# Weak Solutions in Elasticity of Dipolar Porous Materials

Authors: Marin Marin
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

The main aim of our study is to use some general results from the general theory
of elliptic equations in order to obtain some qualitative results in a concrete
and very applicative situation. In fact, we will prove the existence and uniqueness
of the generalized solutions for the boundary value problems in elasticity of initially
stressed bodies with voids (porous materials).


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## Body

## 1. Introduction

The theories of porous materials represent a material
length scale and are quite sufficient for a large number of the solid mechanics
applications.In the following, we restrict our attention to the
behavior of the porous solids in which the matrix material is elastic and the
interstices are voids of material. The intended applications of this theory are
to the geological materials, like rocks and soils and to the manufactured
porous materials.The plane of the paper is the following one. In the
beginning, we write down the basic equations and conditions of the mixed
boundary value problem within context of linear theory of initially stressed
bodies with voids, as in the papers of [1, 2]. Then, we
accommodate some general results from the paper [3], and
the book [4], in order to obtain the existence and uniqueness of a weak
solution of the formulated problem. For convenience, the notations chosen are
almost identical to those of [2, 5].


## 2. Basic Equations

LetB be an open region of three-dimensional
Euclidean space R3 occupied by our porous material at time t=0. We assume that the boundary of the domain B,
denoted by ∂B,
is a closed, bounded and pice-wise smooth surface
which allows us the application of the divergence theorem. A fixed system of
rectangular Cartesian axes is used and we adopt the Cartesian tensor notations.
The points in B are denoted by (xi) or (x).
The variable t is the time and t∈[0,t0).
We will employ the usual summation over repeated subscripts while subscripts
preceded by a comma denote the partial differentiation with respect to the
spatial argument. We also use a superposed dot to denote the partial
differentiation with respect to t.
The Latin indices are understood to range over the integers (1,2,3).In the following, we designate byni the components of the outward unit normal to
the surface ∂B.
The closure of domain B, denoted
by B¯,
means B¯=B∪∂B.Also, the spatial argument and the time argument of a
function will be omitted when there is no likelihood of confusion.The behavior of initially stressed bodies with voids
is characterized by the following kinematic variables:(2.1)ui=ui(x,t),φjk=φjk(x,t),σ=σ(x,t),(x,t)∈B×[0,t0).In our study, we analyze an anisotropic and
homogeneous initially stressed elastic solid with voids. We restrict our
considerations to the Elastostatics, so that the basic equations become as
follows.(i)
The equations of equilibrium
is as follows:(2.2)(τij+ηij),j+ρFi=0,μijk,i+ηjk+uj,iMik+φkiMji−φkr,iNijr+ρGjk=0;(ii)
the balance of the equilibrated forces
is as follows:(2.3)hi,i+g+ρL=0;(iii)
the constitutive  equations
are as follows:(2.4)τij=uj,kPki+Cijmnεmn+Gmnijκmn+Fmnrijχmnr+aijσ+dijkσ,k,ηij=−φjkMik+φjk,rNrik+Gijmnεmn+Bijmnκmn+Dijmnrχmnr+bijσ+eijkσ,k,μijk=uj,rNirk+Fijkmnεmn+Dmnijkκmn+Aijkmnrχmnr+cijkσ+fijkmσ,m,hi=dmniεmn+emniκmn+fmnriχmnr+diσ+gijσ,j,g=−amnεmn−bmnκmn−cmnrχmnr−ξσ+diσ,i;(iv)
the geometric equations are(2.5)εij=12(uj,i+ui,j),κij=uj,i−φij,χijk=φjk,i,σ=ν−ν0.In the above equations we
have used the following notations:(i)
ρ—the constant mass density;(ii)
ui—the components of the displacement field;(iii)
φjk—the components of the dipolar displacement
field;(iv)
ν—the volume distribution function which in
the reference state is ν0;(v)
σ—a measure of volume change of the bulk
material resulting from void compaction or distension;(vi)
τij, ηij, μij—the components of the stress tensors;(vii)
hi—the components of the equilibrated stress;(viii)
Fi—the components of body force per unit mass;(ix)
Gjk—the components of dipolar body force per
unit mass;(x)
L—the extrinsic equilibrated body force;(xi)
g—the intrinsic equilibrated body force;(xii)
εij, κij, χijk—the kinematic characteristics of the strain
tensors;(xiii)
Cijmn, Bijmn,…,Dijm, Eijm,…,aij, bij, cijk, di, ξ represent the characteristic functions of the
material (the constitutive coefficients) and they obey to the following
symmetry relations(2.6)Cijmn=Cmnij=Cijnm,Bijmn=Bmnij,Gijmn=Gijnm,Fijkmn=Fijknm,Aijkmnr=Amnrijk,aij=aji,Pij=Pji,gij=gji.The physical significances of the functionsL and hi are presented in the works [6, 7].The prescribed functionsPij, Mij and Nijk from (2.2) and (2.3) satisfy the following
equations:(2.7)(Pij+Mij),j=0,Nijk,i+Pjk=0.


## 3. Existence and Uniqueness Theorems

In the main section of our paper, we will accommodate
some theoretical results from the theory of elliptic equations in order to
derive the existence and the uniqueness of a generalized solution of the mixed boundary-value
problem in the context of initially stressed bodies with voids.Throughout this section, we assume thatB is a Lipschitz region of the Euclidian
three-dimensional space R3.
We use the following notations:(3.1)W=[W1,2(B)]13,W0=[W01,2(B)]13,with the convention that A13=A×A×⋯×A,
the Cartesian product
is considered to be of 13-times. Also, Wk,m is the familiar Sobolev space. With other
words, W is defined as the space of all u=(ui,φij,σ),
where ui, φij, σ∈W1,2(B) with the norm(3.2)|u|W2=|σ|W1,2(B)2+∑i=13|ui|W1,2(B)2+∑j=13(∑i=13|φij|W1,2(B)2).For clarity and simplification in presentation, we
consider the following regularity hypotheses on the considered functions:(i)
all the constitutive coefficients are functions of
classC2 on B;(ii)
the body loadsFi,Gjk,
and H are continuous functions on B.The ordered array(ui,φjk,σ) is an admissible process on B¯=B∪∂B provided ui, φjk, σ∈C1(B¯)∩C2(B).
Also, the ordered array of functions (τij,ηij,μijk,hi) is an admissible system of stress on B¯ if τij, ηij, μijk, hi∈C1(B)∩C0(B¯) and τij,i, ηij,i, μijk,k, hk,k, h∈C0(B¯).Let∂B=Su∪St∪C be a disjunct decomposition of ∂B,
where C is a set of surface measure and Su and St are either empty or open in ∂B.
Assume the following boundary conditions:(3.3)ui=u˜i,φjk=φ˜jkσ=σ˜onSu,ti≡(τij+ηij)nj=t˜i,μjk≡μijkni=μ˜jk,h≡hini=h˜onSt,where the functions u˜i, φ˜jk, σ˜, t˜i, μ˜jk,
and h˜ are prescribed, u˜i, φ˜jk, σ˜∈W1,2(Su),
and t˜i, μ˜jk, h˜∈L2(St).
Also, we define V as a subspace of the space W of all functions u=(ui,φij,σ) which satisfy the boundary
conditions:(3.4)ui=0,φij=0,σ=0onSu.On the product space W×W,
we consider a bilinear form A(v,u),
defined by(3.5)A(v,u)=∫B{Cijmnεmn(u)εij(v)+Gmnij[εij(v)κmn(u)+εij(u)κmn(v)]+Fmnrij[εij(v)χmnr(u)+εij(u)χmn(v)]+Bijmnκij(v)κmn(u)+Dijmnr[κij(v)χmnr(u)+κij(u)χmn(v)]+Aijkmnrχijk(v)χmnr(u)+Pkiuj,kvj,i−Mik(uj,iψjk+vj,iφjk)+Nrik(uj,kψjk,r+vj,kφjk,r)+aij[εij(v)σ+εij(u)γ)]+bij[κij(v)σ+κij(u)γ)]+cijk[χijk(v)σ+χij(u)γ)]+dijk[εij(v)σ,k+εij(u)γ,k)]+eijk[κij(v)σ,k+κij(u)γ,k)]+fijkm[χijk(v)σ,m+χijk(u)γ,m)]+di[σγ,i+γσ,i]+gijσ,iγ,j+ξσγ}dV,where(3.6)u=(ui,φij,σ),v=(vi,ψij,γ),εij(u)=12(uj,i+ui,j),εij(v)=12(vj,i+vi,j),κij(u)=uj,i−φij,κij(v)=vj,i−ψij,χijk(u)=φjk,i,χijk(v)=ψjk,i.  We assume that the constitutive
coefficients are bounded measurable functions in B which satisfy the symmetries (2.6). Then, by
using relations (3.5) and (2.6) it is easy to deduce that(3.7)A(v,u)=A(u,v).Also, by using symmetries (2.6)
into (3.5), it results in(3.8)A(u,u)=∫B[Cijmnεmn(u)εij(v)+2Gmnijεij(u)κmn(u)+Bijmnκij(u)κmn(u)+2Fmnrijεij(u)χmnr(u)+2Dijmnrκij(u)χmnr(u)+Aijkmnrχijk(u)χmnr(u)+Pkiuj,kuj,i−2Mikuj,iφjk+2Nrikuj,iφjk,r+2aijεij(u)σ+2bijκij(u)σ+2cijkχijk(u)σ+2dijkεij(u)σ,k+2eijkκij(u)σ,k+2fijkmχijk(u)σ,m+2diσγ,i+gijσ,iσ,j+ξσ2]dV,and thus(3.9)A(u,u)=2∫BUdV,where U=ρe is the internal energy density associated to u.We suppose thatU is a positive definite quadratic form, that
is, there exists a positive constant c such that(3.10)Cijmnxijxmn+2Gijmnxijymn+2Fijmnrxijzmnr+Bijmnyijymn+2Dijmnryijzmnr+Aijkmnrzijkzmn+Pkixjixjk−2Mikxjiyjk+2Nrikxjizjkr+2aijxijw+2bijyijw+2cijkzijw+2dijkxijωk+2eijkyijωk+2fijkmzijkωm+2diwωi+gijkωiωj+ξw2≥c(xijxij+yijxij+zijkzijk+ωiωi+w2),for all xij, yij, zijk, ωi,
and w.Now, we introduce the functionalsf(v) and g(v) by(3.11)f(v)=∫Bρ(Fivi+Gjkψjk+Lγ)dV,v∈W,g(v)=∫St(t˜ivi+μ˜jkψjk+h˜γ)dA,v∈W,where v=(vi,ψjk,γ)∈W and ρ, Fi, Gjk, L∈L2(B).Letv=(u˜i,φ˜jk,γ˜)∈W be such that u˜i, φ˜jk, γ˜ on Su may by obtained by means of embedding the space W1,2 into the space L2(Su).The elementv=(ui,φjk,σ)∈W is called weak (or generalized) solution of the
boundary value problem, if(3.12)u−u˜∈V,A(u,u)=f(u)+g(v)hold for each v∈V.
In the above relations, we used the spaces L2(B) and L2(Su) which represent, as it is well known, the
space of real functions which are square-integrable on B,
respectively, on Su⊂∂B.It follows from (3.10) and (3.8) that(3.13)A(v,v)≥2c∫B[εij(v)εij(v)+κij(v)κij(v)+χijk(v)χijk(v)+γiγi+γ2]dV,for any v=(vi,ψjk,γ)∈W.Let us consider the operatorsNkv, k=1,2,…,49,
mapping the space W into the space L2(B),
defined by(3.14)Niv=ε1i(v),N3+iv=ε2i(v),N6+iv=ε2i(v),N9+iv=κ1i(v),N12+iv=κ1i(v),N15+iv=κ1i(v),N18+iv=χ11i(v),N21+iv=χ12i(v),N24+iv=χ13i(v),N27+iv=χ21i(v),N30+iv=χ22i(v),N33+iv=χ23i(v),N36+iv=χ31i(v),N39+iv=χ32i(v),N42+iv=χ33i(v),N45+iv=σ,i(v),N49v=σ(v)(i=1,2,3).It is easy to see that, in fact, the operatorsNkv, k=1,2,…,49,
defined above, have the following general
form:(3.15)Nkv=∑r=1m∑|α|≤krnkrαDαvr,p=|α|,where nkrα are bounded and measurable functions on B.
Also, we have used the notation Dα for the multi-indices derivative, that is,(3.16)Dα=∂|α|∂x1α1∂x2α2∂x3α3.By definition, the operatorsNkv, (k=1,2,…,49) form a coercive system of operators on W if for each v∈W the following inequality takes place:(3.17)∑k=149|Nkv|L2(B)2+∑r=113|vr|L2(B)2≥c1|v|W2,c1>0.In this inequality, the constantc1 does not depend on v and the norms |⋅|L2 and |⋅|W represent the usual norms in the spaces L2(B) and W,
respectively.In the following theorem, we indicate a necessary and sufficient condition for a
system of operators to be a coercive system.Theoem 3.1.
Letnpsα be constant for |α|=ks.
Then the system of operators Npv is coercive on W if and only if the rank of the
matrix (3.18)(Npsξ)=(∑|α|=ksnpsαξα)is equal to m for each ξ∈C3, ξ⁢≠0,
where C3 is the notation for the complex
three-dimensional space, and(3.19)ξα=ξ1α1ξ2α2ξ3α3.The demonstration of this result
can be find in [4].In the following, we assume that for eachv∈W,
we have(3.20)A(v,v)≥c2∑k=149|Nkv|L22,c2>0,where the constant c2 does not depend on v.We denote by𝒫 the following set:(3.21)𝒫={v∈V:∑k=149|Nkv|L22=0},and by V/𝒫 the factor-space of classes v˜,
where(3.22)v˜={v+p,v∈V,p∈𝒫},having the norm(3.23)|v˜|V/𝒫=inf⁡p∈𝒫|v+p|W.In the following theorem, it is indicated a necessary
and sufficient condition for the existence of a weak solution of the
boundary-value problem.Theorem 3.2.
LetA(v,u)=[v˜,u˜] define a bilinear form for each v˜,u˜∈W/𝒫,
where u∈u˜ and v∈v˜.
If it is supposed that the inequalities (3.17) and (3.20) hold, then a
necessary and sufficient condition for the existence of a weak solution of the
boundary value problem is (3.24)p∈𝒫→f(p)+g(p)=0.Moreover, the weak solution, u∈W,
satisfies the following inequality:(3.25)|u|W/𝒫≤c3[|u˜|W+(∑i=1m|fi|L2(B))1/2+(∑i=1m|gi|L2(S))1/2],where c3 is a real positive constant.
Further, one
has(3.26)A(v˜,v˜)≥c4|v˜|W/𝒫,c4>0,for each v˜∈W/𝒫.For the prove of this result, see [3].In the following, we intend to apply the above two
results in order to obtain the existence of a weak solution for the boundary
value problem formulated in the context of theory of initially stressed elastic
solids with voids.Theorem 3.3.
Let𝒫={0}.
Then there exists one and only one weak solution u∈W of our boundary-value  problem.Proof.
Clearly, from (3.13) and (3.14) we immediately obtain
(3.20). The matrix (3.20) has the rank 13 for each ξ∈C3, ξ⁢≠0.
Thus by Theorem 3.1 we conclude that the system of Nk operators, defined in (3.14), is coercive on the
space W.
According to definition (3.21) of 𝒫,
we have that εij(v)=0,κij(v)=0, χijk(v)=0, γi(v)=0, ψ=0 for each v∈𝒫, v=(vi,ψjk,ψ).
So, we deduce that𝒫 reduces to(3.27)𝒫={v=(vi,ψjk,γ)∈V:vi=ai+εijkbjxk,ψjk=εjksbs,γ=c},where ai and bi and c are arbitrary constants and εijk is the alternating symbol.
We will consider two distinct cases. First, we
suppose that the setSu is nonempty. Then the set 𝒫 reduces to 𝒫={0}, and therefore, condition (3.24) is satisfied. By
using Theorem 3.2, we immediately obtain the desired result.In the second case, we assume thatSu is an empty set. Then we have the following
result.Theorem 3.4.
The
necessary and sufficient conditions for the existence of a weak solutionu∈W of the boundary-value problem for elastic
dipolar bodies with stretch, are given by (3.28)∫BρFidV+∫∂Bt˜idA=0,∫Bρεijk(xjFk+Gjk)dV+∫∂Bεijk(xjt˜k+μ˜jk)dA=0,where εijk is the alternating symbol.Proof.
In this case, the boundary value problem𝒫 is given by (3.27), where ai, bi,
and c are arbitrary constants such that we can
apply, once again, Theorem 3.2 to obtain the above result.


## 4. Conclusion

For the considered initial-boundary value problem the
basic results still valid. Now, for different particular cases, the solution
can be found because it exists and is unique.



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