CIALDEA, Alberto

Dolce, Emanuela

LEONESSA, VITA

MALASPINA, Angelica

English

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Reports of Enlarged Session of theSeminar of I. Vekua Instituteof Applied MathematicsVolume 28, 2014ON THE POTENTIAL THEORY IN THE LINEAR THEORY OF VISCOELASTICMATERIALS WITH VOIDSCialdea A., Dolce E., Leonessa V., Malaspina A.Abstract. By using an indirect boundary integral method, the solution of the first (sec-ond) BVP of steady vibrations related to the linear theory of viscoelasticity for Kelvin-Voigtmaterials with voids is represented by means of a simple (double) layer elastopotential.Keywords and phrases: Boundary integral equations, layer potentials, viscoelasticity,Kelvin-Voigt material with voids.AMS subject classification: 31B10, 35C15, 74D05.As is well known, the classical indirect method of Fredholm gives the solution of theDirichlet problem for the n-dimensional Laplacian in terms of a double layer potentialu(x) =∫Σφ(y) ∂∂νys(x − y)dσy, where ν(x) = (ν1(x), . . . , νn(x)) denotes the outwardunit normal vector at the point x = (x1, . . . , xn) ∈ Σ and s is the fundamental solutionof the Laplace equation. There is another approach, which consists in looking for thesolution in the form of a simple layer potential u(x) =∫Σφ(y)s(x− y)dσy; in this casean integral equation of the first kind arises∫Σφ(y)s(x− y)dσy = g(x), x ∈ Σ. (1)Muskhelishvili ([1], p.184) gave a method for solving (1) when n = 2, which leads tothe study of a singular integral equation. Even if such approach is based on the theoryof holomorphic functions of one complex variable and uses the derivative with respectto the arc length, in [2] it was generalized to the case of n real variables by one ofthe authors. The main idea consists in replacing holomorphic functions by conjugatedifferential forms and the derivative with respect to the arc length by the exteriordifferential operator d. This method hinges on the theory of reducible operators andon the theory of differential forms; it does not require the use of pseudo-differentialoperators nor the use of hypersingular integrals. The approach has been applied todifferent BVPs for other PDEs in simply and multiply connected domains (see [3-14]).Here, we consider the application of the method to the study of the two basic BVPsfor the homogeneous equations of steady vibrations in the linear theory of viscoelasticityfor Kelvin-Voigt materials with voids, i.e. for the system{µ1∆u+ (λ1 + µ1)grad div u+ b1grad φ+ ρω2u = 0,(α1∆+ ξ2)φ− ν1div u = 0,(2)where u = (u1, u2, u3) is a complex time-independent vector function, φ is a complextime-independent function, ρ is the reference mass density (ρ > 0), ω is the oscillationfrequency (ω > 0), λ1 = λ − iωλ∗, µ1 = µ − iωµ∗, b1 = b − iωb∗, α1 = α − iωα∗,On the Potential Theory in the Linear Theory of ... 17ν1 = b− iων∗, ξ1 = ξ − iωξ∗, ξ2 = ρ0ω2 − ξ1, ρ0 = ρk, k being the equilibrated inertia(k > 0), and λ, µ, b, α, ξ, λ∗, µ∗, b∗, α∗, ν∗, ξ∗ are (real) constitutive coefficients (see [15]).We denote the matrix of fundamental solution of the homogeneous system (2) byΓ = (Γpq)4×4 (see [15]). Moreover, the fundamental solution of the system{µ1∆u+ (λ1 + µ1)grad div u = 0,α1∆φ = 0is the matrix Ψ = (Ψpq)4×4, whose entries are Ψlj(x) = − 18π(1µ1∆δlj − λ1+µ1µ1µ2∂2∂xl∂xj)|x|(µ2 = λ1 + 2µ1), Ψ44(x) =1α1s(x), Ψl4(x) = Ψ4j(x) = 0, l, j = 1, 2, 3.Lemma 1. ([15], Theorem 4.2) If α1µ1µ2 ̸= 0, then the relationsΨpq(x) = O(1|x|), Γpq(x)−Ψpq(x) = O(1 + |x|),∂m∂xm11 ∂xm22 ∂xm33[Γpq(x)−Ψpq(x)] = O(1|x|m−1)hold in a neighborhood of the origin, where m = m1 + m2 + m3, m ≥ 1, ml ≥ 0,l = 1, 2, 3 and p, q = 1, 2, 3, 4.Therefore, Ψ(x) is the singular part of the matrix Γ(x).In what follows, Ω ⊂ R3 is a bounded simply connected domain (i.e. R3 r Ω isconnected) such that its boundary Σ = ∂Ω is a Lyapunov hypersurface (i.e. Σ has auniformly Hölder continuous normal field of some exponent l ∈ (0, 1]); p indicates areal number such that p ∈ ]1,+∞[. Let us consider the first BVP in the class Sp of thesimple layer elastopotentials U [ϕ](x) =∫ΣΓ(x− y)ϕ(y)dσy with density in [Lp(Σ)]4:U ∈ Sp,A(Dx)U = 0 in Ω,U = F on Σ, F ∈ [W 1,p(Σ)]4(3)(by W 1,p(Σ) we denote the usual Sobolev space), where U = (u, φ) and A(Dx) =(Apq(Dx))4×4 is the matrix whose entries are Alj(Dx)=(µ1∆+ρω2)δlj + (λ1+µ1)∂2∂xl∂xj,Al4(Dx) = b1∂∂xl, A4l(Dx) = −ν1 ∂∂xl , A44(Dx) = α1∆ + ξ2, l, j = 1, 2, 3 (δlj being theKronecker delta).Imposing the boundary condition we get the integral system of the first kind∫ΣΓ(x− y)ϕ(y) dσy = F (x) (4)on Σ. Following the approach introduced in [2], we take the differential of both sidesof (4), obtaining the following singular integral system∫Σdx[Γ(x− y)]ϕ(y) dσy = dF (x). (5)In (5) the unknown is the vector (ϕ1, ..., ϕ4) whose components are scalar functions,while the data is the vector (dF1, . . . , dF4) whose components are differential forms of18 Cialdea A., Dolce E., Leonessa V., Malaspina A.degree 1. We are going to show that the singular integral system (5) can be reducedto an equivalent Fredholm one.It is possible to prove (see ([4], Lemma 5.3)) that the singular integral operator1S0 : [Lp(Σ)]4 −→ [Lp1(Σ)]4, S0(ϕ)(x) =∫Σdx[Ψ(x− y)]ϕ(y)dσycan be reduced on the left. This means that there exists a linear and continuousoperator S ′ : [Lp1(Σ)]4 −→ [Lp(Σ)]4 such that S ′S0 is a Fredholm operator from [Lp(Σ)]4into itself. Let us define the singular integral operatorS : [Lp(Σ)]4 −→ [Lp1(Σ)]4, Sϕ(x) =∫Σdx[Γ(x− y)]ϕ(y)dσy.Since S − S0 is compact by Lemma 1 and we can write S = (S − S0) + S0, we obtainthat S ′S is a Fredholm operator. We thus obtain the next claim.Proposition 1. ([4], Proposition 5.1) The singular integral operator S can bereduced on the left.We deduce that the integral system (5) admits a solution if, and only if,∫Σγi∧dF i =0, i = 1, 2, 3, 4, for every γ ∈ [Lq1(Σ)]4 solution of the homogeneous adjoint systemS∗j γ(x) =∫Σγi(y) ∧ dy[Γij(x− y)] = 0, a.e. x ∈ Σ, j = 1, 2, 3, 4. Ifµ∗ > 0, 3λ∗ + 2µ∗ > 0, α∗ > 0, (3λ∗ + 2µ∗)ξ∗ >34(b∗ + ν∗)2, (6)one can prove that S∗γ = 0 if, and only if, γi is a weakly closed 1-form (see ([4],Theorem 5.1)). Consequently, the singular integral system Sϕ = dF is always solvable.Since any solution of a Dirichlet problem with constant datum can be representedby means of a simple layer elastopotential (see ([4], Lemma 5.5)), we have the followingrepresentation for the solution of the first problem.Theorem 1. ([4], Theorem 5.2) If conditions (6) hold, the first BVP (3) admitsa unique solution U. In particular, the density ϕ of U can be written as ϕ = ϕ0 + ψ0,where ϕ0 solves the singular integral system (5) and ψ0 is the density of a simple layerelastopotential which is constant on Σ.We remark that, the obtained reduction is not an equivalent reduction2. However,we still have an equivalence between (5) and the Fredholm equation S ′Sϕ = S ′(dF ).In fact, as in ([16], pp.253-254), one can show that N(S ′S) = N(S). This implies that,if G is such that there exists a solution g of the system Sg = G, then this system issatisfied if and only if S ′Sg = S ′G. Since we know that the system Sg = dF is solvable,we have that Sg = dF if, and only if, S ′Sg = S ′(dF ) (see ([4], Theorem 5.3)).These results permit also to represent the solution of the second BVP of steadyvibrations related to the linear theory of viscoelasticity for Kelvin-Voight materials1The symbol Lpk(Σ) stands for the space of the differential forms of degree k defined on Σ whosecomponents belong to Lp(Σ) in a coordinate system of class C1 and then in every coordinate systemof class C1.2A left reduction is said to be equivalent if N(S′) = {0}, where N(S′) denotes the kernel of S′. Thisimplies that Sα = β if, and only if, S′Sα = S′β.On the Potential Theory in the Linear Theory of ... 19with voids by means of the double layer elastopotential. For the details we refer to([4], Section 6).R E F E R E N C E S1. Muskhelishvili N.I. Singular Integral Equations. Boundary Problems of Functions Theory andtheir Applications to Mathematical Physics. Revised translation from the Russian, edited by J. R. M.Radok. Reprinted. Wolters-Noordhoff Publishing, Groningen, 1972.2. Cialdea A. On oblique derivate problem for Laplace equation and connected topics. Rend.Accad. Naz. Sci. XL, Serie 5, Mem. Mat. Parte I, 12, (1988), 181-200.3. Cialdea A., Dolce E., Malaspina A., Nanni V. On an integral equation of the first kind arisingin the theory of Cosserat. Intern. J. Math., 24, (2013), 21 pages, doi: 10.1142/S0129167X13500377.4. Cialdea A., Dolce E., Leonessa V., Malaspina A. New integral representations in the lineartheory of viscoelastic materials with voids. Publ. Math. Inst. (Beograd), Nouvelle série, 96, (110),(2014), 49-65, doi: 10.2298/PIM1410049C.5. Cialdea A., Dolce E., Leonessa V., Malaspina A. On the potential theory in Cosserat elasticity.Bull. TICMI, 18, (2014), 67-81.6. Cialdea A., Dolce E., Malaspina A. A complement to potential theory in the Cosserat theory.Math. Methods in the Appl. Sciences, 38, (2015), 537-547, doi: 10.1002/mma.30867. Cialdea A., Hsiao G.C. Regularization for some boundary integral equations of the first kindin Mechanics. Rend. Accad. Naz. Sci. XL, Serie 5, Mem. Mat. Parte I, 19, (1995), 25-42.8. Cialdea A., Leonessa V., Malaspina A. Integral representations for solutions of some BVPs forthe Lamé system in multiply connected domains. Boundary Value Problems 2011, 2011:53, 25 pages,doi:10.1186/1687-2770-2011-53.9. Cialdea A., Leonessa V., Malaspina A. On the Dirichlet and the Neumann problems forLaplace equation in multiply connected domains. Complex Var. Elliptic Equ., 57, (2012), 1035-1054,doi:10.1080/17476933.2010.534156.10. Cialdea A., Leonessa V., Malaspina A. On the Dirichlet problem for the Stokes systemin multiply connected domains. Abstr. Appl. Anal., 2013, Art. ID 765020 (2013), 12 pages,doi:10.1155/2013/765020.11. Malaspina A. Regularization for integral equations of the first kind in the theory of thermoe-lastic pseudo-oscillations. Appl. Math. Inform., 9 (2004), 29-51.12. Malaspina A. On the traction problem in mechanics. Arch. Mech., 57, (2005), 479-491.13. Malaspina A. Regularization of some integral equations of the first kind. AIP ConferenceProceedings, 1281, (2010), 916-918.14. Malaspina A. Integral representation for the solution of Dirichlet problem for the Stokessystem. AIP Conference Proceedings, 1389, (2011), 473-476.15. Svanadze M.M. Potential method in the linear theory of viscoelastic materials with voids. J.Elast. 114, (2014), 101-126.16. Cialdea A. The multiple layer potential for the biharmonic equation in n variables. Atti Accad.Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei, Serie 9, Mat. Appl., 3, (1992), 241-259.Received 23.05.2014; revised 22.11.2014; accepted 29.12.2014.Authors’ address:A. Cialdea, E. Dolce, V. Leonessa, A. MalaspinaDepartment of Mathematics, Computer Science and EconomicsUniversity of BasilicataV.le dell’Ateneo Lucano, 10, 85100 PotenzaItalyE-mail: cialdea@email.it, emanuela.dolce@unibas.itvita.leonessa@unibas.it, angelica.malaspina@unibas.it

On the potential theory in the linear theory of viscoelastic materials with voids

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On the potential theory in the linear theory of viscoelastic materials with voids

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