Article

Viscoelastic-Viscoplastic polymer composites: development and evaluation of two very dissimilar mean-field homogenization models

Abstract and Figures

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 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) transform 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.
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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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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;(ab)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
311;Idev =IIvol (2)
For symmetric second order order tensors, one has:
Ivol :a=1
3amm 1;Idev :a=a1
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(tτ):∂ 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),τt
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=1Giexpt
gi
K(t) = K+J
j=1Kjexpt
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) = 2Gξ
ξ
ξV E (t) + I
i=1s
s
si(t)
σH(t) = 3KεVE
H(t) + J
j=1σH j(t)
(9)
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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 =˙pf
∂ 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
(
Stnσ
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+1ε
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=1Gih1expt
giigi
t
˜
K(t) = K+J
j=1Kjh1expt
kjiki
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
expt
gis
s
si(tn) +
J
j=1
expt
kjσH j(tn)1
(17)
with E=2GIdev +3KIvol 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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INT ERNATIO NAL JO URNAL OF SOLI DS AND STRUCT UR ES (IN PRE SS)
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
expt
gis
s
si(tn) +
J
j=1
expt
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)xωi(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):C1
0(C1C0)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= [AεI]:[C1C0]1:[β
β
β1β
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[CIC0]: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
iωi(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
σiωiωi(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
σiωiωi(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].
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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
nεn+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
,
Stnσ
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
nσ
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+1ε
ε
εn+1=ε
ε
εres
n+
ε
ε
εr
n+1
σ
σ
σn+1=σ
σ
σn+
σ
σ
σn+1σ
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.
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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
ntn
. 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
exptunload
gis
s
si(tn) +
J
j=1
exptunload
kjσH j(tn)1ωi
(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)1ωi
(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+10
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+12˜
G(t)p N
N
Nn+1
p=pn+1pn=gv(σ
σ
σeq
n+1,pn+1)tωi
(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+1σ
σ
σres
n)
(σ
σ
σn+1σ
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+1σ
σ
σres
n
σ
σ
σpred
n+1σ
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+1σ
σ
σres
n)eq
| {z }
σ
σ
σr
n+1eq
+3˜
G(t)pσ
σ
σpred
n+1σ
σ
σres
neq =0
Fp=pgv(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
iωi(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+10
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=
I3˜
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))2p
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) =
13˜
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
res
detailed in the next
subsection 4.5.
res(σ
σ
σn,ε
ε
εn,tunload )Stres
nωi
¯
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:
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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+1ν1
ν0
ε
ε
εr
1n+1
ii.
Secant operators and eigenstress tensors at time
tn+1
are evaluated using the constitutive
box n+1of each material phase (sections 4.2 and 4.3):
n+1(σ
σ
σres
n,ε
ε
εres
n,
ε
ε
εr
n+1)CS
n+1,β
β
βS
n+1ωi
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+1J1
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
res
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
σnσ
σ
σres
n
| {z }
σ
σ
σunload
n
=˜
E(tunload )
| {z }
Cres
n=C(tres
n)
:
ε
ε
εunload
n
+
I
i=11exp(tunload
gi
)si(tn) +
J
j=11exp(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)
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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,CCres
n,β
β
ββ
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
n1: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
nν1
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)
21
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,CE
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(tτ):∂ σ
σ
σ
∂ τ (τ)dτ
=Zt
0
J(tτ):∂ 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(tτ):∂ 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)tt+1
q(τ)(1exp(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)[1H(tτ)] + 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
tn1
ε
ε
ε1n1
| {z }
ε
ε
εnε
ε
εn1
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
εν1
ν0
ε
ε
εI
ii.
The constitutive box
n+1
is called to compute the affine stiffness modulus
C
τ
and
eigenstrain rate ˙
ε
ε
ε0
τrelative to each constitutive phase.
n+1(σ
σ
σn,ε
ε
εn,
ε
ε
ε)C
τ,ε
ε
ε0
τωi
iii.
Isotropic part of
C
τ
is extracted using either general or spectral methods (Doghri and
Ouaar [2003]).
iv. Localization tensors Aεand aεare computed using ˙
ε
ε
ε0
τand isotropic part of C
τ.
AεCiso
0τ,Ciso
1τ
aε(Ciso
0τ,Ciso
1τ,˙
β
β
β0
0τ
|{z}
Ciso
0τ:˙
ε
ε
ε0
0τ
,
Ciso
1τ:˙
ε
ε
ε0
1τ
z}|{
˙
β
β
β0
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+1u):h˙
¯
ε
ε
ε(u)˙
¯
ε
ε
ε0
tn+1(u)idu
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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
i78665U
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 22C. 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
106/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 =1,102,104
and
106/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.
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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 =102/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 =104/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 =106/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 =102/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 =104/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 =106/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 =102
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 =102/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 =104/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 =106/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.
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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 =±102/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 =±104/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)
¯