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VISCOELASTIC-VISCOPLASTIC POLYMER COMPOSITES:

DEVELOPMENT AND EVALUATION OF TWO VERY DISSIMILAR

MEAN-FIELD HOMOGENIZATION MODELS

INT ER NATIONA L JOU RNAL OF SOLIDS AND STRU CT UR ES (I N PR ES S)

Mohamed Haddad

Institute of Mechanics, Materials and Civil Engineering (iMMC)

Université Catholique de Louvain

Bâtiment Euler, 1348 Louvain-la-Neuve, Belgium

mohamed.haddad@uclouvain.be

Issam Doghri∗

Institute of Mechanics, Materials and Civil Engineering (iMMC)

Université Catholique de Louvain, iMMC

Bâtiment Euler, 1348 Louvain-la-Neuve, Belgium

issam.doghri@uclouvain.be

Olivier Pierard

Cenaero

Rue des Frères Wright 29, Gosselies, Belgium

olivier.pierard@cenaero.be

—————————————— Abstract ——————————————

ABSTRACT

This paper deals with the micromechanical modeling of polymer composites with

viscoelastic-viscoplastic (VE-VP) constituents. Two mean-ﬁeld homogenization

(MFH) models based on completely dissimilar theoretical approaches are extended

from elasto-viscoplasticity (EVP) to VE-VP and assessed. The ﬁrst approach is

the incremental-secant method. It relies on a ﬁctitious unloading of the composite

at the beginning of each time step. Then, a thermoelastic-like Linear Comparison

Composite (LCC) is constructed from the computed residual state directly in the

time domain. The method provides naturally isotropic per-phase incremental-secant

operators for isotropic VE-VP constituents. It takes into account both the ﬁrst and

∗Corresponding author. E-mail address : issam.doghri@uclouvain.be

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

the second statistical moment estimates of the equivalent stress micro-ﬁeld. The

second approach is the integral afﬁne method. It starts by linearizing the rates of

viscoplastic strain and internal variables. The linearized constitutive equations are

then recast in a hereditary integral format to which the Laplace-Carson (L-C) trans-

form is applied. A thermoelastic-like LCC is built in the L-C domain, where MFH is

carried out. Finally, the composite’s response in the time domain is recovered by

numerical inversions of L-C transforms. The method is able to overcome the issue

of heterogeneous viscous stresses encountered by time domain MFH models.

The two proposed MFH formulations are able to handle non-monotonic, non-

proportional and multi-axial loading histories. Their accuracy was assessed against

full-ﬁeld ﬁnite element (FE) results for different microstructures and loadings. The

computational cost of both methods is negligible compared to FE analyses. Overall,

the incremental-secant approach is much simpler mathematically and numerically

than the integral afﬁne formulation, its accuracy ranges from acceptable to excellent,

and important improvements can be expected in the future by controlling the virtual

unloading time increment.

Keywords

Micromechanics

·

Mean ﬁeld Homogenization

·

Composites

·

Viscoelasticity-

viscoplasticity ·Second statistical moment

1 Introduction

User-tailored designed structures are gaining popularity with the emergent advances in the

manufacturing techniques and simulation tools. Recently, Ha et al. [2018] and Mohsenizadeh

et al. [2018] proposed miniaturized lattice structures for shock absorption applications. The

simulation of the behavior of such structures with high accuracy is possible with the capture of

featured information at micro-scale. However, some serious difﬁculties need to be addressed

carefully. Usually, reinforced thermoplastics are used in the applications. They exhibit inelastic

and rate-dependent (RD) behavior at all stages of deformation (see Fig.1). When unloading the

material before yielding, it can retrieve its initial state of zero-stress not immediately, as in elasticity,

but after a long waiting time. On the other hand, after yielding, a residual deformation remains,

upon unloading, which may diminish but does not disappear completely. Consequently, the overall

behavior of such materials can be qualiﬁed as coupled viscoelastic-viscoplastic (VE-VP). The main

concern of this paper is the determination of the effective behavior of polymer composites whose

constituents exhibit a coupled VE-VP behavior (typically the matrix phase).

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

Recovery strain

εV P εV E

total strain ε

˙

ε

ε

σ

Figure 1: Illustration of the behavior of VE-VP materials subjected to monotonic tension under

different strain rates. The tension phase is followed by unloading to zero stress.

Direct numerical methods, such as ﬁnite element (FE) method, are well known for their capabilities

to simulate this kind of behavior. Nevertheless, they cannot be envisioned in such a case because

of their prohibitive computational cost. As an alternative, homogenization-based methods (see

Fig.2) seem attractive. Within this framework, a boundary value problem is deﬁned for each

material point of the structure. At ﬁner scale, these macro points are considered as the center

of a representative volume element (RVE) subjected boundary conditions (e.g. corresponding

to a macro strain

¯

ε

ε

ε

). The problem is solved by relating the material tangent operator

¯

Ctg

and

the homogenized stress

¯

σ

σ

σ

and strain

¯

ε

ε

ε

computed as the average of local values over the RVE volume.

~

F

Macro scale problem

Real microstructure Homogenized microstructure

RVE scale

Figure 2: Homogenization-based multiscale methods

An overview of homogenization-based techniques is presented in Kanoute et al. [2009], Geers

et al. [2010b], LLorca et al. [2011], Nemat-Nasser and Hori [2013] and Noels et al. [2016]. They

can be classiﬁed into either purely numerical methods or semi-analytical ones. The numerical

homogenization techniques consist, essentially, on the cell FE method (CFEM) and the Fast Fourier

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

Transform (FFT) method. The CFEM such as the Voronoï cell method (VCFEM) (Ghosh et al.

[1995]) assumes a decomposition of the macro-scale problem into subdomains called Voronoï

cells representing the germination points of heterogeneities. Michel et al. [1999], Feyel [1999],

Miehe [2002] and Kouznetsova et al. [2001] generalized this approach by attributing to each

macro-point a FE model of an RVE leading to the so-called computational homogenization FE

2

.

The FFT method, pioneered by Moulinec and Suquet [1995,1998] and Michel et al. [2001], was

developed to overcome the difﬁculties of meshing the RVE encountered in the FE

2

technique.

In this approach, the RVE is discretized into a ﬁnite number of pixels in 2D or voxels in 3D to

which different mechanical properties can be attributed. Despite their capabilities of handling

general heterogeneous micro-structures and a wide range of non-linear (NL) behaviors, numerical

homogenization techniques are usually avoided owing to their overwhelming computational cost.

Alternatively, the semi-analytical homogenization methods are extensively studied. Among them,

the Transformation-Field Analysis (TFA) proposed by Dvorak et al. [1994]. The method relies

on a partition of the domain into sub-ﬁelds which may contain at most one uniform phase. It

was extended to the Non-Uniform Transformation-Field Analysis (NTFA) by Michel and Suquet

[2003] so that decomposition of one phase into sub-domains is avoided. Moreover, the Mean

Field Homogenization (MFH) method is another semi-analytical technique based on an assumed

relations between volume averages of strain ﬁelds in each constitutive phase of the RVE. Within

this framework, the averaged strains in phases are linked via concentration tensors deﬁned usually

based on the extension of Eshelby [1957]’s solution. Common extensions are Mori and Tanaka

[1973], the self-consistent (Kröner [1958], Hill [1965]) and the double inclusion (Nemat-Nasser

et al. [1996]) schemes.

The MFH represents a very cost-efﬁcient framework for multi-scale modeling of composites with

phases exhibiting either a linear (Pierard et al. [2004]) or a non-linear behavior. In the latter case,

MFH was extensively investigated. Most of the studies in the literature revolve around the deﬁnition

of the Linear Comparison Composite (LCC) (Talbot and Willis [1985,1987,1992], Castañeda

[1991]) through which the nonlinear homogenization problems are brought back to the well-known

range of linear ones via the linearization of phases constitutive equations. Berveiller and Zaoui

[1978] introduced the secant linearization procedure which links the total strain and stress ﬁelds

via pseudo-elastic relation

σ

σ

σ=Csec(ε

ε

ε):ε

ε

ε

, with

Csec(ε

ε

ε)

the secant operator. This linearization

strategy is restricted to monotonic and proportional loadings. The incremental linearization method

(Hill [1965], Pettermann et al. [1999], Doghri and Ouaar [2003]) was developed to circumvent the

limitation of handling only the proportional loading histories in the secant method. It relates the

stress and strain rates via an instantaneous tangent operator Ctan (˙

σ

σ

σ=Ctan :˙

ε

ε

ε).

For elasto-viscoplastic (EVP) microstructures under general loading histories, signiﬁcant MFH

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

models were investigated including the afﬁne interaction law (Molinari et al. [1987], Mercier and

Molinari [2009]), the integral afﬁne method (Masson and Zaoui [1999], Masson et al. [2000],

Pierard and Doghri [2006]), the incrementally afﬁne method (Doghri et al. [2011]) and the

incremental secant approach (Wu et al. [2017]). Sophisticated MFH theories for elasto(visco)plastic

composites were also developed based on the second order estimates of stress or strain microﬁelds,

either through variational formulations (Ponte Castañeda [2002], Brassart et al. [2011,2012],

Suquet and Lahellec [2013], Idiart and Lahellec [2016]) or via direct changes of the ﬁrst moment

MFH methods (Doghri et al. [2011], Wu et al. [2015,2017]). Those theories account for the

variance or ﬂuctuations of the strain or stress ﬁelds in addition to the volume average. They lead

usually to remarkable improvements of predictions accuracy.

In this paper, our interest is in homogenization procedures for RVEs with coupled VE-VP behavior

of constituents, capable of handling multi-axial, non-monotonic and non-proportional loading

histories. A few methods with such capabilities were developed. Among them, the incrementally

afﬁne method generalized by Miled et al. [2013] to VE-VP composites. This approach showed

overstiff results. Based on the interpretations of the authors, the loss of accuracy is caused, mostly,

by the viscoelastic (VE) part of the response due to the per-phase heterogeneous viscous stresses.

In fact, the composite’s response during the VE regime was overestimated, which impacts greatly

the overall behavior.

In order to outperform the incrementally afﬁne approach in VE-VP, a ﬁrst potential MFH

formulation is the integral afﬁne method. It was extended by Pierard and Doghri [2006] for

two-phase EVP composite materials with a full account for internal variables. The method is such

that the linearized constitutive equations rates are recast in thermo-elastic format via transformation

to Laplace Carson (L-C) domain. The correspondence principle is then invoked and homogenization

is conducted in the L-C domain. The L-C based procedure is well-known in the literature for being

very effective in homogenizing VE composites (Friebel et al. [2006], Brenner et al. [2002], Brenner

and Masson [2005]) because of its capabilities to overcome the major issue of heterogeneous

viscous stresses in the time domain. On the other hand, there are cases where the L-C transform can

be inverted analytically (e.g. Ricaud and Masson [2009]). However, in most cases, such as in the

integral afﬁne formulation studied in this article, the L-C inversion has to be carried out numerically,

which is computationally expensive.

A second potential MFH approach is the incremental-secant method. It has been developed for

elastoplastic (EP) and EVP composites with the account for ﬁrst and second moment estimates of

the von Mises equivalent stress (Wu et al. [2013,2015,2017]). This approach uses an intermediate

unloading step before loading to the target state. Its main advantage is that, for isotropic constituents,

the secant operators are isotropic by nature. Hence, the isotropic projection is avoided (unlike the

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

integral afﬁne and the incrementally afﬁne methods). Moreover, its extension to account for the

second moment estimates of per-phase equivalent strain or stress ﬁelds is straightforward.

In the present work, we generalized the incremental-secant and integral afﬁne MFH approaches

to VE-VP composites within the inﬁnitesimal strains framework. These are two completely

dissimilar MFH models based on radically different viewpoints. Our aim is to take advantage of the

capabilities of each formulation in order to accurately predict the homogenized response of the

composite. On the one hand, extension of the integral afﬁne approach seems to be very promising

since the method is, potentially, capable of overcoming the drawbacks of the incrementally

afﬁne formulation. In fact, homogenization in the L-C domain might allow to avoid the issue

of heterogeneous viscous stresses in the time domain encountered with the incrementally afﬁne

approach. On the other hand, we aim to take advantage of the incremental-secant formulation since

it takes into account residual states and enables to construct naturally isotropic secant operator

in each phase. Also, during the extension to VE-VP composites, the unloading scheme will be

dependent on the rate of deformation. Therefore, the method will have more ﬂexibility to improve

the quality of predictions by controlling the stiffness of the unloading path. Furthermore, since this

method operates directly in the time domain, it should be, computationally, much more affordable

than the integral afﬁne approach. Nevertheless, it might suffer from the same limitations of the

incrementally-afﬁne formulation while shifting to VE-VP composites which are the heterogeneous

per-phase viscous stresses.

The paper has the following outline. The main mathematical notations and results needed in the

remainder of this article are presented in Section 2. In Section 3, the local VE-VP constitutive

model is summarized. The key principles of MFH in linear thermo-elasticity are also brieﬂy

recalled. Section 4is devoted to the presentation of the incremental-secant linearization strategy

in VE-VP. The principal modiﬁcations of the integral afﬁne approach are outlined in section 5

while extending the method to VE-VP composites. The two formulations were implemented and

extensively validated against reference full-ﬁeld FE predictions. The results are presented and

discussed in Section 6. Conclusions are drawn on Section 7. Three appendices A-Ccontain details

about the second moment estimates, the L-C transform and its inversion, and the mathematics of

the integral afﬁne method.

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2 Preliminaries

Throughout this text, the following notations and results are used.

Boldface symbols denote tensors of second and fourth order. The former are designated by lowercase

symbols (e.g. σ

σ

σ). The latter are denoted by uppercase symbols(e.g. E).

Einstein’s summation convention over repeated indices (i, j, k or l) is used, unless otherwise

indicated. Tensor operations are expressed as follows:

a:b=ai j bi j;(C:a)i j =Ci jkl alk

C:: D=Ci jkl Dlk j i;(a⊗b)i jkl =ai j bkl

(1)

Symbols

1

and

I

denote the second order and symmetric fourth order identity tensors, respectively.

Ivol and Idev stand for the spherical and deviatoric projectors, respectively deﬁned as:

Ivol =1

31⊗1;Idev =I−Ivol (2)

For symmetric second order order tensors, one has:

Ivol :a=1

3amm 1;Idev :a=a−1

3amm 1=dev(a)(3)

The volume average of ﬁeld •(x)over a domain ωiis deﬁned as :

h•(x)iωi=1

V(ωi)Zωi

•(x)dV (4)

3 Background information

The objective of this section is to present the prerequisites needed to develop the incremental-secant

and the integral afﬁne approaches for VE-VP composites. In particular, the constitutive model

adopted to describe the phases behavior is recalled. Moreover, the key principles of MFH for

two-phase linear thermo-elastic composites are outlined. Finally, evaluation of the second statistical

moment estimates of the equivalent strain and stress is presented.

3.1 Viscoelastic-viscoplastic constitutive model

In semicrystalline polymers, crystal and amorphous phases coexist together. Micromechanical

models (e.g.Nikolov et al. [2002]) and macroscopic models (e.g.Miled et al. [2011]) assume that

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

crystalline lamellae and amorphous chains follow, respectively, VP and VE models assembled

together in series. Therefore, the total strain is deﬁned as a decomposition of VE and VP strains as

follows:

ε

ε

ε(t) = ε

ε

εV E (t) + ε

ε

εV P(t)(5)

3.1.1 VE strain

In the present work, a linear VE model is considered. Within this framework, the VE strain is

governed by Boltzmann [1878] integral law :

σ

σ

σ(t) = Zt

−∞

E(t−τ):∂ ε

ε

εV E

∂ τ (τ)dτ

=hE∗˙

ε

ε

εV E i(τ,t)

(6)

Eq.6relates the Cauchy stress

σ

σ

σ(t)

to VE strains

ε

ε

εV E (τ),τ≤t

via convolutional product

∗

which

confers to the behavior its dependency on the history and the rate of deformations. For isotropic

materials, the relaxation tensor E(t)can be written as

E(t) = 2G(t)Idev +3K(t)Ivol (7)

where

G(t)

and

K(t)

are the shear and bulk relaxation moduli, respectively. For many polymer

materials such as the ones considered in this paper, those moduli are supposed to follow the Prony

series hereafter:

G(t) = G∞+∑I

i=1Giexp−t

gi

K(t) = K∞+∑J

j=1Kjexp−t

kj(8)

G∞

and

K∞

represent the long term shear and bulk relaxation moduli, respectively.

Gi

and

Ki

are the

relaxation weights and giand kjthe relaxation times.

Eqs.8are substituted in Eq.6leading to :

s

s

s(t) = 2G∞ξ

ξ

ξV E (t) + ∑I

i=1s

s

si(t)

σH(t) = 3K∞εVE

H(t) + ∑J

j=1σH j(t)

(9)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

where

s

s

s(t)

and

σH(t)

are the deviatoric and hydrostatic parts of the Cauchy stress, respectively.

ξ

ξ

ξV E

and

εV E

H

are the deviatoric and hydrostatic parts of the VE strain tensor and

s

s

si(i=1..I)

and

σH j(j=1..J)are the deviatoric and hydrostatic viscous stresses:

s

s

si(t) = 2Giexp(−t

gj)Rt

−∞exp(τ

gi)ξ

ξ

ξV E (τ)dτ

σH j(t) = 3Kjexp(−t

kj)Rt

−∞exp(τ

kj)˙

εV E

H(τ)dτ

(10)

3.1.2 VP strain

The VP behavior is described by the classical J2model summarized hereafter.

The VP strain rate obeys the plastic ﬂow rule :

˙

ε

ε

εV P =˙p∂f

∂ σ

σ

σ=˙p N

N

N; with N

N

N=3

2

s

s

s

σ

σ

σeq (11)

where p is the accumulated plasticity,

N

N

N

the plastic ﬂow direction and f is the yield function deﬁned

by :

f(σ

σ

σ,R(p)) = σ

σ

σeq −(R(p) + σy)(12)

σ

σ

σeq

represents the equivalent von Mises stress computed based on either the ﬁrst

ˆ

σ

σ

σeq

or the second

ˆ

ˆ

σ

σ

σeq

moment estimates (more details can be found in paragraph 3.3).

σy

is the initial yield stress and

R(p) the hardening stress.

The visco-plastic multiplier ˙p=2

3˙

ε

ε

εV P :˙

ε

ε

εV P1/2

reads:

˙p=

gV(σ

σ

σeq,p); if f >0

0 ; otherwise

(13)

Here

gv

designates the viscoplastic function. All results presented in Section 6consider the law

deﬁned in Eq.15 for

gv

. A rate independent yield stress is also considered. Moreover, a power law

hardening function is adopted (Eq.14).

R(p) = K pn(14)

gv(σ

σ

σeq,p) = κf

σy+R(p)m

(15)

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where

K

and

κ

are the hardening and the VP moduli, respectively. n and m designate the hardening

and the VP exponent, respectively.

3.1.3 Algorithmic implementation

The constitutive equations from paragraphs 3.1.1 and 3.1.2 need to be discretized. The discretization

method used in this work assumes a constant VE strain rate over a generic time step

[tn,tn+1]

(see

Miled et al. [2011]).

Consider a time interval

[tn,tn+1]

such that the solution at the starting time

tn

denoted by

Stn

(

Stn≡σ

σ

σn,ε

ε

εn,ε

ε

εV P

n,pn)

is assumed to be known. The material is subjected to a strain increment

∆

∆

∆ε

ε

ε=ε

ε

εn+1−ε

ε

εn.

i. Incremental relaxation moduli

The discretization scheme leads to the deﬁnition of incremental relaxation moduli

˜

G(∆t)

and

˜

K(∆t)

which read:

˜

G(∆t) = G∞+∑I

i=1Gih1−exp−∆t

giigi

∆t

˜

K(∆t) = K∞+∑J

j=1Kjh1−exp−∆t

kjiki

∆t

(16)

As a result, the updated stress at time tn+1can be computed as follows:

σ

σ

σ(tn+1) = E∞:ε

ε

εV E (tn) + ˜

E(∆t):∆

∆

∆ε

ε

ε−∆

∆

∆ε

ε

εV P+

I

∑

i=1

exp−∆t

gis

s

si(tn) +

J

∑

j=1

exp−∆t

kjσH j(tn)1

(17)

with E∞=2G∞Idev +3K∞Ivol and ˜

E(∆t) = 2˜

G(∆t)Idev +3˜

K(∆t)Ivol.

Correspondingly, the stress at time tnreads:

σ

σ

σ(tn) = E∞:ε

ε

εV E (tn) +

I

∑

i=1

s

s

si(tn) +

J

∑

j=1

σH j(tn)1(18)

The second and third terms of Eq.18 introduce the effect of viscosity during the VE regime since in

elasticity the material constitutive equation at

tn

is written as

σ

σ

σ(tn) = CEl :ε

ε

εEl (tn)

, where

CEl

is

the elastic operator.

ii. Return mapping algorithm

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Since the material response is nonlinear, the determination of the new solution

Stn+1

at time

tn+1

relies on an iterative process called the return mapping algorithm. This algorithm is based on two

steps, a VE predictor and a VP corrector. The ﬁrst step assumes that the loading increment occurs

in the VE regime (∆

∆

∆ε

ε

ε=∆

∆

∆ε

ε

εV E ). Hence, the predicted stress is written using Eq.17 as:

σ

σ

σpred(tn+1) = E∞:ε

ε

εV E (tn) + ˜

E(∆t):∆

∆

∆ε

ε

ε+

I

∑

i=1

exp−∆t

gis

s

si(tn) +

J

∑

j=1

exp−∆t

kjσH j(tn)1

(19)

If the yield criterion

fσ

σ

σpred(tn+1),R(p(tn)≤0

is satisﬁed, the predicted stress represents indeed

the solution at tn+1. Therefore:

σ

σ

σ(tn+1) = σ

σ

σpred(tn+1)

ε

ε

εV P(tn+1) = ε

ε

εV P(tn)

(20)

Otherwise, VP corrector is required in order to include the VP ﬂow. The predicted solution is then

updated such that:

σ

σ

σ(tn+1) = σ

σ

σpred(tn+1)−˜

E(∆t):∆

∆

∆ε

ε

εV P (21)

Using backward Euler implicit time integration scheme, the new solution is found, iteratively, until

satisfying simultaneously Eq.21 and Eq.22 .

∆

∆

∆ε

ε

εV P =∆p N

N

Nn+1=gv(σ

σ

σeq(tn+1),p(tn+1)) ∆t N

N

Nn+1(22)

3.2 Generalities on MFH in thermo-elasticity

We now address the homogenization of two-phase composites whose constituents exhibit a thermo-

elastic behavior. In the sequel, subscripts 0 and 1 designate matrix and inclusion phases, respectively,

ω0

and

ω1

are the domains occupied by each phase and

ν0

and

ν1

their correspondent volume

fractions. Both phase materials obey a linear thermo-elastic model:

σ

σ

σ(x) = Ci:ε

ε

ε(x) + β

β

βiθ(x)∀x∈ωi(23)

Ci

is the stiffness tensor in phase

ωi

,

β

β

βi=−Ci:αi

where

αi

is the thermal expansion tensor and

θ

designates a change in temperature.

In the multi-scale method depicted in Fig.2, the macro-point

X

is viewed at micro-scale as a

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

center of a representative volume element (RVE) with domain

ω

and boundary

∂ ω

. At this scale,

heterogeneity of the structure is captured. Hill-Mandel condition imposes an energy equivalence

between both scales which implies that, considering adequate boundary conditions, linking the

macro stress

¯

σ

σ

σ

and strain

¯

ε

ε

ε

is equivalent to relating the volume averages of micro-stress

hσ

σ

σiω

and

micro-strain hε

ε

εiωover the RVE. This implies:

¯

σ

σ

σ(X) = ν0hσ

σ

σ(x)iω0+ν1hσ

σ

σ(x)iω1(24)

¯

ε

ε

ε(X) = ν0hε

ε

ε(x)iω0+ν1hε

ε

ε(x)iω1(25)

For simplicity, the volume average of quantity

•

over the phase

ωih•(x)iωi

is denoted by

•i

from

now on. In the remainder, it is assumed that ¯

ε

ε

ε(X)is given and the objective is to compute ¯

σ

σ

σ(X).

The main assumption of MFH in thermo-elasticity, is that the average strains within phases are

related using localization tensors Aεand aεsuch that:

ε

ε

ε1=Aε:¯

ε

ε

ε+aε(26)

Most of the deﬁnitions of localization tensors revolve around the extension of Eshelby [1957]’s

solution for single inclusion to multiple inclusions interacting together in an average manner.

According to numerous authors (e.g. Pierard et al. [2004]) the Mori and Tanaka [1973] (M-T)

scheme is accurate for thermo-elastic composites with moderate values of

ν1

. Consequently, it is

adopted all along the present work. For identical and aligned inclusions, the M-T scheme leads to

the partial strain concentration tensor Bε:

Bε(I,C0,C1) = I+Ξ(I,C0):C−1

0(C1−C0)−1(27)

with

Ξ(I,C0)

representing Eshelby’s tensor dependent on the matrix properties and the inclusions’

shape designated by symbol

I

. Localization tensors from Eq.26 are deﬁned as a function of

Bε

such

that:

Aε=Bε:[ν0I+ν1Bε]−1(28)

aε= [Aε−I]:[C1−C0]−1:[β

β

β1−β

β

β0](29)

The composite’s macroscopic stiffness

¯

C

and eingenstress

¯

β

β

β

tensors are computed from Eqs.30 and

31 hereafter:

¯

C= [ν0C0+ν1C1:Bε]:[ν0I+ν1Bε]−1(30)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

¯

β

β

β=ν0β

β

β0+ν1β

β

β1+ν1[CI−C0]:aε(31)

3.3 Second statistical moments

The VE predictor of the return mapping algorithm (see paragraph 3.1.3) requires the evaluation

of equivalent von Mises stress. Within the framework of ﬁrst order theory, the equivalent stress

is computed as a von Mises measure of the volume average of stress ﬁeld over the considered

material phase. The ﬁrst order estimation can be further enriched, statistically, with account for

ﬁeld ﬂuctuations. This is called the variance of the stress ﬁeld or the second moment estimate.

For composites with NL behavior, the LCC (Talbot and Willis [1985,1987,1992], Castañeda

[1991]) replaces the original composite with the same microstructure but a linearized behavior

through its ﬁctitious elastic operator CLCC and eigenstress tensor β

β

βLCC such that:

∆

∆

∆σ

σ

σ=CLCC

i:∆

∆

∆ε

ε

ε+β

β

βLCC

i∀ωi(32)

Eq.32 is quite identical to the constitutive equation in thermoelasticity (Eq.23). The ﬁctitious

tensors

CLCC

i

and

β

β

βLCC

i

substitute the real thermoplastic properties in the MFH of the LCC, leading

to the homogenized effective tensors of the composite ¯

CLCC and ¯

β

β

βLCC.

The second moment estimate of the equivalent averaged strain increment

∆

∆

∆ˆ

ˆ

ε

ε

εeq

i

over phase

ωi

is

deﬁned as follows :

∆

∆

∆ˆ

ˆ

ε

ε

εeq

i=r2

3Idev :: h∆

∆

∆ε

ε

ε⊗∆

∆

∆ε

ε

εiωi

=rD(∆

∆

∆ε

ε

εeq)2Eωi

∀ωi

(33)

Accordingly, the estimation of the second moment equivalent stress increment ∆

∆

∆ˆ

ˆ

σ

σ

σeq

ireads:

∆

∆

∆ˆ

ˆ

σ

σ

σeq

i=r3

2Idev :: h∆

∆

∆σ

σ

σ⊗∆

∆

∆σ

σ

σiωi∀ωi(34)

The ﬁrst moment estimate of the stress increment

∆

∆

∆ˆ

σ

σ

σeq

is nothing more than the von Mises measure

of this increment.

∆

∆

∆ˆ

σ

σ

σeq

i=r3

2h∆

∆

∆σ

σ

σiωi:Idev :h∆

∆

∆σ

σ

σiωi∀ωi(35)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

Substituting Eq.32 in Eq.34 leads to

∆

∆

∆ˆ

ˆ

σ

σ

σeq

i=9µ2

i(∆

∆

∆ˆ

ˆ

ε

ε

εeq

i)2+6µiD∆

∆

∆ε

ε

ε:Idev :β

β

βLCCEωi

+3

2Dβ

β

βLCC :Idev :β

β

βLCCEωi1

2

∀ωi(36)

where µiis the shear modulus of the LCC which is supposed to be isotropic.

Based on the perturbation method (Buryachenko [2001], Bobeth and Diener [1987])

h∆

∆

∆ε

ε

ε⊗∆

∆

∆ε

ε

εi

reads :

h∆

∆

∆ε

ε

ε⊗∆

∆

∆ε

ε

εiωi=1

νi"∆

∆

∆¯

ε

ε

ε:∂¯

CLCC

∂CLCC

i

:∆

∆

∆¯

ε

ε

ε+∂¯

β

β

βLCC

∂CLCC

i

:∆

∆

∆¯

ε

ε

ε#∀ωi(37)

In the present work, the second moment estimates were developed only for composites containing

spherical particles. The mathematical development is reported in Appendix Abased on the Mori-

Tanaka homogenization scheme.

It should be noted that

∆

∆

∆ˆ

ˆ

σ

σ

σeq

is the second moment estimate of the equivalent stress increment

∆

∆

∆σ

σ

σ=σ

σ

σ(tn+1)−σ

σ

σ(tn)and not the total stress σ

σ

σ(tn+1). The latter satisﬁes:

σ

σ

σ(tn+1) = σ

σ

σ(tn) + ∆

∆

∆σ

σ

σ(38)

The combination of Eq.38 and the deﬁnition Eq.34 leads to the following expression of the second

moment estimate of the equivalent total stress at tn+1:

ˆ

ˆ

σ

σ

σeq

i(tn+1) = rˆ

ˆ

σ

σ

σeq

i(tn)2+3Dσ

σ

σ(tn):Idev :∆

∆

∆σ

σ

σEωi

+ (∆

∆

∆ˆ

ˆ

σ

σ

σeq

i)2(39)

4 Incremental-secant formulation for VE-VP composites

In this section, the principles of the incremental-secant approach for EVP composites are recalled.

Afterwards, the formulation is extended to VE-VP composites.

4.1 Key concepts in elasto-viscoplasticity

The incremental-secant linearization method was pioneered by Wu et al. [2013] for EP composites.

It modiﬁes the original secant formulation (Berveiller and Zaoui [1978]) so that it is able to handle

non-radial and non-monotonic loading histories. It was also extended to account for second

statistical moment estimates in EP and EVP by Wu et al. [2015] and Wu et al. [2017].

14

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

σn

εn

σres

nεres

n

σn+1

εn+1

∆εunload

n∆εn+1

∆εr

n+1

CEl CS

ε

σ

(a) In phases

¯

σn

¯

εn

¯

σn+1

¯

εn+1

∆¯

εunload

n∆¯

εn+1

∆¯

εr

n+1

¯

σres

n=0

¯

CEl ¯

CS

¯

ε

¯

σ

(b) Overall composite

Figure 3: Illustration of the incremental-secant linearization procedure in EVP (a) within the

constitutive phases and (b) within the overall composite.

Let’s consider ﬁrst a single material phase without homogenization. The solution is known until the

beginning of the time step

[tn,tn+1]

(state at time

tn

,

Stn≡σ

σ

σn,ε

ε

εn,ε

ε

εV P

n,pn

is given). The material

is subjected to strain increment ∆

∆

∆ε

ε

εn+1.

Following the incremental-secant procedure, the material is virtually unloaded at time

tn

to an

intermediate residual state Stres

n≡σ

σ

σres

n,ε

ε

εres

n,ε

ε

εV P

n,pn(see Fig.3a). Hence, the problem stated can

be reformulated involving the newly created ﬁctitious state as:

ε

ε

εn+1=ε

ε

εn+∆

∆

∆ε

ε

εn+1→ε

ε

εn+1=ε

ε

εres

n+∆

∆

∆ε

ε

εr

n+1

σ

σ

σn+1=σ

σ

σn+∆

∆

∆σ

σ

σn+1→σ

σ

σn+1=σ

σ

σres

n+∆

∆

∆σ

σ

σr

n+1

(40)

Consider now the overall composite material. Within the framework of the incremental-secant

approach, the LCC is deﬁned from the unloaded state such that the composite is subjected to

homogenized strain increment ∆

∆

∆¯

ε

ε

εr

n+1instead of ∆

∆

∆¯

ε

ε

εn+1(see Fig.3b). This implies:

∆

∆

∆¯

σ

σ

σr

n+1=¯

CS:∆

∆

∆¯

ε

ε

εr

n+1

¯

σ

σ

σres

n=0

(41)

where ¯

CSis the RVE’s secant operator.

15

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

4.2 Extension to VE-VP composites

In this paragraph, each constitutive phase exhibits VE-VP behavior. Thus, the constitutive model

presented in subsection 3.1 is used.

Unlike EVP, the unloading from

Stn

to

Stres

n

is not elastic but VE driven by the unloading time

increment

∆tunload =tres

n−tn

. Consider the unloading strain increment

∆

∆

∆ε

ε

εunload

n=ε

ε

ε(tn)−ε

ε

ε(tres

n)

.

We recall that the predicted stress resulting from the VE predictor of the return mapping algorithm

is given in Eq.19. Likewise Eq.19, the residual stress is evaluated as follows:

σ

σ

σres

n=σ

σ

σ(tres

n) = E∞:ε

ε

εV E (tn) + ˜

E(∆tunload ):∆

∆

∆ε

ε

εunload

n

+

I

∑

i=1

exp−∆tunload

gis

s

si(tn) +

J

∑

j=1

exp−∆tunload

kjσH j(tn)1∀ωi

(42)

Knowing that ∆

∆

∆ε

ε

εr

n+1=∆

∆

∆ε

ε

εn+1+∆

∆

∆ε

ε

εunload

n(see Fig.3a), combination of Eqs.19 and 42 leads to :

σ

σ

σpred(tn+1) = σ

σ

σ(tres

n) + ˜

E(∆t):∆

∆

∆ε

ε

εr

n+1+h˜

E(∆tunload )−˜

E(∆t)i:∆

∆

∆ε

ε

εunload

n

+

I

∑

i=1exp(−∆t

gi

)−exp(−∆tunload

gi

)s

s

si(tn)

+

J

∑

j=1exp(−∆t

kj

)−exp(−∆tunload

kj

)σH j(tn)1∀ωi

(43)

For simplicity, any variable

•

evaluated at time

tn•(tn)

will be designated by

•n

. Idem for variables

at tn+1.

The VE predictor requires the assessment of the yielding function

fpred

n+1

(Eq.12) based on either the

ﬁrst or the second moment estimates of the equivalent predicted stress (subsection 3.3).

If the yield criterion

fpred

n+1≤0

is satisﬁed, the VE predictor is indeed the solution at time

tn+1

.

However, if

fpred

n+1>0

, VP strains have evolved and VP corrector (Eq.20) is needed in order to

update the solution. Consequently, the system of non-linear equations 44 needs to be solved.

σ

σ

σn+1=σ

σ

σpred

n+1−2˜

G(∆t)∆p N

N

Nn+1

∆p=pn+1−pn=gv(σ

σ

σeq

n+1,pn+1)∆t∀ωi

(44)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

N

N

Nn+1

is the plastic ﬂow direction. In the incremental-secant method, it is approximated as follows:

N

N

Nn+1=3

2

Idev :(σ

σ

σn+1−σ

σ

σres

n)

(σ

σ

σn+1−σ

σ

σres

n)eq =3

2

Idev :∆

∆

∆σ

σ

σr

n+1

∆

∆

∆σ

σ

σr eq

n+1

(45)

It satisﬁes

N

N

Nn+1:N

N

Nn+1=3

2

. Eq.45 shows that the direction of viscoplastic ﬂow is collinear to the

deviatoric part of

∆

∆

∆σ

σ

σr

n+1

which represents the ﬁrst order approximation in

∆

∆

∆ε

ε

εV P =∆p N

N

Nn+1

since

it is directed along ∆s

s

sr

n+1and not along s

s

sn+1.

In a similar way, One deﬁnes N

N

Npred

n+1as:

N

N

Npred

n+1=3

2

Idev :σ

σ

σpred

n+1−σ

σ

σres

n

σ

σ

σpred

n+1−σ

σ

σres

neq =3

2

Idev :∆

∆

∆σ

σ

σr pred

n+1

∆

∆

∆σ

σ

σr pred

n+1eq (46)

Using Eqs.44-46, the following result is obtained:

N

N

Nn+1=N

N

Npred

n+1(47)

Deﬁnition 46 and result 47 are substituted in the system 44, leading to :

Fσ= (σ

σ

σn+1−σ

σ

σres

n)eq

| {z }

∆

∆

∆σ

σ

σr

n+1eq

+3˜

G(∆t)∆p−σ

σ

σpred

n+1−σ

σ

σres

neq =0

Fp=∆p−gv(σ

σ

σeq

n+1,pn+1)∆t=0

(48)

The system 48 is solved for

∆

∆

∆σ

σ

σr eq

n+1

and

pn+1

, iteratively, using Newton-Raphson method until

satisfying the convergence criteria on Fσ=0 and Fp=0.

It is important to notice here that, the solution

∆

∆

∆σ

σ

σr eq

n+1

retrieved after convergence represents either

the ﬁrst or the second order evaluation of the equivalent stress increment depending on the moment’s

order used to estimate the equivalent predicted stress

σ

σ

σpred

n+1eq

. It follows that the new material

state at tn+1is constructed by pursuing the steps hereafter:

a. ∆

∆

∆σ

σ

σr

n+1dev =2

3∆

∆

∆σ

σ

σr

n+1eq N

N

Npred

n+1

b. ∆

∆

∆σ

σ

σr

n+1=∆

∆

∆σ

σ

σr

n+1dev +1

3tr(∆

∆

∆σ

σ

σr pred

n+1)1

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

c. σ

σ

σn+1=σ

σ

σres

n+∆

∆

∆σ

σ

σr

n+1

d. ε

ε

εV P

n+1=ε

ε

εV P

n+∆p N

N

Npred

n+1

4.3 Incremental-secant operator and eigenstress tensor

The incremental-secant approach in VE-VP implies that the constitutive equations of each phase of

the composite are linearized such that:

∆

∆

∆σ

σ

σr

n+1=CS

i:∆

∆

∆ε

ε

εr

n+1+β

β

βS

i∀ωi(49)

It can be seen from Eq.49 that this approach leads to a linearized equations format similar to EVP

with an additional term

β

β

βS

i

which is the eigenstress tensor. Expressions of the secant operator

CS

and the eigenstress tensor

β

β

βS

of each averaged material phase material are determined from Eq. 43

and 44.

If the yield criterion

fpred

n+1≤0

is fulﬁlled during the VE predictor, the updated stress veriﬁes Eq.43.

Otherwise, VP corrector leads to Eq.44. As a result, and based on deﬁnition of Eq.49,

CS

n+1

and

β

β

βS

n+1can be deﬁned in a general way as :

CS

n+1=˜

E(∆t)−3˜

G(∆t)∆pIdev :˜

E(∆t)

∆

∆

∆σ

σ

σr pred

n+1eq ∀ωi(50)

β

β

βS

n+1=

I−3˜

G(∆t)∆p

∆

∆

∆σ

σ

σr pred

n+1eq Idev

:"˜

E(∆tunload )−˜

E(∆t):∆

∆

∆ε

ε

εunload

n

+

I

∑

i=1exp(−∆t

gi

)−exp(−∆tunload

gi

)s

s

si(tn) +

J

∑

j=1exp(−∆t

kj

)−exp(−∆tunload

kj

)σH j(tn)1#∀ωi

(51)

Eq.50 shows that in the absence of visco-plastic ﬂow

∆p=0

, expression of

CS

n+1

is simpliﬁed to

CS

n+1=˜

E(∆t)

. Moreover,

CS

n+1

is naturally isotropic. Hence, it can be rewritten as a decomposition

of deviatoric and dilatational parts:

CS

n+1=3KS

n+1Ivol +2GS

n+1Idev ∀ωi(52)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

with

KS

n+1=˜

K(∆t)

GS

n+1=˜

G(∆t)−3(˜

G(∆t))2∆p

∆

∆

∆σ

σ

σr pred

n+1eq

(53)

Similarly, the volumetric and deviatoric parts of the eigenstress tensor read:

1

3tr(β

β

βS

n+1) = ˜

K(∆tunload )−˜

K(∆t)tr(∆

∆

∆ε

ε

εunload

n) + ∑J

j=1hexp(−∆t

kj)−exp(−∆tunload

kj)iσH j(tn)

dev(β

β

βS

n+1) =

1−3˜

G(∆t)∆p

∆

∆

∆σ

σ

σr pred

n+1eq

h2˜

G(∆tunload )−˜

G(∆t)∆

∆

∆ξ

ξ

ξunload

n

+

I

∑

i=1exp(−∆t

gi

)−exp(−∆tunload

gi

)s

s

si(tn)

(54)

Here ∆

∆

∆ξ

ξ

ξunload

nis the deviatoric part of ∆

∆

∆ε

ε

εunload

n.

Furthermore, Eq.51 reveals that the eigenstress tensor

β

β

βS

n+1

depends on the unloading scheme

controlled by

∆tunload

. Thus, if unloading occurs such that

∆tunload =∆t

,

β

β

βS

n+1

will be equal to

zero and homogenization will be done as in linear elasticity similarly to EVP composites not in

thermo-elastic manner.

4.4 MFH algorithm based on incremental-secant formulation

In this subsection, the different steps of MFH based on the incremental-secant linearization pro-

cedure are summarized. For this, assuming that the problem is solved until the beginning of time

interval

[tn,tn+1]

, which means that all state variables at time

tn

are supposed to be determined, the

composite RVE is subjected to macro strain ∆

∆

∆¯

ε

ε

εn+1. The main steps are highlighted hereafter:

a.

The composite material is ﬁctitiously unloaded such that

¯

σ

σ

σres

n=0

. Computation of residual

states in each phase

ωi

is carried out through the residual box

ℜres

detailed in the next

subsection 4.5.

•ℜres(σ

σ

σn,ε

ε

εn,∆tunload )→Stres

n∀ωi

•¯

σ

σ

σres

n=0

•

The macro strain increment

∆

∆

∆¯

ε

ε

εr

n+1

is computed such that:

∆

∆

∆¯

ε

ε

εr

n+1=∆

∆

∆¯

ε

ε

εn+1+∆

∆

∆¯

ε

ε

εunload

n

b. Average strain increment in inclusions ∆

∆

∆ε

ε

εr

1n+1is initialized following:

•∆

∆

∆ε

ε

εr

1n+1is equal to the one of the last time step

c. Iterations until computing the targeted strain increment ∆

∆

∆ε

ε

εr

1n+1:

19

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

i. The average strain in the matrix phase is evaluated such that:

•∆

∆

∆ε

ε

εr

0n+1=1

ν0∆

∆

∆¯

ε

ε

εr

n+1−ν1

ν0∆

∆

∆ε

ε

εr

1n+1

ii.

Secant operators and eigenstress tensors at time

tn+1

are evaluated using the constitutive

box ℜn+1of each material phase (sections 4.2 and 4.3):

•ℜn+1(σ

σ

σres

n,ε

ε

εres

n,∆

∆

∆ε

ε

εr

n+1)→CS

n+1,β

β

βS

n+1∀ωi

iii.

Afterwards, localization tensors (Eqs.28 and 29) are predicted using

CS

n+1

and

β

β

βS

n+1

of

each constitutive phase.

•Aε(CS

0n+1,CS

1n+1)

•aε(CS

0n+1,CS

1n+1,β

β

βS

0n+1,β

β

βS

1n+1)

iv.

The compatibility of average strain increment in inclusions phase

∆

∆

∆ε

ε

εr

1n+1

is checked

by computation of the residual R:

•R=Aε:∆

∆

∆¯

ε

ε

εr

n+1+aε

ε

ε−∆

∆

∆ε

ε

εr

1n+1

•If |R|<tolerance, then the solution was found, loop is exited.

•

Otherwise, new iteration is required,

∆

∆

∆ε

ε

εr

1n+1←∆

∆

∆ε

ε

εr

1n+1−J−1

ac :R

and algorithm

returns back to step c.i. (Jac is the Jacobean matrix).

d.

After convergence, the macroscopic properties

¯

CS

n+1

and

¯

β

β

βS

n+1

are computed using Eqs.30

and 31. Then, macroscopic effective solution ¯

Stn+1= [ ¯

σ

σ

σn+1,¯

ε

ε

εn+1]is determined.

4.5 Residual states

The residual box

ℜres

used in step

a.

of MFH process can be summarized as follows. We recall that

σ

σ

σnand σ

σ

σres

nare deﬁned in Eqs.18 and 42. Combination of the latter two equations leads to:

σ

σ

σn−σ

σ

σres

n

| {z }

∆

∆

∆σ

σ

σunload

n

=˜

E(∆tunload )

| {z }

Cres

n=C(tres

n)

:∆

∆

∆ε

ε

εunload

n

+

I

∑

i=11−exp(−∆tunload

gi

)si(tn) +

J

∑

j=11−exp(−∆tunload

kj

)σ

σ

σH j(tn)1

| {z }

β

β

βres

n=β

β

β(tres

n)

∀ωi

(55)

In Eq.55, the bulk and shear moduli of

Cres

n

are given by Eq.16 after replacing

∆t

with

∆tunload

. This

implies that the relaxation weights

Gi

and

Kj

and times

gi

and

kj

are used for any values of

∆tunload

.

This is a consequence of the linear VE model adopted in subsection 3.1.1 which is based on the

Prony series (Eqs.8).

Since the unloading step is ﬁctitious, the macroscopic residual stress

¯

σ

σ

σres

n=¯

σ

σ

σ(tres

n)

should satisfy:

¯

σ

σ

σres

n=0 (56)

20

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

The problem stated in Eqs.55 and 56 is equivalent to thermo-elastic MFH described in subsection

3.2 with the following substitutions:

σ

σ

σ→∆

∆

∆σ

σ

σunload

n,ε

ε

ε→∆

∆

∆ε

ε

εunload

n,C→Cres

n,β

β

β→β

β

βres

n(57)

Therefore, the residual states are determined following the steps hereafter :

a.

Macro residual operator

¯

Cres

n

and residual eigenstress tensor

¯

β

β

βres

n

are computed using Eqs.30

and 31 with Ci=Cres

n=˜

E(∆tunload )and β

β

βi=β

β

βres

n(i=0;1).

b. Macro unloading strain increment ∆

∆

∆¯

ε

ε

εunload

nis then evaluated :

•∆

∆

∆¯

ε

ε

εunload

n=−¯

Cres

n−1:h¯

σ

σ

σn−¯

β

β

βres

ni

c.

Unloading strain increments in each phase are assessed using

Aε(Cres

0n,Cres

1n)

and

aε(Cres

0n,Cres

1n,β

β

βres

0n,β

β

βres

1n):

•∆

∆

∆ε

ε

εunload

1n=Aε:∆

∆

∆¯

ε

ε

εunload

n+aε

•∆

∆

∆ε

ε

εunload

0n=1

ν0∆

∆

∆¯

ε

ε

εunload

n−ν1

ν0∆

∆

∆ε

ε

εunload

1n

d. Finally, residual stresses within phases are computed using Eq.42.

—————————————-

5 Integral afﬁne formulation for VE-VP composites

Throughout this section, we recall the key principles of the integral afﬁne approach in EVP. After-

wards, the formulation is extended to VE-VP composites .

5.1 Key concepts in elasto-viscoplasticity

Theoretical developments of the integral afﬁne approach were initiated by Masson and Zaoui [1999]

and Masson et al. [2000] for viscoplastic polycrystals. Then, the formulation was extended and

veriﬁed against full-ﬁeld FE for composites with EVP hardening phases by Pierard and Doghri

[2006].

Let’s consider a time step

[tn,tn+1]

such that the problem is solved until the starting time

tn

. A

single internal variable will be considered based on the VP constitutive model of 3.1.2 which is the

accumulated plasticity p. We suppose that both the rate of VP strain and that of pread:

˙

ε

ε

εV P(t) = ˙

ε

ε

εV P(σ

σ

σ(t),p(t)) ; ˙p(t) = ˙p(σ

σ

σ(t),p(t)) (58)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

The ﬁrst step of the integral afﬁne approach (see Fig.4) is the temporal linearization of Eqs.58 at

time t around a time τ:

˙

ε

ε

εV P(t)'˙

ε

ε

εV P(τ) + ∂˙

ε

ε

εV P

∂ σ

σ

σ

|{z}

m(τ)

:[σ

σ

σ(t)−σ

σ

σ(τ)] + ∂˙

ε

ε

εV P

∂p

|{z}

n(τ)

[p(t)−p(τ)] (59)

˙p(t)'˙p(τ) + ∂˙p

∂ σ

σ

σ

|{z}

l(τ)

:[σ

σ

σ(t)−σ

σ

σ(τ)] + ∂˙p

∂p

|{z}

q(τ)

[p(t)−p(τ)] (60)

such that

t,τ∈[tn,tn+1]

. Operators

m

(

τ

),

n

(

τ

),

l

(

τ

) and q(

τ

) are explicitly deﬁned in Appendix

C.1.

An integral form solution of Eq.60 exists and is reported in Appendix C.2.

Secondly, the linearized equations 59 and 60 are inserted in the EVP constitutive model. Conse-

quently, the EVP problem is recast in VE format with an additional term as follows (for the full

mathematical developments see Pierard and Doghri [2006]):

˙

ε

ε

ε(t) = [Jτ˙

σ

σ

σ](τ,t)+˙

ε

ε

ε0

τ(t)(61)

where

Jτ

is the linearized creep operator and

˙

ε

ε

ε0

τ

is the eigenstrain rate tensor of the material.

designates the Stieljes-type convolution product, the derivative of the classical one.

Classically, solving a VE problem relies on its transformation to L-C domain. The latter transforma-

tion is deﬁned as

f∗(s) = sZ+∞

0

f(t)exp(−st )dt (62)

where f(t) is the operator which we want to transform,

f∗(s)

its L-C transform and s the L-C

variable.

The main utility of the L-C transformation is that it converts the Stieljes-type convolution product

to a single contraction product. As a result, the integral equation 61 is written in L-C domain as:

˙

ε

ε

ε∗(s) = J∗

τ(s):˙

σ

σ

σ∗(s) + ˙

ε

ε

ε0∗

τ(s)(63)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

The ﬁctitious relation 63 obtained in L-C domain is form similar to thermo-elastic constitutive

model of subsection 3.2 after the following substitutions :

σ

σ

σ→˙

σ

σ

σ∗,ε

ε

ε→˙

ε

ε

ε∗,C→E∗

τ= [J∗

τ]−1,(β

β

βθ )→ −E∗

τ:˙

ε

ε

ε0∗

τ(64)

Therefore, homogenization is carried out in the L-C domain instead of the time domain. The

effective behavior of the composite is recovered using numerical inversions (see Appendix B).

Real microstructure

Homogenized microstructure

Time domain

H

o

m

o

g

e

n

i

z

a

t

i

o

n

Laplace-Carson domain

Jτ,˙

ε0

τ

Aε

τ,aε

Figure 4: Illustration of the integral afﬁne linearization procedure.

5.2 Extension to VE-VP composites

Complete mathematical developments in EVP are reported with details in Pierard and Doghri

[2006]. We limit ourselves in the present paragraph to present the main modiﬁcations of the integral

afﬁne approach when we extend it to VE-VP composite materials.

The constitutive law (Eq.5) is rewritten, after temporal derivation, under the general form :

˙

ε

ε

ε(t) = ˙

ε

ε

εV E (t) + ˙

ε

ε

εV P(t)(65)

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

Similarly to Eq.6, the VE strain can be written as:

ε

ε

εV E (t) = Zt

−∞

J(t−τ):∂ σ

σ

σ

∂ τ (τ)dτ

=Zt

0

J(t−τ):∂ σ

σ

σ

∂ τ (τ)dτ+J(t):σ

σ

σ(0)

(66)

where J(t) is the creep tensor(J= [E]−1).

Eq.66 is substituted in Eq.65, leading to :

˙

ε

ε

ε(t) = d

dt Zt

0

J(t−τ):∂ σ

σ

σ

∂ τ (τ)dτ+˙

ε

ε

εV P(σ

σ

σ(t),p(t)) + ˙

J(t):σ

σ

σ(0)

= [J˙

σ

σ

σ](τ,t)+˙

ε

ε

εV P(σ

σ

σ,p(t)) + ˙

J(t):σ

σ

σ(0)

(67)

˙

ε

ε

εV P

and

˙p

are linearized as in Eqs.59-60. The overall expression of

˙

ε

ε

εV P

is substituted then in Eq.67

leading to the VE format of Eq.61 with :

Jτ(t) = J(t) + m(τ)t−t+1

q(τ)(1−exp(t q(τ))n(τ)⊗l(τ)

q(τ)(68)

˙

ε

ε

ε0

τ(t) = ˙

ε

ε

εV P(τ)−m(τ):σ

σ

σ(τ) + n(τ)ˆp(τ,t) + ˙e(τ,t)[1−H(t−τ)] + ˆ

ε

ε

ε0(τ,t)(69)

where

H

is the Heaviside step function. These expressions of

Jτ(t)

and

˙

ε

ε

ε0

τ(t)

are valid only for

t>τ. The full expressions of ˙e(τ,t)and ˆ

ε

ε

ε0(τ,t)are reported in Appendix C.3.

The L-C transformation is then invoked. Eqs.68-69 are transformed leading to the deﬁnition of

J∗

τ(s)and ˙

ε

ε

ε0∗

τ(s)whose expressions are given in Appendix C.4.

5.3 MFH algorithm based on the integral afﬁne formulation

Each VE-VP phase of the composite follows the afﬁne relation 63 in L-C domain between

˙

σ

σ

σ∗

and

˙

ε

ε

ε∗

. Thermo-elastic homogenization scheme is applied in order to infer the effective solution of the

overall composite

˙

¯

ε

ε

ε∗

. Then, the new state in the time domain is recovered by numerical inversion

using collocation method (Schapery [1961]).

The iterative process involved in the determination of the new solution at

tn+1

is summarized in the

steps hereafter. We assume that solution at time

tn

is known. The composite material undergoes a

macro strain increment ∆

∆

∆¯

ε

ε

ε.

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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

a. Average strain increment in inclusions ∆

∆

∆ε

ε

ε1is initialized such that:

•∆

∆

∆ε

ε

ε1=∆t

∆tn−1∆

∆

∆ε

ε

ε1n−1

| {z }

ε

ε

εn−ε

ε

εn−1

b. Iterations until ﬁnding the targeted strain in the inclusion phase ε

ε

ε1(tn+1).

i. The average strain increment in the matrix phase is evaluated following:

•∆

∆

∆ε

ε

ε0=1

ν0∆

∆

∆¯

ε

ε

ε−ν1

ν0∆

∆

∆ε

ε

εI

ii.

The constitutive box

ℜn+1

is called to compute the afﬁne stiffness modulus

C∗

τ

and

eigenstrain rate ˙

ε

ε

ε∗0

τrelative to each constitutive phase.

•ℜn+1(σ

σ

σn,ε

ε

εn,∆

∆

∆ε

ε

ε)→C∗

τ,ε

ε

ε∗0

τ∀ωi

iii.

Isotropic part of

C∗

τ

is extracted using either general or spectral methods (Doghri and

Ouaar [2003]).

iv. Localization tensors A∗εand a∗εare computed using ˙

ε

ε

ε∗0

τand isotropic part of C∗

τ.

•A∗εC∗iso

0τ,C∗iso

1τ

•a∗ε(C∗iso

0τ,C∗iso

1τ,˙

β

β

β∗0

0τ

|{z}

−C∗iso

0τ:˙

ε

ε

ε∗0

0τ

,

−C∗iso

1τ:˙

ε

ε

ε∗0

1τ

z}|{

˙

β

β

β∗0

1τ)

v.

Then, localization tensors are numerically inverted using the direct collocation method

presented in Appendix B. This step is the most critical since the L-C inversions are

tricky and usually a source of problems. In fact, the quality of inversion depends

enormously on the number of collocation points used and the interval in which they are

placed. Therefore, the accuracy of the overall response will be highly correlated to the

quality of those inversions.

vi. The new average strain within inclusion phase is evaluated.

•ε

ε

ε1(tn+1) = ε

ε

ε1(0) + Aε∗˙

¯

ε

ε

ε(τ,tn+1)+Rtn+1

0aε(τ,u)du

vii.

The updated value of

ε

ε

ε1(tn+1)

is compared to the value from the previous iteration. If

the deviation is greater than the ﬁxed tolerance, the algorithm has not converged yet.

Therefore, new iteration is performed (return to step

b.i

). Otherwise, the computed

value is indeed the searched solution and the prediction is accepted.

viii. ε

ε

ε0(tn+1)is updated such that: ε

ε

ε0(tn+1) = 1

ν0

¯

ε

ε

ε(tn+1)−ν1

ν0

ε

ε

ε1(tn+1)

cAfter convergence, the macroscopic stress is inferred based on the following expressions:

•¯

σ

σ

σn+1=ν0σ

σ

σ0n+1+ν1σ

σ

σ1n+1

•Or, ¯

σ

σ

σn+1=¯

σ

σ

σ(0) + Rtn+1

0¯

Cτ(τ,tn+1−u):h˙

¯

ε

ε

ε(u)−˙

¯

ε

ε

ε0

tn+1(u)idu

25

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

where

¯

Cτ

and

˙

¯

ε

ε

ε0

tn+1

are the inversions of the macroscopic stiffness modulus and eigenstrain

rate in the time domain. The choice of the method used to determine the macroscopic stress

is further discussed in Pierard and Doghri [2006].

The linearization time τis chosen such that τ=tnfor all the results presented in this paper.

6 Results and discussion

In this section, the incremental-secant and integral afﬁne formulations are assessed against reference

results from full-ﬁeld FE analyses. The two formulations are also compared to the incrementally

afﬁne approach (Miled et al. [2013]).

The FE results were obtained as follows. The geometries and meshes were created using Digimat-FE

. The FE analyses were carried out with Abaqus (2020) linked with Digimat-MF as a user-deﬁned

material (UMAT) because the VE-VP model is unavailable in Abaqus.

Three principal material systems were studied (see Fig.5). Periodic boundary conditions (PBC)

were prescribed. The macroscopic stresses of the microstructures were computed from the reaction

forces at boundary nodes (see Geers et al. [2010a]).

The two developed homogenization procedures were veriﬁed for different loading histories (mono-

tonic, cyclic, relaxation and creep) and triaxiality conditions (uniaxial, biaxial and shear). The CPU

cost of the MFH simulations is negligible compared to full-ﬁeld FE simulations. In fact, the average

CPU time used by the two MFH methods is almost equal to 10 s with 100 time increments using

only one core of the Intel

i7−8665U

processor. Nevertheless, the full-ﬁeld FE simulations were run

on 4 cores simultaneously. They take an average of 4 hours of CPU time for porous matrix, 6 h for

the spherical particle reinforced composite and over 13 h for the short ﬁber reinforced composite.

6.1 Porous polymer matrix

In this example, the incremental-secant and the integral afﬁne formulations are evaluated for a

porous VE-VP matrix. The microstructure considered is pictured in Fig.5a. It is composed of a

polycarbonate matrix with 10

%

of spherical voids. The matrix behavior follows the constitutive

model of subsection 3.1 with model parameters summarized in Table 1. The RVE’s mesh is made

of about 600,000 second order tetrahedral elements.

First, the porous matrix was subjected to uniaxial loading. The effective responses provided by the

integral afﬁne approach with only ﬁrst moment estimates and the incremental-secant approach with

both ﬁrst and second moment estimates of the equivalent von Mises stress, are reported in Fig.6

26

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

(a) Porous matrix RVE (b) Spherical particle reinforced matrix RVE

(c) Short ﬁber reinforced matrix RVE

Figure 5: Microstructures considered in FE analyses. a) Porous polymer matrix including 50

spherical voids (aspect ratio

α=1

) with a volume fraction v

1

= 10

%

. b) Spherical particle

reinforced composite. It includes 50 particles having

α=1

and v

1

=10

%

. c) Short ﬁber reinforced

composite. It contains 30 aligned ellipsoidal particles having α=3 and v1= 15%.

Viscoelastic parameters

Initial shear modulus G0=1074 MPa

Initial bulk modulus K0=3222 MPa

Gi(MPa) gi(s) Kj(MPa) kj(s)

157 0.0021 472 0.007

80 0.00378 242 0.126

37 0.0248 111 0.216

Viscoplastic parameters

Yield stress σy=35 MPa

Hardening function Eq.14 k=150 MPa n=0.43

Viscoplastic function Eq.15 κ=150 /s m =5

Table 1: Parameters of Polycarboante constitutive model at 22◦C. Data from Miled et al. [2013].

27

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

alongside results from the incrementally afﬁne method and reference FE predictions. The ﬁgure

includes curves for four strain rates, varying from

1/s

to the quasi static case at strain rate

10−6/s

.

It appears that, for all strain rates, predictions of our two formulations are in good correlation with

FE results during the VE regime. However, the two MFH methods overestimate reference curves

during VP strain increments and their accuracy is quite comparable to that of the incrementally

afﬁne approach. The use of the second moment estimates in the incremental-secant method permits

to capture the yield point with more accuracy leading to signiﬁcant improvements compared to ﬁrst

moment estimates and a better agreement with FE results.

Moreover, the control of the unloading time increment

∆tunload

during the virtual unloading step

in the incremental-secant formulation allows to further soften the predictions of the method. As

shown in Fig.6, modiﬁed values of

∆tunload

are able to provide predictions that match perfectly FE

results. It should be noted that there is no clear pattern capable of forecasting the values of

∆tunload

for different microstructures and different loading conditions. For instance, the ratios

∆t

∆tunload

used

to perfectly recover FE predictions in Fig.6are equal

∆t

∆tunload =

1.22 , 1.1 , 4.5 and 24 for the strain

rates

˙

ε11 =1,10−2,10−4

and

10−6/s

respectively. These ratios change completely when a biaxial

loading is applied instead of the uniaxial one. All simulations realized using the incremental-secant

method with either the ﬁrst or the second moment estimates were run without modiﬁcation of

∆tunload . Consequently, ∆tunload =∆twas used.

In order to investigate the effects of loading triaxiality on the accuracy of the two proposed MFH

procedures, shear and biaxial loadings were applied. The effective behaviors inferred based

on the integral afﬁne and the incremental-secant methods are depicted in Figs.7-8with FE and

incrementally afﬁne methods results. It is observed throughout Fig.7that the integral afﬁne

formulation predictions ﬁt reference curves, perfectly, for all strain rates when shear loading is

imposed. However, Fig.8shows that this is not the case anymore for biaxial loadings. In fact, the

method is highly dependent on the quality of inversions of localization tensors from L-C to time

domain. The shear test results demonstrate that the choice of the number of collocation points

in addition to the relaxation times spectrum interval is a very critical step in the integral afﬁne

approach since an adequate choice yields to very accurate results as in Fig.7. The conclusion that

can be drawn, concerning the integral afﬁne formulation, is that the method could potentially be

very accurate when inversions recover the target operators in the time domain with a high precision.

Meanwhile, the incremental-secant approach improves slightly the results of the incrementally

afﬁne with ﬁrst and second moment estimates of equivalent stress. But, it still slightly overestimates

reference curves.

28

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

0

10

20

30

40

50

60

70

80

0 0.01 0.02 0.03 0.04 0.05 0.06

60

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

Incr. secantmod

(a) ˙

¯

ε11 =1 /s

0

10

20

30

40

50

60

70

0 0.01 0.02 0.03 0.04 0.05 0.06

60

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

Incr. secantmod

(b) ˙

¯

ε11 =10−2/s

0

10

20

30

40

50

60

70

0 0.01 0.02 0.03 0.04 0.05 0.06

50

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

Incr. secantmod

(c) ˙

¯

ε11 =10−4/s

0

10

20

30

40

50

60

70

0 0.01 0.02 0.03 0.04 0.05 0.06

50

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

Incr. secantmod

(d) ˙

¯

ε11 =10−6/s

Figure 6: Polycarbonate matrix with

10%

of spherical voids. Uniaxial tension loading was applied

at different strain rates. Matrix material parameters are summarized in Table 1.

The macroscopic responses of the RVE under biaxial loading are presented in Fig.8. In this

case, improvements can be seen with our two methods during the VE regime compared to the

incrementally afﬁne formulation of Miled et al. [2013]. The latter authors showed that their

MFH method suffers, mostly, from overestimated results during VE part of responses. Therefore,

the discrepancy propagates to VP strain increments leading to poor quality of effective behavior

predictions. The development of our MFH formulations was based on their capability to provide

improved accuracy during VE time steps. Results of biaxial tests prove that our hypothesis is

validated. This point is emphasized more in subsection 6.3.

The incremental-secant and the integral afﬁne methods were also studied while simulating drastic

changes in loading paths. Predictions of the two formulations for a complete cycle are reported in

Fig.9. Similar conclusions to the monotonic uniaxial case can be drawn here, in terms of agreement

29

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

0

5

10

15

20

25

30

35

40

0 0.01 0.02 0.03 0.04 0.05 0.06

30

0.04 0.05

¯

σ12 (MPa)

¯

ε12 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(a) ˙

¯

ε12 =1 /s

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06

25

0.04 0.05

¯

σ12 (MPa)

¯

ε12 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(b) ˙

¯

ε12 =10−2/s

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06

25

0.04 0.05

¯

σ12 (MPa)

¯

ε12 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(c) ˙

¯

ε12 =10−4/s

0

5

10

15

20

25

30

35

0 0.01 0.02 0.03 0.04 0.05 0.06

25

0.04 0.05

¯

σ12 (MPa)

¯

ε12 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(d) ˙

¯

ε12 =10−6/s

Figure 7: Polycarbonate matrix with

10%

of spherical voids. Shear loading was applied at different

strain rates. Matrix material parameters are summarized in Table 1.

with FE results and comparison with the incrementally afﬁne method predictions, except that the

improvements made during the tensile stage of the cycle with second moment estimates induce

a perfect match of the incremental-secant method results with FE ones during reverse parts of

compression and second tensile stage. In fact, the information related to plastic deformation is

enriched, statically, while accounting for second moment estimates leading to enhanced accuracy

for the overall response.

The capability of our two formulations to capture viscous effects was analysed through relaxation

test (see Fig.10a). The porous matrix is subjected initially to uniaxial tension loading at strain rate

˙

¯

ε11 =10−2

until

t=10

s. After that, the strain is maintained constant

˙

¯

ε11 =0.1

. The predictions

of the incremental-secant linearization method with the ﬁrst and the second moment estimates are

compared to FE results. The ﬁgure reveals that the inclusion of the second moment estimates of the

30

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

0

10

20

30

40

50

60

70

80

90

100

0 0.01 0.02 0.03 0.04 0.05 0.06

80

85

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(a) ˙

¯

ε11 =1 /s

0

10

20

30

40

50

60

70

80

90

0 0.01 0.02 0.03 0.04 0.05 0.06

65

70

75

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(b) ˙

¯

ε11 =10−2/s

0

10

20

30

40

50

60

70

80

0 0.01 0.02 0.03 0.04 0.05 0.06

65

70

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(c) ˙

¯

ε11 =10−4/s

0

10

20

30

40

50

60

70

80

0 0.01 0.02 0.03 0.04 0.05 0.06

60

65

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Incr. afﬁne

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(d) ˙

¯

ε11 =10−6/s

Figure 8: Polycarbonate matrix with

10%

of spherical voids. Biaxial loading was applied at different

strain rates until strain ¯

ε11 =¯

ε22 =0.05. Matrix material parameters are summarized in Table 1.

equivalent stress permits to capture the stress relaxation with more accuracy. The integral afﬁne

method overestimates FE predictions and even the results from the incremental-secant with both

statistical moments. The viscous effects are further investigated through creep test as shown in

Fig.10b. The microstructure undergoes, initially, a uniaxial tensile loading during which the applied

traction increases linearly from 0 to

100

MPa at t = 10 s. Afterwards, the applied load is maintained

constant

σap plied =100

MPa until t = 100 s. It can be observed that the use of the second moment

estimates of the equivalent stress allows to obtain remarkable improvements compared to the ﬁrst

moment estimates. On the other hand, this loading case is very challenging for the integral afﬁne

approach. Indeed, the method did not converge even with the use of different combinations of the

number of collocation points and their spectrum limits.

31

INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)

-100

-80

-60

-40

-20

0

20

40

60

80

-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08

50

60

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(a) ˙

¯

ε11 =±1 /s

-80

-60

-40

-20

0

20

40

60

80

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

50

60

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(b) ˙

¯

ε11 =±10−2/s

-80

-60

-40

-20

0

20

40

60

80

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

40

50

0.04 0.05

¯

σ11 (MPa)

¯

ε11 (%)

Finite element

Integral afﬁne

Incr. secant1st

Incr. secant2nd

(c) ˙

¯

ε11 =±10−4/s

-80

-60

-40

-20

0

20

40

60

80

-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

40

50

0.04 0.05

¯

σ11 (MPa)

¯