Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

doi:10.1016/j.mechmat.2019.103309

Mechanics of Materials

Volume 143, April 2020, 103309

https://doi.org/10.1016/j.mechmat.2019.103309 Get rights and content

Highlights

•
A homogenization approach is provided to compute the strain gradient elasticity coefficients.
•
We modify the method based on quadratic boundary conditions to eliminate the persistence of strain gradient elasticity effects for a homogeneous solid.
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The principle of the approach consists to establish a bridge with the method based on asymptotic series expansion.
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The case of a composite with fibers is considered as an illustration in order to show the improvement of the corrected method.

Abstract

In this paper we deal with the determination of the strain gradient elasticity coefficients of composite material in the framework of the homogenization methods. Particularly we aim to eliminate the persistence of the strain gradient effects when the method based on quadratic boundary conditions is considered. Such type of boundary conditions is often used to determine the macroscopic strain gradient elastic coefficients but leads to contradictory results, particularly when a RVE is made up of a homogeneous material. The resulting macroscopic equivalent material exhibits strain gradient effects while it should be expected of Cauchy type. The present contribution is to provides new relationship to correct the approach based on the quadratic boundary condition. To this purpose, we start from the asymptotic homogenization approach, we establish a connection with the method based on quadratic boundary conditions and we highlight the correction required to eliminate the persistence of the strain gradient effects. An application to a composite with fibers is provided to illustrate the method.

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 $C$ .

⊙_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: $u^{1} = u^{L} - E (x) . y, σ^{0} = σ^{L}$ With this definition, the displacement contains (i) the periodic part u^L 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: ${\begin{matrix} {div}_{y} (σ^{L}) = 0 \\ σ^{L} = C (y) : ɛ_{y} (u^{L}) \end{matrix}$ The usual homogenization problem with the

Discussion

Let us first give the relation that exists between the strain field ε_y(u^Q) computed with the QBC and the first corrector ε¹. For the sake of simplicity, ε^L will stand for ε_y(u^L) and ε^Q for ε_y(u^Q). The first corrective term for the strain is ε¹ that reads (owing to Eq. (5)): $ɛ^{1} = ɛ_{y} (u^{2}) + ɛ_{x} (u^{1})$ Owing to Eq. (25), the displacement u² can be read: $u^{2} = u^{Q} - \frac{\partial u^{1}}{\partial x_{k}} y_{k} - \frac{1}{2} \nabla_{x}^{2} U : (y \otimes y)$ 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.

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Research paperStrain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

Highlights

Abstract

Introduction

Section snippets

Notations

Brief review of the periodic homogenization approach

First order problem

Discussion

Application to a composite with fibers

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Int. J. Solids Struct.

Int. J. Solids Struct.

Int. J. Solids Struct

Int. J. Sol. Struct.

Int. J. Sol. Struct.

Zeitschrift für Physik

J. Mech. Phys. solids

J. Mech. Phys. solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Adv. Appl. Mech.

J. Phys. IV

J. Mech. Phys. Solids

Int. J. Num. Meth. Eng.

Solid Mech. Arch.

Int. J. Engng Sci.

J. Mech. Phys. Solids

Mech. Res. Comm.

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

Research paper
Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods