Silvia Jim enez Department of Mathematics Louisiana State...
Transcript of 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
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
Introduction
Figure: Bone
Figure: Airplane
Heterogeneous materials are abundant in nature, andincreasingly so in manmade systems.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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”.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
Mixtures of N Power Law Materials with Same Exponent
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
Mixtures of N Power Law Materials with Same Exponent
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
Mixtures of N Power Law Materials with Same Exponent
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
Mixtures of N Power Law Materials with Same Exponent
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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 ).
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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
).
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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.
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics
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!
Nonlinear Composites - Silvia Jimenez GEAUX Program, LSU Dept. of Mathematics