silvia jim enez department of mathematics louisiana state...

Nonlinear Composites

Silvia JimenezDepartment of MathematicsLouisiana State University

August 20, 2009

Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics

Research Groups and People

Analysis Research Group at LSU

Partial Differential Equations Group

Scientific Computing and Numerical Analysis

Research Group in Materials Science at LSU


https://www.math.lsu.edu/dept/research/analysis

https://www.math.lsu.edu/dept/research/pdes

https://www.math.lsu.edu/dept/research/scicomp

https://www.math.lsu.edu/dept/grad/material

Outline

Introduction

What is the goal and what do we need?

Possible Courses

Mixtures of N Power Law Materials with Same Exponent

Homogenization Theorem (Fusco and Moscariello)

Corrector Theorem (Dal Maso and Defranceschi)

Lower bound on Field Concentrations


Introduction

Figure: Bone

Figure: Airplane

Heterogeneous materials are abundant in nature, andincreasingly so in manmade systems.


Introduction

Often the distribution of hetereogeneity is such that thematerial appears to be homogeneous at a large enough lengthscale.

What are the ”average” properties of such heterogeneousmaterials?

”Effective properties” of such ”composite materials,” directlyfrom the properties of their ”microstructure”.


Introduction

In my research, I work with composites with nonlinearconstitutive behaviour, called power law materials.

Used to describe several phenomena ranging from plasticity tooptical nonlinearities in dielectric media.


Introduction

Important to provide insight into the statistical distribution oflocal quantities, such as averages or higher moments of thestrains and stresses in each phase.

These local quantities are extremely useful for understandingthe evolution of nonlinear phenomena such as plasticity ordamage.


Introduction

Research on composite materials with nonlinear constitutivebehaviour can be traced back to the classical work of Taylor(”Plastic Strain in Metals”, 1938) on the plasticity ofpolycrystals


Introduction

In composites, failure initiation is a multiscale phenomena.

A load applied at the structural scale is often amplified by themicrostructure creating local zones of high field concentration

Properties of local fields inside mixtures of two nonlinearpower law materials are studied.

This work addresses a prototypical problem in the scalarsetting.


What is the goal and what do we need?

The goal is to develop new multiscale tools to bound the localsingularity strength inside micro-structured media in terms ofthe macroscopic applied fields.

The research carried out in this project draws upon themathematical theory of Elliptic PDEs, Corrector theory, Youngmeasures, and Homogenization methods.


Possible courses

MATH 4999-5: Vertically Integrated Research: ComplexMaterials and Fluids.Instructor: Prof. Lipton with S.Armstrong: Introduction to the field of homogenizationtheory, guide to the current research literature useful forunderstanding the mathematics and physics of complexheterogeneous media.

7380-1 Numerical Optimization. Prof. Zhang.: Basicnonlinear optimization theories and algorithms.

7380-3 Partial Differential Equations. Prof. Shipman.: Widevariety of topics in partial differential equations with adetailed understanding of illuminating examples.

SPRING 2010: 7390 Topic TBA in Materials Science. Prof.Lipton



We consider N nonlinear power law materials periodicallydistributed inside a domain Ω.

Ω is an open bounded subset of Rn

The length scale of the microstructure relative to the domainis denoted by ε.

We describe the geometry of the mixture through thecharacteristic functions χεi , i = 1, ...,N, corresponding to eachof the materials.

For i = 1, ...,N, χεi = 1 in the i-th phase and zero outside.



Since the mixture is periodic, we use the unit period cell Y todefine χεi , i = 1, ...,N.

For i = 1, ...,N, the indicator function of the i-th phase in theunit cell Y is χi (y).N∑

i=1

χi (y) = 1.

The ε periodic mixture inside Ω is described by

χεi (x) = χi (x/ε) for i = 1, ...,N.



Let p ≥ 2 and let q such that

1

p+

1

q= 1.

The piecewise power law material is defined by theconstitutive law A : Rn × Rn → Rn given by

A(y , ξ) =N∑

i=1

χi (y)ai |ξ|p−2 ξ,with ai ≥ 0,

and the constitutive law for the ε-periodic composite is givenby

Aε(x , ξ) = A(x

ε, ξ)

, for every ε > 0.



For a given source term f ∈W−1,q(Ω), consider the Dirichletproblem

−div (Aε (x ,∇uε)) = f on Ω,

uε ∈W 1,p0 (Ω).

where

Aε(x , ξ) =N∑

i=1

χεi (x)ai |ξ|p−2 ξ

and1

p+

1

q= 1.


Mixtures of N Power Law Materials with Same Exponent -Homogenization Theorem (Fusco and Moscariello)

As ε→ 0, the solutions uε converge strongly to u in Lp(Ω), and∇uε converges weakly to ∇u in Lp(Ω; Rn), where u is solution of

−div (b (∇u)) = f on Ω,

u ∈W 1,p0 (Ω);

where b : Rn → Rn is defined for all ξ ∈ Rn by

b(ξ) =

∫Y

A(y , p(y , ξ))dy ,

where p(y , ξ) = ξ +∇υ(y), where υ is the solution to the cellproblem:∫

Y (A(y , ξ +∇υ),∇w) dy = 0 for every w ∈W 1,pper (Y ),

υ ∈W 1,pper (Y ).


Mixtures of N Power Law Materials with Same Exponent -Homogenization Theorem (Fusco and Moscariello)

W 1,pper (Y ) denotes the set of all functions u ∈W 1,p(Y ) with

mean value zero which have the same trace on the oppositefaces of Y .

As before, for ε > 0, we rescale and define

pε(x , ξ) = p(x

ε, ξ)

= ξ +∇υξ(x

ε

).


Mixtures of N Power Law Materials with Same Exponent -Corrector Theorem (Dal Maso and Defranceschi)

Y iε = ε(i + Y ), where i ∈ Zn.

Iε =i ∈ Zn : Y i

ε ⊂ Ω

.

Let ϕ ∈ Lp(Ω,Rn) andMεϕ : Rn → Rn be a functiondefined by

Mε(ϕ)(x) =∑i∈Iε

χY iε(x)

1

|Y iε |

∫Y i

ε

ϕ(y)dy .

Mε takes the average of the vector field in every cube.


Mixtures of N Power Law Materials with Same Exponent -Corrector Theorem (Dal Maso and Defranceschi)

Theorem (Corrector Theorem)∥∥∥p (x

ε,Mε(∇u)(x)

)−∇uε(x)

∥∥∥Lp(Ω,Rn)

→ 0, as ε→ 0.


Lower bound on Field Concentrations

We use the Corrector Theorem and Young Measures to studythe behavior of gradients of solutions (∇uε) of the Dirichletproblem.

These tools allow us to bound nonlinear quantities of thegradients from below in terms of the local solution p and thehomogenized gradient.

Numerical Methods for the Homogenized correctors aresimpler than the original ε problem. This is a multiscale lowerbound.


Mixtures of N Power Law Materials with Same Exponent -Lower bound on Field Concentrations

Theorem (Lower Bound of Field Concentrations)

We have that for all Caratheodory function ψ ≥ 0 and E ⊂ Ωmeasurable∫

D

∫Yψ(x , (p(y ,∇u(x))))dydx ≤ lim inf

ε→0

∫Dψ(x ,∇uε(x))dx .

In particular, for r > 1,∫D

∫Y|p(y ,∇u(x))|r dydx ≤ lim inf

ε→0

∫D|∇uε(x)|r dx .

Functions of the form ψ are often used in failure criteria(Tsai-Hahn/Tsai-Hill/Tsai-Wu).

If the sequence ψ (x ,∇uε(x)) is weakly convergent in L1(Ω),then the inequality becomes an equality.


Mixtures of N Power Law Materials with Same Exponent -Lower bound on Field Concentrations

In particular, for r > 1,∫D

∫Y|p(y ,∇u(x))|r dydx ≤ lim inf

ε→0

∫D|∇uε(x)|r dx .

A use of this inequality is found in the Weibull Theory ofFailure.

Probability of failure in D = 1− e−cRD |∇uε|rdx ,

where r= scatter, which is a material property. So we get anlower bound for this quantity.

Thank you!


silvia jim enez department of mathematics louisiana state...

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