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Semi-inverse solution for Saint-Venant’s problem in the theory of porous elastic materials

- Extended form of the paper -

Ionel-Dumitrel Ghiba

Department of Mathematics, University “A.I. Cuza” of Ia¸si, 700506 Ia¸si, Romania, Blvd. Carol I, no. 11, 700506 Ia¸si, Romania &

Octav Mayer Institute of Mathematics, Romanian Academy, 700505 - Ia¸si, email: [email protected]

Abstract

The purpose of this research is to study the Saint-Venant’s problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids.

We reformulate the equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant’s problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant’s problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.

Keywords: Saint-Venant’s problem; Semi-inverse solution; Anisotropic elastic materials with voids

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Contents

1 Introduction 2

2 Materials with voids 3

2.1 Equations of the nonlinear theory . . . 3 2.2 The linear theory of materials with voids . . . 7 3 Some mathematical results in the equilibrium theory 11 3.1 Reciprocity and uniqueness theorems . . . 11 3.2 Existence and uniqueness theorems . . . 13 4 The generalized plane strain state 15

5 Saint-Venant’s problem 16

5.1 Analysis of Saint-Venant’s problem by plane section solutions . . . 18 5.2 The relaxed Saint-Venant’s problem . . . 24 6 Extension, bending and torsion of isotropic cylinder 28 7 Flexure of isotropic porous elastic cylinder 33

8 Saint-Venant’s principle 35

1 Introduction

The theory of elastic materials with voids has been established by Nunziato and Cowin (1979).

In this theory, the bulk density is written as the product of two fields, the matrix material density field and the volume fraction field. The first investigations in the theory of thermoelastic materials with voids are due to Nunziato and Cowin (1979) and Ie¸san (1986). The intended applications of the theory are to geological materials and to manufactured porous materials. A presentation of this theory can be found in (Ie¸san and Ciarletta, 1993) and in (Ie¸san, 2004).

Cowin and Nunziato (1983) solved the problem of pure bending of an isotropic and homoge- neous porous beam of rectangular cross section. In (Ie¸san and Ciarletta, 1993) is presented the problem of extension and bending of right cylinders made of an isotropic elastic material with voids and in a recent paper Ie¸san and Scalia (2007) investigates the problem of extension and bending of right cylinder made of inhomogeneous and orthotropic elastic materials with voids.

A study of Saint-Venant’s problem for homogeneous and isotropic porous elastic cylinders has been presented by Dell’Isola and Batra (1997). The Saint-Venant’s principle in the theory of elastic materials with voids has been studied by Batra and Yang (1995).

In this paper we consider the Saint-Venant’s problem for right cylinders made of an inhomo- geneous anisotropic elastic materials with voids. For the treatment of this problem, we resort to the method used by Chirit¸˘a (1992), (see also (Ie¸san, 1987)), for the study of Saint-Venant’s problem in linear viscoelasticity. In the first part of the paper we formulate the Saint-Venant’s problem for an anisotropic elastic cylinder with voids and we define the generalized plane strain for the cross section of the considered cylinder. Then, we study the possibility to reduce the

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problem to some generalized plane strain problems, which are more tractable. For this, we treat the Saint-Venant’s problem by reformulating the equilibrium equations with the axial variable playing the role of a parameter. Further, we point out two classes of semi-inverse solutions in the set of solutions of Saint-Venant’s problem that may be expressed in terms of a state of generalized plane strain. These classes are relevant in obtaining a semi-inverse solution to the relaxed Saint-Venant’s problem. We use the results obtained in the anisotropic case to solve the extension, bending, torsion and flexure problem of isotropic porous elastic circular cylinders for the relaxed Saint-Venant’s problem.

An analysis and important proprieties of the solutions of the relaxed Saint-Venant’s problem in the classical elasticity have been presented in (Ie¸san, 1987). Ie¸san (2007) study the extension of an isotropic circular cylinder with a special kind of inhomogeneity. Follow the method used in (Chirit¸˘a, 1992), Gale¸s (2000) studied the Saint-Venant’s problem in micropolar viscoelaticity.

The deformation of isotropic microstretch elastic cylinders has been studied by Ie¸san and Nappa (1995) and by De Cicco and Nappa (1995) and the extension, bending and torsion of anisotropic microstretch elastic cylinders have been studied by Scalia (2000).

The present paper can be useful for the study of orthotropic porous material, for porous material with different types of inhomogeneous and for functionally graded porous materials.

2 Materials with voids

2.1 Equations of the nonlinear theory

We consider a body made by porous thermoelastic materials, which at the timet0 occupies the region B of the three-dimensional Euclidian space E3. In the following, the configuration of the body at the initial time t0 is considered as the reference configuration.

We refer the motion of the body to a fixed system of rectangular axes OXK (K = 1,2,3).

We denote byXK, (K = 1,2,3) the coordinates of a generic material pointM0, of the domain B. We suppose that after the deformation process, at the time t, the body occupies a new domain B which has the boundary ∂B, the material point M0 arriving in the position M. We will refer the configuration B of the body at the time t to a new fixed system of rectangular axes oxi (i= 1,2,3). Thus, the coordinates of the position M are denoted xi, (i= 1,2,3). In the rest of this book, X denotes the vector with the components (X1, X2, X3), and x denotes the vector (x1, x2, x3)

We assume that the body should not penetrate itself and hence there is a one–to–one applications between B and B. Let us consider a fixed time interval I = [t0, t1), wheret0 ≥0, while t1 >0 can be infinite.

The deformation of the body will be described by the relation

x=x(X, t), (X, t)∈B¯×I. (2.1) These applications are of class C2 on ¯B ×I and we have

det ∂xi

∂XK

>0. (2.2)

The coordinates XK will be called material coordonates, while the coordonates xi will be called spatial coordinates.

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Throughout this book (when we do not specify else) Latin subscripts take the values 1,2,3 whereas the Greek indices are understood to range over the integers (1,2), summation is carried out over repeated indices. Typical conventions for differential operations are implied such as a superposed dot or comma followed by a subscript to denote the partial derivative with respect to the time or to the corresponding cartesian coordinate.

We say that a function f defined onB ×(t0, t1) is of classCM,N, wherem and n are given positive number, if the functions

mf(n) ≡ ∂m

∂XP∂XQ· · ·∂XKnf

∂tn

, m∈ {0,1, ...M}, n ∈ {0,1, ..., N}, m+n ≤max{M, N},

exist and are continuous on B×(t0, t1).

In order to introduce the theory of elastic materials with voids, Nunziato and Cowin [216]

used the concept of a distributed body previously introduced by Goodman and Cowin [140] in the formulations of a theory of granular materials.

According with the paper of Goodman and Cowin [140], adistributed bodyis a one-parameter family {Bt}, 0< t <∞, of regions of Euclidean three space such that

(a) for any times t and t0, the regionBt is homeomorphic to the region Bt0, and

(b) for each t, the region Bt is endowed with a structure given by two real valued set of applications mt and vt subjected to the following axioms:

(b1) mt and vt are non-negative measures defined for all Borel subsets Pt ⊂Bt, (b2) vt(Pt)≤v(Pt), for all Pt⊂Bt (v is the Lebesque volume measure),

(b3) mt is absolutely continuous with respect to vt.

It is easy to see that, Bt represents the configuration of the distributed body at the time t, Pt is subset ofBt,mt and vt are thedistributed volume and distributed mass, respectively.

For the case of porous bodies, the volume is always less than or equal to the total volume of any part of the body. The axiom (b2) takes into account this fact. Axiom (b3) implies that, in the theory of Goodman and Cowin [140] the void mass of a porous or granular material are neglected.

Because, from axiom (b2) it follows that the distributed volume measure vt is absolutely continuous with respect to the Lebesgue volume measure, by the Radon-Nikodym theorem, there exists a real valued Lebesgue integrable function ν(x, t) defined on Bt such that for any part Pt ⊂Bt

vt(Pt) = Z

Pt

νdv. (2.3)

The function ν will be called the volume distribution function (0 < ν ≤ 1) and represent a measure of the change of the bulk materials which results from the presence of voids.

Similarly, from axiom (b3), it follows that there is an essentially bounded vt–integrable function, defined on Bt such that

mt(Pt) = Z

Pt

ˆ

ρdvt for almost all Pt⊂Bt. (2.4)

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We call the function ˆρ distributed mass density or simply the distributed density. From the absolute continuity of distributed volume with respect to Lebesgue volume, we have

mt(Pt) = Z

Pt

ˆ

ρνdv. (2.5)

This relation suggest the introduction of the function

ρ=νρ,ˆ (2.6)

which is called is called the bulk density of the distributed body.

Clearly, we have this representation in the reference configuration B, that is

ρ00ρˆ0, (2.7)

where ρ0, ˆρ0 and ν0 are the bulk density, the distributed density and the volume distribution function in the reference configuration.

The theory of Nunziato and Cowin [216] is based on this decomposition which introduced a new kinematic variable ν. In the rest of this paper, we will suppose that ν is of class C2 on B×I and ρ0 is a strictly positive continuous function on B×I.

In what follows, we present a short path in the deduction of the nonlinear equations which describe the thermal bahavior of the materials with voids. For this we use the way described by Ie¸san in [161, 173], using the procedure of Green and Rivlin [143].

Let us consider an arbitrary regular subset P of the body which after the deformation process becomes P.

We have the following relation which postulate the conservation of energy for the regular subset P of B and for every time, in the form [173]

d dt

Z

P

ρ0

e+ 1

2x˙2+ 1 2κν˙2

dV =

Z

P

ρ0(f ·x˙ +`ν˙ +S)dV +

Z

∂P

(T·x˙ +Hν˙ +Q)dA,

(2.8)

wheredV anddAare element of volume and area in the reference configuration,eis theinternal energy per unit mass,f is thebody force per unit mass,` is theextrinsic equilibrated body force per unit mass, S is the heat supply per unit mass, T is the stress vector associated with the surface ∂P, but measured per unit area of the surface ∂P,H is theequilibrated stress with the surface ∂P, but measured per unit area of the surface∂P,Q is theheat flux across the surface

∂P, but measured per unit area of the surface ∂P, κ is the equilibrated inertia. We suppose that f,` andS are continuous functionsB×(t0, t1),e is of classC0,1 onB×(t0, t1),T,H and Q are of class C1,0 on B ×(t0, t1) and continuous on ¯B ×[t0, t1), and κ is a strictly positive known function on B.

We assume that ρ0, ˙e, f, `, T, H and ˙ν are not affected when we superposed a given rigid motion with constant translation velocity and with constant angular velocity. Following the procedure of Green and Rivlin [143], we have that there is the tensorTAi calledPiola-Kirchhoff stress tensor, the vector QK calledheat flux vector and the vector HK calledequilibrated stress vector such that

Ti =TKiNK,(H−HKNK) ˙ν+Q−QKNK = 0, (2.9)

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where NK is the unit outward normal vector to the surface ∂P.

Also, in our hypothesis, from the relation (2.8) we have the equations of motion

TAi,A0fi0i, (2.10) the balance of equilibrated forces

HK,K +g+ρ0` =ρ0κν¨ (2.11)

and theenergy equation

ρ0e˙=TKivi,K+HKν˙,K−gν˙ +QK,K0S, (2.12) onB×(t0, t1), whereg is a dependent constitutive variable which is called intrinsic equilibrated body force.

As in classical theory, we consider the Piola-Kirchhoff stress tensor TAB given by

TAi =xi,BTAB. (2.13)

and theLagrangian strain tensor

EKL = 1

2(xi,Kxi,L−δKL), (2.14)

where δKL is the Kronecker’s delta.

Assuming the invariance of the quantities ρ0, e, T˙ KL, HK, QK, S, g and ˙ν to a superposed uniform rigid body motion with a constant angular velocity, we have

TKL =TLK (2.15)

and hence, the energy equation can be written in the following form

ρ0e˙=TKLKL+HKν˙,K −gν˙ +QK,K0S. (2.16) We have not insisted here on the complete proof of the above results. More details about the deduction of the above equations can be found in the books by Ie¸san [165, 173], while the physical significance of the quantities of the present theory can be found in the works of Goodman and Cowin [140], Nunziato and Cowin [216], Cowin and Leslie [69].

According with the classical thermodynamics theory (see Truesdell and Noll [261]), for every subset P of the body B and for every time, we must have

Z

P

ρ0ηdV − Z

P

1

0SdV − Z

∂P

1

TQdA≥0, (2.17)

where η is theentropy per unit mass and T is the absolute temperature. We consider that T is a positive function of class C2,1 and η is of classC0,1 onB ×(t0, t1).

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We say that a media is a thermoelastic materials with voids if the following constitutive equations hold

ψ = ψ(Ee KL, T, T,B, ν, ν,M, XN), TKL = TeKL(EKL, T, T,B, ν, ν,M, XN),

η = eη(EKL, T, T,B, ν, ν,M, XN), QK = QeK(EKL, T, T,B, ν, ν,M, XN), HK = HeK(EKL, T, T,B, ν, ν,M, XN),

g = eg(EKL, T, T,B, ν, ν,M, XN), Q = Q(Ee KL, T, T,B, ν, ν,M, XN), H = H(Ee KL, T, T,B, ν, ν,M, XN),

(2.18)

where

ψ =e−T η (2.19)

is the Helmholtz free-energy. We assume that the response functions are of class C2 on their domain.

In view of the constitutive hypothesis we obtain

H =HKNK, Q=QKNK, (2.20)

and from the entropy inequality (2.17), as in classical theory, we have that the constitutive equations of the thermoelastic materials with voids are

W = ˆW(EKL, T, ν, ν,M, XN), TKL = ∂W

∂EKL, ρ0η=−∂W

∂T HK = ∂W

∂ν,K, g =−∂W

∂ν , QK = QˆK(EKL, T, T,Mν, ν,M, XN),

(2.21)

where W =ρ0ψ and

QKT,K ≥0. (2.22)

In conclusion, the basic equations of the nonlinear theory of thermoelastic materials with voids consists of equations of motion (2.10), the balance of equilibrated forces (2.11), the energy equation (2.16), the constitutive equations (2.21) and the geometrical equations (2.14), on B ×(t0, t1). We must adjoin to these equations boundary conditions and initial conditions.

The boundary conditions can be of Dirichlet type, of Neumann type or we can have mixed boundary conditions.

2.2 The linear theory of materials with voids

In this section we give the equations of the linear theory of thermoelastic and of the linear theory of elastic materials with voids. We use the following notations XiiAXA, f,i = ∂f

∂Xi

and we introduce the displacement vector u defined by

ui =xi−Xi. (2.23)

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We denote by T0 and ν0 the absolute temperature and the volume distribution function in the reference configuration and we suppose that are given positive constants. We also suppose that in the natural state, the body is free from stresses and has zero intrinsic equilibrated body force and entropy.

Let us consider now, the changes of the temperature and the variation of the volume dis- tribution function from the reference configuration by

ϕ =ν−ν0, θ=T −T0. (2.24)

In the linear theory, we can suppose thatu=εu0, ϕ=εϕ0 andθ=εθ0 whereεis a constant small enough to haveεn'0, for n ≥2.

As in classical theory, the Lagrangian strain tensor reduces to eij = 1

2(ui,j+uj,i). (2.25)

The linear theory of thermoelastic materials with voids

In the case of linear theory of thermoelastic materials with voids, the response functional can be consider functions of eij, θ, θ,i, ϕ, ϕ,k and Xm. Using the notations

tjijKTKi, hiiKHK, qiiKQK, from the relations (2.21), we can write

tij = ∂W

∂eij, ρ0η =−∂W

∂θ , hi = ∂W

∂ϕ,i, g =−∂W

∂ϕ (2.26)

and

qiθ,i≥0. (2.27)

In the linear approach, we have W = 1

2(Cijrseijers−2βijeijθ−aθ2+Aijϕ,iϕ,j+ 2Bijϕeij +2Dijkeijϕ,k+ 2diϕϕ,i+ξϕ2−2mθϕ−2aiϕ,iθ),

(2.28) where the constitutive coefficients are functions on Xm and have the following symmetries

Cijrs =Crsij =Cjirs, βijji, Dijk =Djik,

Aij =Aji, Bij =Bji. (2.29)

Thus, the constitutive equations of the linear theory of anisotropic elastic materials with voids are

tij = Cijrsers+Bijϕ+Dijkϕ,k−βijθ, hi = Aijϕ,j+Drsiers+diϕ−aiθ,

g = −Bijeij −ξϕ−diϕ,i+mθ, ρ0η = βijeij+aθ+mϕ+aiϕ,i

qi = kijθ,j

(2.30)

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The inequality (2.27) implies that the conductivity tensor kij is positive semi-definite, i.e.

kijξiξj ≥0 for all ξs.

In view of the assumption of the linear theory, the equations which describe the bahavior of the thermoelastic materials with voids are

tji,j0fi =ρ¨ui, (2.31)

hi,i+g+ρ0`=ρ0κϕ,¨ (2.32)

ρ0T0η˙ =qi,i0S, (2.33)

and the components of the surface traction t, the equilibrated surface traction h and the heat flux q at regular points of ∂B are given by

ti =tjinj, h=hini, q=qini, (2.34) where nj = cos(nx, Oxj) and nx is the unit vector of the outward normal to ∂B atx.

We consider that the constitutive coefficients are known and have the properties:

Cijrs, Bij, Dijk, βij, Aij, di, ai, and kij are function of class C1 on B and a, ξ, and m are continuous on B.

We recall that the unknowns are the displacement vector, the variation of the volume dis- tribution function (which will be called in the followingthe porosity function) and the variation of the temperature from the reference configuration. In terms of this unknowns, the equations of the linear theory are

(Cijrsur,s+Bijϕ+Dijkϕ,k−βijθ),j0fi0i, (Aijϕ,j+Drsiur,s+diϕ−aiθ),i−Bijui,j −ξϕ−diϕ,i+ +mθ+ρ0` =ρ0κϕ,¨ (kijθ,j),i−T0βiji,j−T0aθ˙−T0mϕ˙ −T0aiϕ˙,i =−ρ0S,

(2.35)

onB ×(0, t1).

In the case of isotropic materials, we have only nine constitutive coefficients, λ, µ, b, β, α, ξ, m, a,k, and the constitutive equations are

tij = λerrδij + 2µeij +bϕδij −βθδij, hi = αϕ,i,

g = −berr−ξϕ+mθ, ρ0η = βerr+aθ+mϕ,

qi = kθ,i

(2.36)

and the internal energy density W is positive defined quadratic form if and only if

µ >0, α >0, ξ >0, 3λ+ 2µ >0, (3λ+ 2µ)ξ > 3β2. (2.37) It follows that the basic equations of homogeneous and isotropic materials are

µ∆ui+ (λ+µ)uj,ji+bϕ,i−βθ,i0fi0i, α∆ϕ−buj,j−ξϕ+mθ+ρ0` =ρ0κϕ,¨ k∆θ−T0βu˙j,j−T0aθ˙−T0mϕ˙ =−ρ0S,

(2.38)

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where ∆ is the Laplace operator.

We consider the following mixed boundary conditions

ui = uei on S1×(t0, t1), tjinj =eti pe S2×(t0, t1), ϕ = ϕe on S3×(t0, t1), hini =eh on S4×(t0, t1),

θ = θe on S5×(t0, t1), qini =qe on S6×(t0, t1),

(2.39)

whereSr,(r= 1,2, ...,6),are subsets of∂Bso thatS1∪S2 =S3∪S4 =S5∪S6 =∂B, S1∩S2 = S3 ∩S4 = S5 ∩ S6 = ∅. The functions uei, ϕe and θe are prescribed continuous functions on S1×(t0, t1), S3×(t0, t1), S5×(t0, t1), respectively, whileeti,eh andqeare prescribed continuous functions on S2×(t0, t1), S4×(t0, t1), S6×(t0, t1), respectively.

The initial conditions are

u(X, t0) = u0(X), u(X, t˙ 0) = v0(X), ϕ(X, t0) =ϕ0(X),

˙

ϕ(X, t0) = ζ0(X), θ(X, t0) =θ0(X), X∈B, (2.40) with u0, v0, ϕ0, ζ0 and θ0 are prescribed continuous functions on B;

For more details regarding the equations of the linear theory, the readers can use the book of Ie¸san [173].

The linear theory of elastic materials with voids. Equilibrium equations

In the isothermal linear theory of elastic materials with voids (see [165]), the constitutive functionals depend only on eij, ϕ, ϕ,k and Xm. In consequence, we obtain

W = 1

2(Cijrseijers+Aijϕ,iϕ,j

+2Bijϕeij + 2Dijkeijϕ,k+ 2diϕϕ,i+ξϕ2),

(2.41) and the constitutive equations becomes

tij = Cijrsers+Bijϕ+Dijkϕ,k, hi = Aijϕ,j+Drsiers+diϕ,

g = −Bijeij−ξϕ−diϕ,i.

(2.42) The equations which describe the bahavior of the elastic materials are

tji,j0fi0i (2.43)

and

hi,i+g+ρ0`=ρ0κϕ.¨ (2.44) We must adjoin boundary conditions and initial conditions which are similar with (2.39)1−4

and (2.40)1−4.

When we study the equilibrium of elastic materials, we assume that all the quantities of the equations are time-independent quantities.

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Thus, the above equations are reduced to the equilibrium equations tji,j+Fi = 0,

hi,i+g+L= 0 (2.45)

inB, whereFi0fi and L=ρ0` are continuous functions in B.

The mixed boundary value problem of the equilibrium theory of elastic materials with void consists in finding the functionsui and ϕ which satisfy the equations (2.43), (2.44), (2.25) and (2.42) and the boundary conditions

ui = uei on S1×(t0, t1), tjinj =eti pe S2×(t0, t1),

ϕ = ϕe on S3×(t0, t1), hini =eh on S4×(t0, t1), (2.46) whereSr,(r = 1,2, ...,6),are subsets of∂Bso thatS1∪S2 =S3∪S4 =∂B, S1∩S2 =S3∩S4 =∅.

The functionseui, ϕeandθeare prescribed continuous functions onS1, S3, respectively, whileeti,eh and qeare prescribed continuous functions on S2, S4, respectively.

3 Some mathematical results in the equilibrium theory

3.1 Reciprocity and uniqueness theorems in the equilibrium theory

In this section we consider the equilibrium problem defined by the equations (2.45) and the boundary conditions (2.46). For this problem we present a uniqueness theorem and a reciprocity theorem.

We consider two sets of external dataI(α) = (fi(α), `(α),ue(α)i , ϕe(α), et(α)i , eh(α)), (α = 1,2).For this external data, let us consider the corresponding solutions s(α) = (u(α)i , ϕ(α), e(α)ij , t(α)ij , h(α)i , g(α)).

We have the following result [165].

Theorem 3.1 (Reciprocity theorem) If an elastic material with void occupies the domain B and it is subjected to the following external dataI(α),(α= 1,2), then the corresponding solutions s(α) satisfy the following reciprocity relation

Z

B

ρ0(fi(1)u(2)i +`(1)ϕ(2))dv+ Z

∂B

(t(1)i u(2)i +h(1)ϕ(2))da

= Z

B

ρ0(fi(2)u(1)i +`(2)ϕ(1))dv+ Z

∂B

(t(2)i u(1)i +h(2)ϕ(1))da,

(3.1)

where

t(α)i =t(α)ji nj, h(α)=h(α)i ni, (α= 1,2).

Proof. Let us consider

2Wαβ =t(α)ij e(β)ij +h(α)i ϕ(β),i −g(α)ϕ(β). From the relation (2.42) we deduce that

2Wαβ = Cijrse(β)ij +Bij(e(β)ij ϕ(α)+e(α)ij ϕ(β)) +Dijr(e(β)ij ϕ(α),r +e(α)ij ϕ(β),r )

+di(α)ϕ(β),i(β)ϕ(α),i ) +Aijϕ(α),i ϕ(β),i +ξϕ(α)ϕ(β).

(3.2)

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It is easy to see that, from the symmetries of the coefficients, we have W12=W21.

Using the relations (2.42), we deduce

2Wαβ = (t(α)ij u(β)i ),j+ (h(α)i ϕ(β)),i−t(α)ji,ju(β)i −(h(α)i,i +g(α)(β). (3.3) By a direct integration of this relation, and taking into account the boundary conditions we deduce the desired result.

Now, let us considerI(1) = (fi, `,eui,eti,ϕ,e eh) and s(1) = (ui, ϕ, eij, tij, hi, g). Then, from (3.2) and (3.3) we obtain

2 Z

B

W11dv = Z

B

(tijeij +hiϕ,i−gϕ)dv = 2 Z

B

Wdv,

where W is given by (2.41). With the help of the previous relation, we deduce that 2

Z

B

Wdv= Z

∂B

(tiui+hϕ)da+ Z

B

ρ0(fiui+`ϕ)dv. (3.4) A direct consequence of these relations is the following result:

Theorem 3.2 (Uniqueness theorem). If the internal energy W is a positive defined quadratic form in terms of eij, ϕ and ϕ,k for every X ∈ B, then two solutions of the mixed boundary problems defined by (2.45) and (2.46) are equals, excepting a rigid motion. Moreover, if S1 is not empty, then the considered mixed boundary problem has at most one solution.

Proof. Let us consider two solutions, (u0i, ϕ0) and (u00i, ϕ00), of the mixed boundary problem and u= (ui, ϕ) their difference.

In view of the linearity of the problem, (ui, ϕ) is solution of the problem defined by the equilibrium equations (2.45) and by null boundary conditions. Moreover, we have uiti+hϕ= 0 on ∂B. From (3.4) we deduce that

Z

B

Wdv= 0,

where W is the internal energy corresponding to (ui, ϕ). Because W is a positive defined quadratic form, we have that eij = 0, ϕ= 0 and, hence

ui =aiijrbjxr, ϕ = 0,

where ai and bi are arbitrary constants. If S1 is not empty, then ai = bi = 0, which complete the proof.

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3.2 Existence and uniqueness theorems in the equilibrium theory

We consider the equilibrium problem of elastic materials with voids. We assume that the domainB is bounded by a Lipschitz surface. Let consider that∂B =Su∪St∪C, where C is a null measurable subset of the boundary and Su andSt are empty or open subsets. We consider the boundary conditions

ui =eui, ϕ=ϕeon Su, tijnj =eti, hini =eh on St, (3.5) where eui,ϕe∈W1,2(B) and eti, eh∈L2(St). We define the space

W(B) ={U= (ui, ϕ); ui, ϕ ∈W1,2(B)}, (3.6) and the following norm

||U||2W(B) =||ϕ||2W1,2(B)+

3

X

i=1

||ui||2W1,2(B). (3.7) We consider the space V(B) defined as the subspace of W(B) for which

ui = 0, ϕ = 0 pe Su, (3.8)

in the trace sense. We also consider the bilinear formA(V,U) defined on W(B)×W(B) by A(V,U) =

Z

B

[Cijrseij(v)eij(u) +Bij(eij(v)ϕ+eij(u)ψ) +Dijr(eij(v)ϕ,r+eij(u)ψ,r)

+di(ϕψ,i,iψ) +Aijϕ,iψ,j+ψϕψ]dv,

(3.9)

where U= (ui, ϕ), V = (vi, ψ), eij(u) = 1

2(ui,j+uj,i).

In the following, the constitutive coefficients are considered measurable bounded functions in B which satisfy the symmetry relations (2.29). We have

A(U,V) = A(V,U), A(U,U) = 2 Z

B

Wdv, (3.10)

where W is the internal energy associated to U.

Let us define the functionals f, g :W(B)→Rby f(V) =

Z

B

ρ0(fivi+`ψ)dv, g(V) =

Z

St

(etivi+ehψ)da, V= (vi, ψ)∈W(B).

(3.11)

We assume that ρ0, fi, `∈L2(B).Let be ˜U= (eui, ϕ) such thate uei,ϕeonSu are restrictions on Su of some functionsuei,ϕe∈W1,2(B), in the trace sense.

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We say thatU∈W(B) is a solution of the boundary value problem defined by the relations (2.45) and (3.5) if U−Ue ∈V(B) and verifies

A(V,U) = f(V) +g(V), (3.12)

for every V ∈V(B).

We suppose that there is a positive constant α for which A(U,U)≥α

Z

B

[eij(u)eij(u) +ψ2,iψ,i]dv, (3.13) for all U∈W(B).

We define the operators Ns :W(B)→L2(B),(s= 1,2, ...,10) by N1U=u1,1, N2U= 1

2(u1,2+u2,1), N3U= 1

2(u1,3+u3,1), N4U=u2,2, N5U= 1

2(u2,3+u3,2), N6U=u3,3, N7U=ψ,1, N8U=ψ,2, N9U=ψ,3, N10U=ψ,

(3.14)

for all U= (ui, ψ)∈W(B). In view of the relation (3.13) the following inequality occurs A(U,U)≥2α

10

X

s=1

||NsU||2L2(B). (3.15) Let prove that this operators form acoercive operator system onW(B), i.e. there is a constant k >0 such that [165], [214]

||U||L2(B)+

10

X

s=1

||NsU|2L2(B) ≥k||U||2W(B), ∀ U∈W(B). (3.16) Using the results derived by Neˇcas [214], we can say that the above inequality is true if the operator systemNs has the order 4. These can be prove if we take into account the definitions given to the operators Ns.

We consider the subspace P of V(B) defined by P ={V∈V(B);

10

X

s=1

||NsV||2L2(B)= 0}. (3.17) In view of the definitions of the operators Ns we deduce that

P ={V = (vi, ψ)∈V(B);vi =aiirsbrxs, ϕ= 0}, (3.18) where ai and bi are arbitray constants, and let us consider the factor space W(B)/P of the classes of equivalence ˆU={U+p; p∈ P}, U∈W(B), endowed with the norm

||U||ˆ W(B)/P = inf

p∈P||U+p||W(B). (3.19)

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We remark that for everyp ∈ P and U∈W(B) we have A(U,p) = 0

and we can define the bilinear form A on the product space W(B)/P ×W(B)/P by

A( ˆU,V) =ˆ A(U,V), U∈U,ˆ V ∈V.ˆ (3.20) If Su is not empty, then P ={0} and we have the following result

Theorem 3.3 Let consider that P = {0}. In this case the boundary value problem has a uniqueness weak solution.

If Su empty then:

Theorem 3.4 Necessary and sufficient conditions for the existence of a weak solution u∈W of the traction problem are

Z

B

ρ0fidv+ Z

∂B

etida= 0, Z

B

ρ0εijrxjfrdv+ Z

∂B

εijrxjetrda = 0.

(3.21)

From the above theorem, we can conclude that to have a solution of the traction problem, the forces must have vanishing resultant and torque. This results are established using the method established in the paper by Hlav´aˇcek ¸si Neˇcas [150] (see also Fichera [107]) for elliptic equations.

4 The generalized plane strain state

Throughout this section, B denotes the interior of a right cylinder of length L, occupied by a porous elastic material. We choose a rectangular cartesian systemOx1x2x3 so that theOx3 axis is parallel with the generator of the cylinder and one of the ends lies in the x1Ox2 plane. We denote by∂B the boundary ofB and byD(x3)⊂R2 the interior of the bounded cross section at distance x3 from the x1Ox2 plane. The boundary, ∂D, of each cross section is assumed sufficiently smooth to admit application of the divergence theorem in the plane cross section.

The lateral boundary of the cylinder isπ =∂D×[0, L].

Moreover, we assume that the porous materials which made the cylinder are cross-section inhomogeneous. Thus, we have

Cijrs = Cijrs(x1, x2), Aij =Aij(x1, x2), Bij =Bij(x1, x2),

Dijk = Dijk(x1, x2), di =di(x1, x2), ξ=ξ(x1, x2). (4.1) Following Ie¸san [160] and Chirit¸˘a [34] we define the state of generalized plane strain for the interior of the cross section domain,D⊂R2, of the considered cylinder to be the state in which the displacement field w and the volume distribution ψ depend only on x1 and x2

wi =wi(x1, x2), ψ =ψ(x1, x2), (x1, x2)∈D. (4.2)

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In this case, the components of the stress tensor, the components of the equilibrated stress vector and the intrinsic equilibrated body force are function of x1 and x2, i.e. Tij =Tij(x1, x2), Hi =Hi(x1, x2) ¸si G=G(x1, x2). Moreover, we have

Tij(W) = Cijkαwk,α+Bijψ+Dijαψ, Hi(W) = Aψ+Drαiwr,α+diψ,

G(W) = −Bwi,α−ξψ−dαψ,

(4.3)

where W= (wi, ψ).

Given the body force, the tensions and the extrinsic equilibrated force, the plane strain problem for D∪ ∂D consists in finding a solution w, ψ of the boundary value problem S defined by the equations

Tαi,α(W) +Fi = 0, Hα,α(W) +G(W) +L= 0 in D, (4.4) and by the boundary conditions

Tαi(W)nα=Tei, Hα(W)nα =He on ∂D. (4.5) We substitute the relation (4.3) into (4.4) and (4.5) and we write the boundary value problem S in the following form

[Ciαkβwk,β+Bψ+Diαβψ]=−Fi,

[Aαβψ+Drβαwr,β+dαψ]−Bwi,α−ξψ−dαψ =−L in D, (4.6) and

[Ciαkβwk,β+Bψ+Diαβψ]nα =Tei,

[Aαβψ+Drβαwr,β +dαψ]nα =He on ∂D. (4.7) The generalized plane strain state was studied in various papers (see for example the book by Ie¸san [162] and by Leknitski [199]).

We assume that Fi, L∈C(D) andTei,He ∈C(∂D). Following an appropriate method as that used in [162] and in view of the Theorem 3.4, we have that the necessary and sufficient conditions for the existence of a solution of the generalized plane problem [165] are given by

Z

D

Fida+ Z

∂D

Teids= 0, Z

D

ε3αβxαFβda+ Z

∂D

ε3αβxαTeβds= 0, (4.8) where εijk is the alternating symbol. The assumptions of regularity imply that the solution belong to C(D) (cf. [107]).

5 Saint-Venant’s problem

As in the previous section, we will consider that the domainB is the interior of a right cylinder of length L. All the notations and assumptions introduced in previous section will be used in this section.

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We recall that the component of the tension vector and the equilibrated tension in a regular point of the frontier ∂B are given by

ti(U) = tij(U)nj, h(U) = hj(U)nj, (5.1) where nj nj are the components of the outward unit normal to∂B and U= (u, ϕ).

The equations of equilibrium of porous continua, in the absence of body loads, are given by tji,j(U) = 0, hj,j(U) +g(U) = 0. (5.2) Saint-Venant’s problem forB consists in the determination of the displacement field u and the volume distribution function ϕ onB, subjected to the requirements

ti(U) = 0, h(U) = 0 pe π, ti(U) = t(1)i , h(U) =h(1) onD(0), ti(U) = t(2)i , h(U) =h(2) onD(L),

(5.3)

where t(α)i and h(α) are functions preassigned on D(0) andD(L), respectively.

Following the results presenting in the Section 3.2, necessary and sufficient conditions for the existence of a solution to this problem are given by

Z

D(0)

t(1)i da+ Z

D(L)

t(2)i da= 0, Z

D(0)

εijkxjt(1)k da+ Z

D(L)

εijkxjt(2)k da = 0.

(5.4)

This equations imply that, for the equilibrium of cylinder, the tractions at the bases have vanishing resultant and torque.

The displacement fielduand the volume distribution function ϕ satisfy the boundary value problem defined by the equations

Ti(U)≡[Cijrsur,s+Bijϕ+Dijkϕ,k],j = 0,

H(U)≡[Aijϕ,j+Drsiur,s+diϕ],i−Bijui,j−ξϕ−diϕ,i= 0, (5.5) onB =D×(0, L), the lateral boundary conditions

Bi(U) ≡ [Ciαrsur,s+Bϕ+Diαkϕ,k]nα = 0,

G(U) ≡ [Aαjϕ,j+Drsαur,s+dαϕ]nα= 0, (5.6) on∂D×(0, L), and the end boundary conditions

t3i(U) =t(1)i , h3(U) =h(1) on D(0),

t3i(U) = t(2)i , h3(U) = h(2) on D(L). (5.7) In the study of Saint–Venant’s problem [127, 130], we assume that the internal energy density associated to the solution of the boundary value problemT, defined by the equations (5.5) and the boundary conditions (5.6)–(5.7), is positive defined.

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5.1 Analysis of Saint-Venant’s problem by plane section solutions

In this subsections, following thesemi-inverse method used by Ie¸san [160, 163] and [34, 36], we study the possibility to reduce the system (5.5) and the lateral boundary conditions (5.6) to a generalized plane problem.

For this aim, we consider the system (5.5) and the boundary condition (5.6) on the cross section D∪∂D. More exactly, we consider the plane boundary value problem

Ti(U) = 0, H(U) = 0 ˆın D (5.8)

and

Bi(U) = 0, G(U) = 0 pe ∂D, (5.9)

considered x3 ∈ (0, L) as parameter. We formulate the following problem: when the solution of the Saint-Venant’s problem described above, by the relations (5.5)–(5.7), can be reduced to the case of generalized plane strain problem associated to the cross section of the cylinder?

The components of the resultant force and of the resultant moment of the traction about the origin of the rectangular cartesian system Ox1x2x3, acting on the cross-section D of the cylinder are defined by

Ri(U) = Z

D

t3i(U)da, Mi(U) = Z

D

εijkxjt3k(U)da. (5.10) We will have a answer to the above questions, in terms of the vector-valued linear functionals R and Mi. We remark that

Mα(U) = ε3αβ Z

D

xβt33(U)da−x3ε3αβRβ(U), M3(U) = ε3αβ

Z

D

xαt(U)da.

(5.11)

We write the plane boundary value problem (5.8) and (5.9) in the form [Ciαkβuk,β+Bϕ+Diαβϕ]

+[Ciαk3uk,3+Diα3ϕ,3]+t3i,3(U) = 0, [Aαβϕ+Drβαur,β+dαϕ]

−Bui,α−ξϕ−dαϕ−Bi3ui,3−d3ϕ,3+

+[Aα3ϕ,3+Dr3αur,3]+h3,3(U) = 0 in D,

(5.12)

and

[Ciαkβuk,β+Bϕ+Diαβϕ]nα=−[Ciαk3uk,3 +Diα3ϕ,3]nα,

[Aαβϕ+Drβαur,β+dαϕ]nα =−[Aα3ϕ,3+Dr3αur,3]nα (5.13) on ∂D, and we see the boundary value problem defined by (5.12) and (5.13) as a generalized plane strain boundary value problem, with

Fi = [Ciαk3uk,3+Diα3ϕ,3]+t3i,3(U),

L = [Aα3ϕ,3+Dr3αur,3]−Bi3ui,3−d3ϕ,3+h3,3(U), Tei = −[Ciαk3uk,3+Diα3ϕ,3]nα,

He = −[Aα3ϕ,3+Dr3αur,3]nα.

(5.14)

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The necessary and sufficient conditions (4.8), show us that we must have Z

D

t3i,3(U)da= 0, Z

D

ε3αβxαt3β,3(U)da= 0. (5.15) From (4.1) and (2.42) we can write

t3i,3(U) =t3i(U,3), (5.16)

so that the relations (5.15) take the form Z

D

t3i(U,3)da = 0, (5.17)

Z

D

ε3αβxαt(U,3)da = 0. (5.18) We observe, in view of relations (5.15), (5.10) ¸si (5.11), that a sufficient condition for expressing a solution of Saint-Venant’s problem in terms of a generalized plane strain is the fact that Ri(U) and M3(U) are independent of x3.

We are thus led to the following result.

Proposition 5.1 Let U be a solution to Saint-Venant’s problem. If Ri(U) and M3(U) are independent of x3 then U can be expressed in term of a state of generalized plane strain.

Corollary 5.1 Let U be a solution to Saint-Venant’s problem for which Ri(U) and M3(U) are independent of x3. Then Mα(U) are independent of x3.

Proof. In view of equation (5.2) and the boundary condition (5.3), we get Z

D

xαt33(U,3)da= Z

D

xαt33,3(U)da=− Z

D

xαtβ3,β(U)da

=− Z

D

(xαtβ3(U)) da+ Z

D

tα3(U)da

=− Z

∂D

xαtβ3(U)nβds+Rα(U) =Rα(U).

(5.19)

On the other hand, by means of relation (5.11), we have Mα,3(U) = ε3αβ

Z

D

xβt33(U)da

,3

−ε3αβRβ(U)−x3ε3αβRβ,3(U) (5.20) and the proof is complete.

Remark 5.1 Relation (5.19) combined with the fact that Rα(U) is independent of x3, yields Z

D

xαt33(U,33)da = 0. (5.21)

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