RESULTS ON THE CONVERGENCE OF THE WEAK SOLUTION FOR THE STATIC PROBLEM OF PERIODIC
ELASTIC MEDIA PROBLEMS
Ene Remus-Daniel and Bˆınzar Tudor
Abstract. The static equations governing the stress state and the de- formation of a periodic medium are established based on the homogenization method. Properties of the stress and elasticity tensors are obtained. The weak convergence of the solution of the microscopic problem is proven.
2000 Mathematics Subject Classification: 35J20, 74A10, 74A15.
1. The phisical problem: homogenized equations
When the material has a strong heterogeneous structure, it is practically impossible to determine its properties in each point. As a result, the porous media study is performed by approximation with periodic media which allow to the microscopic equations to had through homogenization to macroscopic laws and in certain conditions the solution of the microscopic problem is convergent to the solution of the macroscopic one.
In [5] Sanchez-Palencia applies the homogenization method to the case of composite materials. He investigates static problems such as: the thermal conduction problem in a periodic medium with Dirichlet and Neumann type boundary conditions, the elasticity problem with Dirichlet type boundary con- ditions. For the thermal conduction equation he proved a weak convergence of the weak solution of the microscopic problem to the weak solution of the macroscopic one.
Based on Sanchez-Palencia techniques, for the linear elasticity static prob- lem, the macroscopic equations for porous media are established and the weak
solution convergence of the microscopic problem toward the weak solution of the macroscopic one is investigated.
This procedure implies that the medium is considered as having a periodic structures.
The homogenization method in elasticity problems is used also by Abdulle [1] for a polygonal domain with the numerical investigation of the homogenized equations based on the finite elaments method (FEM).
Let us consider a porous body which occupies the region Ω ⊂R3 (having the characteristic dimension L) with a piecewise smooth boundary ∂Ω. The coordinates of a certain point in the domain x = (x1, x2, x3) ∈ Ω are written with respect to the well known coordinate system R={O;e1, e2, e3}. Follow- ing [3], [5], Ω is assumed to be periodically domain. The microstructure is defined by periodic cell P (having the characteristic dimensionl) with a solid part PS and a empty part PV. For example, see the next figure:
• Nomenclature:
(a) the porous medium is enclosed in Ω = (0, L)3;
(b) the microstructure is defined by ωl= (0, l)3 with a solid partωSl (ωSl ⊂ ωl) and a empty part ωVl =ωl−ωsl;
(c) the cube Ω has N cells (ωl), i.e., Ω = SNj=1ωjl, where ωjl = xj +ωl, N = Ll33 ∈N,ωSl
j =xj +ωSl and ωVl
j =xj +ωVl ;
(d) denote = Ll = √31
N <<1, Ω =
N
[
j=1
ωSl
j, Γ=
N
[
j=1
∂ωSl
j; We have ∂Ω= ΓS∂Ω;
(e) The basic cell is Y = 1ωl = (0, L)3 with the solid part YS = 1ωSl and the empty part YV = 1ωlV;
(f) In dimensionless variables x0i = xLi, i= 1,2,3, we have:
Ω→Ω0 = L1Ω = (0,1)3, Ω →Ω0 = L1Ω, Γ →Γ0 = L1Γ, ωl →ω0 = L1ωl = (0, )3, ωSl →ω0S = L1ωSl, ωlV →ω0V = L1ωVl ,
Y →Y0 = L1Y = (0,1)3, YS →Y0S = L1YS, YV →Y0V = L1YV; (g) The following relations are valid:
Ω0 =SNj=1ω0j, where ω0j = L1xj+ω0 and
ω0Sj = L1xj +ω0S ω0Vj = L1xj+ω0V
,
Ω0 =SNj=1ω0Sj, Γ0=SNj=1∂ω0Sj, where
∂ω0Sj = L1xj +∂ω0S, ∂Ω0 = Γ0∪∂Ω0, ∂Y0S = 1∂ω0S; (h) The porosity is denoted by ΠY =|Y0V|.
The dimensionless microscopic problem is written:
1 L
∂
∂xj[1
Laijklekl(u0)] =−ρ0f0i(x0) in Ω0, i= 1,2,3. (1) Let us assume the asymptotic expansion:
u0=u00(x0, y0) +u01(x0, y0) +2u02(x0, y0) +..., (2) withy0 = x0 ∈YS0,x0 ∈Ω0, and the functions u00,u01, ... are periodic functions in y0.
The small deformations tensor can be written:
ekl(u0) = 12∂u∂x00k l + ∂u∂x00l
k
= 12∂u∂x000k l + ∂u∂x000l
k
+112∂u∂y000k l +∂u∂y000l
k
+ +12∂u∂x010k
l + ∂u∂x001l k
+12∂u∂y010k l +∂u∂y001l
k
+2 12∂u∂x020k l +∂u∂x020l
k
+12∂u∂y020k l +∂u∂y020l
k
+...=
= 112∂u∂y000k l +∂u∂y000l
k
+e0kl(x0, y0) +e1kl(x0, y0) +...
(3) where the folowing notation were introduced:
e0kl(x0, y0) =eklx0(u00) +ekly0(u01) e1kl(x0, y0) =eklx0(u01) +ekly0(u02) e2kl(x0, y0) =eklx0(u02) +ekly0(u03) ...
(4)
with
eklx0(v) = 12∂v∂xk0 l
+∂x∂v0l k
ekly0(v) = 12∂v∂yk0 l
+∂y∂vl0 k
, ∀ v =v(x0, y0). (5) With (4) the small deformation tensor (eij(u0))3i,j=1 has the form:
eij(u0) = 1
eijy0(u00) +e0ij(x0, y0) +e1ij(x0, y0) +..., i, j = 1,2,3, (6) leading to the stress tensor (σij)3i,j=1:
σij =aijklh1ekly0(u00) +e0ij(x0, y0) +e1ij(x0, y0) +2e2ij(x0, y0) +...i=
= 1aijklekly0(u00) +σij0(x0, y0) +σij1(x0, y0) +2σ2ij(x0, y0) +...
(7) where
σ0ij(x0, y0) =aijkle0kl(x0, y0) σ1ij(x0, y0) =aijkle1kl(x0, y0)
...
. (8)
Let us assume the operator dxd0 i
= ∂x∂0 i
+ 1∂y∂0 i
, i= 1,2,3.
Replacing (7) in (1) we get:
1 L2
∂
∂x0j +1∂y∂0 j
1
aijklekly0(u00) +σij0(x0, y0) +σij1(x0, y0)+
+2σij2(x0, y0) +...=−ρ0fi0, in Ω0, i= 1,2,3. (9) In (9) considering the coefficients of −i,i= 2,1,0, we have:
∂
∂y0j
aijklekly0(u00)= 0, ˆın YS0, i= 1,2,3, (10)
∂
∂x0j
aijklekly0(u00)+ ∂
∂yj0
σ0ij(x0, y0)= 0, ˆın YS0, i= 1,2,3, (11) 1
L2
∂
∂x0j
σij0(x0, y0)+ 1 L2
∂
∂y0j
σij1(x0, y0)=−ρ0fi0, ˆınYS0, i= 1,2,3. (12) Relation (10) shows the dependence ofu00 only on the macroscopic variable x0.
Equations (11), (12) can be rewritten:
∂
∂x0j
aijklekly0(u00)+∂y∂0 j
aijkleklx0(u00) +aijklekly0(u01)= 0, ˆın YS0, u01 Y0−periodic˘a
1
|Y0|
Z
YS0
u01 dy0 = 0
(i= 1,2,3) (13) and
1 L2
∂
∂x0j
aijkleklx0(u00)+aijklekly0(u01)+ 1 L2
∂
∂y0j
aijkleklx0(u01)+aijklekly0(u02)=−ρ0fi0, (14) in YS0, i= 1,2,3, respectively.
The homogenized elasticity tensor can be evaluated from the simplified form of (13):
∂
∂y0j
aijklekly0(u01)=−∂a
ijkl
∂yj0 eklx0(u00), in YS0 u01 Y0−periodic
1
|Y0|
Z
YS0
u01 dy0 = 0
(i= 1,2,3). (15)
Based on the liniarity of (15)1 we seak for a solution of the form :
u01 =wmhemhx0(u00), ˆınYS0, (16) with wmh Y0- periodic functions. Let use assume fixed m, h ∈ {1,2,3} such that emhx0(u00) = 1. We have (15) and the variational formulation:
Findwmh∈Hfper1 (Y0) such that
Z
YS0
aijklekly0(wmh)·eijy0(v) dy0 =
Z
YS0
∂aijmh
∂yj0 ·vi dy0, ∀v ∈Hfper1 (Y0) (i= 1,2,3).
(17) From relations (8)1, (4)1 and (16) we obtain:
σ0ij(x0, y0) =aijklhδkmδlh+ekly0(wmh)iemhx0(u00), in which mediation relative to the Y0 cell, we obtain the relation:
σeij0(x0) = 1
|Y0|
Z
YS0
σij0(x0, y0)dy0 =
= 1
|Y0|
Z
YS0
aijklhδkmδlh+ekly0(wmh)idy0emhx0(u00) =
=aHijmhemhx0(u00), x0 ∈Ω0, i, j = 1,2,3,
(18)
where
aHijmh = 1
|Y0|
Z
YS0
aijklhδkmδlh+ekly0(wmh)i dy0, (19) (i, j, m, h= 1,2,3) represent the homogenized elasticity tensor.
Integrating the equation (14) on YS0 and using the Y0−periodically condi- tion of the functionsu01 andu02, we obtain the homogenized equations (macro- scopic equations):
∂
∂xj
h(1−ΠY)aijmhemhx(u0) + 1
|Y0|
Z
YS0
aijklekly0(wmh) dy0emhx(u0)i=−fei, (20) ˆın Ω, i = 1,2,3, in the dimensional form, for that |Y|YS00|| = |Y0|Y|−|Y0|V0| = 1−
|YV0|
|Y0| = 1 −ΠY, where fei0 = 1
|Y0|
Z
YS0
ρ0fi0 dy0. These equations describes the macroscopic laws.
2. aHijkl, wmh, σHij - properties
Following the ideas from [4], we are interested in establishing some proper- ties of the homogenized tensor in order to characterize σijH. These properties are then used for the homogenized equations study.
Remark 1. The following properties are valid:
a)
aHijkl = (1−ΠY)aijkl, ∀i, j, k, l∈ {1,2,3}, with i6=j. (21) b)
Z
YS0
aiimhemhy0(wii)dy0 =
Z
YS0
akkmhemhy0(wkk)dy0, ∀i, k = 1,2,3, (22)
Z
YS0
aiimhemhy0(wii)dy0 =
Z
YS0
aiimhemhy0(wkk)dy0, ∀i, k= 1,2,3, (23)
Z
YS0
∇y0 ·w11dy0 =
Z
YS0
∇y0 ·w22dy0 =
Z
YS0
∇y0 ·w33dy0. (24) c)
σijH(x, t) = (1−ΠY)aijkleklx(u0) +bϕ(x, t)δij, (25) with b=λ+ 23µ(bulk modulus) and ϕ(x, t) = 1
|Y0|
Z
YS0
∇y0 ·wdy0(∇x·u0).
3. The microscopic problem solution’s convergence Some complementary results are presented in this section.
Remark 2. The characteristic function ϕ of YS0 belongs to L2(Y0) and its mean value has the form:
ϕe= 1
|Y0|
Z
Y0
ϕ dy0 = 1
|Y0|
Z
YS0
ϕ dy0 = |YS0|
|Y0| = 1−ΠY.
Similarly, ϕ(x0) = ϕ(y0), x0 =y0, the characteristic function of ω0 belongs to L2(ω0S). This function can be extended to ϕe ∈L2(Ω0) ofϕ weak convergent in L2(Ω0) toward ϕe= 1−ΠY (conform [2]).
Let us the Hilbert space V = (H1(Ω0))3 with the scalar product (u, v)V =
3
X
i=1
(∇ui,∇vi)L2(Ω0), u= (u1, u2, u3), v = (v1, v2, v3)∈ V (26)
and the norm
kvkV =
3
X
i=1
k∇vik2L2(Ω0)
12
, v = (v1, v2, v3)∈ V. (27) Let us assume that f0 ∈ V0.
The microscopic problem for homogeneous boundary conditions is defined:
1 L
∂
∂xj[L1aijklekl(u0)] = −ρ0f0i(x0) in Ω0 njL1aijklekl(u0)= 0 on∂Ω0\Γ0
(i= 1,2,3). (28)
The variational formulation on V, is corresponding by:
Findu0 ∈ V such that 1
L2
Z
Ω0
aijklekl(u0)eij(v) dx0 =
Z
Ω0
ρ0f0ivi dx0 ∀v ∈ V , (29) which is equivalent to
( Find u0 ∈ V such that
a(u0, v) =< F, v >, ∀v ∈ V, (30) where
a(u, v) = 1 L2
Z
Ω0
aijklekl(u)eij(v) dx0, ∀u, v ∈ V, (31) is a bilinear elliptic form on V × V [3], and
< F, v >=
Z
Ω0
ρ0f0ivi dx0, (32) a linear continuous functional on V with
kFkV0 ≤ kρ0f0kV0 ≤ kρ0f0k(L2(Ω0))3. (33) Theorem 1. Letf0 ∈ V0. The problem (29) has a unique solutionu0 ∈ V.
Moreover, there exist a positive constant C1 such that
ku0kV ≤C1kρ0f0k(L2(Ω0))3. (34) Proof. The hypothesis of the Lax-Milgram theorem are satisfied, i.e. the functional F is continuous on V, the bilinear form a = a(u, v) is continuous
and V - eliptic, so the problem (29) has a unique solution u0 ∈ V. The Korn’s type inequality is valid:
ku0k(H1(Ω0))3 ≤CKhku0k(L2(Ω0))3 +
Z
Ω0
|eij(u0)|2 dx01/2i, (35) where CK = cK(Ω0). The elasticity tensor (aijkl) is positive definite one, so there exist a constant C >0 such that
aijklekl(u0)eij(u0)≥Ceij(u0)eij(u0).
We obtain:
a(u0, u0)= 1 L2
Z
Ω0
aijklekl(u0)eij(u0)dx0≥C 1 L2
Z
Ω0
eij(u0)eij(u0) dx0≥
≥ L2C·CKku0k2V,
(36) and
|< F, u0 >| ≤ kFkV0 · ku0kV ≤ kρ0f0k(L2(Ω0))3 · ku0kV, (37) respectively. The inequalities (36), (37) yield
ku0k2V ≤ L2CK
C ·a(u0, u0)= L2·CK
C ·|< F, u0 >| ≤ L2·CK
C ·kρ0f0k(L2(Ω0))3·ku0kV, so
ku0kV ≤C1· kρ0f0k(L2(Ω0))3, C1 = L2CK C . The proof is complete.
For l, m∈ {1,2,3}the functions Plm(y0) = Pklm(y0)3
k=1 are defined Pklm(y0) = ym0 δkl, k= 1,2,3. (38) The solution χlm(y0) =χlmk (y0)3
k=1 of the system
∂
∂y0j
aijkl∂(χlmk −Pklm)
∂yh0
= 0, inYS0, i= 1,2,3
χlmk Y0−periodics
1
|Y0|
Z
YS0
χlmk dy0 = 0,
(39)
and wkh (x0),
wkh (x0) = Pkh(x0)−χkh(x0
), x0 ∈Ω0, k, h = 1,2,3, (40) are considered as in Cior˘anescu D. [3], also
(ηkl(x0))ij =aijmh(x0)emhy0(wkl) [?]. (41) Theorem 2. Let f0 ∈ V0. Then, there exists an extension u0p of the solu- tion u0 of the problem (29), from Ω0 to Ω0 and a sequence n →0 such that:
i.u0pn * u00 in (H1(Ω0))3, ii.u0pn →u00 in (L2(Ω0))3,
iii.aijkln ekl(u0pn)* aHijkleklx0(u00) in (L2(Ω0))3×3,
where u00 = (u001, u002, u003) is the unique solution in (H1(Ω0))3 of the homoge- nized problem
1 L2
∂
∂x0j
aHijkleklx0(u00)=−fei0, in Ω0
1
LaHijkleklx0(u00) = 0, on ∂Ω0
, i= 1,2,3, (42)
and (aHijkl) defined by (19).
Proof. From Theorem 1, we have
ku0k(H1(Ω0))3 ≤C1kρ0f0k(L2(Ω0))3.
Forσ = (σij)3i,j=1 defined by σij =aijklekl(u0) the following inequality stands [3]:
kσk(L2(Ω0))3 ≤C.
All these estimations [2], Lemma 3, lead the conclusion: there exists an extension u0p of u0 of (29), from Ω0 to Ω0 and a sequence n→0 such that:
• ku0pk2H1(Ω0) ≤c2ku0pk2H1(Ω0)≤c2C12kρ0f0k2(L2(Ω0))3,
i. u0pn * u00, in (H1(Ω0))3, ii. u0pn →u00, in (L2(Ω0))3, iii. σpn *σe0, in (L2(Ω0))3×3,
(43)
respectively, where σpn = (σij pn ), withσij pn =aijkln ekl(u0pn).
The variational formulation of (28), point out that the tensor (σijn) verifies the equation:
1 L
Z
Ω0n
σijneijx0(v) dx0 =< ρ0f0, v >V0,V, ∀v ∈ V. (44) Let us evaluate all the terms in (44):
1 L
Z
Ω0n
σijneijx0(v)dx0 = 1 L
Z
Ω0
ϕenσijneijx0(v) dx0 → 1 L
Z
Ω0σeij0eijx0(v) dx0 (45) for n→0. The right side of the equation (44) can be written for n→0:
< ρ0f0, v >V0,V=
Z
Ω0n
ρ0fi0vi dx0 =
Z
Ω0
ϕenρ0fi0vi dx0 →<fe0, v >, (46) Taking into account (43)3, we get:
1 L
Z
Ω0σe0ijeijx0(v) dx0 =<fe0, v >, ∀v ∈(C0∞(Ω0))3. (47) The proof is completed if:
σe0ij =aHijkleklx0(u00). (48) Let φ∈ C0∞(Ω0),φwkln test function in (44) and v =φu0n in relation
Z
Ω0
ηkleklx0(v) dx0 = 0, ∀v ∈(H01(Ω0))3. We have:
1 L
Z
Ω0n
aijmhn emh(u0n)eij(wkln)φ dx0+ 1 2L
Z
Ω0n
σijnh(wkln)i ∂φ
∂x0j + (wkln)j ∂φ
∂x0i
i dx0
=< ρ0f0, φwkln >
Z
Ω0n
(ηkln)ijeij(u0n)φ dx0+ 1 2
Z
Ω0n
(ηkln)ij
hu0in ∂φ
∂x0j +u0jn ∂φ
∂x0i
i dx0 = 0.
(49) The symmetry of (aijkln ), that assumes
σijneij(wkln) = (ηkln)ijeij(u0n).
The equation (49) are equivalent to
1 2L
Z
Ω0n
σijnh(wkln)i ∂φ
∂x0j + (wkln)j∂φ
∂x0i
i dx0−
− 1 2L
Z
Ω0n
(ηkln)ijhu0in ∂φ
∂x0j +u0jn ∂φ
∂x0i
i dx0 =< ρ0f0, φwkln >,
(50)
i.e.,
1 2L
Z
Ω0
ϕenσijnh(wkl
n)i ∂φ
∂x0j + (wkl
n)j∂φ
∂x0i
i dx0−
− 1 2L
Z
Ω0
ϕen(ηkl
n)ijhu0in ∂φ
∂x0j +u0jn ∂φ
∂x0i
i dx0 =
Z
Ω0
ϕenρ0f0iφ(wkl
n)i dx0. (51) For n→0, in (51), becomes:
1 2L
Z
Ω0σeij0hy0lδki ∂φ
∂x0j +yl0δkj∂φ
∂x0i
i dx0−
− 1 2L
Z
Ω0
aHijklhu00i ∂φ
∂x0j +u00j ∂φ
∂x0i
i dx0 =Dfe0, φPklE.
(52)
which can be rewritten as:
1 L
Z
Ω0σeij0eij(φPkl)dx0− 1 L
Z
Ω0σekl0φ dx0 + 1 2L
Z
Ω0
aHklijeij(u00)φ dx0 =Dfe0, φPklE. (53) If we write (47), for the test function φPkl, the main result of the theorem is
obtained Z
Ω0σekl0φ dx0 =
Z
Ω0
aHklijeij(u00)φ dx0 φ∈ C0∞(Ω0), (54) and
σekl0 =aHklijeij(u00).
The proof is complete.
Acknowledgement. This work was supported by CNCSIS-UEFISCDI, project number PNII - IDEI code 1081/2008 No. 550/2009.
References
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[3.] Doina Cior˘anescu, Patrizia Donato, An introduction to homogeniza- tion, Oxford Lecture Series in Mathematics and its Application, 17, 2000.
[4.] Remus Ene, Adara Blaga, Applications of the biharmonic equations to the study of porous plates, Annals of the West University of Timisoara, Ser.
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Ene Remus-Daniel and Bˆınzar Tudor Department of Mathematics
University of Politehnica
Address: Romania, 300006 Timisoara, Pta Victoriei no. 2 email:[email protected]
