# Explicit Solution of Elastostatic Boundary Value Problems for the Elastic Circle with Voids

Authors: Ivane Tsagareli
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We solve the static two-dimensional boundary value problems for an elastic porous circle with voids. Special representations of a general solution of a system of differential equations are constructed via elementary functions which make it possible to reduce the initial system of equations to equations of simple structure and facilitate the solution of the initial problems. Solutions are written explicitly in the form of absolutely and uniformly converging series. The question pertaining to the uniqueness of regular solutions of the considered problems is investigated.


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## Body

## 1. Introduction

The physicomathematical foundations of the linear theory of elastic materials with voids or empty pores and applications of this theory to some technological problems were originally proposed in the work Cowin and Nunziato [1]. Such materials include, in particular, rocks and soils, granulated and some other manufactured porous materials.This theory differs essentially form the classical theory of elasticity in that the volume fraction corresponding to the void volume is considered as an independent variable. Voids have no mechanical or energetic meaning.In recent years, problems of elasticity for materials with voids were investigated by many authors. Below we mention only a few works where the bibliographical information can also be found.In [2], the traditional problems of thick-walled spherical and circular cylindrical shells under the action of internal pressure are solved in the framework of the theory of linear elastic materials with voids. Some fundamental solution existence and uniqueness theorems, duality relations, and variational characterization of a solution are considered in terms of the aforementioned theory in [3]. In [4], solutions are obtained for equations of the classical theory of elastodynamics of homogeneous and isotropic elastic materials with voids. Using the results established by Kupradze et al. [5] in classical elasticity, in [6] boundary value problems of stationary oscillations are studied in terms of the linear theory of homogeneous and isotropic materials with voids, while [7] is concerned with the study of the behavior of plane harmonic waves in an elastic material with voids. The basic properties of these waves are established. Also established are the existence and uniqueness of solutions for external problems. Existence theorems in equilibrium theory are proved in [8], and in [9] the fundamental solution is obtained for a differential system of micropolar elasticity with voids in the case of stationary vibrations. The duality properties are also studied.The linear theory of thermoelastic materials with voids was for the first time considered in the work Ieșan [10]. Solutions of Galerkin type and uniqueness theorems in the theory of thermoelasticity for materials with voids are proved in [11, 12]. Problems of steady vibrations of elastic and thermoelastic materials with voids are investigated in [13]. The linear theory of micropolar thermoelasticity is considered for materials with voids in the papers [14, 15]. Fundamental solutions of systems of differential equations of steady-state oscillations [14] and diffusion [15] are constructed in terms of elementary functions. The nonlinear theory of elastic porous materials with voids was proposed in the work Nunziato and Сowin [16].Along with the generalization and development of the linear theory of elasticity for materials with voids in various directions, a great deal of attention is attached to the mathematical investigation and construction of explicit solutions of boundary value problems for concrete domains.In the present paper, the special representations of a general solution of a system of differential equations of the theory of elastic materials with voids are constructed by using harmonic, biharmonic, and metaharmonic functions which make it possible to reduce the initial system of equations to equations of simple structure and facilitate the solution of initial problems. These representations are used in the paper to solve problems for an elastic circle with voids. The solutions are written explicitly in the form of absolutely and uniformly converging series.The uniqueness of regular solutions of the considered problems is also investigated.


## 2. Formulation of Boundary Value Problems

Let the isotropic elastic circleD, consisting of empty pores, be bounded by the circumference K of radius R. Denote the area of the macropoint of x=(x1,x2) by S(x), and the area of pores at this point by Sp(x). The value s(x), which is defined by the equality s(x)=Sp(x)/S(x), is called the relative area of pores. Generally speaking, as a result of deformation of the body the relative area of pores changes, too. We denote this change by φ(x).A system of equations of the linear theory of elastic materials with voids has the following form [1, 6]:(1)μΔux+λ+μgraddivux+βgradφx=0,(2)αΔ-ξφx-βdivux=0,x∈D,where u(x)=(u1,u2) is the displacement vector, and φ(x) is a change with respect to the pore area; λ and μ are the Lamé constants; α, β, and ξ are the constants characterizing the body porosity.Let us now formulate the boundary value problems. Find, in the circleD, a regular vector U(x)=(u(x),φ(x)) that satisfies (1) and (2) and, on the boundary K, one of the following conditions:(3)uz=fz,φz=f3zin  the  ProblemI;P1∂z,nUz=fz,α∂nφz=f3zin  the  ProblemII,where (x)∈C1(D-)∩C2(D), z=(z1,z2)∈K, n(z)=(n1(z),n2(z)) is the external normal with respect to K at the point z; f=(f1,f2), f1, f2, and f3 are functions given on K;(4)P∂x,n=Tij∂x,nβni0α∂n3×3is the stress operator in the theory of elasticity for porous bodies with voids [1], i,j=1,2. Stress in the vector form can be written as(5)P∂x,nUx=P1∂x,nUxα∂nφx,where (6)P1∂x,nUx=T∂x,nux+βnxφx,and(7)T∂x,nux=μ∂nux+λnxdivux+μ∑i=12nixgraduixis the stress vector in the classical theory of elasticity [17], ∂n=(n·grad).


## 3. General Representations of a System of Equations

Applying the operatordiv to (1)1, from (1) and (2) we obtain a system of equations with respect to divu and φ:(8)μ0Δdivu+βΔφ=0,βdivu-αΔ-ξφ=0,where μ0=λ+2μ. The determinant of system (8) is equal to -μ0αΔ(Δ+s12), where(9)s12=-μ0ξ-β2μ0α.Assume(10)λ>0,μ>0,α>0,μ0ξ>β2.From (10) it follows that ξ>0. Thus s12<0, where s1=i(μ0ξ-β2)/μ0α=is0,i=-1.Since system (8) is homogeneous, we write(11)ΔΔ+s12divu=0,ΔΔ+s12φ=0.Taking (11) into account, from (1) we get μΔ2(Δ+s12)u=0.From this equation and also from the second equation (11), we conclude that the representation of a solution of u(x) contains a harmonic, biharmonic, and a metaharmonic functions, while the representation of φ(x) contains a harmonic and a metaharmonic functions.Solution of system consisting of (1) and (2) are written in the following form:(12)ux=c0u0x+c1u1x,φx=φ1x+φ2x,where φ1 is a harmonic function, Δφ1=0, and φ2 is a metaharmonic function with the parameter s12, (Δ+s12)φ2=0; c0 and c1 are the unknown functions for the time being. A general solution u0=(u10,u20) of the homogeneous equation, corresponding to the nonhomogeneous equation (1) with respect to u(x), is represented as follows [18]:(13)u0x=gradΦ1x+Φ2x+rotΦ3x+c2Γx,where functions Φ2 and Φ3 are interrelated by(14)μ0gradΔΦ2x+μrotΔΦ3x=0;ΔΦ1(x)=0, ΔΔΦ2(x)=0, ΔΔΦ3(x)=0, Φ1, Φ2, Φ3 are scalar functions, Γ=(Γ1,Γ2); Γ1=x2, Γ2=-x1, divΓ=0; c2 is the sought coefficient, rot=-∂/∂x2,∂/∂x1.u 1   = ( u 1 1 , u 2 1 ) is one of the particular solutions of (1):(15)u1x=-βμ0grad-1s12φ2+φ0,where φ0 is chosen such that Δφ0=φ1. It is obvious that φ0 is a biharmonic function: ΔΔφ0=Δφ1=0. For simplicity, the function φ1 is chosen such that φ1=divu0≡ΔΦ2. Then we can take φ0=Φ2.Let us calculate the values of the coefficientsc0 and c1 in representation (12). We apply the operator div to the first equality in (12) and compare the obtained expression with divu defined by (2). Using (9), we obtain (16)c0=-μ0ξ-β2μ0β,c1=1.By an immediate verification we make sure that representations (12) satisfy (1) and (2).


## 4. Uniqueness Theorems

For a regular solutionU(x) of (1) Green’s formula [5],(17)∫DEu,u+βφdivudx=∫KuT∂y,nu+βφndyK,is valid, where(18)Eu,u=λ+μ∂u1∂x1+∂u2∂x22+μ∂u1∂x1-∂u2∂x22+μ∂u1∂x2+∂u2∂x12,is the nonnegative quadratic form [17] when conditions (10) are used.On the other hand, we multiply (2) by φ(x) and integrate in D. Applying the equality(19)∫DφΔφdx=∫Kφ∂φ∂ndyK-∫Dgradφ2dx,we obtain the following formula:(20)∫Dαgradφ2+ξφ2+βφdivudx=∫Kαφ∂φ∂ndyK.Using formulas (17) and (20), we write(21)∫DEu,u+αgradφ2+ξφ2+2βφdivudx=∫KuT∂y,nu+βφn+αφ∂φ∂ndyK.Let us assume that each of the above-stated problems has two solutions. To define their difference, the right-hand parts of formulas (17) and (20) are assumed to be equal to zero on K:(22)∫DEu,u+βφdivudx=0,∫Dαgradφ2+ξφ2+βφdivudx=0.From these equalities, taking (10) into account, we obtain φ1(x)=k, φ2(x)=0,(23)Eu,u+βkdivu=0.where k is arbitrary constant.LetU′=u′,φ′ and U′′=u′′,φ′′ be two arbitrary solutions of any of problems I and II. Then the differences U=u,φ=U′-U′′ are the solutions of the corresponding homogeneous problems:(24)μΔux+λ+μgraddivux=0,x∈D,uz=0,φz=0,in  the  problemI;T∂z,nUz+βnzφz=0,α∂nφz=0,z∈K,in  the  problemII.For homogeneous boundary conditions, for Problem I, we obtain from (23): E(u,u)=0. The solution of the above equation has the following form [17]:(25)u1x=-px2+q1,u2x=px1+q2,where p, q1, q2 are arbitrary constants. In addition, in Problem I, by virtue of homogeneous boundary condition, we get p=q1=q2=0, and hence, u(x)=0,x∈D.The solution of the homogeneous Problem II, which also satisfies (23), has the following form:(26)ux=lx+m,where l=βk/2(λ+μ), m=(m1,m2), m1, m2, k are arbitrary constants.Thus, the following assertions are true.Theorem 1.
Problem I has a unique solution.Theorem 2.
Two arbitrary solutions of Problem II may differ from each other only in equality (26) which express the rigid displacement of the body as a whole.


## 5. Solution of the Problems

Problem I.Let us rewrite representations (12) in terms of polar coordinates r and ψ as normal and tangent components:(27)un=∂rc0Φ1+c3Φ2+c4φ2-c0r∂ψΦ3,us=1r∂ψc0Φ1+c3Φ2+c4φ2+c0∂rΦ3-c2r,φ=φ1+φ2,where c3=-ξ/β, c4=β/μ0s12, r2=x12+x22.Using formula (14) and the equality φ1=ΔΦ2, the harmonic and biharmonic functions and also the metaharmonic function contained in (27) are represented in the circular disc D as series [19, 20]:(28)φ1x=∑m=0∞rRmXm1·νmψ,Φ2x=R24∑m=0∞1m+1rRm+2Xm1·νmψ,Φ3x=μ0R24μ∑m=0∞1m+1rRm+2Xm1·smψ,Φ1x=∑m=0∞rRmXm3·νmψ,φ2x=∑m=0∞Ιms0rXm2·νmψ,where Χmk is the sought two-component vector, k=1,2,3, x=(r,ψ); νm(ψ)=(cos⁡mψ,sin⁡mψ), sm(ψ)=(-sin⁡mψ,cos⁡mψ), Ιm(s0r)- is the Bessel function of the imaginary argument.Let us write the boundary conditions of ProblemΙ in the form of normal and tangent components: (29)unz=fnz,usz=fsz,φz=f3z.Let functions fn, fs, and f3 be expanded into Fourier series:(30)fnz=α02+∑m=1∞αm·νmψ,fsz=β02+∑m=1∞βm·smψ,f3z=γ02+∑m=1∞γm·νmψ,where αm, βm, and γm are the Fourier coefficients of the functions fn,fs, and f3, respectively; αm=(αm1,αm2), βm=(βm1,βm2), γm=(γm1,γm2), m=0,1,….Let us substitute expressions (28) into (27) and pass to the limit as r→R. Taking (30) into account, we substitute the results into (29). For m=0 we obtain the linear algebraic system:(31)c3Χ01+2c4s0Ι0′s0RΧ02=α0,c0μ0R2Χ01-2μRΧ03=β0μ,Χ01+Ι0s0RΧ02=γ02.where, for convenience, we have introduced the notation Χ03≡c2.Form=1,2,… for each m we obtain(32)R4m+1c3m+2+c0mμ0μΧm1+c4s0Ιm′s0RΧm2+c0mRΧm3=αm,Rc3mμ+c0μ0Rm+24μm+1Χm1+c4mRΙms0RΧm2+c0mRΧm3=βm,Χm1+Ιms0RΧm2=γm,m=1,2,….We solve systems (31) and (32). The obtained values of vectors Χmk are substituted into (28). Now, using formulas (13), (15), and (12), we obtain the solution of the considered problem.Problem II. Passing in (5) to normal and tangent components, we obtain (33)P1∂x,nxUxn=μ0∂runx+λr∂ψusx+βφx,P1∂x,nxUxs=μ∂rusx+1r∂ψunx,∂nφx=∂rφx,where un, us, and φ are defined in (27). Rewrite boundary conditions (3)2 in the form(34)P1∂z,nzUzn=fnz,P1∂z,nzUzs=fsz,α∂rφz=f3z.Substitute (27) and (28) into (34). Using (30) and passing to the limit as r→R, from (30) we obtain the algebraic system. For m=0 we have(35)μ0c32-λμ0Rc02μ+βX01+μ0c4s02I′′s0R+βI0s0RX02=α02,μ0Rc02X01-μX03=β02,s0I0′s0RX02=γ02α.For m=1,2,… we get(36)μ0m+2c34+μ0mc04μR-λm2c0R2-λμ0Rm+2c04μm+1+βXm1+μ0s02Im′′s0Rc4-λm2c4R2+βI0s0RXm2+μ0mm-1c0R2Xm3=αm,μ0s02Im′′s0Rc4-λm2c4R2+βI0s0RXm1+2ms0c4RIm′s0R-mc4R2Ims0RXm2+mc0R22m-1Xm3=βmμ,mRXm1+s0Im′s0RXm2=γmα.Solving these systems and substituting the obtained values of the vectors Χmk into (28), by formulas (13), (15), and (12) we obtain the solution of Problem ΙΙ.For the obtained series to converge absolutely and uniformly it suffices to require the following.In Problem I,fj∈C3(K), where j=1,2,3.In Problem II,fj∈C2(K), where j=1,2,3.



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