization (ROH) methods

Nowadays, studies on the mechanism of macro-scopic nonlinear behavior of materials by accumulation of micro-scopic degradation are attracting more attention from researchers. Among numerous approaches, multiscale methods have been proved as powerful and practical approaches in predicting macro-scopic material status by averaging and homogenizing physical information from associated micro-scopic material behavior. Usually in mechanical problem, the stress, consistent material modulus, and possible material state variables are quantities in interest through the upscaling process. However, the energy-related quantities are not studied much. Some initiative work has been done in the early year including but not limited to the Hill–Mandel condition in multiscale framework, which gives that the macro-scopic elastic strain energy density can be computed by volumetric averaging of that in the micro-scale. However, in the nonlinear analysis, the energy dissipation is an important quantity to measure the degradation status. In this manuscript, two typical multiscale methods, the first-order computational homogenization (FOCH) and reduced-order homogenization (ROH), are adopted to numerically analyze a fiber-reinforced composite material with capability in material nonlinearity. With numerical experiments, it can be shown that energy dissipation is the same for both approaches.

Two versions of the extended Hill’s lemma for non-Cauchy continua based on the couple stress theory

Two versions of the extended Hill’s lemma for non-Cauchy continua satisfying the couple stress theory are proposed. Each version can be used to determine two effective elasticity (stiffness) tensors: one classical and the other higher order. The classical elasticity tensor relates the symmetric part of the force stress to the symmetric strain, whereas the higher-order elasticity tensor links the deviatoric part of the couple stress to the non-symmetric curvature. Four sets of boundary conditions (BCs) are identified using Version I, and three sets of BCs are obtained from Version II of the extended Hill’s lemma, which can all satisfy the Hill–Mandel condition. For each BC set selected, admissibility and average field requirements are checked. Furthermore, the equilibrium is examined for the cases with the kinetic BCs, and the compatibility is checked for the cases with the kinematic BCs. To illustrate the two newly proposed versions of the extended Hill’s lemma, a homogenization analysis is conducted for a two-phase composite using a meshfree radial point interpolation method. The effective elastic constants obtained in this analysis are compared with those predicted by the Voigt and Reuss bounds and computed through a finite element model constructed using COMSOL, which verifies and supports the current method.

Energetically-consistent multiscale analysis of fracture in composites materials

Two distinct length scale transition methodologies are developed to establish effective traction-separation relations for fracture in composite materials within a hierarchical multiscale framework. The two methodologies, one kinetics-based and the other kinematics-based, specify effective fracture properties that satisfy a surface-based Hill-Mandel consistency condition. Correspondingly, the total amount of energy dissipated is the same whether a crack is described in detail with micro quantities or in terms of an effective macroscopic crack. Though both methods guarantee consistency in terms of energy rates across length scales, they provide in general distinct effective traction-separation relations. Several representative samples of fiber reinforced composites are analyzed numerically, including the formation and propagation of cracks at mid-ply locations as well as (idealized) ply interfaces. Through post-processing of the microscale results, it is shown that the kinematics-based averaging method provides a macroscopic traction that is prone to rapid fluctuations while the kinetics-based averaging method shows a more smooth response but with openings that can deviate from the surface average of the microscale openings. The two methods are also compared with a previously-proposed scale transition methodology, which is a hybrid method that only satisfies the Hill-Mandel condition approximately. The suitability of the three methods is discussed in light of the results obtained from the simulations.

FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

In the current work, the response of heterogeneous magnetorheological elastomers (MREs) which are loaded by external magnetic fields in the absence and also in the presence of free space is studied. A fully-coupled two-scale finite element computational homogenization procedure is used to derive the material response at the macro-scale from the averaged response of the underlying micro-scale problem. Different combinations of boundary conditions, that satisfy the Hill-Mandel condition and are based on the primary variables of the magneto-elastic enthalpy and energy functionals are applied to solve the micro-scale boundary value problem. Furthermore, the influences of various microstructures on the macroscopic response of the MREs are investigated. The results indicate that the choice of microscopic boundary conditions and microstructure types can significantly affect the macroscopic responses of MREs.

Extended Hill’s lemma for non-Cauchy continua based on a modified couple stress theory

Two extended versions of Hill’s lemma for non-Cauchy continua are provided using a modified couple stress theory, which includes a constraint relation between the displacement gradient and the micro-rotation, unlike the Cosserat (micro-polar) elasticity theory. The first version is used to obtain the classical elasticity tensor that relates the Cauchy (force) stress to the strain, while the second version is employed to determine the couple stress elasticity tensor that links the couple stress to the curvature. The kinematic (displacement- or strain-based) boundary conditions used in each version of the extended Hill’s lemma are then modified to accommodate composites with periodic microstructures by introducing periodic parts to the displacement and micro-rotation fields and reconstructing the Hill–Mandel condition. To illustrate the two newly proposed versions of the extended Hill’s lemma, homogenization analyses are performed for a two-phase composite using a meshfree radial point interpolation method incorporating the extended Hill’s lemma and the modified couple stress theory, which is newly developed in the current study.

Numerical prediction of orthotropic elastic properties of 3D-printed materials using micro-CT and representative volume element

The mechanical property of 3D-printed components often exhibits anisotropic behaviors and a strong dependence on printing orientations and process parameters. In this study, computational models based on microstructures of 3D-printed ABS polymers are developed using micromechanics of a representative volume element (RVE) to investigate the orthotropic elastic properties. Two finite element models, based on Micro-CT or CAD geometry, with different raster angles: $$0^{\circ }$$, $$30^{\circ }$$, $$45^{\circ }$$, and $$60^{\circ }$$ (corresponding to the layups of $$0^{\circ }/90^{\circ }$$, $$30^{\circ }/{-}60^{\circ }$$, $$45^{\circ }/{-}45^{\circ }$$, and $$60^{\circ }/{-}30^{\circ }$$), are considered. The Micro-CT model useds the realistic geometry of a 3D-printed cube reconstructed from Micro-CT scans. The CAD model useds periodic layers and filaments with sizes specified according to the dimensional statistics from the Micro-CT model, including bond width, layer height, filament width, and total porosity. All models are subjected to six independent load cases of macroscopically uniform boundary conditions (BCs) admitted by the Hill–Mandel condition to obtain a fully orthotropic elastic stiffness matrix. Scale-dependent bounds from those BCs are used to establish the convergence of RVE responses. The theoretical method of Mori–Tanaka homogenization is also employed to verify numerical predictions. The results of both Micro-CT and CAD models are close to each other and agree well with experimental results in the literature. Parametric studies are further performed in CAD models by varying layer height, filament width, and bond width to investigate their effects on the orthotropic elastic properties. It is found that the porosity plays a significant role, and more porosity leads to larger elastic anisotropy in 3D-printed materials.