A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

Transcription

1 Comput Mech (214) 54: DOI 1.17/s ORIGINAL PAPER A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials Xikui Li Yuanbo Liang Qinglin Duan B.A. chrefler Youyao Du Received: 17 February 214 / Accepted: 11 July 214 / Published online: 29 July 214 pringer-verlag Berlin Heidelberg 214 Abstract A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts. Keywords Mixed finite element Gradient Cosserat continuum Discrete particle assembly Granular material Computational homogenisation X. Li (B) Y. Liang Q. Duan Y. Du The tate Key Laboratory of tructural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 11624, People s Republic of China xikuili@dlut.edu.cn B. A. chrefler Department of Civil, Environmental and Architectural Engineering, University of Padua, via Marzolo 9, Padua, Italy 1 Introduction Granular materials are highly heterogeneous and discontinuous media at the meso-scale and consist of a large number of particles and voids. These materials exist widely in both nature and engineering practices, such as soils, clays, soil-rock mixtures, and concretes. Fibrous materials in plane stain conditions can also be dealt with as such material, see e.g. some cable in conduit superconductors. Different homogenisation techniques have been proposed to build continuum description of discrete materials, including gradientenhanced continuum modeling [1 7]. Equivalent material properties of heterogeneous materials, such as composite materials, meso-structurally porous materials and granular materials, that are modelled as elastic Cauchy or Cosserat continua at the macro-scale have been obtained as results of analytical or semi-analytical homogenisation techniques [7 12]. Jenkins and co-workers [13,14] proposed a theoretical model to study the elastic response of a granular material idealized as a dense, randomly packed aggregate of elastic, frictional spheres and to derive an analytical relationship between the overall stress and strain increments for the pre-failure regime. It was reported that the effective elastic moduli predicted by the proposed theoretical model are in excellent agreement with those measured in numerical simulations [14]. When significant meso-structural evolution in the granular material develops for the post-failure regime characterized by the strain softening phenomena of the material, severe dissipative sliding and rolling frictions and loss of contacts for a reference packed aggregate with its immediate neighbors occur and the damage-plastic dissipation of effective continuum is to be taken into account [15,16]. Alternatively, computational multiscale approaches [17 26] may be adopted to establish the corresponding meso-

2 1332 Comput Mech (214) 54: mechanically informed non-linear macroscopic constitutive relationships. One promising method among computational multiscale approaches is the computational homogenisation approach (the global-local nested analysis scheme) [24 29] developed in recent years. The approach is based on averagefield theory and the concept of the representative volume element (RVE) under the category of concurrent methods [17]. The approach enables the incorporation of large deformations and to upscale nonlinear material behaviours of well-characterized meso-structure, while does not require the macroscopic constitutive relationship to be specified a priori at selected macroscopic points. A hierarchical method for elasto-plastic analysis at macroscopic level, based on unit cells at microscopic level containing elastic or inelastic disks in contact has been developed in [3,31]. Most existing computational homogenisation procedures developed in these approaches are first-order methods. Their main disadvantages lie in the intrinsic assumption of the uniformity of the macroscopic stress-strain fields attributed to the boundaries of each meso-structural RVE, so that the meso-structural size effect and the gradients of macroscopic strain field along the boundaries of each RVE cannot be included. ome second-order computational homogenisation schemes have been proposed to overcome the disadvantages of first-order computational homogenisation methods [28,29,32 36]. These schemes have been developed for heterogeneous materials modelled with gradient continua, such as the Toupin Mindlin [28,29,35,36] and micropolar continua [33], at the macroscopic scale. One feature of second-order computational homogenisation schemes is the adoption of gradient continuum models at the macroscale, in which strain gradients and energyconjugated stress moments are introduced. As a consequence, both macroscopic strains and their gradients defined at a local material point of macroscale continua are transited to prescribe proper kinematic boundary conditions on a meso-structural RVE attributed to the macroscopic material point. In the frame of computational homogenisation, granular material should be homogenised with the Cosserat continuum rather than the Cauchy continuum [1 6,26] due to the nature of the granular medium modelled as a discrete particle assembly at the meso-scale, in which each discrete particle possesses independent rotational and translational degrees of freedom (DOF) kinematically and is capable of bearing and transmitting couples from one particle to the other in contact. The structure of the derived mesomechanically informed degraded isotropic constitutive relationships of granular materials also demonstrated the rationality of adopting Cosserat continuum models for granular materials [37]. Kruyt presented a theoretical framework for the statics and kinematics of discrete Cosserat-type granular materials [38]. Zhang et al. [39] extended the unit cell of the above mentioned hierarchical method [3,31] to include multiple Cosserat bodies. The reason for this was to introduce an internal length scale when analysing materials with granular or fibrous structure. However the length scale works only for shear behaviour in localisation problems. Chang et al. [2] applied the principle of virtual work to treat a discrete particle assembly as an equivalent continuum. They demonstrated that contact moments are disallowed and do not contribute work in a classical Cauchy continuum; a granular medium consisting of a contiguous collection of discrete particles should be treated as an equivalent discrete classical Cosserat continuum or an equivalent discrete higher-order Cosserat continuum, in the latter of which the gradients of derivatives of translational displacements and their energy-conjugated stress moments are introduced. The downscaling rule that determines the microscale boundary value problem for the RVE with the governing macroscopic kinematical quantities is laid down by the Hill s lemma [32,4] formulated according to the adopted micromacro continuum models for the homogenization, which conducts properly enforcement of the boundary conditions to the RVE to ensure satisfaction of the Hill Mandel energy condition. Even a high-order continuum structure characterized by the Cosserat curvature and coupled stress is introduced into the classical Cosserat continuum model, the admissible microscale boundary conditions derived from the generalized Hill s lemma are prescribed by a uniform macroscopic strain field attributed to each RVE if the classical Cosserat continuum is assumed for both the macroscale and all RVEs in the first order macro-micro homogenization [26]. It implies that the macroscopic strain field passed down to each RVE is restricted to be uniform, and no bending (high-order) deformation modes of the RVE due to gradients of the macroscopic strain field over the spatial length scale associated to the RVE size could be represented. However, those bending deformation modes physically may exist and even play a decisive role particularly as the absolute size of the RVE and involvement of macroscopic gradients of strains are significant with respect to the meso-structure s response. The macroscopic energy product will be incorrectly predicted due to the absence of the bending deformation modes in a homogenisation procedure using the classical Cosserat continuum model at the macroscopic level. Indeed, the rationality of the uniform macroscopic strain field attributed to each RVE relies on the concept of separation of scales. The uniformity assumption is not appropriate in critical regions of high strain gradients, where the macroscopic strain and stress fields vary considerably. To remedy the defects of the first order homogenization mentioned above it is required to develop the second order multi-

3 Comput Mech (214) 54: Gradient-enhanced Cosserat continuum ampling point RVE c x i c f i c n i RVE i c x b x b Δ i c r b V Discrete particle assembly Cosserat continuum equivalent Peripheral particles Interior particles Fig. 1 Micro-macro homogenization of granular materials: discrete particle assembly Cosserat continuum modeling scale computational homogenization for granular materials, in which the gradient (high-order) Cosserat continuum model and the classical Cosserat continuum model are assumed for both the macro-scale and all RVEs respectively [32]. According to the generalized Hill s lemma derived for the second order computational homogenization the non-uniform macroscopic strain field with macroscopic strain gradients may be transited to each RVE, meanwhile satisfaction of the Hill Mandel energy condition is ensured. The macroscopic strain field is no longer restricted to be uniform over the spatial length scale associated to the RVE size, the bending (the second-order) deformation modes of the RVE may be represented. Consequently, the second-order computational homogenization is appropriate in critical regions of high gradients such as in the regions of strain localization and the scale separation assumption is no longer required for the proposed two-scale algorithm. To formulate the generalized Hill s lemma each RVE is modelled as a classical Cosserat continuum. On the other hand, the RVE is also modelled as a discrete particle assembly as shown by Fig. 1 for a detailed description of the RVE microstructure and its evolutions. An equivalence between the discrete particle assembly model and the discrete classical Cosserat continuum model at the RVE scale was established by Chang and Kuhn [2] in the sense of the equivalence between the virtual work of the discrete system and the virtual work of the classical Cosserat continuum approximately restricted to continuous virtual displacement fields with a polynomial series containing quadratic terms.

4 1334 Comput Mech (214) 54: With this equivalence the non-uniform macroscopic strain field derived from the generalized Hill s lemma to prescribe the boundary condition on the RVE of classical Cosserat continuum is imposed to deform the peripheral particles on the boundary of the RVE of discrete particle assembly in the proposed second-order computational homogenization procedure [41]. The boundary value problem for the RVE of discrete particle assembly is well determined and performed by the discrete element method (DEM). The mesomechanical behavior is then identified with the DEM solutions for the BVP of the RVE, in which the discontinuity (loss and re-generation of contacts) and dissipative interparticle displacements (according to the Coulomb law of friction) involved at the meso-scale and their evolutions with respect to the loading history are taken into account. The meso-mechanically informed macroscopic stress measures and the informed macroscopic rate stress-strain constitutive equations are extracted for upscale transition to the macroscale with the meso-mechanical quantities resulting from the DEM solutions for the BVP of the RVE at each incremental load step [41]. Then the first purpose of the present work is to develop the second-order computational homogenisation method and corresponding global-local FEM-DEM nested analysis scheme for granular material, in which the material is modelled as a gradient Cosserat continuum using the FEM and a discrete particle assembly using the DEM at the macro- and micro-scales, respectively. The inclusion of gradients would also introduce an internal length scale [42,43] which is effective for localisation analysis in all situations. To the authors knowledge, there have been no published reports of such a computational homogenisation procedure for heterogeneous granular materials modelled as a gradient Cosserat continuum at the macroscale and the related FEM-DEM nested scheme. An important task in the development of the proposed second-order computational homogenisation for granular materials with a related FEM-DEM nested scheme is to construct a new finite element (FE) for the gradient Cosserat continuum, as both the third-order non-symmetric strain gradient tensor and the microcurvature tensor are included. The key issue in the construction of the new FE is attaining C 1 continuity of the interpolation of displacement for a displacement-based FE; both translational displacements and their first-order derivatives should be continuous across inter-element boundaries [44]. The direct strategy for achieving the C 1 -continuity requirement is to use Hermitian interpolation. Hermitian FEs are truly C 1 continuous, in which the displacement field is the only field that needs to be discretised using a C 1 -continuity element [45]. To circumvent the C 1 -continuity requirement in the development of displacement-based FEs with high-order displacement interpolations, some authors have developed mixed FEs, that fulfill continuity requirements only in a weak form, for different strain gradient continua using couple stress theory [46,47] and Toupin Mindlin theory [28,48 5]. An overview of existing FE formulations and implementations for gradient elasticity in statics and dynamics was given by Askes and Aifantis [51]. The second purpose of the present work is the development of the mixed FE and related FE procedures for the gradient Cosserat continuum in the frame of second-order computational homogenization and the related global-local FEM-DEM nested scheme. The element formulations are derived based on the weak form of the Hu Washizu variational principle and previous work on the Cauchy continuum using Toupin Mindlin gradient theory [48]. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are interpolated as independent variables. The displacement-strain gradient compatibility condition of the symmetric part of the strain gradients defined in gradient Cosserat continuum theory is enforced in the weak form. The meso-mechanically informed macroscopic stress-strain constitutive relationship derived from the second-order computational homogenisation is also satisfied in the weak form. The patch tests are designed and performed to validate the present mixed FE formulations for the gradient Cosserat continuum in the frame of the second-order computational homogenisation procedure using meso-mechanically informed macroscopic constitutive relationships. To the authors knowledge, there are no published reports of FEMs for the gradient-enhanced Cosserat continuum and the related FE procedure in the frame of second-order computational homogenisation. In addition to the numerical example for the patch tests, two numerical examples for different purposes are specially designed and performed. The purpose of the first example is to demonstrate the advantage of the proposed second-order computational homogenisation procedure using the gradient Cosserat continuum model for granular materials in capturing the meso-structural size effect over the first-order computational homogenization procedure that uses the classical Cosserat continuum model at the macro-scale. The second example, a square panel example problem, is performed to demonstrate the performance of the present mixed FE constructed in the frame of the second-order computational homogenization procedure of granular materials in simulating strain softening and localisation phenomena and to reveal the meso-mechanical mechanisms of the macroscopic failure of granular materials, while without the need to specify the macroscopic constitutive relationship and material failure model. In addition, good convergence of the present mixed FE procedure is illustrated. The proposed second-order computational homogenization procedure is also validated with a reference solution of the DEM simulation performed for the whole domain of the panel. Finally a numerical study of

5 Comput Mech (214) 54: the influence of the RVE size on the macroscopic mechanical behavior is briefly provided. 2 Classical and gradient Cosserat continuum models 2.1 Classical Cosserat continuum model [52] In the Cosserat continuum model, the independent kinematic DOFs are the displacements u i and microrotations ω i. Accordingly, independent couple stresses μ ji are introduced in addition to the classical Cauchy stresses σ ji.thestrain measures ε ji and κ ji (curvatures) are defined as ε ji = u i, j e kji ω k (1) κ ji = ω i, j (2) where e kji is the permutation tensor. The equilibrium conditions can be expressed as σ ji, j = b f i (3) μ ji, j + e ijk σ jk = bi m (4) where b f i and b m i are the body force and body moment per unit volume of the medium, respectively. Equilibrium on the surface of the medium yields t i = σ ji n j (5) m i = μ ji n j (6) where t i is the surface traction, m i is the surface couple, and n j is the unit vector normal to the surface. 2.2 Gradient Cosserat continuum model [2], [32] In addition to the strain measures ε ji and κ ji defined by Eqs. (1) and (2), the strain gradient, denoted by a third-order tensor E ljk, is defined in the gradient-enhanced Cosserat continuum model and can be decomposed into two parts, expressed as E ljk = ε jk x l = Ê ljk + Eljk (7) in which, using Eq. (1), one may express Ê ljk, Eljk as Ê ljk = u k, j x l = u k, jl, Eljk = e ijk ω i,l (8) The strain gradient Ê ljk is a symmetric third-order tensor in the sense that Ê ljk = u k, j x l = u k, jl = u k,l x j = u k,lj = Ê jlk (9) Accordingly, the stress moment ljk energy-conjugated with the strain gradient E ljk is introduced and decomposed into symmetric and skew-symmetric parts: ljk = ˆ ljk + ljk (1) where the symmetry of ˆ ljk and skew-symmetry of ljk are defined and expressed as ˆ ljk = ˆ jlk, ljk = jlk (11) The equilibrium equations for the gradient Cosserat continuum are expressed as (σ ji kji,k ), j = b f i (12) μ ji, j + e ijk (σ jk ljk,l ) = bi m (13) Consider the weak form of the equilibrium equations (12) and (13) over the domain of the gradient Cosserat continuum in consideration, δu i [(σ ji kji,k ), j b f i ]d + δω i [μ ji, j + e ijk (σ jk ljk,l ) b m i ]d = (14) one may perform an integration by parts on Eq. (14) and replace it with an alternative statement of the form δε ji σ ji d + δκ ji μ ji d + δe kji kji d = (δu i t i + δω i m i + δu i, j g ji )d + where (δu i b f i + δω i b m i )d (15) t i = (σ ji kji,k )n j (16) m i = (μ ji e ilk jlk )n j (17) g ji = kji n k (18) i.e. equilibrium on the surface of the gradient Cosserat continuum yields the surface traction t i, surface couple m i, and surface high-order generalized traction g ji associated with the virtual displacement δu i, virtual microrotation δω i, and virtual displacement gradient δu i, j, respectively, on the surface of the medium. It is noted that the integral δu i, j g ji d in Eq. (15) is only applicable to the surface, for which the unit normal vector n to the differential of surface d is identical with one of normalized base vectors of the Cartesian coordinates,

6 1336 Comput Mech (214) 54: i.e. n i n j = δ ji,δ ji is the Kronecker delta. Generally the integral δu i, j g ji d is inapplicable due to the existence of the curved surface, on which the orientation of the unit normal vector n to d varies and generally disagrees with the orientations of the Cartesian coordinates. To correctly identify the boundary condition related to the surface integral δ u i, j g ji d in the weak form (15) one needs to resolve δ u i, j into a surface gradient D j δ u i and a normal gradient n j Dδu i [53], δu i, j = D j δu i + n j Dδu i (19) where D = n i, x i D j = (δ ji n j n i ) x i (2) With the use of Eq. (19), Eq. (15) can be re-expressed as δε ji σ ji d + δκ ji μ ji d + δe kji kji d = (δu i ti + δω i m i )d + [r i (Dδu i )]d + (δu i b f i + δω i b m i )d (21) where the generalized surface traction t i and the double stress traction r i in Eq. (21) on the surface have their expressions in the forms as follows ti = t i + n k n j kji (D p n p ) D j (n k kji ) (22) r i = n k n j kji (23) If n i n j = δ ji, one has t i = t i and [r i(dδu i )]d = δ u i, j g ji d, Eqs.(15) with (16) (18) are retrieved. 3 Mesomechanically informed constitutive model of the gradient Cosserat continuum To develop the second-order computational homogenisation procedure for granular material, downscaling and upscaling rules to perform the two-way coupling between the macroscale and the micro-scale should be specified. The downscaling determines the micro-scale boundary value problem of the RVE with the given macroscopic strain measures via enforcement of the RVE boundary conditions in light of the Hill s lemma. On the other hand, the upscaling transited back the stress measures of the higher scale and the meso-mechanically informed macroscopic constitutive model from volume average of the sub-scale solution to the gradient Cosserat continuum. 3.1 Generalised Hill s lemma and RVE boundary conditions downscaling The generalised Hill s lemma for second-order computational homogenisation of the heterogeneous gradient Cosserat continuum is σ ji ε ji + μ ji κ ji σ ji ε ji μ ji κ ji lji E lji = 1 1 (n k σ ki n k σ ki )(u i u i, j x j V 2 u i, jlx j x l )d r r + 1 V (n k μ ki n k μ ki )(ω i ω i ω i,l x l )d r (24) r where x j and x l are coordinates of a point located on the boundary r of the RVE with the volume V, σ ji ε ji and μ ji κ ji are defined as the volume averages of the microscopic products σ ji ε ji and μ ji κ ji over the RVE, respectively. In ect. 3 of this paper, ( ) is defined to denote the volume average of a microscopic variable (*) over the domain of the RVE and should be prescribed to equal its macroscopic measure at the local material point to which the RVE is attributed. The Hill Mandel energy condition for the gradient Cosserat continuum can be directly extracted from the Hill s lemma (24) and given in the form σ ji ε ji + μ ji κ ji = σ ji ε ji + μ ji κ ji + lji E lji (25) provided that the RVE boundary conditions are prescribed in such a way that the following two boundary integrals for the RVE are enforced to vanish: 1 V r (n k σ ki n k σ ki )(u i u i, j x j 1 2 u i, jlx j x l )d r = (26) 1 (n k μ ki n k μ V ki )(ω i ω i ω i,l x l )d r = (27) r To solve the RVE boundary value problem for the average stress measures and to derive the meso-mechanically informed macroscopic rate constitutive equations, the following translational and microrotational displacement boundary conditions at the RVE are prescribed: u i r = u i, j x j u i, jlx j x l, ω i r = ω i + ω i,l x l (28) 3.2 Average stress measures and meso-mechanically informed macroscopic constitutive model upscaling The Hill Mandel condition (25) is re-written in the following two forms:

7 Comput Mech (214) 54: σ ji ε ji + μ ji κ ji = σ ji Ɣ ji + T k ω k + μ ji κ ji + ˆ jlk E jlk (29) σ ji ε ji + μ ji κ ji = σ ji Ɣ ji + T k ω k + μ ji κ ji + ˆ jlk Ê jlk (3) The macro internal torque T k and derivatives of translational displacements Ɣ ji are defined as T k = e kji σ ji, Ɣ ji = 1 V V u i, j dv (31) and the generalised couple stresses are defined as μ ji = μ ji e ikl jkl (32) μ ji = μ ji e ikl ˆ jkl = μ ji e ikl jkl (33) The boldfaced forms of macroscopic (average) stress measures σ ji, T k, ˆ jlk, μ ji, μ ji shown in Eqs. (29) and (3) can be expressed as below in the form of boundary integrals along the RVE boundary. With the equivalence established between the discrete classical Cosserat model and the discrete particle assembly model at the RVE scale [2], those boundary integrals are further discretised into discrete quantities assigned at the N c contact points of the peripheral particles of the particle assembly with the RVE boundary, i.e. σ = 1 1 N c x td = xi c ti c V V r,i = 1 N c xi c fi c V i=1 i=1 r (34) T = e : σ = 1 N c V e : xi c fi c (35) i=1 ˆ = 1 1 x x td = 2V 2V r N c N c xi c xi c ti c r,i i=1 = 1 xi c xi c fi c (36) 2V i=1 μ = 1 1 x md + x x td : e V 2V r r = 1 V μ = 1 V N c i=1 N c i=1 x c i m ext i + 1 2V N c xi c xi c fi c : e (37) i=1 x c i m ext i (38) where xi c and ti c are the position vector and the surface traction, respectively, defined at the ith contact point of peripheral particles with the outside material, fi c and mi ext are the traction force and couple moment applied to the peripheral particle via the ith contact point by the outside material, respectively, r,i is the length of the boundary segment associated with the ith contact point, and e is the boldfaced form of the third-order permutation tensor e kji. The discrete counterpart of the boundary conditions given by Eq. (28), which are prescribed on the N c peripheral particles through their contacting points with the RVE boundary, can be written in the rate form for a typical contacting point b as u c b = xc b Ɣ (xc b xc b ) : Ê (39) ω b = ω + x c b κ (4) where Ɣ, Ê, ω, κ are boldfaced forms of Ɣ ji, Ê lji ( u i, jl ), ω i, κ ji ( ω i, j ), respectively, u b c is the rate translational displacement of the contact point with the position xb c of peripheral particle x b, and ω b is the microrotation rate of the peripheral particle b contacting with the RVE boundary. Equations (28), (39), (4) transit the macroscopic strain measures and their gradients down to each RVE via its boundary in the downscaling process. The DEM solver is then used to numerically solve for the boundary value problem (BVP) of the discrete particle assembly within each mesostructural RVE. The non-linear meso-mechanical behavior of the well-characterized meso-structure is identified with the DEM solutions of the BVP of the RVE, in which the discontinuity (loss of contacts) and dissipative relative movements between each two immediate neighboring particles involved at the meso-scale by contact conditions are taken into account. The upscale transition for the macroscopic stress measures shown by Eqs. (34) (37) is fulfilled by the volume averages of the microscopic stress measures within each RVE and the meso-mechanical quantities of the RVE of discrete particle system resulting from the DEM solutions for the BVP of the RVE. With the transformation of the RVE boundary conditions from the contact points of the peripheral particles on the RVE boundary to the centres of those peripheral particles, the discrete element modelling of the particle assembly within the RVE results in the relationship of ḟi c and ṁi ext shown in Eqs. (34) (38) with u c j, ω j, which are expressed in Eqs. (39) (4) as u b c, ω b respectively, i.e. [41] N p [ ] ḟi c = (Kuu b ) ij u c j + (Kbc uω ) ij ω j j=1 ṁ ext i = N p j=1 [ ] (Kωu b ) ij u c j + (Kbc ωω ) ij ω j (41) (42)

8 1338 Comput Mech (214) 54: where (Kuu b ) ij,(kuω bc ) ij,(kωu b ) ij,(kωω bc ) ij are tangent substiffness tensors condensed to the peripheral particles of the RVE of discrete particle assembly [41]. With imposition of RVE boundary conditions Eqs. (39) and (4), the substitution of Eqs. (41) and (42) into the rate forms of Eqs. (34) (38) results in meso-mechanically informed macroscopic constitutive relationships that relate the macro-stress rates to the macro-strain rates for the gradient-enhanced Cosserat continuum modelling of granular materials as follows: σ = D σɣ : Ɣ + D σω ω + D σ Ê. Ê + D σκ : κ (43) T = D T Ɣ : Ɣ + D T ω ω + D T Ê. Ê + D T κ : κ (44) ˆ = D ˆƔ : Ɣ + D ˆω ω + D ˆÊ. Ê + D ˆκ : κ (45) μ = D μɣ : Ɣ + D μω ω + D μê. Ê + D μκ : κ (46) The sixteen meso-mechanically informed macroscopic tangent modular tensors shown in Eqs. (43) (46) and their dependence on the meso-structure of the RVE of discrete particle assembly and its evolution are formulated in [41] and not be given here owing to the limitation of space. The meso-mechanically informed macroscopic constitutive relationships (43) (46) derived for the gradient Cosserat continuum demonstrate that the Cauchy stresses are constitutively related to not only the strains and microrotations but also to the curvatures and strain gradients; likewise, the couple stresses are constitutively related to not only the curvatures but also to the strains and strain gradients. It is noted that the rate stress measures σ, ˆ, Ṫ, μ expressed by Eqs. (43) (46) are not objective. ome objective rates associated with σ, ˆ, Ṫ, μ known as co-rotational rates should be formulated. Alternatively, we resort to a co-rotational approach for continua [54 56]. tensor (third-order tensor) are asymmetric since the microrotations and their derivatives (curvatures) are introduced into the Cosserat continuum. econd, the meso-mechanically informed macroscopic rate constitutive relations (43) (46) imply that the compatible strain and strain gradient measures defined in the gradient Cosserat continuum theory are adopted in the homogenization, however, the strain gradients defined in the FE frame are approximately interpolated with independent nodal primary variables and are incompatible with the assumed displacement field. Let us recall the weak form of equilibrium equations (12) and (13) for the gradient Cosserat continuum δu i [(σ ji kji,k ), j b f i ]d + δω i [μ ji, j + e ijk (σ jk ljk,l ) b m i ]d = (47) in which is the domain of the macroscopic gradient Cosserat continuum. Tensors σ ji, kji,μ ji in Eq. (47) are macroscopic stress measures. The over-bars, applied to denote macroscopic variables in the frame of computational homogenization in ect. 3, are dropped for macroscopic stress measures in Eq. (47) and other subsequent macroscopic strain and stress measures for clarity in the following derivation of FE formulations. As the C 1 -continuity requirement is presented from the interpolation approximation of second-order derivatives of the assumed displacement field, the displacement gradient is still determined with u and is compatible with the assumed displacement field. Here, denotes the gradient operator. A second-order tensor is the nodal primary variable and is introduced to approximately interpolate the second-order derivatives of translational displacements defined as a thirdorder symmetric tensor η(η kji = η jki ) so that at any point in a FE mesh: 4 Mixed finite element procedure 4.1 Weak form of the Hu Washizu variational principle To circumvent the C 1 -continuity requirement of displacement-based elements due to the appearance of second-order derivatives of translational displacements, a mixed FE procedure using C -continuity shape functions only for the gradient Cosserat continuum will be developed in this paper based on the mixed FEs derived by hu et al. [48] for the gradient Cauchy continuum using Toupin Mindlin strain gradient theory. The main issues of the proposed mixed FE procedure distinct from existing ones for gradient continua [48] stem from two aspects. First, the strain tensor and gradient strain η kji = 1 2 (ψ ki, j + ψ ji,k ) (48) The symmetric part of the strain gradient will no longer be defined as the second-order derivative of the assumed displacement field, which is the compatible strain gradient measure Ê kji = u i, jk in the frame of the proposed mixed FE procedure. Instead, with the Hu Washizu variational principle, the third-order symmetric tensor η is introduced as the incompatible symmetric part of the strain gradient, and the relationship between η and compatible symmetric part Ê of the strain gradient is enforced in the weak form given by δ ˆ kji (η kji Ê kji )d = (49)

9 Comput Mech (214) 54: With Eqs. (9) and (48) and the symmetry of ˆ kji defined by Eq. (11)-1, Eq. (49) can be re-written as δ ˆ kji,k ( ji u i, j )d δ ˆ kji ( ji u i, j )n k d = (5) One may alternatively assume to enforce the relationship δ ˆ kji,k ( ji u i, j )d = (51) Equation (51) provides a constraint to enforce the equality ψ ji = u i, j (52) in the volume-averaged sense with an arbitrary variation δ ˆ kji,k and also ensures that the boundary integral in Eq. (5) is negligible, i.e. δ ˆ kji,k ( ji u i, j )d δ ˆ kji ( ji u i, j )n k d = δ ˆ kji,k ( ji u i, j )d = (53) ˆ kji,k in Eqs. (53) and (51) are additional primary variables to be determined in the proposed mixed FE procedure. In the following expressions, ˆ kji,k will be defined as the Lagrange multipliers and denoted by the tensor ρ ji for clarity. Eq. (51) is then re-written as δρ ji ( ji u i, j )d = (54) Noticing the incompatible symmetric part of the strain gradient, η kji defined by Eq. (48) for the proposed mixed FE and the compatible symmetric part of the strain gradient, Ê ljk (Eq. (8)-1) defined in the gradient Cosserat continuum, and using integration by parts, the virtual work δê kji ˆ kji d is expressed as δê kji ˆ kji d = δη kji ˆ kji d + (δ ji δu i, j )ρ ji d (δ ji δu i, j ) ˆ kji n k d (55) The meso-mechanically informed macroscopic rate constitutive relationships, i.e. Eqs. (43) (46), are used to define the assumed stress rates in terms of Ɣ ji, Ê lji ( u i, jl ), ω i, κ ji at the discrete integrating points. The constitutive relationships of rate stress measures with strain components Ɣ ji, ω i, κ ji are satisfied in a point-wise manner. Therefore, the weak form of the rate stress-strain constitutive relationships in the sense of the Hu Washizu variational principle for the proposed mixed FE procedure of the gradient Cosserat continuum implies δɣ ji D σ Ê jinml ( Ê nml η nml )d =, δη kji D ˆÊ kjinml ( Ê nml η nml )d = δω i D T Ê inml ( Ê nml η nml )d =, δκ ji D μê jinml ( Ê nml η nml )d = (56) The validity of Eq. (56) can be proven as follows: δη kji D ˆÊ kjinml ( Ê nml η nml )d = δη kji,n D ˆÊ kjinml ( ml Ɣ ml )d δη kji D ˆÊ kjinml ( ml Ɣ ml )n n d = (57) because variation δη kji,n, which is the variation of the second-order derivatives of ji, may be assumed negligible and δη kji D ˆÊ kjinml ( ml Ɣ ml )n n d is validated to be null, as indicated and discussed by Eqs. (5) (53). The weak form of the Hu Washizu variational principle for the gradient Cosserat continuum comprises the weak forms of equilibrium equations (47), the deformation compatibility condition (54) and the stress-strain constitutive relationships (56). It can be expressed in the following form: δu i [(σ ji kji,k ), j b f i ]d + δω i [μ ji, j + e ijk (σ jk ljk,l ) b m i ]d δρ ji ( ji u i, j )d = (58) Although Eq. (56) does not explicitly appear in Eq. (58), it will be used later in the derivation of Eqs. (69) (72), in which Ê in Eqs. (43) (46) are replaced by η. Eq.(58) is the starting point for the derivation of the proposed mixed FE formulations for the gradient Cosserat continuum in this paper.

10 134 Comput Mech (214) 54: With further expansions of the integrals in Eq. (47), δu i σ ji, j d = δu i, j σ ji d + δu i kji,kj d = δê kji kji d δu i σ ji n j d (59) + δu i, j kji n k d δu i kji,k n j d (6) δω i μ ji, j d = δω i, j μ ji d + δω i μ ji n j d δω i e ijk ljk,l d = δ Ekji kji d (61) δω i e ijk ljk n l d (62) and using Eqs. (1) and (2), Eq. (58) can be re-written as δε ji σ ji d δκ ji μ ji d + δe kji kji d δρ ji ( ji Ɣ ji )d= (δu i t i + δω i m i + δu i, j g ji )d (δu i b f i + δω i b m i )d (63) δɣ ji σ ji d + δω k T k d + + δη kji ˆ kji d + (δ ji δɣ ji )ρ ji d + δρ ji ( ji Ɣ ji )d = (δu i t i + δω i m i +δ ji g ji )d + δκ ji (μ ji e ikl ˆ jkl )d (δu i b f i + δω i b m i )d (65) For an incremental procedure that omits rate virtual kinematic quantities, such as δ Ɣ ji,δ ω k,δ κ ji,δ η kji,δ ji, δ ρ ji,δ u i, in the derivation of tangent stiffness matrices, the rate form of Eq. (65) can be written as δɣ ji σ ji d + + δω i Ṫ i d + δκ ji ( μ ji e ikl ˆ jkl )d δη kji ˆ kji d + (δ ji δɣ ji ) ρ ji d + δρ ji ( ji Ɣ ji )d = (δu i ṫ i + δω i ṁ i + δ ji ġ ji )d Finite element procedure (δu i ḃ f i + δω i ḃ m i )d (66) where the surface traction t i, surface couple m i and surface high-order generalised traction g ji are defined in Eqs. (16) (18), respectively. To use the meso-mechanically informed macroscopic constitutive relationships given by Eqs. (43) (46)forthe FE procedure, Eq. (63) is re-written as δɣ ji σ ji d + + δω k T k d + δκ ji (μ ji e ikl ˆ jkl )d δê kji ˆ kji d + δρ ji ( ji Ɣ ji )d = (δu i t i + δω i m i + δu i, j g ji )d + (δu i b f i + δω i b m i )d (64) ubstitution of Eq. (55) for the integral δê kji ˆ kji d into Eq. (64) results in FE formulations are typically expressed in boldfaced vectormatrix forms for the convenience of their implementation. The kinematic and kinetic variables in Eq. (66) are expressed in their vector forms, and thus, Eq. (66) can be re-expressed as δɣ T σ d + + δω T Ṫd + δη T ˆd δκ T ( μ H ˆ)d + (δ T δɣ T ) ρd + δρ T ( Ɣ)d = (δu T ṫ + δω T ṁ + δ T ġ)d + (δu T ḃ f + δω T ḃ m )d (67) in which, for the two-dimensional gradient Cosserat continuum, the 2 2 second-order tensors Ɣ ji, ji and σ ji, ρ ji in Eq. (66) are expressed as 1 4 row vectors Ɣ T, T and 4 1 column vectors σ, ρ; second-order tensors κ ji and μ ji

11 Comput Mech (214) 54: ( j = 1, 2; i = 3) are degraded into vectors and denoted as the 1 2 row vector κ T and 2 1 column vector μ, respectively. The third-order tensors η kji, ˆ kji are expressed as the 1 8 row vector η T and the 8 1 column vector ˆ. Particularly, the third-order permutation tensor e ikl (i = 3, k, l = 1, 2) applied to the third-order tensor ˆ jkl is expressed as the matrix H, defined by [ ] e311 e 312 e 321 e 322 H = e 311 e 312 e 321 e 322 (68) With the use of the weak form of constitutive relationship given by Eq. (49), the constitutive relationships (43) (46) expressed in the tensor form are re-written in the vectormatrix form below: σ = D (ij)(kl) σɣ Ṫ = D (z)(ij) T Ɣ ˆ = D (ijk)(lm) ˆƔ μ = D (iz)( jk) μɣ Ɣ + D (ij)(z) σω Ɣ + D (z)(z) T ω Ɣ + D (ijk)(z) ˆω Ɣ + D (iz)(z) μω ω + D (ij)(klm) η + D (ij)(kz) σ Ê σκ κ (69) ω + D (z)(ijk) η + D (z)(iz) T Ê T κ κ (7) ω + D (ijk)(lmn) ˆÊ ω + D (iz)( jkl) μê η + D (ijk)(lz) κ ˆκ (71) η + D (iz)( jz) μκ κ (72) Equations (69) (72) can be further expressed in the compact form given by ℵ =D I with ℵ T = σ T T T ˆ T μ T, I T = [ Ɣ T ω T η T κ T ] (73) The virtual variables on the left side of Eq. (67) can be denoted with a vector δ = [ δɣ T δω T δη T δκ T (δ T δɣ T ) δρ T ] T and transformed to (74) with the relationship δ = A δ (76) The rate variables on the left side of Eq. (67) can be denoted with a vector [ = σ T Ṫ T ˆ T (μ H ˆ) ] T T ρ T ( T Ɣ T ) (77) and transformed to [ = σ T Ṫ T ˆ T μ T ρ T ] T ( T Ɣ T ) (78) with the relationship = A (79) With the use of Eqs. (68) (79), the left side of Eq. (67) can be written in the compact form δ T d = δ T A T A d = δ T A T A D d = δ T ˆD d (8) where A and A can be obtained by comparing Eqs. (74) and (77) with Eqs. (75) and (78), respectively, and D σɣ D σω D σ Ê D σκ D T Ɣ D T ω D T Ê D T κ D = D ˆƔ D ˆω D ˆÊ D ˆκ D μɣ D μω D μê D μκ, I 4 I 4 I 4 [ ] ˆD ˆD = A T A D = 11 ˆD 12 (81) ˆD 21 ˆD 22 δ = [ δɣ T δω T δη T δκ T δ T δρ T ] T (75) with D σɣ D σω D σ Ê D σκ ˆD 11 = D T Ɣ D T ω D T Ê D T κ D ˆƔ D ˆω D ˆÊ D (82) ˆκ HD ˆƔ + D μɣ HD ˆω + D μω HD ˆÊ + D μê HD ˆκ + D μκ I 4 [ ] [ ] ˆD 12 =, I4 ˆD 21 =, ˆD I 4 22 = (83) I 4

12 1342 Comput Mech (214) 54: The D σɣ etc. expressed in Eqs. (81) and (82) are expressed such that they correspond to D (ij)(kl) σɣ etc., using the same subscripts but dropping the superscripts in Eqs. (69) (72) for clarity. The translational displacements u, microrotations ω,displacement gradients, and Lagrange multipliers ρ are the primary field variables defined in the macroscopic continuum to be interpolated in the proposed mixed FE procedure. Denoting their nodal values by u, ω,, ρ, one may use them to interpolate u, ω,, ρ at any point within the FE mesh and expressed as follows: u = N u u, ω = N ω ω, = N, ρ = N ρ ρ (84) where N u, N ω, N, N ρ are shape functions for u, ω,, ρ, respectively. The spatial derivatives Ɣ, κ, η of the primary variables u, ω,, which are present as sub-vectors of the vector in Eq. (75) at any point within the FE mesh, can be expressed in terms of u, ω,, respectively. Thus, Ɣ = B u u, η = B, κ = B ω ω, (85) where B u, B, B ω are the spatial derivatives of shape functions N u, N, N ω, respectively. The interpolation functions N u, N ω, N, N ρ for primary variables u, ω,, ρ, respectively, depend on construction of the multivariable mixed FEs under consideration. The present paper focuses on the development of the mixed FE procedure for the gradient Cosserat continuum in the frame of secondorder computational homogenisation. A particular multivariable mixed FE is designed and presented; it is termed as quadrilateral element QU38L4, as shown in Fig. 2, with 38 nodal DOFs and four Lagrange multipliers: eighteen DOFs for translational displacements defined at nine nodes, four DOFs for microrotations, and sixteen DOFs for displacement gradients defined at four corner nodes, with four Lagrange multipliers defined at the element centre. Eq. (84)-4 then implies a quasi-interpolation of the Lagrange multipliers and actually represents a uniform distribution of the Lagrange multipliers over one element, with their values defined at the element centre. The element matrices for the proposed element QU38L4 are numerically integrated with the 2 2 Gauss quadrature scheme. Patch tests are specially designed for the element in the gradient-enhanced Cosserat continuum and will be performed and discussed later in ect With denotation U defined as the nodal primary unknown vector, U T [ ] = u T ω T T ρ T T (86) the spatial FE discretisation of vector can be expressed by = BU (87) with B u N ω B = B B ω N N ρ (88) ubstitution of Eq. (87) into Eq. (8) with the use of Eqs. (81) (83) yields δ T d = δu T B T ˆDBd U = δu T T k t d U = δu Kt U (89) where K t is the tangent stiffness matrix of the proposed mixed FE, in which the integrand matrix k t is given by k uu k uω k u k uρ k t = k ωu k ωω k ω k ωρ k u k ω k k ρ k ρu k ρω k ρ k ρρ B u T D Bu σɣb T D σωn ω u +Bu T D Bu σκb T D σ Ê B Bu T N ρ ω Nω T D Nω T ƔB T (D T ωn ω + D T κ B ω ) u = +Bω T (D μɣ HD ˆƔ )B +Bω T (D μω HD ˆω )N Nω T D T Ê B ω u +Bω T (D μκ HD ˆκ )B +Bω T (D μê HD ˆÊ )B ω B T D ˆƔ B B T D ˆω N ω+ u B T D ˆκ B B T D ˆÊ B ψ N T N ρ ω Nρ T B u Nρ T N (9)

13 Comput Mech (214) 54: DOF at DOF at DOF at ui ω ij u ( i = 1,2) i ui ij ψ ( i, j = 1,2 ) ρ ( i, j = 1,2 ) Fig. 2 ketch of mixed finite element QU38L4 for gradient Cosserat continuum ThesymmetryofK t and its integrand matrix k t given by Eq. (9), can be easily verified with the definitions of those sub-tensors listed in Eq. (81)-1 given by Li et al. [41]. 5 Numerical examples 5.1 Meso-structural size effects in granular materials A major advantage of the second-order computational homogenisation approach over the first-order one is the capability of taking into account the absolute size of the mesostructure and subsequently exhibiting the meso-structural size effects. It is noted that the term of meso-structural size effect in the context of multi-scale homogenization is known from the size effect in the sense of dependence of material properties on the size of the structure. In this subsection, this capability is particularly demonstrated for the second-order computational homogenisation procedure proposed and implemented for granular materials via an example composed of a set of five RVEs with the same meso-structure. The set of RVEs with two symmetric axes x 1, x 2 are initially morphologically identical. Each RVE consists of 4 uniform round particles collocated with the same regular pattern but different absolute sizes. They are arrayed in Fig. 3 according to increasing particle radii:.5,.1,.2,.4,.8 m. The undeformed profiles of the five RVEs are square with the size L L, in which L = , , , , m respectively. Figure 3 presents the five deformed RVEs as the macroscopic strain gradient u 1,12 =.6/m and macroscopic strains u 1,1 =.15 and u 2,2 =.5 are imposed as the external loads on the boundaries of the RVEs. Namely, the translational displacements of the discrete contacting points of the peripheral particles with the RVE boundary are prescribed according to Eq. (39) with the given values of u 1,12, u 1,1, u 2,2. The material parameters for the round particles are listed in Table 1. The DEM proposed in [57], a linearspring-dashpot (LD) contact model taking into account dissipative sliding and rolling frictions, is adopted to simulate the meso-mechanical behaviours of the RVEs. The details of the model will not be introduced here owing to the limitation of space. Figure 3 illustrates that the deformation modes depend on the absolute size of the meso-structure as the strain gradient is included. The four terms on the right side of Eq. (3) are divided into two parts and denoted by J 1 = σ ji Ɣ ji + T k ω k +μ ji κ ji and J 2 = ˆ kji Ê kji. They are used as indices for the low- and high-order energy densities of the RVE with the dimension of joule per cubic metre (J/m 3 ), respectively, in the gradient Cosserat continuum. J 1 = σ ji Ɣ ji + T k ω k + μ ji κ ji is also valid as the index for the energy product when the gradient Cosserat continuum degrades to the classical Cosserat continuum. Figures 4 and 5 plot J 1, averaged Cauchy stress σ 11, and J 2 of the RVE versus the radius of the particle obtained by the first-order homogenization and the proposed second-order homogenization procedure as the RVEs are subjected to a varying macroscopic strain gradient: u 1,12 =.2,.4,.6/m with u 1,1 =.15 and u 2,2 =.5. The influences of the high-order deformation mode u 1,12 imposed to the RVE boundary on values of J 1,σ 11 and J 2 are illustrated. The first type of meso-structural size effect concerns the influence of the absolute size of the characteristic length of the meso-structure, which is the particle radius in the present example, on the resulting macroscopic responses. The curves in Fig. 4 show that the values of J 1 and σ 11 resulting from the proposed gradient Cosserat continuum model at the macroscale for granular materials decrease with increasing absolute size of the meso-structural RVE. The meso-mechanical mechanism of the size effect of meso-structural constituents can be detected from Fig. 3. The red solid dots in Fig. 3 denote the slide yielding contact points, at each of which the relative sliding friction tangential force between two contacting particles reaches its maximum value according to the Coulomb law of friction and the dissipative inter-particle movements occur. The blue solid dots denote the contact points with no slide yielding. Each green solid dot represents a pair of material points located at two immediate neighboring particles respectively that contact each other initially but currently lose their mutual contact. Note that (1) there are no red or green solid dots in the first deformed RVE (with the smallest absolute size); (2) the number of red solid dots increases with increasing meso-structural absolute sizes from the second to the fifth RVE; (3) the green solid dots appear only at the fifth RVE with the largest meso-structural absolute size. The mesomechanical mechanisms of the reductions of values of both

14 1344 Comput Mech (214) 54: Fig. 3 Dependence of the overall deformation mode and evolution of inter-particle contacts on the size of the microstructure. The five deformed RVEs with particle radii:.5,.1,.2,.4,.8 m respectively as macroscopic strain gradient u 1,12 in addition to macroscopic strains u 1,1 and u 2,2 is imposed. Types of inter-particle contact: blue: normal contacts with no slide yielding, red: contacts with slide yielding, green: loss of contacts. (Color figure online) x 2 x 1 J 1 and σ 11 with increasing meso-structural absolute size due to the macroscopic strain gradient lie in the dissipative interparticle displacements followed by loss of particle contacts. When the macroscopic uniform strains u 1,1 =.15 and u 2,2 =.5 are only prescribed to the RVEs, as in the first-order homogenisation procedure, a constant averaged Cauchy stress results for each RVE that is independent from the absolute size of the meso-structure. In addition, the resulting inter-particle force distributions over the domain of RVEs (and the distributions of the resulting external forces applied in the direction of axis x 1 along axis x 2 via the contact points of the peripheral particles with both the left and right boundaries of each RVE) are symmetric about axis x 1 of the RVEs. The macroscopic strain gradient u 1,12 =.6/m applied to the RVEs as a bending deformation mode will tend to generate, in addition to a uniform compression in the direction of axis x 1 of the RVEs due to u 1,1 =.15, both additional inter-particle tensile forces in the direction of axis x 1 Table 1 Material parameters used in the DEM simulation of RVEs. Parameters elected value tiffness coefficient of normal force (k n ) (N/m) tiffness coefficient of sliding force (k s ) (N/m) tiffness coefficient of rolling force (k r ) (N/m) tiffness coefficient of rolling moment (k θ ) 1. (Nm/rad) Damping coefficient of normal force (c n ).4 (Ns/m) Damping coefficient of sliding force (c s ).4 (Ns/m) Damping coefficient of rolling force (c r ).4 (Ns/m) Damping coefficient of rolling moment (c θ ).4 (Nms/rad) liding friction force coefficient (μ s ).5 Rolling friction force coefficient (μ r ) Rolling friction moment coefficient (μ θ ).2 on the top layers of particles of the RVEs and additional interparticle compressive forces in the direction of axis x 1 on the bottom layers of particles of the RVEs. Consequently, more