A degenerating Robin-type traction problem in a periodic domain

    Matteo Dalla Riva Affiliation
    ; Gennady Mishuris   Affiliation
    ; Paolo Musolino   Affiliation


We consider a linearly elastic material with a periodic set of voids. On the boundaries of the voids we set a Robin-type traction condition. Then, we investigate the asymptotic behavior of the displacement solution as the Robin condition turns into a pure traction one. To wit, there will be a matrix function b[k](·) that depends analytically on a real parameter k and vanishes for k = 0 and we multiply the Dirichlet-like part of the Robin condition by b[k](·). We show that the displacement solution can be written in terms of power series of k that converge for k in a whole neighborhood of 0. For our analysis we use the Functional Analytic Approach.

Keyword : Robin boundary value problem, integral representations, integral operators, integral equations methods, linearized elastostatics, periodic domain

How to Cite
Dalla Riva, M., Mishuris, G., & Musolino, P. (2023). A degenerating Robin-type traction problem in a periodic domain. Mathematical Modelling and Analysis, 28(3), 509–521.
Published in Issue
Sep 4, 2023
Abstract Views
PDF Downloads
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.


H. Ammari and H. Kang. Polarization and moment tensors, volume 162 of Applied Mathematical Sciences. Springer, New York, 2007. With applications to inverse problems and effective medium theory.

I. Andrianov, Gluzman S. and Mityushev V. Chapter 1 - L.A. Filshtinsky’s contribution to applied mathematics and mechanics of solids. In Mechanics and Physics of Structured Media, pp. 1–40. Academic Press, 2022.

Y.A. Antipov, O. Avila-Pozos, S.T. Kolaczkowski and A.B. Movchan. Mathematical model of delamination cracks on imperfect interfaces. International Journal of Solids and Structures, 38(36):6665–6697, 2001.

R. Bailey and R. Hicks. Behaviour of perforated plates under plane stress. Journal of Mechanical Engineering Science, 2(2):143–165, 1960.

M. Dalla Riva and M. Lanza de Cristoforis. Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach. Eurasian Math. J., 1(2):31–58, 2010.

M. Dalla Riva and M. Lanza de Cristoforis. A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach. Analysis (Munich), 30(1):67–92, 2010.

M. Dalla Riva, M. Lanza de Cristoforis and P. Musolino. Singularly perturbed boundary value problems–a functional analytic approach. Springer, Cham, 2021.

M. Dalla Riva, G. Mishuris and P. Musolino. Integral equation method for a Robin-type traction problem in a periodic domain. Trans. A. Razmadze Math. Inst., 176(3):349–360, 2022.

M. Dalla Riva and P. Musolino. A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite. SIAM J. Appl. Math., 73(1):24–46, 2013.

M. Dalla Riva and P. Musolino. A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach. Math. Methods Appl. Sci., 37(1):106–122, 2014.

P. Drygaś, S. Gluzman, V. Mityushev and W. Nawalaniec. Applied analysis of composite media–analytical and computational results for materials scientists and engineers. Elsevier/Woodhead Publishing, Cambridge, MA, 2020.

R. Falconi, P. Luzzini and P. Musolino. Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problem. Math. Methods Appl. Sci., 44(2):2111–2129, 2021.

D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

S. Gluzman, V. Mityushev and W. Nawalaniec. Computational analysis of structured media. Mathematical Analysis and Its Applications. Academic Press, London, 2018.

J.E. Goldberg and K.N. Jabbour. Stresses and displacements in perforated plates. Nuclear Structural Engineering, 2(4):360–381, 1965.

D. Gómez, S.A. Nazarov and M.E. Pérez. Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys., 69(2):35, 2018.

D. Gómez, S.A. Nazarov and M.-E. Pérez-Martínez. Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. J. Elasticity, 142(1):89–120, 2020.

E. Hille and R.S. Phillips. Functional analysis and semi-groups. American Mathematical Society, Providence, R.I., 1974.

G. Horvay. The Plane-Stress Problem of Perforated Plates. Journal of Applied Mechanics, 19(3):355–360, 04 2021.

R.C.J. Howland and L.N.G. Filon. Stresses in a plate containing an infinite row of holes. Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences, 148(864):471–491, 1935.

D. Kapanadze, G. Mishuris and E. Pesetskaya. Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials. Complex Var. Elliptic Equ., 60(1):1–23, 2015.

A. Klarbring and A.B. Movchan. Asymptotic modelling of adhesive joints. Mechanics of Materials, 28(1):137–145, 1998.

P. Luzzini. Regularizing properties of space-periodic layer heat potentials and applications to boundary value problems in periodic domains. Math. Methods Appl. Sci., 43(8):5273–5294, 2020.

G.W. Milton. The theory of composites, volume 88. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2023.

G. Mishuris. Interface crack and nonideal interface concept (Mode III). International Journal of Fracture, 107(3):279–296, 2001.

G. Mishuris. Imperfect transmission conditions for a thin weakly compressible interface. 2D problems. Arch. Mech., 56(2):103–115, 2004.

V.V. Mityushev, E. Pesetskaya and S.V. Rogosin. Analytical Methods for Heat Conduction in Composites and Porous Media, pp. 121–164. John Wiley & Sons, Ltd, 2008.

A.B. Movchan, N.V. Movchan and C.G. Poulton. Asymptotic models of fields in dilute and densely packed composites. Imperial College Press, London, 2002.

P. Musolino and G. Mishuris. A nonlinear problem for the Laplace equation with a degenerating Robin condition. Math. Methods Appl. Sci., 41(13):5211–5229, 2018.

V.Ya. Natanson. On stresses in an extended plate weakened by equal holes in chessboard arrangement. Mat. Sb., 42:616–636, 1935.

M. Sonato, A. Piccolroaz, W. Miszuris and G. Mishuris. General transmission conditions for thin elasto-plastic pressure-dependent interphase between dissimilar materials. International Journal of Solids and Structures, 64-65:9–21, 2015.

Y. Xu, Q. Tian and J. Xiao. Doubly periodic array of coated cylindrical inclusions model and applications for nanocomposites. Acta Mechanica, 231(2):661 – 681, 2020.