Day4 2 Micromechanics Formulas

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:


L0

Solution procedure: I.  Remove the inclusion and allow it

to undergo a stress-free strain

rr


L0



II.  Apply a surface traction to the inclusion

rr r


L0



II.  Apply a surface traction to the inclusion Put it back in the matrix

rr r

r


L0

rr

r

r



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


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


r

Lr

L0

r

L0


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


If the prescribed tractions at the boundary are such that is constant on the surface S:


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


12

.. N

..

r

inhomogeneities

0

matrix

Lr

L0

Definition of effective modulus of heterogeneous media A straightforward approach of the effective stiffness tensor:


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:


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


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


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:


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 stress concentration tensor is thus written:

and:


The Mori-Tanaka Approximation

12

.. N

..

r r

t =σn

inhomogeneity

0

matrix

Lr

n

t0 =σ 0n

n

Lr

L0 L0


Each phase is supposed to be embedded in an infinite matrix where the boundary conditions depends the average strain in the matrix:


From the average strain theorem:

Using again the localization equation:



The expression of the concentration tensors are identified:

and:


Each phase is supposed to be embedded in an infinite matrix where the boundary conditions depends on to the average stress in the matrix:


From the average stress theorem:

Using again the localization equation combined with the Hooke’s law:


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


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:


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 stress concentration tensor is thus written:

and:



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= =

Day4 2 Micromechanics Formulas

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