Research paperStrain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods
Introduction
The derivation of strain gradient elasticity theory in the framework of homogenization approaches has been the subject of intense research during the last years but, in fact, has begun earlier with the works of Diener et al. Diener, Raabe, Weissbarth, 1981, Diener, Raabe, Weissbarth, 1982, Diener, Hurrich, Weissbarth., 1984. Starting from the variational principles of Hashin, Shtrikman, 1962, Hashin, Shtrikman, 1962, Dederich and Zeller (1973) and Kröner and Koch (1976) derived new bounds for the elastic coefficients when a random medium is subjected to non homogeneous mean fields. As a consequence the homogenized elastic coefficients depend on the wave vector of the mean field and, due to the emergence of internal lengths, the macroscopic behavior turns out to be non-local. The link with strain-gradient elasticity has been established later by Drugan and Willis (1996) who assumed a polynomial representation for the mean field. By doing so, the overall behavior shows dependence on the gradient(s) of the macroscopic strain and, based on the usual energy principle, the authors derive the elastic coefficients of the Toupin (1962) and Mindlin (1964) strain-gradient model. However, such an approach only provides bounds for the higher-order elastic coefficients and not their exact values, and is restricted to some particular microstructures (examples can be found in Drugan, 2000, Drugan, 2003).
Strain gradient elasticity models have also been derived using asymptotic series expansion methods. Periodic homogenization approaches have been introduced and developed by Auriault and Sanchez-Palencia (1977), Bensoussan et al. (1978), Sanchez-Palencia (1980), Suquet (1982)... They are based on the introduction of a small parameter, the scale factor, and the determination of the solution by means of polynomial expansion series in power of this small parameter. By accounting for the second-order term of the series, Bakhvalov and Panasenko (1989), Gambin and Kroner (1989), Boutin (1996), Triantafyllidis and Bardenhagen (1996) found that the local solution exhibits a dependence with the macroscopic strain gradient (and also higher-order derivatives of the macroscopic strain gradient when accounting for higher-order local solutions). The connection with the Toupin and Mindlin strain gradient elasticity model has been really established by Smyshlyaev and Cherednichenko (2000), Peerlings and Fleck (2001), Tran et al. (2012). The strain gradient elasticity coefficients are computed by solving unit cell problems with Periodic Boundary Conditions (PBC). The first problem provides the fundamental solution and, by only keeping this term in the series expansion, the macroscopic description of the material remains of Cauchy type. Higher-order cell problems introduce the microstructural effects, i.e. the macroscopic description depends on an internal length scale that is characteristic of the microstructure and of the dependence of the elastic relationship with higher order derivatives of the macroscopic strain. The numerical implementation of the higher-order cell problems and the determination of strain-gradient elasticity coefficients have been already addressed in Peerlings and Fleck (2001) using the Finite Element Method (FEM) or in Tran et al. (2012) considering iterative schemes based on Fast Fourier Transform (FFT). It has been observed that when the scaling of elastic coefficients of the constituents have the appropriate order with respect to the scale factor, the effective material may exhibit strong strain gradient effects. This effect has been evidenced in the case of elastic composites reinforced by highly rigid slender inclusions by Pideri, Seppecher, 1997, Pideri, Seppecher, 1997 and Boutin and Soubestre (2011). More recently, it has been demonstrated that the effective elastic energy of lattice materials having soft modes of deformation can be of generalized type Alibert et al. (2003). Some general results have been obtained in Abdoul-Anziz, Seppecher, 2018, Abdoul-Anziz, Seppecher, 2018 combining asymptotic analysis and Γ-convergence method. However even if a method to determine the higher-order parameters was proposed in this last reference it fails when the first order elasticity is not degenerated.
Another attractive method for the computation of strain-gradient elasticity coefficients is based on the use of the Quadratic Boundary Conditions (QBC) on the Representative Volume Element (RVE). It generalizes the classic approach based on Uniform Strain Boundary Conditions (USBC) by applying a displacement that has a quadratic dependence with the position vector. The enforcing term is the macroscopic gradient of strain (or equivalent the macroscopic double gradient of displacement). The method is generally attributed to Besdo, Dorau, 1985, Besdo, Dorau, 1988 but has been really introduced in the context of homogenization by Gologanu et al. (1997) who derived the strain-gradient version of the Gurson model Gurson (1977). As already mentioned in Fleck and Hutchinson (1997), the results are however somewhat surprising in that the dominant effect of strain-gradient effects is not influenced by the void volume fraction or void spacing and the strain gradient effects persist even when the void volume fraction tends to zero. Indeed, the strain gradient effects are due to the presence of a microstructure at a lower scale and when the volume fraction tends to zero, the solid becomes homogeneous and no microstructural effects should be observed. Later, the method of Gologanu et al. (1997) has been applied in the context of elasticity by Zybell et al. (2009) with the same result: when radii of the inclusions tend to zero, the resulting microstructural effects still persist instead of vanishing as they should. The method has been later used in the case of elasticity and non linear constituents Forest (1998); Forest and Trinh (2011); Bouyge, Jasiuk, Ostoja-Starzewski, 2001, Bouyge, Jasiuk, Ostoja-Starzewski, 2002; Kouznetsova, Geers, Brekelmans, 2002, Kouznetsova, Geers, Brekelmans, 2004 but in each case, the persistence of the strain gradient is retrieved for a homogeneous material at the local scale. This has been again reported in Yuan et al. (2008) who suggested that the method surely over evaluates the macroscopic energy density and probably exaggerates the strain gradient effects. It is important to notice that such observation does not concern the periodic homogenization based on asymptotic series expansion. Indeed, when the medium is homogeneous at the local scale, the source term in the second order cell problem, which depends only on the fluctuation, vanishes and the macroscopic behavior remains Cauchy elastic. Note that another approach based on QBC has been used by Bigoni and Drugan Bigoni and Drugan (2007), Bacca et al. Bacca, Bigoni, Corso, Veber, 2013, Bacca, Bigoni, Corso, Veber, 2013. However their method is quite different in the sense that it is based on a small perturbation of the elastic properties and by approximating its effects on the macroscopic behavior by a strain gradient term. The approach is more related to the statistical method developed in Drugan and Willis (1996).
The paper is organized as follows. To evidence the issue with the QBC based method, we start from the periodic homogenization with the asymptotic series expansion and we establish the connection between the two methods. To this end, we give a brief overview of the homogenization method based on asymptotic series expansion in Section 3. The first two unit cell problems, that lead to the reference solution and the first corrector, are detailed. In Section 4, using an appropriate change of variables, a bridge between the two homogenization procedures is established. In Section 5, we discuss the results and we provide a new method to determine correctly the strain gradient elastic coefficients in the QBC based approach. Finally, in Section 6, we propose an illustration for a composite with fibers. FEM computations with QBC are compared to FFT simulations for the problem with PBC.
Section snippets
Notations
PBC: Periodic Boundary Conditions.
QBC: Quadratic Boundary Conditions.
USBC: Uniform Strain Boundary Conditions.
RVE: Representative Volume Element.
FFT: Fast Fourier Transform.
Tensorial notations:
Vectors, second order and third order tensors are denoted by bold letters, usually lower case for microscopic quantities u, and upper case for the associated macroscopic ones U. Fourth- and higher-order tensors are denoted by blackboard letters such as .
⊙n stands for the generalized dot product between
Brief review of the periodic homogenization approach
Asymptotic methods in periodic homogenization, initially introduced by Sanchez-Palencia Sanchez-Palencia (1974), Bensoussan et al. Bensoussan et al. (1978), have been later considered by Bakhvalov and Panasenko Bakhvalov and Panasenko (1989), Gambin and Kroener Gambin and Kroner (1989), Boutin Boutin (1996) to evaluate the deviation from the standard elastic description. The rigorous convergence of the method in the classical situation has been demonstrated by Allaire in Allaire (1992). In this
First order problem
Consider the first order problem, defining the displacement u1 (see equations (7) with (8)). Let us apply the following change of variable:With this definition, the displacement contains (i) the periodic part uL due to the local heterogeneities (the latter vanishes for a homogeneous material at the local scale) and (ii) a linear displacement E(x).y. With this change of variable, the first local problem becomes:The usual homogenization problem with the
Discussion
Let us first give the relation that exists between the strain field εy(uQ) computed with the QBC and the first corrector ε1. For the sake of simplicity, εL will stand for εy(uL) and εQ for εy(uQ). The first corrective term for the strain is ε1 that reads (owing to Eq. (5)):Owing to Eq. (25), the displacement u2 can be read:Replacing in Eq. (33) the displacements u1 and u2 by Eq. (15) and Eq. (34) respectively, we deduce after some elementary
Application to a composite with fibers
As an illustration purpose, we consider a composite with fibers. The fibers are aligned in the x3 direction and are periodically distributed in the (x1, x2) plane. By R we denote the radius of the fiber and by h the distance between two neighboring fibers (the period). The strain-gradient elastic properties are determined by considering the two kind of boundary conditions, periodic and quadratic. When the periodic boundary conditions are used, the solution is computed with the FFT method
Conclusion
In this paper, we have provided a unified approach to the determination of the strain-gradient elasticity properties. Two methods have been investigated. The first is based on asymptotic series expansion within the framework of periodic homogenization, the second consider the quadratic condition (QBC) on the boundary of the representative volume element (RVE). A bridge between the two methods has been established and a modification of the QBC-based method has been provided in order to be
CRediT authorship contribution statement
Vincent Monchiet: Writing - original draft, Conceptualization, Methodology, Software. Nicolas Auffray: Writing - review & editing, Conceptualization, Software. Julien Yvonnet: Writing - review & editing, Conceptualization.
Declaration of Competing Interest
The authors declare have no conflict of interest.
References (49)
- et al.
Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part I: Closed form expression for the effective higher-order constitutive tensor
Int. J. Solids Struct.
(2013) - et al.
Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites. Part II: Higher-order constitutive properties and application cases
Int. J. Solids Struct.
(2013) - et al.
Establishment of strain gradient constitutive relations by using asymptotic analysis and the finite element method for complex periodic microstructures.
Int. J. Solids Struct
(2018) Microstructural effects in elastic composites
Int. J. Sol. Struct.
(1996)- et al.
Generalized inner bending continua for linear fiber reinforced materials
Int. J. Sol. Struct.
(2011) - et al.
Variational treatment of the elastic constants of disordered materials
Zeitschrift für Physik
(1973) - et al.
Bounds for the non-local effective properties of random media
J. Mech. Phys. solids
(1981) - et al.
Bounds for the non-local effective properties of random media ii
J. Mech. Phys. solids
(1982) Micromechanics-based variational estimates for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites
J. Mech. Phys. Solids
(2000)Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials
J. Mech. Phys. Solids
(2003)
A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites
J. Mech. Phys. Solids
Strain gradient plasticity
Adv. Appl. Mech.
Mechanics of generalized continua: construction by homogenization
J. Phys. IV
A variational approach to the theory of the elastic behavior of polycrystals
J. Mech. Phys. Solids
Multi-scale constitutive modeling of heterogeneous materials with gradient enhanced computational homogenization scheme
Int. J. Num. Meth. Eng.
Effective properties of disordered materials
Solid Mech. Arch.
Comportements local et macroscopique d’un type de milieux physiques hétérogènes
Int. J. Engng Sci.
Nonhomogeneous Media and Vibration Theory
Plasticité et homogénéisation
The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models
J. Mech. Phys. Solids
A micromechanical approach of nonlocal modeling for media with periodic microstructures
Mech. Res. Comm.
Constitutive equations for porous plane-strain gradient elasticity obtained by homogenization
Arch. Appl. Mech.
Homogenization of periodic graph-based elastic structures
Journal de l’Ecole Polytechnique-Mathématiques
Strain gradient and generalized continua obtained by homogenizing frame lattices
Math Mech Complex Syst
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