Homogenization in Elasticity

Transcription

1 uliya Gorb November 9 13, 2015

2 uliya Gorb Homogenization of Linear Elasticity problem We now study Elasticity problem in Ω ε R 3, where the domain Ω ε possess a periodic structure with period 0 < ε 1 and has a smooth boundary Ω = Γ D Γ N, where Γ D Γ N = Consider the reference periodic cell = [0,1] 3 Assume that elastic coeff s C ε (x) = {C ε ijkl(x)} i,j,k,l=1,2,3 are periodic functions of x Ω ε with period 0 < ε 1, defined as follows: Let C(y) = {C ijkl (y)} i,j,k,l=1,2,3 be bounded, -periodic function, extended by periodicity to whole R 3 and that for all y satisfies and Define and study C ijkl = C jikl = C klij (1) m > 0 s.t. τ (τ ij = τ ji) : C ijkl τ kl τ ij mτ ijτ ij (2) ( x C ε (x) = C ε)

3 uliya Gorb Homogenization of Linear Elasticity problem (cont.) Classical Formulation of Linear Elasticity problem: σij ε +f i(x) = 0, x Ω ε x j ( i {1,2,3} : σij ε x ) = C ijkl e kl (u ε ) ε ui ε (x) = 0, x Γ D σijn ε j = 0, x Γ N (3) where n = (n 1,n 2,n 3) is the unit outer normal to Γ N Definition Homogenization of (3) consists in finding the limit of the solution u ε of (3) as ε 0

4 uliya Gorb Two-scale Asymptotic Method Look for expansions of ui ε (x) and σij(x) ε in the forms: ( ui ε (x) = ui 0 x, x ) ( +εui 1 x, x ) ( +ε 2 ui 2 x, x ) +..., ε ε ε σij(x) ε = 1 ( ε σ 1 ij x, x ) ( +σij 0 x, x ) ( +εσij 1 x, x ) +... ε ε ε (4) where u m i (x,y) and σ m ij (x,y) are the -periodic functions of y = x ε, i {1,2,3}, m {0,1,...} Substitute (4) back to (3) using e ij(u) = 1 ε ey ij (u0 )+ε 0[ ] e y ij (u1 )+eij(u x 0 ) +..., where eij(v) x = 1 ( ) vi + vj, e y ij 2 x j x (v) = 1 ( ) vi + vj i 2 y j y i Then σij(x,y) ε = 1 ε C ijkl(y)e y ij (u0 )+ε 0[ ] C ijkl (y)e y ij (u1 )+C ijkl (y)eij(u x 0 ) +...,

5 uliya Gorb Homogenization of Linear Elasticity problem (cont.) Hence, σ 1 ij (x,y) = C ijkl (y)e y kl (u0 ) [ ] (5) σij(x,y) 0 = C ijkl (y) e y kl (u1 )+ekl(u x 0 ) Now plug (4.1) into (3) and collect the terms of the same powers of ε to obtain σ 1 ε 2 ij : (x,y) = 0, y y j ε 1 σij 0 : (x,y) = σ 1 ij (x,y), y (6) y j x j ε 0 σij 1 : (x,y) = σ0 ij (x,y) f i(x), y y j x j The suitable space for variational formulation for u 0 and u 1 is { } W = v H0(Ω) 1 3 : v is periodic, v(y)dy = 0 equipped with the norm [ v V = ( v 2 + v 2 ) dx Ω ] 1/2

6 Determination of u 0 uliya Gorb From (6.1) we have σ 1 ij y j (x,y) = 0, y σ 1 ij (x,y) is periodic in y Variational Formulation for (6.1) where Find u 0 W s.t. v W : a (u 0,v) = 0, (7) a (u 0,v) = C ijkl (y)e kl (u 0 ) e ij(v) dy It is easy to see that the solution u 0 of (10) is a constant w.r.t. y, that is, u 0 (x,y) = u 0 (x)

7 Determination of u 1 uliya Gorb From (6.2) we use the following ansatz for solution u 1 (x,y) = λ mn(y)e x mn(u 0 )+û 1 (x) (8) that we substitute back in (6.2) to obtain the Cell Problem i,m,n {1,2,3} : Select P mn(y) = ( P 1 mn(y),p 2 mn(y),p 3 mn(y) ) where [C ijkl (y)e y kl (λmn)] = Cijmn(y), y y j y j λ mn(y) is periodic (9) P i mn(y) = y nδ im, i,m,n {1,2,3} where δ im the Kronecker delta then (9) becomes {C ijkl (y)e y kl [λmn +Pmn(y)]} = 0, y i,m,n {1,2,3} : y j λ mn(y) is periodic

8 uliya Gorb Determination of u 1 (cont.) Variational Formulation for (6.1) Find λ mn W s.t. v W : a (λ mn +P mn,v) = 0, m,n {1,2,3} (10) where a (ξ,η) = C ijkl (y)e kl (ξ) e ij(η) dy

9 Effective Elasticity uliya Gorb We now determine equation satisfied by u 0 Substitute (8) into (6) σ 0 ij (x,y) = C ijkl(y) [ e y kl (λmn)+δ mkδ nl ] e x mn (u 0 ) = [ C ijkl (y)e y kl (λmn)+c ijmn(y) ] e x mn (u0 ) Now average σij(x,y) 0 over : { ˆσ ij(x) 0 1 := [C ijkl (y)e y kl }e (λmn)+cijmn(y)] dy mn(u x 0 ) =: C ijmnemn(u x 0 ), where Effective Elasticity Remark: C ijmn = 1 [C ijkl (y)e y kl (λmn)+cijmn(y)] dy (11) Expression ˆσ 0 ij(x) = C ijmne x mn(u 0 ) is the stress-strain relation for the homogenized body

10 Macroscopic Equation uliya Gorb The macroscopic balance equation is obtained from (6.3) by integrating it over : σij 1 σ 0 ij (x,y) dy = (x,y)dy f i(x)dy y j x j Using the periodicity of σij(x,y) 1 in y and that f is a function of x Ω: σij(x,y) 0 dy +f i(x) = 0, x Ω x j Hence, Macroscopic Equation: i {1,2,3} : where C is given by (11) ˆσ 0 ij (x)+f i(x) = 0, x j ˆσ ij(x) 0 = C ijmnemn(u x 0 ) u 0 i (x) = 0, x Ω x Γ D ˆσ 0 ij(x)n j = 0, x Γ N That is, the homogenized material is linearly elastic

11 Macroscopic Equation uliya Gorb Remark: The macroscopic stress tensor ˆσ ij 0 := σ 0 ij(x,y)dy is defined as the volumeric mean of σij(x,y). 0 Below we show that it also can be given via surface mean σik 0 due to (6.1): (x,y) = 0, y y k Multiply the above equation by y j and integrate the result over : σik 0 (x,y)y j dy = 0 y k The integration by parts yields: Macroscopic Stress ˆσ ij(x) 0 = σ 0 ik(x,y)n k y j ds y (12)

12 uliya Gorb Properties of Homogenized Coefficient As in the previous cases of homogenization of diffusion and Stokes flow, it is easy to check that the effective elasticity tensor C is given by (11) is 1 symmetric: C ijkl = C jikl = C klij 2 and positive definite: Remark: m > 0 s.t. τ (τ ij = τ ji) : C ijklτ kl τ ij mτ ijτ ij where constant m is the same as in (2) Even if the original (heterogeneous) material is isotropic the homogenized one is anisotropic! Indeed, in (3) consider C ijkl (y) = λ(y)δ ijδ kl +µ(y)(δ ik δ jl +δ il δ jk ) and substitute this into (11), and obtain C ijkl = λ δ ijδ kl +µ (δ ik δ jl +δ il δ jk )+δ ij λ(y)emm(λ y kl ) dy+2 where λ = λ(y) dy, µ = µ(y) dy µ(y)e y ij (λ kl) dy