Multi‐scale modelling of fibre reinforced composite with non‐local damage variable
L. Wu, L. Noels,
L. Adam and I. Doghri
June 5 ‐ 10, 2011
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
1
/Introduction
• Materials are multiscale in nature:
Metals
10‐9 m 1 m
Polymers
Molecular Dynamics
Monte Carlo Method, Statistical Mechanics …
Continuum Mechanics
Why Multiscale?
2
/Introduction
• Multiscale Methods for CompositesFor material design: These effective properties are difficult or expensive to measure.For composite structures analysis: • Continuum mechanics analysis at Macroscale
Accuracy!• Take into account the individual component properties and
geometrical arrangements.Expensive, unreachable!
• Solution:The engineering problems are solved at macroscopic scale with the homogenized properties.The homogenized properties are obtained from the individual component properties and their microstructure.
Why Multiscale?
3
/Introduction
• Finite element solutions for strain softening problems suffer from: – The loss the uniqueness and strain localization– Mesh dependence
Problem in finite element simulations
The numerical results change with the size of mesh and direction of meshHomogenous unique solution
Lose of uniqueness
Strain localized
The numerical results change without convergence
4
/Introduction
Multiscale Methods have this problem too!
•Solution:Introduce high order term in the continuum
description
Strain gradient model, nonlocal model…
Problem in finite element simulations
5
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
1
/Mean Field Homogenization
• At the macroscale, the problem is a classical continuum mechanics problem (Finite Element method).
• At a macroscopic material point the properties of the material correspond to a representative volume element (RVE) of the microstructure.
Macro Macro
Macro
MicroMFH
nε ε∆C σ
Basing on The marco strain and stress
equal the average strain and stress over a RVE
εσ
σε
∫= V VaVa d)(1 X
6
/Mean Field Homogenization
• How to get in RVE? Such that– Direct finite element simulation – Semi‐analytical mean field homogenization models
( Voigt, Reuss,Mori‐Tanaka, Double‐Inclusion, Self‐Consistent …)
• Two‐phase composite– Volume fraction
C εCσ :=
110 =+ vv
Subscription: 0(matrix) and 1(inclusion)
1010 ωωσσσ vv +=
1010 ωωεεε vv +=
11:1 ωω εCσ =
00:0 ωω εCσ =
???
7
/Mean Field Homogenization
• Single inclusion problem
is single inclusion strain concentration tensor (numerical, analytical)
is Eshelby’s tensor
• Multiple inclusion problem
Mori‐Tanaka model:
and
01:
ωε
ωεε B=
εH
S
101
10 )](::[
−− −+= CCCSIH ε
εε HB =8
:1
εε εω
A=
∞= εε εω
:),,( 101
CCIH
0ωεε =∞
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
1
/Nonlocal Approach
• DescriptionSome variables (a ) are replaced by their weighted (w
) average over a characteristic volume (V c) to reflect the interaction between neighboring material points.
The state variable a can be strains, internal variables (eg. accumulated plastic strain, damage….)
Problem: Weight function w ?? Characteristic volume V c ??
∫=cVc
awdVV
a 1
9
/Implicit Gradient Formulation
• Gradient model– Derived from non‐local models by expanding the integration of in Taylor series.
– The coefficients c1, c2... depend on the weight function and the characteristic volume V c.Explicit gradient formulation:
c has the dimension of length squared.
a...42
21 +∇+∇+= acacaa
acaa 2∇+=
10
/Implicit Gradient Formulation
• Implicit gradient formulation *: Green’s function G(y; x) weight function w(y; x)
The natural boundary condition:
aaca =∇− 2
0=∂∂
=∂∂
ii x
anna
How can we use it in Mean Field Homogenization ?
11* Peerlings et al., 1996
∫=cVc
awdVV
a 1
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
1
/Ductile Damage in the Matrix of Composite
• Damage in matrix only (neglect the Damage in fiber)• Lemaitre ‐ Chaboche ductile damage mode:
where S0 and n are the material parameters
Y is the strain energy release rate
is the accumulate plastic strain ,
• Nonlocal damage:
pSYD n && )(
0
=
e0
e ::21 εEε=Y
2/1pp ]:32[ εε &&& =pp ∫= tpp d&
ppcp =∇− 2
12
pSY
pcpcpSYD
n
n
&
&&&&
)(
...)()(
0
42
21
0
=
+∇+∇+=
/Ductile Damage in the Matrix of Composite
• Considering the damage in matrix, the incremental form of stress in composite*:
0011 σσσ δυδυδ +=
DD δδδ 00alg00 ˆ:)1( σεCσ −−= )1/(ˆ 00 D−=σσ
DDa δυδυδυδ 000alg00I
lgI1 ˆ:)1( σεCεCσ −−+=
ppDDa δυδδ∂∂
−= 00lg ˆ: σεCσ
13
Mori‐Tanaka
Subscription: 0(matrix) and 1(inclusion)
* Doghri I. et al., 2003
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
• For implicit gradient enhanced elastic‐plasticity
where ‐ the body force vector;
‐ the characteristic length of matrix material.
• Discretization (in each element)
Governing equations
⎩⎨⎧
=∇−=+∇
only materialmatrix for material comopsitefor
22 pplp0fσ
fl
/Finite Element Implementation
uNU u= pN pp =uBε u= pBpN ppp =∇=∇ ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
−−
=⎥⎦
⎤⎢⎣
⎡
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ppppup
puuu
dd
FFFF
pu
KKKK intext
14
/Outline
• Introduction• Mean Field Homogenization• Nonlocal Approach and Implicit Gradient Formulation
• Ductile Damage in the Matrix of Composite• Finite Element Implementation• Validation and Simulation
1
/Validation and Simulation
• Polyamide matrix reinforced by short glass fibers (15.7 %, ellipsoidal, AR = 15)Matrix: E = 2.1 GPa, v = 0.3, σY = 29 MPa, R(p) = h1 p+ h2(1‐exp(‐mp)), h1 = 139 MPa,
h2 = 32.7 MPa and m = 319; LC‐Damage: S0 = 2.0 MPa, n = 0.5, p0 = 0.;
Inclusions: E = 72 GPa, v = 0.22;
15
Verification: FEM implementation
/Validation and Simulation
16
• Unidirectional fiber reinforced compositeEpoxy Matrix: E = 2.89 GPa, v = 0.3, σY = 35 MPa, R(p) =h(1‐exp(‐mp))
h = 73.0 MPa and m = 60; LC Damage: S0 = 2.0 Mpa, n = 0.5, p0 = 0.
Carbon fiber: E = 238 GPa, v = 0.26;
Validation: DNS vs. FE/MFH
/Validation and Simulation
• Unidirectional fiber reinforced compositeTransverse Stress‐stain:
17
Validation: DNS vs. FE/MFH
/Validation and Simulation
• Notched sampleCharacteristic length l =0.0002 m
Characteristic length l =0.001 m
18
Simulation
Thank you!
/Validation and Simulation
• Elasto‐plastic matrix, elastic inclusions (15 %, spherical)Matrix: E = 100 GPa, v = 0.3, σY = 75 MPa, R(p) = hpm , h=400 MPa, m = 0.4;
Validation: material law vs. MFH code
Damage parameters:
LC: S0 = 2.0 MPa, n = 0.5,
p0 = 0.01;
LIN: p0 =0.01, pC =0.2.
Inclusions: E = 200 GPa,
v = 0.2;