day4 2 micromechanics formulas
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formulas de micromecanciaTRANSCRIPT
Micromechanics based on the Eshelby solu4on: Concepts and formulas
2nd IIMEC Winter school 2013 College Station, Texas, USA
Yves Chemisky and Fodil Meraghni
Arts et Métiers ParisTech, France
Motivations
Properties
Process
Microstructure Effect of injection process on the orientation of fibers
Mat
eria
l by
desi
gn
Materials
Courtesy of P. Chinesta
Local microstructures
Motivations
Effect of injection process on the orientation of fibers
Local microstructures
Properties
Process
Microstructure
Mat
eria
l by
desi
gn
Materials
Courtesy of P. Chinesta
Microstructure Variability: Distribu2ons of fiber volume frac2on, fiber orienta2on and length induced by the
injec2on process.
Need to predict the overall behavior of automo4ve component integra4ng the process induced microstructure of the composite
(courtesy of Plas4c Omnium)
Automo4ve Industry Requirements:
Ex. Tailgate (rear closure) made of discon4nuous fiber composite (SMC)
5
t =σn
Equivalent homogeneous material
n
The problem of homogenization
12
.. N
..
r
t =σn
inhomogeneities
0
matrix
n
Lr
L0
?
Find an equivalent homogeneous material that has the same macroscopic behavior.
Given an heterogeneous material
Fundamental micromechanics problem
Inclusion : Part of an elastic medium with same elastic properties
r
Lr
L0
r
L0
L0
Inhomogeneity: Part of an elastic medium with different elastic properties than the surrounding medium
r
L0
L0
Inclusion problem definition: A region in an infinite elastic medium undergoes a change of shape and size by introduction of an eigenstrain What is the stress state of the inclusion and the surrounding matrix?
rr
r
L0
L0
?
Fundamental micromechanics problem
The Eshelby solution
r
L0
L0
Solution procedure:
The Eshelby solution
L0
Solution procedure: I. Remove the inclusion and allow it
to undergo a stress-free strain
rr
The Eshelby solution
L0
Solution procedure: I. Remove the inclusion and allow it
to undergo a stress-free strain
II. Apply a surface traction to the inclusion
rr r
The Eshelby solution
L0
Solution procedure: I. Remove the inclusion and allow it
to undergo a stress-free strain
II. Apply a surface traction to the inclusion Put it back in the matrix
rr r
r
The Eshelby solution
L0
rr
r
r
Solution procedure: I. Remove the inclusion and allow it
to undergo a stress-free strain
II. Apply a surface traction to the inclusion Put it back in the matrix
III. Cancel those tractions by applying the opposite tractions on the surface of the inclusion
Eshelby solution
r
L0
Inclusion problem solution: Solving the following boundary value problem (e.g. with Green’s functions) to find stress and strain state induced by the effect of the eigenstrain
L0
r
The Eshelby solution Eshelby fundamental results: - In an ellipsoidal inclusion, the total strain induced by the appearance of
a uniform eigenstrain is uniform - The uniform total strain can be expressed as a function of the
eigenstrain per:
Fourth order Eshelby tensor
The Eshelby tensor depends on the material properties and the shape of the inclusion (i.e., aspect ratio) Analytical expressions can be found for isotropic linear materials for some specific shapes (spheres, cylinders, …)
Constitutive law for linear elasticitty:
Considering two vectors and , defined below: σ ε
The constitutive equation for an anisotropic material can be written:
Where the components of the 6*6 matrix L are:
The uniform total strain can be expressed as a function of the eigenstrain per
Where the components of the 6*6 matrix S are:
2 2
2
Analytical solution for Spheres and cylinders
For a homogeneous, isotropic linear elastic behavior of the media: Spherical inclusion:
2"1"
3"
Analytical solution for Spheres and cylinders
For a homogeneous, isotropic linear elastic behavior of the media: Cylindrical inclusion (axis of revolution 1):
1"
3"
2"
Numerical estimation of the Eshelby tensor
For a homogeneous, anisotropic behavior. { }
1 2
3 min1 0
1 ( ) ( )8
mijkl mnkl imjn jS C d G G d
π
ζ ζ ζ ωπ
+
−
= +∫ ∫Eshelby Tensor
Eshelby’s Equivalence principle
r
inhomogeneity
LrL0
Total stress in the inhomogeneity: Principle of superposition:
Eshelby’s Equivalence principle Total stress in the inclusion, subjected to an arbitrary eigenstrain: Principle of superposition:
r
L0
Eshelby’s Equivalence principle
r
Lr
L0
r
L0
An inhomogeneity can be treated as an inclusion, with a prescribed eigenstrain (to be defined) that corresponds to the elastic stiffness mismatch These two situations are equivalent if the stress state in the inclusion is identical
Eshelby’s Equivalence principle
r
Lr
L0
r
L0
Eshelby’s Equivalence principle
Making use of the Eshelby solution:
The eigenstrain can found as a function of the stiffness tensors and the Eshlby tensor:
The total strain in the inclusion is expressed:
Where:
or:
If the prescribed tractions at the boundary are such that is constant on the surface S and without the presence of body forces:
Average theorems
12
.. N
..
r
inhomogeneity
0
n
matrix
The average stress is defined as
S
Let a domain D of volume V being subjected to prescribed tractions over its entire boundary S
Average theorems Demonstration
Using the product rule of derivatives
Making use of the Divergence theorem:
Conservation law of linear momentum without body forces
Average theorems Demonstration
If the prescribed tractions at the boundary are such that is constant on the surface S:
Average theorems Demonstration
Using again the divergence theorem:
If the prescribed displacement at the boundary is such that is constant on the surface S:
Average theorems
12
.. N
..
r
inhomogeneity
0
n
matrix
The average strain is defined as
S
Let a domain D of volume V being subjected to prescribed displacement over its entire boundary S
Demonstration
Average theorems
Making use of the Divergence theorem:
If the prescribed displacement the boundary is such that is constant on the surface S:
Demonstration
Average theorems
Using again the divergence theorem:
Demonstration
Average theorems
Hill-Mandel theorem
Evaluation of the strain energy (per unit volume) of a heterogeneous material:
Hill-Mandel theorem
Could this be compared to the strain energy of an “equivalent material”, i.e. a media with the following strain energy:
Hill-Mandel theorem
Rearranging these expression yields an important form of the Hill lemma:
For homogeneous boundary conditions, in terms of displacement or tractions, i.e:
The Hill-Mandel theorem yields:
or
Definition of effective modulus of heterogeneous media
Consider an heterogeneous media composed of N distinct phases. The stiffness of each phase r is defined by its stiffness tensor Lr
L1
..
..
inhomogeneity
L0
matrix
L2
Lr
LN
grain
L1
L2
Lr
LN ..
..
Applicable to the 2 main types of microstructures, i.e composites and polycrystals:
How to define the effective modulus ?
Definition of effective modulus of heterogeneous media
Consider an heterogeneous media composed of N distinct phases. The stiffness of each phase r is defined by its stiffness tensor Lr
Equivalent homogeneous material
12
.. N
..
r
inhomogeneities
0
matrix
Lr
L0
Definition of effective modulus of heterogeneous media A straightforward approach of the effective stiffness tensor:
Equivalent homogeneous material
12
.. N
..
r
inhomogeneities
0
matrix
Lr
L0
And of the effective compliance tensor:
Definition of effective modulus of heterogeneous media Equivalence of the strain energy of the heterogeneous (composite) media and the homogeneous media:
Using the Hill theorem:
The heterogeneous and homogeneous media are subjected to a prescribed displacement at the boundary, such that is constant on the surface Sc and Sh. Therefore, and:
Definition of effective modulus of heterogeneous media Equivalence of the strain energy of the heterogeneous (composite) media and the homogeneous media:
Using the Hill theorem:
The heterogeneous and homogeneous media are subjected to prescribed tractions at the boundary, such that is constant on the surface Sc and Sh. Therefore, and:
Localization laws
Consider an heterogeneous material. Each local components (stress, strain but also other quantities) can be related to the prescribed/average quantities:
Heterogeneous media
A and B are referred as strain concentration tensor and stress concentration tensor, respectively
Localization laws
If the material is composed of N distinct phases, each per-phase average components (stress, strain but also other quantities) can be related to the prescribed/average quantities:
A and B are referred as strain concentration tensor and stress concentration tensor, respectively
12
.. N
..
r
inhomogeneity
0
matrix
Concentration tensors and effective modulus If the material is composed of N distinct phases:
The average strain
becomes
Thus:
Concentration tensors and effective modulus If the material is composed of N distinct phases:
The average stress
becomes
Thus:
Concentration tensors and effective modulus For an heterogeneous material:
From the definition of the effective stiffness tensor:
Local constitutive law
Localization law
Concentration tensors and effective modulus If the material is composed of N distinct phases:
From the definition of the effective stiffness tensor:
Local constitutive law
Localization law
12
.. N
..
r
0
Approximation methods to determine the effective properties
The effective modulus requires the definition of the concentration tensors. The methods based on the Eshelby solution are always constructed with the same spirit: The expression of the concentration tensors as a function of the interaction tensor.
The Dilute Approximation
12
.. N
..
r r
inhomogeneity
0
matrix
Lr
L0
Lr
L0
The dilute approximation : 1 inhomogeneity, same volume fraction of the phases
t =σn
n
t =σn
n
The Dilute Approximation
Consider that the composite is subjected to the following prescribed displacement boundary condition:
The effective stiffness tensor is obtained from the expression of the concentration tensors since
Since each phase is considered as a single inhomogeneity:
and:
The Dilute Approximation
Consider that the composite is subjected to prescribed tractions such that :
At the boundary, the strain is expressed using the Hooke’s law:
Making use of the Hook’e law again:
thus
The Dilute Approximation
The stress concentration tensor is thus written:
and:
The effective stiffness tensor is obtained from the expression of the concentration tensors since
The Mori-Tanaka Approximation
12
.. N
..
r r
t =σn
inhomogeneity
0
matrix
Lr
n
t0 =σ 0n
n
Lr
L0 L0
Consider that the composite is subjected to the following prescribed displacement boundary condition:
Each phase is supposed to be embedded in an infinite matrix where the boundary conditions depends the average strain in the matrix:
The Mori-Tanaka Approximation
From the average strain theorem:
Using again the localization equation:
The Mori-Tanaka Approximation
The effective stiffness tensor is obtained from the expression of the concentration tensors since
The expression of the concentration tensors are identified:
and:
Consider that the composite is subjected to the following prescribed displacement boundary condition:
Each phase is supposed to be embedded in an infinite matrix where the boundary conditions depends on to the average stress in the matrix:
The Mori-Tanaka Approximation
From the average stress theorem:
Using again the localization equation combined with the Hooke’s law:
The Mori-Tanaka Approximation
The effective compliance tensor is obtained from the expression of the concentration tensors since
The expression of the concentration tensors are identified:
and:
The Self-Consistent Approximation
12
.. N
..
r r
t =σn
inhomogeneity
0
matrix
Lr
n n
Lr
L0
t =σn
The SC approximation : Inhomogeneities, same volume fraction of the phases
N times Equivalent homogeneous material
Consider that the composite is subjected to the following prescribed displacement boundary condition:
The localization tensor is obtained from the effective elastic properties of the medium:
Each phase is considered as a single inhomogeneity embedded in the effective medium:
and:
The Self-Consistent Approximation
Consider that the composite is subjected to prescribed tractions such that :
At the boundary, the strain is expressed using the Hooke’s law:
Making use of the Hook’e law again:
thus
The Self-Consistent Approximation
The stress concentration tensor is thus written:
and:
The effective stiffness tensor is obtained from the expression of the concentration tensors since
The Self-Consistent Approximation
Summary Refr r
r r
r r
T
A
B
ε ε
ε ε
σ σ
=
=
=
1 2
.. N
.
. r r
inhomogeneity
0
matrix
Lr
L0RefLRefε
Eshelby Dilute Solution (EDS)
Mori-Tanaka Solution (MTS)
Self-Consistent Solution (SCS)
0Refε ε=0
Refε ε ε= =
0RefL L=
0RefL L=
RefL L=
Prescribed uniform strain
Average strain in matrix
Average strain of the composite
0Refε ε ε= =
0EDS EDSr r rA Aε ε ε= =
Eshelby Dilute Solution (EDS) 0RefL L=
0Refr r rT Tε ε ε= =
0EDS EDSr r rB Bσ σ σ= =
110 0( )EDS
r r rr T I S L L LA−−⎡ ⎤= = + −⎣ ⎦
0EDS
r rr L T MB =0 0
0r r r r r r rL L T L T Mσ ε ε σ= = =
0Refε ε ε= =
0RefL L=Mori-Tanaka Solution (MTS) 0Refε ε=
1 1
0 0
N NMTS EDS
r r r r rr rr r
T c T c TA A− −
= =
⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦∑ ∑
1 110
0 0
N NMTS EDS
r r r r r rr rr r
L T c T M c TB B− −
−
= =
⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦∑ ∑
It can be shown that:
1
00
NMTS
r rr
c TA−
=
⎡ ⎤= ⎢ ⎥⎣ ⎦∑
1
000
NMTS
r rr
L c TB−
=
⎡ ⎤= ⎢ ⎥
⎣ ⎦∑
0with T I=
11( )SCS
r r rr T I S L L LA−−⎡ ⎤= = + −⎢ ⎥⎣ ⎦
SCSrrr L T MB =
Self-Consistent Solution (SCS) RefL L=0Refε ε ε= =