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J. Mech. Phys. Solids 160 (2022) 104759

Available online 12 January 2022

0022-5096/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/).

Contents lists available at ScienceDirect

Journal of the Mechanics and Physics of Solids

journal homepage: www.elsevier.com/locate/jmps

Particle size effects in ductile composites: An FFT homogenization

study

M. Magri a,∗, L. Adam b, J. Segurado c,a,∗∗

aIMDEA Materials Institute, C/ Eric Kandel 2, 28906, Getafe, Madrid, Spain

be-Xstream Engineering, Axis Park-Building H, Rue Emile Francqui 9, B-1435 Mont-Saint-Guibert, Belgium

cDepartamento de Ciencia de Materiales, E. T. S. de Ingenieros de Caminos, Universidad Politécnica de Madrid, Calle Profesor Aranguren

3, 28040 Madrid, Spain

ARTICLE INFO

Keywords:

Strain gradient plasticity

Ductile damage

Particle reinforced composites

Implicit gradient regularization

FFT homogenization

ABSTRACT

We present a computational homogenization study on the particle size effect in ductile

composites. The micromechanical formulation is based on non-local models through (i) the

incorporation of a lower-order strain gradient plasticity model and (ii) the application of

an implicit gradient regularization technique to the Gurson–Tvergaard–Needleman ductile

damage model for metals. In this way, the extended model is equipped with two length-scale

parameters, one for each non-local extension, which modulate the size dependent character of

the formulation. The problem consists of a system of partial differential equations in which two

Helmholtz-type equations for the damage regularization are coupled with the balance of linear

momentum through the stress, which depends on the non-local variables and on the plastic

strain gradient.

A series of numerical simulations are conducted to investigate the behavior of three-

dimensional microstructures representative of particle reinforced metal matrix composites. The

change in strengthening and ductility, as a function of the particle size, is first analyzed

by means of a parametric study in which the considered non-local extensions act both

independently and together. Finally a comparative study with experimental results demonstrates

that the particle size induced strengthening in metal matrix composites can be quantitatively

captured by the considered model.

1. Introduction

Particle reinforced metal matrix composites consist of a metal matrix whose stiffness and strength is enhanced by the addition

of hard reinforcing particles, resulting in advanced materials commonly used in the automotive and aerospace industries (Ibrahim

et al.,1991). In such composite structures, the mechanisms controlling the overall strength and ductility typically operate at different

length scales since heterogeneities of different size characterize their material microstructure (Babout et al.,2001,2004;Weck et al.,

2008). In particular, the overall mechanical response of particle reinforced metal matrix composites shows a prominent size effect

connected to the size of the reinforcing particles (Lloyd,1994;Gustafson et al.,1997;Dai et al.,2001;Milan and Bowen,2004;

Liu et al.,2014). For instance, at a constant particle volume fraction it has been observed an increase in strength with decreasing

∗Corresponding author.

∗∗ Corresponding author at: Departamento de Ciencia de Materiales, E. T. S. de Ingenieros de Caminos, Universidad Politécnica de Madrid, Calle Profesor

Aranguren 3, 28040 Madrid, Spain.

E-mail addresses: marco.magri@imdea.org (M. Magri), javier.segurado@imdea.org (J. Segurado).

https://doi.org/10.1016/j.jmps.2021.104759

Received 5 July 2021; Received in revised form 30 November 2021; Accepted 18 December 2021

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

2

M. Magri et al.

particle size. In addition to that, the size of the particles influences the ductility of the composite material, although in this case

bigger particles can increase or decrease the ductility depending on the dominant damage mechanism (Lloyd,1994).

Different factors contribute to these prominent size effects such as the distribution of the particles, the local stress triaxiality

as well as the mechanical properties of both metal matrix and reinforcements. As a result, the size dependence of the mechanical

response in particle reinforced metal matrix composites is particularly complex and a quantitative understanding is still lacking.

The rise of computational power, parallelization methods, and the development of more efficient solvers have increased the

potential of micromechanics based predictive models that, in conjunction with experiments, can speed-up significantly the design of

advanced materials with complex microstructures (Matouš et al.,2017). In computational micromechanics, the mechanical behavior

of composite materials is predicted at a microscopic scale within a suitable Representative Volume Element (RVE), i.e. a statistically

representative digital description of the material microstructure. In the context of metal matrix composites, numerous studies have

analyzed the impact of the composite microstructure on the mechanical response either by considering different failure mechanisms

separately – e.g. ductile failure in the matrix material (LLorca and Segurado,2004), debonding (Segurado and LLorca,2005), and

particle fracture (Böhm et al.,2004) – or by accounting for the combination of multiple damaging processes (Huber et al.,2005;

Shakoor et al.,2018;Dorhmi et al.,2020). Nevertheless, such models rely on conventional local mechanics, so the influence of size

on the plastic-damage processes cannot be accounted as the relevant governing equations are insensitive to dimensional changes.

In addition, the simulation of damage in continuum mechanics gives rise to a well-known pathological grid dependence of the

numerical solution, thus limiting the validity of the results to the adopted level of the spatial discretization of the RVE (Jirasek,

2007).

The characteristic size dependence of plastically deformed metals at micron and submicron scales can be effectively captured by

the so-called Strain Gradient Plasticity (SGP) models (Aifantis,1987;Voyiadjis and Song,2019). Despite the lack of a unified general

theory, such formulations are generally categorized in the following two distinct groups: (i) higher-order SGP (Fleck et al.,1994;

Fleck and Hutchinson,2001;Gurtin and Anand,2005;Gudmundson,2004) and (ii) lower-order SGP models (Bassani,2001;Nix and

Gao,1998). In the former class, strain gradients are included in the description of the continuum kinematics so that, according to

the principle of virtual work, an additional microforce balance equation, involving higher order stresses, is coupled with the classical

balance of linear momentum. On the other hand, lower-order strain gradient theories include the effect of strain gradients only at

the constitutive level by prescribing strain gradient dependent hardening moduli. The main benefit of this second approach is that

the structure of conventional continuum mechanics is preserved and, therefore, classical numerical algorithms, such as traditional

Finite Elements (FE), can be still employed with little modifications. In both cases, the model includes a length scale parameter

which controls the influence of the plastic strain gradient in the material response.

When damage and fracture of ductile metals are considered, an additional size dependence of the problem is related to the

absolute size of the region in which microscopic damage mechanisms take place. The use of classical continuum damage mechanics

for modeling this process typically leads to the loss of ellipticity of the relevant governing equations. As a consequence, their

numerical solution produces non-objective results since damage localizes in a band whose size depends on the discretization

level (Eringen,1966). The regularization of such a damage localization through non-local models suppresses this problem by

introducing a length parameter that controls damage localization and, therefore, rules the size effect when modeling damage.

Among different non-local theories (Jirasek,2007), the so-called implicit gradient regularization enhances the constitutive equations

through some non-local fields, typically a scalar measure of the plastic strain, which are the solution of additional partial differential

equations (Peerlings et al.,1996). In this way, the resulting formulation is an enriched problem where the classical balance of linear

momentum is fully coupled with a Helmholtz-type equation for each of the non-local variables. The numerical implementation of this

coupled problem is, therefore, much more expensive than the one of classical damage mechanics and the simulation of regularized

damage models in complex multi-phase microstructures is nowadays limited by the use of traditional FE solvers.

As a result of the aforementioned limitations involved with the implementation of SGP formulations and non-local damage

models, only a few numerical works have studied the effect of particle size on the strengthening and ductility in metal matrix

composites. For example, the strengthening size effect was analyzed via the implementation of a lower-order SGP model in two-

dimensional (Yan et al.,2007;Zhou et al.,2011) and three-dimensional (Zhang et al.,2019) microstructures using FE. The influence

of damaging processes was also considered in order to study the particle size effect on both ductility and strengthening in Weng et al.

(2019) and Ban et al. (2020). In particular, Weng et al. (2019) presented three-dimensional simulations accounting for particle size

effects in the strength through the use of a lower-order SGP model coupled with interface damage. Similarly, Ban et al. (2020) studied

the coupled size effects of plasticity and damage by means of idealized single-particle axisymmetric simulations including particle–

matrix debonding and ductile failure in the matrix material. This last model combines size effects in strengthening and ductility by

making the intrinsic length scale of the adopted SGP formulation as a function of damage evolution. However, the model still relies

on a classical (local) version of the Lemaitre damage model (Lemaitre,1985). In general, the use of non-local damage mechanics in

micromechanical simulations of particle reinforced composites is very limited because of some intrinsic difficulties, e.g. the extension

of existing non-local formulations to heterogeneous media and the high computational cost of the resulting models. Therefore, the

available numerical studies employ simple damage models (Drabek and Böhm,2006,2005) or are restricted to two-dimensional

setting (Reusch et al.,2008). Only recently, the numerical implementation of regularized damage models in heterogeneous media

via solvers based on Fast-Fourier-Transforms (FFT) has shown promising perspectives for the simulation of non-local problems with

complex microstructures (Magri et al.,2021).

In this work, we present a numerical study on the size dependent mechanical response of particle reinforced metal matrix

composites in the field of computational homogenization. As discussed in Section 2, to account for size effects, the damage model for

microvoid nucleation, growth, and coalescence of Tvergaard and Needleman (1984) is purposely extended via (i) the incorporation

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

Fig. 1. Schematic of the microscopic arrangement of a particle reinforced metal matrix composite.

of the lower-order SGP model of Huang et al. (2000) and (ii) the application of the implicit non-local damage regularization

for heterogeneous materials derived in Magri et al. (2021). For simplicity, alternative damaging mechanisms, e.g. particle–matrix

debonding or particle fracture, are not included here. The resulting formulation is a non-conventional mechanical problem consisting

of three coupled partial differential equations. This problem is numerically solved via the iterative staggered algorithm described

in Section 3that exploits the sequential usage of FFT-Galerkin and conjugate gradient based solvers. The impact of the non-

local extensions on the composite mechanical response is analyzed in Section 4through a series of parametric studies. Finally, a

comparison with experimental results demonstrates the capability of the present formulation of capturing quantitatively the typical

particle size dependent strengthening in metal matrix composites.

2. Model description

A metal matrix composite is here idealized as a biphasic medium consisting of ceramic reinforcements embedded in a metallic

matrix, as represented schematically in Fig. 1. The RVE, with volume , is then characterized by the presence of a reinforcing

material (denoted by S), which occupies an overall volume , and a matrix material (denoted with M) with volume . The

condition =∪thus holds. Mechanical equilibrium is prescribed by the local form of the balance of linear momentum in

the absence of inertial and body forces

div[𝝈] =

0,(1)

where 𝝈is the Cauchy stress tensor.

The reinforcements are assumed to behave as elastic solids insensitive to damage. Their mechanical behavior corresponds to the

one of an elastic isotropic solid with Young modulus and Poisson coefficient . On the other hand, the material model of the

metallic matrix incorporates the physical processes that characterize ductile fracture of metals, namely void nucleation, growth,

and coalescence. In this regard, we select the phenomenological extension proposed by Tvergaard and Needleman (1984) (known

as the GTN model) of the physically-based model for void growth of Gurson (1977). Two extensions of the classical version of the

GTN model are introduced here. Firstly, to deal with a well-posed failure model, a non-local regularization of the GTN model will

be considered. Secondly, a lower-order strain gradient plasticity model will be incorporated in the evolution of the flow stress to

capture the size dependent strengthening induced by the reinforcing particles.

2.1. The GTN model

The classical version of the GTN model is characterized by the definition of the so-called effective porosity ∗, which quantifies

the level of damage induced by the presence of voids in the material matrix. The mechanical degradation related to the presence

of porosity is taken into account by the definition of the following yield surface

(,

0, ∗) =

2

+ 2 ∗1cosh −3

2

2

−1 + 2

12

∗,(2)

where

0is the matrix equivalent plastic strain and the matrix flow stress. The parameters 1,2are phenomenological adjusting

coefficients, while and refer to the Mises equivalent stress and hydrostatic pressure, respectively, i.e.

=3

2𝒔∶𝒔and = −1

3tr [𝝈],

being 𝒔=dev [𝝈]the deviatoric stress tensor. Eq. (2) implies that the plastic behavior is pressure dependent for non-zero values of

the effective porosity. The latter affects the yield surface by decreasing the set of admissible stress states as ∗increases, leading

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

eventually to a complete loss of bearing capacity of the material matrix. This condition is reached for a limit value of the effective

porosity ∗

that can be calculated by imposing = 0 at zero stress, namely

∗

=1

1

.

The stress tensor 𝝈is defined according to an isotropic linear elastic law in rate form

𝝈=tr

𝜺−

𝜺𝑰+ 2 dev

𝜺−

𝜺,(3)

where and are the bulk and shear moduli. Symbols 𝜺and 𝜺refer to the total and plastic strains, while a superposed dot indicates

a partial time (or more precisely pseudo-time) derivative. In the realm of small strain analysis, a standard additive decomposition

of the rate of strain tensor of type

𝜺=

𝜺+

𝜺

is adopted, where 𝜺is the elastic strain. The evolution of the plastic strain tensor 𝜺is derived from application of the normality

rule

𝜺=

𝝈=

𝝈+

𝝈,(4)

where identifies the plastic multiplier. In standard rate independent plasticity (Simo and Hughes,2006), the plastic multiplier is

calculated by exploiting the so-called persistency condition, i.e. by imposing

= 0 if > 0. The inelastic material behavior is then

completed by the evolution of the internal variables

0and ∗. In the GTN model, the evolution of the matrix equivalent plastic

strain reads1

0=𝝈∶

𝜺

(1 − )

,(5)

where is the actual void volume fraction defined in the spirit of Gurson (1977). The constitutive definition of the matrix flow

stress will be discussed in Section 2.3 along with the incorporation of strain gradient effects. The kinetics of evolution of

the effective porosity ∗is established to model the physical processes taking place in ductile fracture of metals, namely (i) void

nucleation, (ii) void growth, and ultimately (iii) formation of macro cracks due to void coalescence. The latter process is prescribed

using the following phenomenological law

∗() =

if < ,

+∗

−

−

(−)if < ,

∗

if ,

(6)

where is a critical void volume fraction above which void coalescence activates, while refers to the void volume fraction at

fracture. Finally, the evolution of reads

=

+

,(7)

being

and

the time variation of the porosity due to the mechanisms of void nucleation and growth, respectively.

According to Nguyen et al. (2020), these functions have to be properly defined in order to deal with an objective formulation

of ductile failure. Such aspects will be summarized next.

2.2. Non-local regularization of the GTN model

A suitable regularization of the GTN model requires the kinetics of nucleation and growth processes to be established in terms

of non-local plastic variables, as follows

=

0

0,and

=(1 − )

,

where is a strain rate controlled nucleation rate, while

0and

are the non-local counterparts of the matrix equivalent plastic

strain and the volumetric part of the plastic strain in the porous material,

0and

=tr [𝜺]respectively. According to Chu and

Needleman (1980), the void nucleation function reads

0=

2

exp

−1

2

0−

2

.(8)

1For the sake of clarity, it is remarked that symbol 𝜺identifies the plastic strain tensor in the porous metal while the slightly different symbol

0refer to

the scalar matrix equivalent strain.

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

where in Eq. (8),,and are material parameters. represents the volume fraction of void nucleating particles while

and correspond to the mean value and standard deviation of the strain necessary to nucleate a void. Note that the classical

specifications of

and

, as given in Tvergaard and Needleman (1984), can be recovered by replacing the non-local variables

with their local counterparts. The introduction of the non-local variables in the evolution of damage is employed in order to avoid

strain localization during fracture propagation.

The definition of the non-local variables and

depends on the particular type of regularization employed. In this work, an

implicit gradient approach has been chosen because it results in additional partial differential equations which allow the formulation

of the regularized problem with local equations. Moreover, due to the form of the additional field equations, the implicit gradient

model is particularly suitable for the numerical solution through FFT-based algorithms, as shown by Magri et al. (2021), whose

formulation will be followed here.

In the implicit gradient regularization, the non-local plastic strain variables are solutions of the following partial differential

equations of Helmholtz type

0−div 𝓁2

∇

0=

0,(9a)

−div 𝓁2

∇

=

,(9b)

where 𝓁is a material parameter with the unit of length that introduces an internal length scale related to the damage process.

From a strictly mathematical point of view, 𝓁can be treated exclusively as a parameter related to the non-local regularization

of damage (i.e. to state a well-posed boundary value problem). To produce an effective non-local regularization, the magnitude of

𝓁has to be selected at least as the size of a couple of voxels. Nevertheless, 𝓁might be connected to the physical mechanisms

underlying ductile fracture. More specifically, 𝓁can be interpreted in terms of the average distance between micro-voids in the

materials, i.e. the one that are described in a homogenized sense – or the size of a process zone – i.e. the thickness of the damage

bands. Following this path of reasoning, 𝓁shall not be treated as an intrinsic material parameter, but rather identified based on

the specific material microstructure and on the scale at which the material is idealized (e.g. the size of microvoids is much smaller

than the reinforcing particles). Therefore, for the scope of this study, it seems legitimate to select 𝓁with the same order magnitude

of the reinforcing particles since the thickness of the fracture zone is expected to be of that size.

In the context of FFT-based solvers, the non-local averaging equations are resolved in the whole domain and, although

the source of and

are restricted to the matrix, the resulting non-local fields can enter the regions occupied by the elastic

reinforcements . Moreover, the parameter 𝓁has to be assigned to the elastic phases as well. As proven in Magri et al. (2021),

if 𝓁is specified as

𝓁( ) = 𝓁

if ∈,

𝓁

otherwise ,(10)

with 𝓁

∕𝓁

→0, a free-Neumann interface condition between reinforcing particles and metal matrix is recovered. In fact, setting

𝓁

∕𝓁

1prevents spurious diffusion of non-local variables, and thus the propagation of fracture, from the damageable matrix

to the elastic phases.

2.3. Mechanism-based strain gradient plasticity

In order to study particle size-effects in metal matrix composites, the GTN model is extended to include the effects of strain

gradients in the work hardening law. Due to its simplicity and physical background, we selected the so-called mechanism-based

strain gradient plasticity model (MSGP) developed by Nix, Gao, Huang and co-workers (Nix and Gao,1998;Gao et al.,1999;Huang

et al.,2000,2004). Unlike other strain gradient theories of phenomenological nature, the MSGP introduces plastic strain gradients in

the work hardening law based on microscopic dislocation mechanisms via the notion of geometrically necessary dislocations (GNDs).

Since the effect of the strain gradients is introduced only at the constitutive level, the thermodynamic consistency of the model is

satisfied without the introduction of higher order stresses. This model falls in the class of the so-called lower-order strain gradient

plasticity models as the conventional structure of continuum mechanics is preserved. As a consequence, no additional boundary

conditions need to be set to prescribe flux of dislocations through the boundaries (e.g. micro-hard or micro-free conditions) and

plastic strain gradients develop naturally.

The starting point of the MSGP is the crystal level, where the model assumes that the critical resolved shear stress for plastic

flow in a slip system, , is a function of the dislocation density, following the Taylor model (Taylor,1934,1938)

= +,(11)

where ≃ 0.3is an empirical coefficient, the shear modulus, the magnitude of Burgers vector, while symbols and denote

the densities of statistically stored dislocations (SSDs) and GNDs, respectively. According to Nye (1953) and Ashby (1970), the density

of geometrically necessary dislocations is linked to a suitable scalar measure of the plastic strain gradients, , as follows

=

,(12)

being ≃ 1.9the so-called Nye factor defined in Arsenlis and Parks (1999). The macroscopic tensile flow stress of a polycrystal,

, is then proportional to through the Taylor factor ≃ 3.06, namely

= ,

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

which, substituted into Eq. (11) and in view of Eq. (12), gives

= +

.(13)

In the particular case of uniaxial tension, the plastic strain is uniform and the plastic strain gradients vanish. Accordingly, is

equivalent to the conventional flow stress 0, namely

0= ,(14)

so that the density of SSDs can be simply estimated as function of 0as

=0

2

.(15)

In a general scenario, the conventional flow stress is expressed in terms of the equivalent plastic strain as

0=

0,(16)

where denotes a reference stress and

0is a non-dimensional hardening function that will be specified in the numerical

examples that follow (see Eqs. (30) and (34)). Therefore, by combination of Eqs. (13),(15),(16), and after some algebra, the

following flow stress results

= 2

0+𝓁,(17)

where

𝓁=2 2

2

≃ 18 2

2

, (18)

is a parameter with the units of length that introduces another intrinsic material length scale in the model. According to Eq. (18)

and for ∕ ≃ 100, typical values of 𝓁are of the orders of microns, which is in fact the length scale where size effects are

not negligible in experimental evidence. Eq. (17) introduces a correction of the conventional tensile flow stress dependent on the

product 𝓁. In the case where the length scale of the plastic deformation gradients is much larger than 𝓁,is much smaller

than the plastic internal length, so the product 𝓁is negligible and →0. On the other hand, if the variations of the plastic

strain field are not negligible at a length of the order of 𝓁(or less), > 0leading to a size-dependent strengthening effect.

According to Huang et al. (2004), a suitable expression for the effective plastic strain gradient can be derived from the Nye’s

dislocation tensors 𝝌. This tensor measures the incompatibility of the plastic deformation as follows

𝝌= −curl 𝜺.

The equivalent plastic strain gradient can be then defined as the quadratic invariant of 𝝌, i.e.

=𝝌∶𝝌.(19)

3. Numerical implementation

The problem stated in the previous section consists of a system of three partial differential equations, the balance of linear

momentum Eq. (1) and the two averaging Eqs. (9). An iterative staggered approach is used to solve the governing coupled equations.

Under this framework, for each time increment, the partial differential equations are solved sequentially assuming that they are

uncoupled. In this way, the solution of each equation provides a correction for its relevant primary unknown (e.g. the strain field

for Eq. (1) and the non-local plastic variables for Eqs. (9)). This sequential solution is carried out iteratively, employing a fixed-

point iteration scheme, until the correction of the unknown fields between two consecutive iterations becomes smaller than a small

tolerance, such that the solution coincides with the one obtained with a monolithic approach (Steinke et al.,2017). As demonstrated

in Boeff et al. (2015) and Magri et al. (2021), combining iterative staggered schemes with solvers based on Fast Fourier Transforms

provides efficient algorithms for the solution of non-local damage mechanics in the field of computational homogenization. FFT

based homogenization, firstly introduced in the 90s (Moulinec and Suquet,1994,1998), have demonstrated to be an efficient

alternative to the Finite Element Method. Typical advantages related to FFT-based solvers are that periodic boundary conditions are

imposed intrinsically and digital images of real microstructures can be directly used without the need of meshing. In addition, direct

comparisons with FE solvers revealed that FFT solvers can be computationally faster especially in problems characterized by a large

number of unknowns (Lucarini and Segurado,2019c). In this work, the iterative FFT-algorithm proposed in Magri et al. (2021)

is first extended to incorporate the considered strain gradient plasticity model and is used thereon as the numerical framework to

study the size effects in particle reinforced metal matrix composites. The solving scheme including the gradient plasticity model

implementation will be presented in this section along with the discretized form of the governing equations.

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

3.1. Governing equations in incremental form

The rate form of the constitutive equations of the modified GTN model is integrated following the Euler Method. Assuming a

loading history of the problem defined on the time interval [0, ], we then divide it in increments, = (+1 −), so that the

discretized fields read

(∙)=(∙)=, (∙)=(∙)+1 −(∙)= 0,1,…− 1.(20)

For any time (or load) increment , the incremental form of the problem yields to these balance equations

div 𝝈+1=

0,(21a)

0+1 −div 𝓁2

∇

0+1=

0+1 ,(21b)

+1 −div[𝓁2

∇

+1] =

+1 ,(21c)

with the constitutive laws

𝝈+1 =𝝈+tr 𝜺−𝜺𝑰+ 2 dev 𝜺−𝜺,(22a)

𝜺=

+1

𝝈+1

+

+1

𝝈+1 ,(22b)

=𝝈+1 ∶𝜺

1 − +1

0+1 , ,(22c)

=(

0+1)

0+1 − +1

,(22d)

and such that

=

+1

0+1 ,

2

+ 2 ∗+1 1cosh

−3

2

2+1

0+1 ,

−1 + 2

1∗2

+1= 0.(22e)

The problem is completed by the choice of the following initial conditions

𝝈=0=𝝈0,𝜺=0=𝜺

0,

0=0

=

0, =0=0.

Note that the functional dependence of is written in terms of instead of +1, which renders the numerical method

semi-implicit. This approximation is generally taken in the numerical solution of lower-order SGP models since it allows for a

straightforward implementation of Eq. (21a) using similar solving algorithms to the one employed for local plasticity models

(see Haouala et al. (2020) for instance). Eqs. (21) are solved for a prescribed mixed loading history given by a combination of

components of the macroscopic strain 𝐄+1 and stress 𝜮+1 tensors, such that

+1 = ()=1

+1 d , +1 = ()=1

+1 d ,

where , and , refer, respectively, to the components in which macroscopic strain or stress are prescribed (with , ∩ , = ∅).

In addition, periodic boundary conditions are imposed for all the solution variables

𝜺() = 𝜺( +),(23a)

0() =

0( +),(23b)

() =

( +),(23c)

with referring to any vector with components obtained as the product of an integer number by the periodicity of the RVE, i.e. .

3.2. The iterative staggered algorithm

At increment , assuming the relevant field variables to be known, the incremental problem (21) consists in finding 𝜺+1,

0,

and

with a macroscopic prescribed state given by a combination of strain and stress components +1 and +1 . The adopted

iterative staggered algorithm schematizes as follows

Initialize the solution fields:

–𝜺(0)

+1 =𝜺,

0(0)

+1 =

0,

(0)

+1 =

Compute the effective plastic strain gradient, , from 𝜺, Eq. (19) (Section 3.4)

Iterate over = 0,1,…until convergence

Journal of the Mechanics and Physics of Solids 160 (2022) 104759

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M. Magri et al.

(a) solve the purely mechanical problem via the FFT-Galerkin algorithm:

– assign

0+1 =

0()

+1 and

+1 =

()

+1 in Eq. (22d)

– compute 𝜺(+1)

+1 by solving (21a) along with Eqs. (22) and Eq. (17)

– save the local plastic variables

0(+1)

+1 and

(+1)

+1

(b) solve the non-local averaging equations in FFT:

– assign

0+1 =

0(+1)

+1 in Eq. (21b)

– assign

+1 =

(+1)

+1 in Eq. (21c)

– compute

0(+1)

+1 by solving Eq. (21b)

– compute

(+1)

+1 by solving Eq. (21c)

(c) check convergence:

– if (max{1, 2, 3}<tolerance): increment completed

– else: =+ 1 and go to (a)

where

–1=max𝜺(𝒌+𝟏)

𝒏+𝟏−𝜺(𝒌)

𝒏+𝟏

𝜺(𝒌+𝟏)

𝒏+𝟏

–2=

0(+1)

+1 −

0()

+1

0(+1)

+1

–3=

(+1)

+1 −

()

+1

(+1)

+1

At the core of the algorithm two distinct spectral solvers are used for the solution of the governing equations of the problem. The

first type of solver is an FFT-Galerkin algorithm (Vondřejc et al.,2014;Zeman et al.,2017) exploited for the solution of the non-linear

Eq. (21a), which has the mathematical structure of a conventional mechanical problem within this iterative solver. This Fourier–

Galerkin solver is complemented with the technique proposed by Lucarini and Segurado (2019a) for prescribing, without requiring

additional iterations, macroscopic loading states that combine strain and stress components. On the other hand, the averaging

equations ((21)b-c) are both solved numerically in Fourier space with a conjugate-gradient based algorithm (Magri et al.,2021).

Further details on the solving algorithms are left in the Appendix for completeness.

3.3. Spatial discretization and discrete derivative rule

The spatial discretization of Eqs. (21) follows the standard procedure adopted for spectral solvers. In micromechanical analysis,

the simulation domain is typically a periodic RVE embedded in a cuboidal domain with edges lengths 1, 2, 3. Such an RVE is

discretized with a regular array of 1×2×3voxels, where each voxel belongs to any of the phases represented. The unknown

discrete fields correspond to their values at the center of each cell, with position given by

= (1

2+)

, = 1,2,3and ∈ [0, − 1] .

The Fourier transform of the discrete field is taken as the discrete Fourier transform. The corresponding 1×2×3discrete

frequencies in Fourier space,

, are given by

=2

−(− 1)

2if odd

−

2if even

for = 1,2,3and ∈ [0, − 1] .(24)

Using the properties of the Fourier transforms, the partial derivative of a given generic function in the real space can be easily

calculated given its Fourier transform []as

= i [],(25)

where is the Fourier transform operator while iindicates the imaginary unit. If the field exhibits strong spatial discontinuities,

the usage of the standard derivative rule Eq. (25) might give rise to noisy solutions. Such a behavior is typical of problems with

large phase property contrasts, especially near the phase boundaries, or in the case where the field to be derived drops sharply to

zero in some regions, as it happens for instance for the plastic strain in the elastic reinforcements of a composite materials.

To alleviate this numerical noise, we adopt the modified derivative rule proposed by Willot (2015) for a purely mechanical

problem. This alternative definition – which is based upon the application of a finite difference scheme in the real space for a

spatial derivative – substitutes the classical continuum derivative Eq. (25) with

= i (

)[],(26)

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M. Magri et al.

where

is a modified version of the frequency vector that reads

(

) = 1

4tan

2(1 +

1)(1 +

2)(1 +

3),

with

=∕(2). This modified derivative is exploited for the solution of Eq. (21a) only, (see Appendix A). On the other hand,

the numerical solution of Eqs. ((21b)-c) in Fourier space is carried out using the standard form Eq. (25).

3.4. Calculation of the equivalent plastic strain gradient

In the MSGP model considered here, size effects in the mechanical response are introduced through the dependence of , the

scalar measure of the plastic strain gradient, in the modified flow stress Eq. (17).is selected here as a quadratic invariant of the

Nye’s dislocation tensor 𝝌in Eq. (19). According to the geometrical considerations of Nye (1953), 𝝌quantifies the incompatibility

of the plastic strain tensor and takes the following expression

𝝌= −curl 𝜺,

which is computed operatively as

𝝌 = −

,(27)

where 𝝐denotes the Levi-Civita permutation tensor. The calculation of the components of the plastic strain gradient can be easily

carried out in Fourier space by using the discrete derivative rule (26) in order to increase the accuracy in the calculation of 𝝌and

limit the emergence of Gibbs oscillations at phase interfaces (see Haouala et al. (2020), Berbenni et al. (2014), Brenner et al. (2014),

Lebensohn and Needleman (2016) and Marano et al. (2019) for instance). However, the calculation of spatial derivatives using a

finite difference formula should be handled carefully if applied to discontinuous fields. The numerical derivative computed with this

strategy indeed depends on the spatial discretization leading to a non-objective calculation of 𝝌as demonstrated in Appendix C.

Therefore, the implementation of SGP models in FFT-based solvers can lead to inaccurate results depending on how plastic strain

gradients are computed numerically.

In this paper, we propose an alternative numerical strategy for the calculation of plastic strain gradients that produces grid

independent results. The idea is to apply an implicit gradient regularization to each component of the plastic strain gradient and

then apply the curl operation to these fields. The regularization of each plastic strain gradient component follows Eq. (9) as

−𝓁2

∇2

=

,(28)

where 𝓁is the relevant characteristic length of the regularization, which is assumed to be homogeneous throughout the domain.

The components of the plastic strain gradient appearing on the right-hand side of Eq. (28) are calculated in Fourier space using the

centered discrete derivative rule (see Willot (2015) for details)

=−1 isin

,

Finally, the equivalent plastic strain gradient is estimated from the Nye’s dislocation tensor

𝝌 = −

,(29)

This alternative strategy for computing the dislocation tensor is objective and its use in the considered MSGP model produces

grid independent numerical results as demonstrated by the convergence study reported in Appendix C. It is worth noting that

Eq. (28) introduces a further length parameter into the model. Nevertheless, contrary to 𝓁and 𝓁,𝓁merely represents a numerical

parameter for the objective calculation of plastic strain gradients and is not associated to any physical process. Its numerical value,

in the simulations that follow, will be selected equal to the size of a few voxels in order for the implicit regularization to be effective

and, at the same time, to provide a limited smoothening of the original discrete derivatives. The numerical solution of Eq. (28) is

carried out in Fourier space, as pursued for the Helmholtz-type equations ((21b)-c). In this case, the choice of a homogeneous 𝓁

speeds up significantly the calculations since it allows for a direct solution of the problem without the need of resorting to iterative

solvers.

4. Numerical results

4.1. Representation of the microstructure and material parameters

The generation of the three-dimensional RVEs used in the numerical simulations follows the methodology described in Segurado

and Llorca (2002). A cubic unit cell of size ××is filled with 30 non-overlapping identical spheres representing the reinforcement.

This number of particles is sufficient to have a statistically isotropic arrangement of the spherical inclusions. The position of the

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Fig. 2. Periodic multi-particle RVEs adopted in the 3D numerical simulations in the case of (a) 15% and (b) 20% volume fraction of reinforcing particles. The

grid resolution is 1003voxels for a total of 1 × 106grid points.

particles is generated using a Random Sequential Adsorption algorithm, in which the centers of spheres of radius are introduced

sequentially and randomly. In this algorithm, a new generated particle is accepted if the distance between its center and the one

of all the previous particles is greater that 2.07, in order to have approximately a minimum of two voxels representing the matrix

between two particles.

A voxel-based digital representation of the RVE is then generated by rastering the considered microstructure using the same

number of grid points along the three principal axis. The digital voxel-based unit cells used in this study are reported Fig. 2.

The material parameters are set to mimic the response of an Aluminum alloy reinforced with SiC particles. In particular, the

elastic properties of the matrix are selected as = 70 GPa and = 0.33 while = 427 GPa and = 0.17 are assigned to the

elastic reinforcements. The parameters of the GTN model for the composite matrix are set as 1= 1.5,2= 1,= 0.15,= 0.25,

= 0.08,= 0.1, and = 0.05. The initial void volume fraction is zero and microvoids will nucleate according to Eq. (8). The

non-dimensional hardening function

0is specified as a power law

0=

1 +

0

,(30)

with = 200 MPa, = 250 MPa, and = 0.1.

In the GTN model, the local bearing capacity of the matrix material is lost as soon as the damage indicator ∗equals ∗

. If such a

condition is attained, the convergence of the FFT-Galerkin scheme used to solve Eq. (21a) is compromised since the contrast between

the stiffness of matrix and inclusions becomes too high. Therefore, for numerical convenience, the upper limit of the damage variable

is limited to ∗= 0.95 ∗

= 0.63. Accordingly, a low residual stress capacity is left to the ‘‘fully damaged’’ material matrix. To prevent

spurious propagation of inelastic non-local variables in the region occupied by the elastic particles when solving Eqs. (9), the length

parameter in the elastic phases 𝓁

is taken such that 𝓁

∕𝓁

= 0.02. In this way, according to Eq. (10),𝓁is defined as function

of 𝓁

only. Therefore, for the sake of readability, symbol 𝓁will be directly used in place of 𝓁

in what follows, being the latter

the only relevant parameter for the non-local regularization of damage. The length-parameter introduced in the definition of the

objective plastic strain gradient is taken as 𝓁= 0.02for all the simulations. This value corresponds to two voxels length, which is

the minimum number of elements suggested in Miehe et al. (2015), in the framework of phase-field fracture, for the fracture length

scale parameter in order to represent accurately the solution of the Helmholtz equation.

Unless otherwise specified, in the numerical simulations that follow, the RVE is subjected to a uniaxial tensile loading in which

a macroscopic strain is prescribed in the direction 1, while a stress free condition is enforced in the remaining components

𝑬=

11 ∗ ∗

∗ ∗ ∗

∗ ∗ ∗

and 𝜮=

∗00

0 0 0

0 0 0

.

4.2. Parametric study of the size dependent strengthening and ductility

The goal of this section is to show the influence of the considered non-local extensions on the proposed model, i.e. the modified

MSGP flow stress (17) and the Helmholtz-type Eqs. (9). Such non-local equations introduce multiple length-scales into the model

through the intrinsic length parameters 𝓁and 𝓁. The former one modulates the plastic strain gradient induced strengthening in

the material matrix. On the other hand, 𝓁controls the spreading of the damage related plastic variables thus influencing the overall

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Fig. 3. (a) Plot of the average stress component 11 against the average strain component 11 for different values of the adimensional parameter 𝓁∗

. (b) Log–log

plot of the relative strengthening 11 introduced by the considered MSGP model as a function of the adimensional parameter 𝓁∗

=𝓁∕. The reported results

are obtained for a RVE containing 20% volume fraction of reinforcing particles.

ductility of the composite. To provide a general analysis of the impact of the two non-local mechanisms separately, the first two

sets of numerical simulations that follow will be parametrized with respect to the non-dimensional parameters

𝓁∗

=

𝓁

and 𝓁∗

=𝓁

,(31)

with indicating the particle diameter. Therefore, the change in strengthening and ductility promoted by the considered non-local

theories will be directly linked to the particle size as usually assessed in the experimental studies.

4.2.1. Particle size induced strengthening

The impact of the MSGP model on the composite material is firstly analyzed by neglecting the mechanical degradation associated

with damage. To suppress damage nucleation, parameter is set to zero and, therefore, the composite matrix behaves as an elasto-

plastic material without softening. Fig. 3a represents the stress strain curves obtained from the simulations for different values of

the adimensional parameter 𝓁∗

. The numerical results highlight a prominent size effect in the macroscopic response of the metal

matrix composite in view of Eq. (31): the lower the particle size (i.e. higher 𝓁∗

), the larger the simulated macroscopic stress. By

comparing the reference solution 𝓁∗

= 0 with the other curves, it can be seen that the particle size basically induces a negligible

strengthening if the particle diameter is much larger than the parameter 𝓁(i.e. > 𝓁∕10). On the other hand, when the particle

diameter approaches 𝓁the composite strength increases significantly up to about 20% in case of 𝓁= 2. A quantitative evaluation

of the strengthening induced by the MSGP model can be calculated as the relative increment in simulated stress (at the end of each

simulation) with respect to the one obtained in the case of conventional plasticity, i.e.

11(𝓁∗

) =

11(𝓁∗

) − 11(𝓁∗

= 0)

11(𝓁∗

= 0) .(32)

These results are represented in Fig. 3b using a − graph, showing that the strengthening scales linearly with 𝓁∕, similarly

to the results obtained with other SGP models (see Bassani et al. (2001) for instance).

Fig. 4 represents the spatial distribution of the equivalent plastic strain gradient ∗in the case of the maximum 𝓁∗

simulated,

which corresponds to the smallest particle size =𝓁∕2. It can be observed that the plastic strain gradient localizes in regions

touching the particle interfaces. In the opposite case, when 𝓁∗

= 0, plastic strain gradient effects are neglected in the modified flow

stress.

Fig. 5 reports the spatial distribution of the equivalent Mises stress in the material matrix for different values of 𝓁∗

at the end of

each simulation. As observed in Fig. 3a, the RVEs containing particles with small diameters are capable of sustaining higher stress

states. In addition to that, a smaller particle diameter induces a different stress distribution in the material matrix with a higher

stress concentration. By comparing Fig. 5c with Fig. 4, i.e. the one reporting the distribution of the equivalent plastic strain gradient

for the same 𝓁∗

, it can be observed that the distribution of the Mises stress basically coincides with the one of . Therefore, for

small particles the equivalent plastic strain gradient produces a dominant effect in the modified flow stress Eq. (17).

4.2.2. Size dependent ductility

In this second set of numerical simulations we aim at evaluating the impact of the non-local regularization of damage defined

by Eqs. (9). In order to focus specifically on damage regularization, strain gradient plasticity effects are switched off by prescribing

𝓁= 0.Fig. 6a collects the macroscopic stress strain curves obtained with different values of the non-dimensional parameter 𝓁∗

.

Note that the material behavior of the composite matrix in the case of 𝓁∗

= 0 coincides with the classical GTN model (Tvergaard

and Needleman,1984) since

0=

0and

=

. As expected, parameter 𝓁∗

influences significantly the overall ductility of the

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Fig. 4. Spatial distribution of the non-dimensional equivalent plastic strain gradient ∗obtained as ∗= for 𝓁∗

= 2.0(the black arrow indicates the loading

direction). The reported results are obtained for a RVE containing 20% volume fraction of reinforcing particles.

Fig. 5. Mises equivalent stress in the RVE at 11 = 0.1for (a) 𝓁∗

= 0.0, (b) 𝓁∗

= 1.0, and (c) 𝓁∗

= 2.0. Note that colorbars have been reported with different

scales to better capture the stress distribution in each case. The black arrow indicates the loading direction.

metal matrix composite: a higher ductility results for higher values of 𝓁∗

. From the reported results, it can be noted that all the

simulations stop before the final failure of a material point is reached, i.e. before the macroscopic stress tends to zero. This occurs

because of lack of convergence of the FFT solver after a significant damage evolution in the matrix in spite of the restriction made

on ∗

. As demonstrated in Magri et al. (2021), a more relaxed (but less accurate) choice of ∗

would allow for the simulation of the

complete failure of the RVE induced by the propagation of damage following a percolation path compatible with the reinforcements.

Therefore, the assumption made here on ∗

is a compromise between accuracy of the solution – since ∗

has not been penalized

excessively – and numerical convergence – as the FFT solver allows capturing the stress softening regime corresponding to incipient

macroscopic fracture.

A quantitative comparison of the simulated composite ductility, as a function of 𝓁∗

, can be drawn by computing the so-called

ultimate composite ductility . The latter is defined as the macroscopic strain corresponding to the maximum simulated composite

stress , i.e. the one identified by the condition d11∕d11 = 0.Fig. 6b plots the increment, using a log–log scale, in the ultimate

composite ductility calculated relative to the case with 𝓁∗

= 0

(𝓁∗

) = (𝓁∗

) − (𝓁∗

= 0)

(𝓁∗

= 0) .(33)

Fig. 7 reports the spatial distribution of the non-local equivalent plastic strain in the matrix,

0, for three different particles

sizes (similar considerations can be made for

). For smaller 𝓁∗

(larger size of the reinforcements) the spatial spreading of the

plastic variable is small (Eq. (9a)) and damage bands are small relative to the particle size (see Fig. 8a–b). In the case of 𝓁∗

= 0

the non-local plastic equivalent plastic strain equals the discrete local field so damage bands correspond to the size of a voxel.

Accordingly, the mechanical degradation associated with the evolution of inelastic variables localizes in a small fraction of the

composite matrix leading to an early macroscopic softening. On the other hand, a remarkable diffusion of

0results for higher

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Fig. 6. (a) Plot of the average stress component 11 against the average strain component 11 for different values of the adimensional parameter 𝓁∗

. (b) Log–log

plot of the change in ductility induced by the adopted damage regularization as a function of the adimensional parameter 𝓁∗

=𝓁∕. The reported results

are obtained for a RVE containing 20% volume fraction of reinforcing particles.

Fig. 7. Non-local equivalent plastic strain in the RVE at the end of each simulation for (a) 𝓁∗

= 0, (b) 𝓁∗

= 0.2, and (c) 𝓁∗

= 0.4. Note that colorbars have been

reported with different scales to better capture the strain distribution in each case. The black arrow indicates the loading direction.

values of 𝓁∗

(smaller particles size). A larger portion of the RVE matrix is, therefore, involved with damage induced mechanical

degradation and the composite is capable of dissipating more mechanical energy. These conclusions are supported by the results

reported in Fig. 8, which collects the distribution of the effective porosity ∗for different values of 𝓁∗

. Damage localizes in small

regions in correspondence of the particle–matrix interfaces while the rest of the composite matrix experiences a limited amount of

nucleated voids for small 𝓁∗

(see Fig. 8a). Conversely, a higher 𝓁∗

promotes the development of damage in a larger volume fraction

of the RVE as shown in Fig. 8b–c.

4.2.3. Combined SGP and non-local damage

In this section, we evaluate the mechanical response of the metal matrix composite in the case where both the SGP model and

the non-local damage regularization are active. To analyze the size dependent response of the model in terms of strengthening

and ductility, the value of the characteristic length in SGP, 𝓁, is fixed and the numerical simulations that follow are parametrized

with respect to the particle diameter and the internal length scale 𝓁.Fig. 9 collects the simulated macroscopic strain stress

curves for different particle diameters in case of 𝓁= 1 μm and 𝓁= 5 μm. In both graphs, 𝓁= 10 μm is prescribed. A significant

increase in strengthening and ductility is obtained for decreasing particle size for the two considered internal lengths of the damage

regularization. Therefore, at least for the considered range of particle sizes and material parameters, the coupling between the

considered non-local theories promotes a mechanical size effect of type: the smaller the reinforcements, the higher the simulated

strengthening and ductility. All the experimental studies show a clear particle size effect on the strength, as captured by the combined

model. On the other hand, the experimental trend observed for the ductility is not so clear and, although in many cases the ductility

increases with the particle size – contrary to our model prediction – other results show almost size independent (or even decreasing)

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Fig. 8. Effective porosity in the RVE at the end of each simulation for (a) 𝓁∗

= 0, (b) 𝓁∗

= 0.2, and (c) 𝓁∗

= 0.4. The black arrow indicates the loading direction.

Fig. 9. Simulated macroscopic stress strain curves for different diameters of reinforcing particles for 𝓁= 10 μm and in the case of (a) 𝓁= 1 μm and (b)

𝓁= 5 μm. The reported results are obtained for a RVE containing 20% volume fraction of reinforcing particles.

ductility for increasing particle size (e.g. see Al/SiC at 100 ◦C in Lloyd (1994)). These difference in the ductility trend can be

explained by looking at the dominant damage mechanism in the composite. If either particle fracture or debonding are the main

mechanisms, the broken particles and the detached regions act as large voids which promote damage growth. In this case, particles

with smaller sizes lead to en early evolution of damage in the composite since the stress in the particles and at the particle/matrix

interface is higher. This is the typical case at room temperature (Lloyd,1994) where the size effect is of type ‘‘smaller is weaker’’.

On the contrary, when the matrix yield stress is small enough, particle cracking and debonding is suppressed so that their effect

in the fracture process reduces to the generation of stress concentration areas at the particle/matrix interface leading to microvoid

nucleation, growth, and coalescence. This might occur, for instance, at high temperature and in this case, even if the particle size

effect is small, it can show a trend of type ‘‘the smaller, the more ductile’’ in agreement with the predictions of this model.

Fig. 10 plots the relative ultimate composite stress and ductility obtained from the coupling between MSGP and implicit

gradient regularization. By comparison between Fig. 3b and Fig. 10a, it results that the magnitude of the strengthening induced by

the coupling between the MSGP model and the damage regularization is higher than the one obtained by considering the MSGP

model only. Therefore, if coupled with a strain gradient model, the damage regularization impacts significantly on the composite

strengthening as well, inducing a higher strengthening for increasing 𝓁. In addition, it can also be noted that the strengthening

in Fig. 10a grows slightly faster with 1∕than the one in Fig. 3b. Similar considerations on the simulated composite ductility: the

simulated composite ductility changes considerably if the MSGP model is considered along with the non-local damage regularization

(cf. Fig. 6b and Fig. 10b). In particular, the composite ductility increases if strain gradients effects are included.

4.3. Impact of stress triaxiality

For the sake of completeness, a brief analysis of the impact of the prescribed stress triaxiality – in view of the considered damage

regularization – is reported in this section. To this end, three different macroscopic loading conditions have been applied to the

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Fig. 10. Log–log plot of (a) the relative strengthening and (b) ductility as a function of 1∕induced by the fully-coupled model including MSGP and

the damage regularization. The reported results are obtained for a RVE containing 20% volume fraction of reinforcing particles and by taking 𝓁= 10 μm.

Fig. 11. (a) Macroscopic stress/strain condition used for the numerical analysis aiming at showing the impact of stress triaxiality. (b) Evolution of the average

microporosity and average effective microporosity ∗as a function of the load parameter . (c) Plot of the macroscopic stress components versus the load

parameter (in parenthesis is reported the specific stress component printed in each curve). The reported results are obtained for a RVE containing 20% volume

fraction of reinforcing particles along with 𝓁∗

= 0.1and 𝓁∗

= 0.

particle composite as schematized in Fig. 11a. The considered mixed macroscopic loadings aim at mimic the cases of pure shear

(corresponding to macroscopic triaxiality = 0), uniaxial tension with = 1∕3, and biaxial tension with = 2∕3.Fig. 11b plots the

evolution of the void volume fraction and average effective porosity ∗as a function of the load parameter . The reported

results reproduce the trends found in the local version of the GTN model, a higher macroscopic triaxiality promotes a faster microvoid

growth. This induces a more ductile response for lower applied triaxiality, as usually expected. In addition, high stress triaxiality

results in higher values of in correspondence of the macroscopic coalescence condition, i.e. where and ∗bifurcate. Similar

conclusions on the simulated ductility can be drawn by the analyzing the evolution of the macroscopic stress components, as reported

in Fig. 11c. The latter, confirms that higher composite ductility is obtained for lower stress triaxialities.

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M. Magri et al.

Fig. 12. Comparison between the experimental results of Lloyd (1994) and the numerical outcomes obtained with the considered extended model (a) without

and (b) with the impact of damage.

4.4. Comparison with experiments

The numerical outcomes of the proposed model will be now compared with some of the experimental results reported in the

paper by Lloyd (1994). The mechanical tests considered therein were obtained on particle reinforced light metal composites made

up of an A365 aluminum alloy matrix and SiC reinforcing particles. The composite material was fabricated with 15% volume fraction

of reinforcing particles. The particle size induced strengthening was evaluated by comparing the mechanical response for different

sizes of the reinforcement, i.e. 7.5 μm and 16 μm. To fit the typical strain hardening of the A365 aluminum matrix, the function

0was selected as a modified Voce function as follows

0=1

+

0+∞1 − exp −

0,(34)

where = 105 MPa,= 600 MPa,∞= 90 MPa, and = 150 are hardening parameters, while = 250 MPa is taken as the

reference stress.

Fig. 12a shows the comparison between the considered experimental results and the outcomes of the numerical analyses assuming

the matrix undamageable. This particular condition served as a reference case to adjust the parameters of the SGP model according

to the experimental curves for moderate applied strain, i.e. in the regime where the impact of damage is supposed to be negligible.

In particular, it turned out that the size dependent strain hardening of the composite is well captured for 𝓁= 40 μm in the strain

regime 11 <0.05. On the contrary, for a higher magnitude of the applied macroscopic strain, the numerical simulations clearly

overestimate the composite stress as no damage mechanisms has been included.

Damage evolution was then activated in the subsequent simulations in order to evaluate the numerical results of the complete

extended non-local model. In this case, the numerical analyses were parametrized with respect to the parameters and 𝓁while

the other material constants of the extended GTN model were selected as in Section 4.1. A first set of numerical results, not reported

here for the sake of brevity, showed a negligible impact of parameter 𝓁(taken in the range 0.1÷5 μm) for the considered magnitude

of the applied strain. On the other hand, parameter affects significantly the magnitude of the simulated macroscopic stress as

reported in Fig. 12b. In particular, the comparison between Fig. 12a and 12b remarks a significant reduction of the composite stress

for the damageable matrix in the regime 11 >0.05. In the case of = 0.04, the experimental results for the 16 μm particle

size composite are well captured in the whole strain range while the curves of the 7.5 μm composite are overestimated. A different

scenario is obtained for = 0.08 in which the numerical analyses slightly overestimate and underestimate the composite stress for

7.5 μm and 16 μm reinforcement size, respectively.

5. Summary and conclusions

In this paper we analyze the reinforcement size effect in a particle reinforced composite by means of computational homog-

enization along with the use of non-local versions of plasticity and ductile damage. To this end, the lower-order SGP model of

Huang et al. (2004) and the implicit gradient regularization of damage for multi-phase media (Magri et al.,2021) are employed

to enhance the classical GTN model. As a result of that, two length scale parameters are introduced in the extended model, one

for each non-local extension, which modulate the size sensitivity of the problem. The resulting formulation consists of a coupled

system of three partial differential equations (i.e. the classical balance of linear momentum and the two Helmholtz-type equations

of the damage regularization) which are solved via the FFT-based algorithm proposed in Magri et al. (2021). The latter exploits

an iterative staggered scheme that incorporates an FFT-Galerkin solver for the solution of the purely mechanical problem, and a

conjugate gradient spectral solver for the Helmholtz-type equations.

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A series of numerical simulations are performed on three-dimensional microstructures of particle reinforced metal matrix

composites. The impact of each non-local extension of the GTN model is first analyzed separately as a function of each relevant

length parameter. The obtained numerical results show that the SGP model induces a particle size dependent strengthening in the

composite, resulting in a stronger response for smaller particles. This reproduces the experimental trends shown in most studies. In

addition, the non-local regularization of the damage affects the ductility of the homogenized response inducing a higher ductility

for a smaller size of the reinforcements. The latter result can be explained by the fact that microvoid damage is the only degradating

mechanism considered in this numerical study. This experimental trend has been observed only for high temperatures (Lloyd,1994),

where it is expected that the particle debonding is hindered by the low yield stress and, as in this model, ductile damage in the matrix

is the only relevant damage mechanism. The combined effect of the considered non-local models is also investigated by studying

the change in the resulting strengthening and ductility as a function of the particle size of the composite. Finally, a comparative

study with experimental results confirms that the particle size induce strengthening in metal matrix composites can be quantitatively

captured by the considered model.

Future developments will focus on adding alternative damaging mechanisms in the composite material within this non-local

setting, e.g. particle–matrix debonding and particle fracturing. It is, indeed, well known that the mechanical response of metal matrix

composites results from the competition of multiple degradation mechanisms acting on different length scales. These improvements

will potentially allow us to get more insights on the physical nature of the length parameters involved with non-local models through

a more reliable comparison with experimental results.

CRediT authorship contribution statement

M. Magri: Conceptualization, Methodology, Software, writing – original draft. L. Adam: Funding acquisition. J. Segurado:

Conceptualization, Methodology, Funding acquisition, writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared

to influence the work reported in this paper.

Acknowledgment

The authors gratefully acknowledge the support provided by the Luxembourg National Research Fund (FNR), Reference No.

12737941. JS and MM also acknowledge the European Union’s Horizon 2020 research and innovation programme for the project

‘‘Multi-scale Optimization for Additive Manufacturing of fatigue resistant shock-absorbing MetaMaterials (MOAMMM)’’, grant

agreement No. 862015, of the H2020-EU.1.2.1. - FET Open Programme.

Appendix A. FFT-Galerkin solver

The solution of equation ((21)a) is calculated using the Galerkin FFT method of Vondřejc et al. (2014) and Zeman et al. (2017)

together with the procedure to incorporate stress and mixed control proposed by Lucarini and Segurado in Lucarini and Segurado

(2019a). Such a control technique sets a combination of components of the macroscopic strain (, ) and/or the macroscopic stress

(, ) history, i.e. ()and (), respectively. The boundary value problem is then solved expressing the weak form of the

linear momentum balance for the current time increment

G∗∗𝜻() ∶ 𝝈𝜺(), d= 0 ,

= +1 ,

=+1 ,

(35)

where ⋅represents the volume average, G∗stands for the projector operator that enforces the compatibility of the test functions,

𝜻()are second order tensor test functions, ∗is the convolution operation, and the indices of the macroscopic stress and strain obey

, ∩, = ∅.In(35) the original projection operator for small strain proposed in de Geus et al. (2017) is replaced by a modified

G∗that includes modified zero frequencies to enforce the value of stress averages and has a closed-form expression in Fourier space

G∗(Lucarini and Segurado,2019a). Moreover, in order to reduce numerical noise, the modified frequencies Eq. (26) resulting from

the use of the discrete derivation rule presented in Section 3.3 are used for the definition of this operator.

After performing the spatial discretization described in Section 3.3, the weak form of the equilibrium Eq. (35) results in an

algebraic system of non-linear equations (Vondřejc et al.,2014)

∗(𝝈)∶= −1

G∗∶𝝈𝜺(), =𝟎,(36)

with symbol indicating the Fourier transform. A Newton–Raphson algorithm is adopted for the solution of the resulting non-linear

problem so that the stress tensor is linearized with respect to the total strain as

𝝈(+1) =𝝈()+𝝈

𝜺∶𝜺(+1) .(37)

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M. Magri et al.

where 𝜺(+1) is the strain correction for Newton–Raphson iteration + 1. A linear problem is finally obtained by substitution of (37)

into (36)

∗𝝈

𝜺∶𝜺(+1)= −∗𝝈()−𝜮+1 ,

where 𝜮+1 is a tensor containing the non-zero components of the imposed stress at time +1 . Such a linear problem is solved

using the conjugate gradient method, whose convergence rate, efficiency, and memory allocation are optimal for this problem.

Appendix B. FFT solver for the Helmholtz-type equations

The Helmholtz-type equations ((21)a–c) can be rewritten in abstract setting as

−div 𝓁2

() ∇= ,

where is a generic local variable while is the non-local counterpart of . This problem can be rephrased in a more convenient

form as

Find such that = , (38)

where =(∙)−div 𝓁2

() ∇ (∙)is a linear differential operator. In spectral solvers, problem (38) can be easily solved in the frequency

domain because it reduces to the following linear problem

=()(39)

where symbol indicates the Fourier transform operator. The derivation of the left-hand side of Eq. (39) is straightforward in

Fourier space and holds

=+

⋅𝓁2

()−1

,

where

is the frequency vector. Therefore, the linear problem (39) rewrites in Fourier space as

=(),

with

=(∙)+

⋅𝓁2

()−1

(∙).

Such an equation is solved numerically exploiting a conjugate gradient algorithm. Following the approach proposed in Lucarini

and Segurado (2019b) for a displacement based FFT homogenization algorithm the usage of the following preconditioner

=1 +

𝓁

+𝓁

2

⋅

−1

speeds up significantly this iterative algorithm.

Appendix C. Objective calculation of plastic strain gradients

The purpose of this section is to justify the need of an objective numerical derivative for the implementation of SGP models in

spectral solvers and show how the approach proposed is able to provide discretization independent results for the gradient term.

In the case of heterogeneous media, where different material phases constitute the material microstructure, some of the field

variables of the problem might be discontinuous at the phase interfaces. If the constitutive theory depends on the spatial derivative

of such fields, as it happens in strain gradient plasticity, the numerical solutions through FFT-solvers might lead to numerical noise

or to a spurious grid dependence depending on how the derivative is performed numerically. For instance, it is well known that the

derivative of a step-like function (see Fig. 13) using the standard derivative in discrete Fourier space leads to numerical noise. In

order to alleviate this problem, discrete differences discrete derivative rules are used instead. The result is the elimination of the

noise, but paying the prize of introducing a prominent grid dependence, as it can be observed in Fig. 14a when the discrete derivative

in Eq. (26) is applied to the step-like function . Therefore, this numerical derivative cannot be regarded as an objective operator.

As discussed in Section 3.4, one possible strategy for alleviating this grid dependence is to apply the implicit gradient regularization

Eq. (28) to the original non-objective derivative. Fig. 14b supports this choice since the proposed alternative operator produces

converging results for a decreasing grid size. It is worth recalling that the magnitude of this objective derivative depends on the

length-parameter 𝓁. In particular, the higher the constant 𝓁, the lower the magnitude of the derivative.

In order to prove that the proposed objective derivative produces grid independent results, a convergence study for the considered

MSGP model is reported here. This study is carried out by simulating the homogenized mechanical response of a selected RVE

topology with different digital resolutions. The reference RVE is generated with a low digital resolution, i.e. with 323voxels, and

then discretized also with 643and 963voxels in a way such that the RVE topology is conserved (for instance compare the shape

of the reinforcements for the three different resolutions in Fig. 17). The simulations are then performed by accounting for the

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M. Magri et al.

Fig. 13. Plot of the step function used for comparing the considered numerical derivatives.

Fig. 14. Comparison between the spatial derivative obtained (a) by application of the original discrete differential operator Eq. (26) and (b) by using the

objective discrete derivative Eq. (28). For both graphs are reported the results obtained with different digital resolutions along the axis, i.e. 33, 66, 99, 132

voxels. The length-parameter used to generate the results reported in (b) is selected as 𝓁= 2∕33 for each grid discretization.

Fig. 15. Comparison between the numerical predictions of the considered SGP model for different RVE resolutions in the case of defined (a) by the original

discrete derivative Eq. (27) and (b) by the objective one proposed here (see Eq. (29)). The simulations were carried out by taking 𝓁∗

= 2 and 𝓁= 2∕32. The

dashed curve represents the reference, grid independent, case in which strain gradient effects are suppressed by assigning 𝓁∗

= 0.

considered MSGP model as conducted in Section 4.2.1.Fig. 15 plots the homogenized response for the different grid resolutions in

the case where is computed via (i) the original discrete derivative (see Fig. 15a) and (ii) using the proposed objective derivative

(see Fig. 15b). The comparison between these two cases clearly shows that the proposed alternative derivative rule provides grid

independent results. Moreover, due to the smoothening operated by Eq. (28) on the components of the plastic strain gradient, the

strengthening induced by the MSGP is smaller than the one obtained using the original discrete derivative. This last result arises

because parameter 𝓁is somehow large, compared to the RVE size, since it is selected based on the coarsest grid resolution.

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M. Magri et al.

Fig. 16. Spatial distribution of the equivalent plastic strain gradient at the end of the simulation obtained using the original discrete differential operator

Eq. (27). The results are reported for different RVE resolutions, i.e. (a) 323, (b) 643, and (c) 963voxels. For each image, the black line draws the particle/matrix

interface.

Fig. 17. Spatial distribution of the equivalent plastic strain gradient at the end of the simulation obtained using the proposed objective numerical derivative

Eq. (29). The results are reported for different RVE digital, i.e. (a) 323, (b) 643, and (c) 963voxels. For each image, the black line draws the particle/matrix

interface.

Figs. 16 and 17 report the spatial distribution of at the end of each simulation for the different derivative operators. If the

original discrete derivative is considered, the distribution of is remarkably different for the considered grid resolutions. Moreover,

the relevant plots show a sort of strain gradient localization far from the reinforcing particles that does not seems to have a clear

physical background. On the other hand, a grid independent distribution of is obtained if the proposed objective derivative is

employed. In this case, the regularization operated by Eq. (28) effectively suppresses spurious strain gradient localizations far from

the reinforcing particles providing a more physical distribution of .

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