Local elds in Nonlinear Power Law Materialsmath.colgate.edu/~sjimenez/Talks/SES2008.pdfLocal elds in...

Local fields in Nonlinear Power Law Materials

Silvia Jimenez

Louisiana State University

Symposium: Local Field Properties, Microstructure, andMultiscale Phenomena in Heterogeneous Media

October 13th, 2008

Silvia Jimenez Bolanos Society of Engineering Science - Annual Meeting, UIUC 2008

Outline

Motivation

Theory

Result

Example

Future work

Outline

Motivation

Theory

Result

Example

Future work

Outline

Motivation

Theory

Result

Example

Future work

Outline

Motivation

Theory

Result

Example

Future work

Outline

Motivation

Theory

Result

Example

Future work

Motivation

Composites are materials madefrom two or more constituentmaterials with significantly differentphysical or chemical properties andwhich remain separate and distincton a macroscopic level within thefinished structure.

Fiber reinforced epoxy( Boeing 777 ).

Nonlinear elasticity is often usedto model failure.

Motivation

Theory

The goal is to characterize Local Fields inside NonlinearPower Law Materials.

This research develops new multiscale tools to bound thesingularity strength inside micro-structured media in terms ofthe macroscopic applied fields.

The research carried out in this project draws upon themathematical theory of Elliptic partial differential equations,Corrector theory, Young measures, and Homogenizationmethods.

Theory

Behavior of gradients of solutions to nonlinear PDEs withhighly oscillatory coefficients

Consider the Dirichlet problem−div (Aε (x ,∇uε)) = f on Ω,

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

2 < α1 ≤ α2 with 1α1

+ 1β2

= 1 and 1α2

+ 1β1

A : Rn × Rn → Rn is defined by

A (x , ξ) = α1χ1 (x) |ξ|α1−2 ξ + α2χ2 (x) |ξ|α2−2 ξ.

For every ε > 0, we define for all x ∈ Rn and for all ξ ∈ Rn,

Aε(x , ξ) = A(x

ε, ξ)

and χεi (x) = χi

)(i = 1, 2).

A describes the physical properties of a composite materialobtained by mixing two different nonlinear power-lawmaterials with different exponents.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

2 < α1 ≤ α2 with 1α1

+ 1β2

= 1 and 1α2

+ 1β1

Aε(x , ξ) = A(x

ε, ξ)

and χεi (x) = χi

)(i = 1, 2).

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

2 < α1 ≤ α2 with 1α1

+ 1β2

= 1 and 1α2

+ 1β1

Aε(x , ξ) = A(x

ε, ξ)

and χεi (x) = χi

)(i = 1, 2).

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

2 < α1 ≤ α2 with 1α1

+ 1β2

= 1 and 1α2

+ 1β1

Aε(x , ξ) = A(x

ε, ξ)

and χεi (x) = χi

)(i = 1, 2).

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

2 < α1 ≤ α2 with 1α1

+ 1β2

= 1 and 1α2

+ 1β1

Aε(x , ξ) = A(x

ε, ξ)

and χεi (x) = χi

)(i = 1, 2).

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

Ω is a piece/sample of the material.

f load.

ε > 0 is length scale of the composite microstructure which issignificantly smaller than the length scale of the load.

uε temperature/elastic displacement.

Scalar but it follows for elastic.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

f load.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

f load.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

f load.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

f load.

uε ∈W 1,α10 (Ω); f ∈W−1,β2(Ω).

f load.

Homogenization Theorem

Have uε converges to u strongly in Lα1(Ω) and ∇uε convergesweakly to ∇u in Lα1(Ω,Rn), as ε→ 0,

where u is solution of−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

where the monotone map b : Rn → Rn(independent of f and Ω) isdefined for all ξ ∈ Rn by

b(ξ) =

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,α1per (Y ),

υ ∈W 1,α1per (Y ).

Y = (0, 1)n: unit cube in Rn.

Have uε converges to u strongly in Lα1(Ω) and ∇uε convergesweakly to ∇u in Lα1(Ω,Rn), as ε→ 0, where u is solution of

−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

b(ξ) =

A(y , p(y , ξ))dy ,

υ ∈W 1,α1per (Y ).

−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

b(ξ) =

A(y , p(y , ξ))dy ,

υ ∈W 1,α1per (Y ).

−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

b(ξ) =

A(y , p(y , ξ))dy ,

where p(y , ξ) = ξ +∇υ(y),

where υ is the solution to the cellproblem:∫

υ ∈W 1,α1per (Y ).

−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

b(ξ) =

A(y , p(y , ξ))dy ,

υ ∈W 1,α1per (Y ).

−div (b (∇u)) = f on Ω,

u ∈W 1,α10 (Ω);

b(ξ) =

A(y , p(y , ξ))dy ,

υ ∈W 1,α1per (Y ).

Y = (0, 1)n: unit cube in Rn.Silvia Jimenez Bolanos Society of Engineering Science - Annual Meeting, UIUC 2008

Corrector Theorem

Y iε = ε(i + Y ), where i ∈ Zn.

Iε =i ∈ Zn : Y i

ε ⊂ Ω

Let ϕ ∈ Lα2(Ω,Rn)

andMεϕ : Rn → Rn be a functiondefined by

Mε(ϕ)(x) =∑i∈Iε

χY iε(x)

|Y iε |

∫Y i

ϕ(y)dy .

If we take ϕ = ∇uε, Mε takes the average of the field in everycube.

Corrector Theorem

Iε =i ∈ Zn : Y i

ε ⊂ Ω

Let ϕ ∈ Lα2(Ω,Rn)

andMεϕ : Rn → Rn be a functiondefined by

χY iε(x)

|Y iε |

∫Y i

ϕ(y)dy .

Corrector Theorem

Iε =i ∈ Zn : Y i

ε ⊂ Ω

Let ϕ ∈ Lα2(Ω,Rn) andMεϕ : Rn → Rn be a functiondefined by

χY iε(x)

|Y iε |

∫Y i

ϕ(y)dy .

Corrector Theorem

Iε =i ∈ Zn : Y i

ε ⊂ Ω

Let ϕ ∈ Lα2(Ω,Rn) andMεϕ : Rn → Rn be a functiondefined by

χY iε(x)

|Y iε |

∫Y i

ϕ(y)dy .

Corrector Theorem

Since we use power law materials, we use the following norm.

Orlicz Norm:

‖f ‖Orlicz(Ω) =

[∫Ωχ1(x) |f (x)|α1 dx

] 1α1

[∫Ωχ2(x) |f (x)|α2 dx

] 1α2

We construct a family of correctors which permit one toexpress ∇uε in terms of ∇u up to a remainder whichconverges to 0 strongly in the Orlicz norm.

Corrector Theorem

Orlicz Norm:

[∫Ωχ1(x) |f (x)|α1 dx

] 1α1

[∫Ωχ2(x) |f (x)|α2 dx

] 1α2

Corrector Theorem

Orlicz Norm:

[∫Ωχ1(x) |f (x)|α1 dx

] 1α1

[∫Ωχ2(x) |f (x)|α2 dx

] 1α2

Corrector Theorem

Orlicz Norm:

[∫Ωχ1(x) |f (x)|α1 dx

] 1α1

[∫Ωχ2(x) |f (x)|α2 dx

] 1α2

Corrector Theorem

If the structure is disperse, we have

∥∥∥p (x

ε,Mε(∇u)(x)

)−∇uε(x)

∥∥∥Orlicz(Ω)

→ 0,

as ε→ 0.

Corrector Theorem

If the structure is disperse, we have∥∥∥p (x

ε,Mε(∇u)(x)

)−∇uε(x)

∥∥∥Orlicz(Ω)

→ 0,

as ε→ 0.

Motivation

In heterogeneous media the initiation of failure is a multi-scalephenomena.

If you apply a load at the structural scale, the load is oftenamplified by the microstructure creating local zones of highfield concentration.

Motivation

In heterogeneous media the initiation of failure is a multi-scalephenomena.

If you apply a load at the structural scale, the load is oftenamplified by the microstructure creating local zones of highfield concentration.

Lower bound

By means of Young Measures and the previous Corrector Theorem,we obtain

∫Yφ (p(y ,∇u(x))) dydx ≤ lim inf

ε→0

∫Dφ (∇uε(x)) dx .

where D ⊂ Ω measurable, for all φ: failure criteria.In particular, if φ(x) = |x |p, p > 1, we have∫

∫Y|p(y ,∇u(x))|p dydx ≤ lim inf

ε→0

∫D|∇uε(x)|p dx .

Lower bound

By means of Young Measures and the previous Corrector Theorem,we obtain∫

ε→0

where D ⊂ Ω measurable, for all φ: failure criteria.

In particular, if φ(x) = |x |p, p > 1, we have∫D

ε→0

Lower bound

By means of Young Measures and the previous Corrector Theorem,we obtain∫

ε→0

where D ⊂ Ω measurable, for all φ: failure criteria.In particular, if φ(x) = |x |p, p > 1, we have∫

ε→0

Example

uin solves σ2−α12−α2 ∆α1u = 0

umid solves ∆u = 0.

uout solves σ∆α2u = 0.

At r = a:

uin = umid ,

σ2−α12−α2 n · |∇uin|α1−2∇uin = n · ∇umid .

At r = b:

umid = uout ,

n · ∇umid = σn · |∇uout |α2−2∇uout .

At r = c :

uout = Ec cos θ; where |E | = σ

12−α2 ,

σn · |∇uout |α2−2∇uout = E cos θ.

Example

At r = a:

uin = umid ,

At r = b:

umid = uout ,

At r = c :

12−α2 ,

Example

At r = a:

uin = umid ,

At r = b:

umid = uout ,

At r = c :

12−α2 ,

Example

At r = a:

uin = umid ,

At r = b:

umid = uout ,

At r = c :

12−α2 ,

Example

At r = a:

uin = umid ,

At r = b:

umid = uout ,

At r = c :

12−α2 ,

Example

At r = a:

uin = umid ,

At r = b:

umid = uout ,

At r = c :

12−α2 ,

Example

Solution:u = Er cos θ,

where |E | = σ1

2−α2 .

Example

where |E | = σ1

2−α2 .

Example

where |E | = σ1

2−α2 .

Example

where |E | = σ1

2−α2 .

Example

where |E | = σ1

2−α2 .

Neutral Inclusion of Nonlinear Materials

Example

where |E | = σ1

2−α2 .

Neutral Inclusion of Nonlinear Materials

Future work

What about random materials?

What if the structure is not disperse?

Study the case of Linear Laminates: Even thought they arenot disperse maybe the same results can be obtained.

Future work

Local elds in Nonlinear Power Law Materialsmath.colgate.edu/~sjimenez/Talks/SES2008.pdfLocal elds in...

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