Homogenization of ductile porous materials with pressure-sensitive matrix: theoretical formulation and numerical bounds.

1. HOMOGENIZATION OF DUCTILE POROUS MATERIALS
WITH PRESSURE-SENSITIVE MATRIX:
THEORETICAL FORMULATION AND NUMERICAL BOUNDS
, . , . .
Institut Jean Le Rond d’Alembert, UMR 7190 CNRS, UPMC, France
La Sapienza, Roma - April 26, 2017
CHENG et al. (LML, France) Presentation EMMC14 - Goteborg EMMC14 - Goteborg, Aug. 27-29, 2014 1 / 18

2. Outline
Introduction and motivation
1 Limit analysis of porous material matrix 2
Conclusions
CHENG et al. (LML, France) Presentation EMMC14 - Goteborg EMMC14 - Goteborg, Aug. 27-29, 2014 2 / 18

3. Introduction
Physical mechanisms of ductile fracture of metals
Ductile fracture in metals
Modifications of the Gurson’s model (Tvergaard, 1982; Tvergaard and Needleman, 1984; Needleman and Tvergaard, 1985).
Weck and Wilkinson (2008). In-situ SEM images of the deformation sequence of an aluminum alloy 5052.
Modeling ductile behavior of porous materials
Limit analysis and non-linear homogenization approaches
Gurson, (1977); Ponte Castaneda, (1991); Michel and Suquet, (1992); Guo et al., (2008); Anoukou et al., (2016).
Gurson’s model (1977)
F =
⌃2
eq
2
0
+ 2p cosh
✓
tr⌃
2 0
◆
1 p2
D =
@F
@⌃
˙p = (1 p)trDwith
(
= 0 if F(⌃ < 0)
0 if F(⌃ = 0)
Strength criterion Normality rule Evolution law
nucleation & growth of voids porosity
(Puttick, 1960; Rogers, 1960, McClintock 1968, Rice and Tracey 1969)

4. Carte min ral Carte de porosit
<5
12,5
20
27,5
(-)
250 µm
LML, UMR8107 CNRS, USTL Colloque SOIZE 2010 01-02 July 18 / 18
porosity plays a crucial role
Geomaterials : example of argillite with a matrix (known to be
pressure sensitive)

5. Presentation of the model
Experimental evidences of the behavior of chalk materials
30
35
40
Contrainteisotrope(MPa)
0
5
10
15
20
25
0 10 20 30 40 50 60 70 80
Déformation axiale (10-3)
Contrainteisotrope(MPa)
phase "élastique"
phase plastique
phase "élastique"
"pore collapse"
q
l' espace des contraintes p' / q
p' (contrainte effective moyenne) = (σ'1 + 2 σ'3) / 3
q (contrainte déviatorique) = σ'1 - σ'3
cisaillement
"pore collapse"
p'
traction
Complex phenomenological multi-surface-based plasticity models
are usually considered, but they are not easy for numerical implementation
Challenge : proposal of a constitutive model based on a micromechanically
derived single yield surface

6. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
‚ Geomaterials and some ductile metals pressure-sensitive and
Lode angle dependen (see for instance Xue 2007 ; Bai and
Wierzbicki 2008 and 2010)
‚ Guo et al. 2008, have propose a strength criterion of a porous solid
with pressure-sensitive dilatant matrix
‚ theoretical modelling of porous solids with a
matrix obeying a pressure and Lode angle dependent plastic
criterion : Mohr-Coulomb criterion
F(‡) = ‡eq ` –(◊L )(‡m ´ H) § 0
–(◊L ) =
sin(„)
’(◊L )
, H = C cot(„)
’(◊L ) = 1?
3
cos(◊L ) ´ 1
3
sin(„) sin(◊L )
◊L = ´1
3
arcsin
´?
27J3
2‡3
eq
¯
´30˝
§ ◊L § 30˝
Kokou Anoukou EMMC-14 Sweden 3 / 16

7. Kinematical Limit Analysis as an appropriate tool to predict strength
spherical void
eR
iR
.v xD
Perfectly plastic matrix
Strength of the solid phase defined by a the convex set Gs of admissible stress states
Gs
= {σ, fs
(σ) ≤ 0}
Dual definition by means of the support function πs(d) of Gs
πs
(d) = sup(σ : d, σ ∈ Gs
)
πs(d) represents the maximum "plastic" dissipation
Séminaire LMA - Marseille (USTL) Micromécanique et Milieux poreux 02/02/2010 4 / 25

8. Methodology
Kinematical Limit Analysis again
Macroscopic counterpart of πs(d) in the absence of interface effect
Πhom
(D) = (1 − f) inf
v∈V(D)
πs(d)
s
with d =
1
2
(grad v +t
grad v)
Support function of the domain Ghom of macroscopic admissible stresses
Πhom
(D) = sup(Σ : D, Σ ∈ Ghom
)
Ghom : domain of macroscopic admissible stresses
limit stress states at the macroscopic scale : Σ = ∂Πhom/∂D
Séminaire LMA - Marseille (USTL) Micromécanique et Milieux poreux 02/02/2010 5 / 25

9. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Mohr-Coulomb criterion and the associated plastic dissipation function
– Perfect plasticitywith Mohr-Coulomb
strength criterion:
In terms of principal stresses the M-C criterion reads
F(‡) = sup
i,j P t1,2,3u
t| ‡i ´ ‡j | `(‡i ` ‡j ) sin(„)u ´ 2C cos(„) = 0
Kokou Anoukou EMMC-14 Sweden 4 / 16
By means of the stress invariants
F(‡) = ‡
s
eq ` –(◊L )(‡m ´ H) § 0
–(◊L ) =
1
3
’
in
(
(
◊
„
)
)
, H = C cot(„)
’(◊L ) = ?1
3
cos
L
(◊L ) ´
´?
sin
27
(
J
„)
¯
sin(◊L )
1
3
◊L = ´ arcsin 3
2‡3
eq
´30˝
§ ◊L § 30˝
fi(d) = `8, if tr(d) † (|d1 | ` |d2 | ` |d3 |) sin(„)
fi(d) = H.tr(d), if tr(d) • (|d1 | ` |d2 | ` |d3 |) sin(„)

10. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Atrialvelocityfieldforthekinematicallimitanalysis
A first step of the analysis: ice of a relevant trial velocity
field that is one which must be kinematically admissible (K.A) and
comply with the plastic admissibility (P.A.) condition.
A -parameter velocity field basedon , CR Mecanique is considered:
v(x) = A0
´b
r
¯3—
x ` A¨x
with — =
3 ´ ‘ sin(„)
3(1 ` ‘ sin(„))
, and A = A1
`
efl befl ` eÏ beÏ
˘
` A2ez bez (9)
‚ Kinematic admissibility (K.A.) condition : v(x = ber ) = D.x
#
fl(A0 ` A1) = flDfl
z(A0 ` A2) = zDz
ñ
$
&
%
A1 = Dm ´
Deq
2
sign(J3) ´ A0
A2 = Dm ` Deqsign(J3) ´ A0
‚ Let us recall the plastic admissibility (P.A.) condition
tr(d) • (|d1 | ` |d2 | ` |d3 |) sin(„)
Kokou Anoukou EMMC-14 Sweden 6 / 16

11. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Minimization problem and resolution (see Anoukou et al. JMPS
Part I, 91, 145-171
– Minimization problem
P :=
$
’’’’’&
’’’’’%
⇧(D) = min
v
xfi(d)y⌦
s.c. A1 = Dm ´
Deq
2
sign(J3) ´ A0 (a)
A2 = Dm ` Deqsign(J3) ´ A0 (b)
dm • Gi (c)
which by considering the limit load relation ⌃:D =⇧(D), translates in the following
auxiliary minimization problem P˚
:
P˚
:=
$
’’’’’&
’’’’’%
min
v
`
xfi(d)y⌦
´ 3⌃mDm ` ⌃eqDeq
˘
s.c. A0 ` 1
3
(2A1 ` A2) = Dm (a)
2
3
(A1 ´ A2) = Deqsign(J3) (b)
xdmy⌦´Ê • xGiy⌦´Ê (c)
with
xfi(d)y⌦
= 3(1 ´ f )H xdmy⌦´Ê = 3H
”
Dm(1 ´ f ) ´ A0
`
f 1´—
´ f
˘ı
Kokou Anoukou EMMC-14 Sweden 8 / 16

12. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Minimization problem : Lagrangian method and Karush-Kuhn-Tucker
(KKT) conditions
– Lagrangian method and Karush-Kuhn-Tucker (KKT) conditions
`
xdmy⌦´Ê´xGiy⌦´Ê
˘
L(v, ⁄) = ´3(1´f )H xdmy⌦´Ê`3⌃mDm`⌃eqDeq`⁄(1-f)
where ⁄ is the Lagrange-KKT multiplier.
or explicitly using (1 ´ f ) xdmy⌦´Ê =
“
Dm(1 ´ f ) ´ A0
`
f 1´—
´ f
˘‰
, we have :
L(A0, Dm, Deq, ⁄) = ´(3H ´ ⁄)
´
(1 ´ f )Dm ´ (f 1´—
´ f )A0
¯
` 3⌃mDm`
⌃eqDeq ´ ⁄Ji(A0, Deq)
with Ji(A0, De) = (1 ´ f ) xGiy⌦´Ê
The KKT conditions are given in system (S) as follows :
(S) :=
$
’’’’’’&
’’’’’’%
@L
@A0
= 0,
@L
@Dm
= 0,
@L
@Deq
= 0
@L
@⁄
• 0 ô xdmy⌦´Ê • xGiy⌦´Ê
⁄ • 0
⁄(xdmy⌦´Ê ´ xGiy⌦´Ê) = 0 ô ⁄ = 0 or xdmy⌦´Ê = xGiy⌦´Ê
Kokou Anoukou EMMC-14 Sweden 9 / 16

13. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Parametric form of the strength criterion and void growth evaluation
– Parametric form of the strength criterion
F(Σm, Σeq) :=
$
’’’’’’&
’’’’’’%
Σm
C
=
`
1 ´ f
˘Ψm
3γ
`
f 1´β ´ f
˘
` Ψm
Σeq
C
=
`
f 1´β
´ f
˘Ψeq
γ
`
f 1´β ´ f
˘
` Ψm
with
Ψm =
ω
1
β
´1
2
˜
1
β
´
I(ω) ` m
¯
` ωI1
(ω)
¸
Ψeq =
ω
1
β
4β
˜
β ´ 1
β
´
I(ω) ` eq
¯
´ ωI1
(ω)
¸
– Void growth : evolution law
9f = 3
Deqω
2β
´
f 1´β
´ f
¯
or 9f = 3Dm(1 ´ f ) ´
3Deqω
1
β
4β
I(ω)
Kokou Anoukou EMMC-14 Sweden 11 / 16

14. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Some features of the obtained criterion
f =25%
f =10%
f =5%
C = 1, f = 20 °
-15 -10 -5 0
-4
-2
0
2
4
6
Sm
Sr-Sz
f=0.01°
f=1°
f=10°
C = 1, f = 25 %
-2 -1 0 1
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Sm
Sr-SzC = 1, f = 10 %, f = 10 °
J3 < 0 HqL = 30 °L
J3 > 0 HqL = -30 °L
-5 -4 -3 -2 -1 0 1 2
0.0
0.5
1.0
1.5
2.0
Sm
Seq
Kokou Anoukou EMMC-14 Sweden 12 / 16
Few comments on cap models

15. Numerical assessment

16. Introduction – motivation Derivation of the macroscopic strength criterion Results and discussion Conclusion and perspectives
Numerical bounds (Pastor et al. 2010 6 JMPS, Part II)
– Finite element discretization combined with a non-linear optimization
procedure using MOSEK software
– Numerical results
The lower and upper
bounds of the
macroscopic criterion are
Kokou Anoukou EMMC-14 Sweden 13 / 16

17. 10 8 6 4 2 0 2 4
2
1
0
1
2
3
= 5
f = 1%
f = 5%
f = 10%
⌃m/c
⌃gps/c
Lower bound
Upper bound
New criterion
(a)
16 14 12 10 8 6 4 2 0 2 4
4
2
0
2
4
= 10
f = 1% f = 25%
⌃m/c
⌃gps/c
Lower bound
Upper bound
New criterion
(b)
Comparison of the derived strength criterion predictions
with the numerical bounds for two fixed values of
friction angle and several values of porosity: (a) = 5 ,
f = 1%, 5% and 10%, (b) = 10 and f = 1% and 25%.

18. 16 14 12 10 8 6 4 2 0 2 4 6
4
2
0
2
4
f = 1%
= 10 = 5
⌃m/c
⌃gps/c
Lower bound
Upper bound
New criterion
(a)
12 10 8 6 4 2 0 2
4
2
0
2
4
f = 10%
= 20 = 5
⌃m/c
⌃gps/c
Lower bound
Upper bound
New criterion
(b)
Comparison of the derived strength criterion predictions
with the numerical bounds, for fixed porosity f = 10%and
two friction angles =5 and 20 .

19. Conclusions
micromechanical procedure -
.
The approach
The criterion illustrated and validated by
comparison with .
Extension to saturated geomaterials by means of an effective stress concept
Implemen t and structural computations with the complete model
Extension to nanoporous materials (joint work with Prof. G. Vairo (Roma
2) : co-supervised PhD Thesis of Stella Brach (MOM, 2016; IJP, 2017)
CHENG et al. (LML, France) Presentation EMMC14 - Goteborg
EMMC14 - Goteborg, Aug. 27-29, 2014 18 /
18

20. Introduction
Can porous strength criteria be effectively used at sub-micron lengthscales?
Hutchinson (2000), International Journal of Solids and Structures, 37, pp. 225-238.
At nanoscales, mechanical features are dramatically different
from those of the same material at a larger lengthscale
Biener et al., (2005), (2006); Cheng et al., (2013); Hakamada et al., (2007); Hodge et al., (2007)

21. SEM of a nanoporous alumina (Kustandi et al., 2010)
Organic or inorganic solid matrix
a  100 nmUniform void size
High specific surface area
[Hodge et al. (2007)]
[Volkerta et al. (2006)]
Nano-indentation test
Fan and Fang, (2009)
Ligament size [nm]
Yieldstrength[GPa]
Void-size effects
Ligament size
a  100 nm
Some experimental observations on anoporous materials
Zhang et al., (2007, 2008, 2010); Goudarzi et al., (2010); Moshtaghin et al., (2012); Dormieux and Kondo, (2013).
Limit analysis
Dormieux and Kondo, (2010); Monchiet and Kondo, (2013).
Non-linear homogenization
Strength Properties of Nanoporous materials : A molecular Dynamics based approach
by S. Brach, L. Dormieux, D. Kondo & G. Vairo (work with Univ. Tor Vergata; MOM 2016)

22. Stella Brach PhD dissertation, 29 November 2016
Introduction Non-linear homogenization Molecular Dynamics Limit Analysis Conclusions
Objective: strength properties nanoporous samples at the nanoscale and void-size effects.
State-of-the-art: Marian et al. (2004, 2005), Traiviratana et al. (2008), Zhao et al. (2009), Bringa et al. (2010),
Tang et al. (2010), Mi et al. (2011).
Material strength under uniaxial/hydrostatic tests;
Few attention has been paid to void-size effects.
X Y
Z
x
y
z
a0
a0/2
< 100 >
[100]
[010]
[001]
b
L
2R
L
L
a0/2
Single-crystal/single-void FCC Aluminium
Fixed values of porosity and void sizes in
0.271 nm - 3.247 nm
Periodic boundary conditions
Different deformation paths and triaxiality levels
Simulations carried out in LAMMPS
Brach, S., Dormieux, L., Kondo, D., & Vairo, G. (2016b). Mechanics of Materials, 101, 102-117.
16 / 39
Molecular Dynamics

23. Stella Brach PhD dissertation, 29 November 2016
Conclusions
Void-size effects: overall expansion of strength domain as void size reduces
A significant dependency on the three stress invariants I1, J2, ✓⌃
−6 −4 −2 0 2 4 6 8 10 12 14 16 18
0
1
2
3
4
5
6
7
8
9
10
[GPa]
r[GPa]
Bulk
L/B=30
L/B=40
L/B=70
L/B=100
L/B=110
Bulk
TXE ( =0)
SHR ( = /6)
TXC ( = /3)
R = 0.812 nm
R = 1.082 nm
R = 1.894 nm
R = 2.706 nm
R = 2.977 nm
Void-size effects: shape transition in deviatoric profiles
r = 2 GPa
r = 4 GPa
r = 6 GPa
r = 8 GPa
L/B=20
L/B=40
L/B=60
L/B=70
SHR⌃
SHR⌃
SHR⌃
SHR⌃
SHR⌃
SHR⌃
TXC⌃
TXC⌃
TXC⌃ TXE⌃
TXE⌃
TXE⌃
✓D =
⇡
6
✓D =
⇡
3
✓D = 0
✓⌃ = ⇡/3
✓⌃ = ⇡/6
✓⌃ = 0
r=2 GPa
r=4 GPa
r=6 GPa
r=8 GPa
210
60
90
120
150
180
L/B=20
L/B=40
L/B=60
L/B=70
r=2 GPa
r=4 GPa
r=6 GPa
r=8 GPa
210
60
90
120
150
180
L/B=20
L/B=40
L/B=60
L/B=70
r=2 GPa
r=4 GPa
r=6 GPa
r=8 GPa
60
90
120
150
180
L/B=20
L/B=40
L/B=60
L/B=70
r=2 GPa
r=4 GPa
r=6 GPa
r=8 GPa
210
60
90
120
150
180
L/B=20
L/B=40
L/B=60
L/B=70
R = 1.082 nm
R = 1.624 nm
R = 2.165 nm
R = 0.541 nm= 1.353 nm, p = 1%
⇣ =
I⌃
1
p
3
, r =
q
2J⌃
2 , cos 3✓⌃ =
3
p
3J⌃
3
2J⌃
2
3/2
Brach, S., Dormieux, L., Kondo, D., & Vairo, G. (2016b). Mechanics of Materials, 101, 102-117.
Strength domain of a nanoporous nanoscale sample