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Mathematical models for nonlocal elastic composite materials. (English) Zbl 1373.35300

Summary: In this paper we derive and solve nonlocal elasticity a model describing the elastic behavior of composite materials, involving the fractional Laplacian operator. In dimension one we consider in (\(\mathcal D\)) the case of a nonlocal elastic rod restrained at the ends, and we completely solve the problem showing the existence of a unique weak solution and providing natural sufficient conditions under which this solution is actually a classical solution of the problem. For the model (\(\mathcal D\)) we also perform numerical simulations and a parametric analysis, in order to highlight the response of the rod, in terms of displacements and strains, according to different values of the mechanical characteristics of the material. The main novelty of this approach is the extension of the central difference method by the numerical estimate of the fractional Laplacian operator through a finite-difference quadrature technique. For higher dimensions \(N\geq 2\) we study more general problems for which the existence of weak solutions is proved via variational methods. The obtained results provide an original contribute in the knowledge of composite materials with properties of nonlocal elasticity.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35J60 Nonlinear elliptic equations
74B20 Nonlinear elasticity
74E30 Composite and mixture properties

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[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.; Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[2] T. M. Atanackovic and B. Stankovic, Generalized wave equation in nonlocal elasticity, Acta Mech. 208 (2009), 1-10.; Atanackovic, T. M.; Stankovic, B., Generalized wave equation in nonlocal elasticity, Acta Mech., 208, 1-10 (2009) · Zbl 1397.74100
[3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699-714.; Autuori, G.; Fiscella, A.; Pucci, P., Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125, 699-714 (2015) · Zbl 1323.35015
[4] G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 977-1009.; Autuori, G.; Pucci, P., Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20, 977-1009 (2013) · Zbl 1273.35137
[5] G. Autuori and P. Pucci, Entire solutions of nonlocal elasticity models for composite materials, in preparation.; Autuori, G.; Pucci, P., Entire solutions of nonlocal elasticity models for composite materials · Zbl 1375.35536
[6] J. L. Bassani, A. Needleman and E. Van der Giessen, Plastic flow in a composite: A comparison of nonlocal continuum and discrete dislocation predictions, Int. J. Solids Struct. 8 (2001), 833-853.; Bassani, J. L.; Needleman, A.; Van der Giessen, E., Plastic flow in a composite: A comparison of nonlocal continuum and discrete dislocation predictions, Int. J. Solids Struct., 8, 833-853 (2001) · Zbl 1004.74006
[7] Z. P. Bazant, Why continuum damage is nonlocal: Micromechanics arguments, J. Eng. Mech. 117 (1991), 1070-1087.; Bazant, Z. P., Why continuum damage is nonlocal: Micromechanics arguments, J. Eng. Mech., 117, 1070-1087 (1991)
[8] P. W. R. Beaumont, On the problems of craking and the question of structural integrity of engineering composite materials, Appl. Compos. Mater. 21 (2014), 5-43.; Beaumont, P. W. R., On the problems of craking and the question of structural integrity of engineering composite materials, Appl. Compos. Mater., 21, 5-43 (2014)
[9] M. J. Beran and J. J. McCoy, Mean field variations in a statistical sample of heterogeneous linearly elastic solids, Int. J. Solids Struct. 6 (1970), 1035-1054.; Beran, M. J.; McCoy, J. J., Mean field variations in a statistical sample of heterogeneous linearly elastic solids, Int. J. Solids Struct., 6, 1035-1054 (1970) · Zbl 0217.23501
[10] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.; Brézis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011) · Zbl 1220.46002
[11] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490.; Brézis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, 486-490 (1983) · Zbl 0526.46037
[12] A. Carpinteri, P. Cornetti and A. Sapora, Static-kinematic fractional operators for fractal and nonlocal solids, Z. Angew. Math. Mech. 89 (2009), 207-217.; Carpinteri, A.; Cornetti, P.; Sapora, A., Static-kinematic fractional operators for fractal and nonlocal solids, Z. Angew. Math. Mech., 89, 207-217 (2009) · Zbl 1159.74001
[13] W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Amer. 115 (2004), 1424-1430.; Chen, W.; Holm, S., Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Amer., 115, 1424-1430 (2004)
[14] P. Cornetti, A. Carpinteri, A. G. Sapora, M. Di Paola and M. Zingales, An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus, XIX Congress AIMETA (Ancona 2009), Italian Association for Theoretical and Applied Mechanics, Ancona (2009), 14-17.; Cornetti, P.; Carpinteri, A.; Sapora, A. G.; Di Paola, M.; Zingales, M., An explicit mechanical interpretation of Eringen non-local elasticity by means of fractional calculus, XIX Congress AIMETA, 14-17 (2009)
[15] G. Cottone, M. Di Paola and M. Zingales, Elastic waves propagation in 1D fractional nonlocal continuum, Phys. E 42 (2009), 95-103.; Cottone, G.; Di Paola, M.; Zingales, M., Elastic waves propagation in 1D fractional nonlocal continuum, Phys. E, 42, 95-103 (2009) · Zbl 1181.74011
[16] D. C. de Morais Filho, M. A. S. Souto and J. M. do Ó, A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones 19 (2000), 1-17.; de Morais Filho, D. C.; Souto, M. A. S.; do Ó, J. M., A compactness embedding lemma, a principle of symmetric criticality and applications to elliptic problems, Proyecciones, 19, 1-17 (2000)
[17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573.; Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023
[18] M. Di Paola, G. Failla, A. Pirrotta, A. Sofi and M. Zingales, The mechanically based non-local elasticity: An overview of main results and future challenges, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 371 (2013), Article ID 20120433.; Di Paola, M.; Failla, G.; Pirrotta, A.; Sofi, A.; Zingales, M., The mechanically based non-local elasticity: An overview of main results and future challenges, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 371 (2013) · Zbl 1327.74023
[19] M. Di Paola and M. Zingales, Long-range cohesive interactions of nonlocal continuum faced by fractional calculus, Int. J. Solids Struct. 45 (2008), 5642-5659.; Di Paola, M.; Zingales, M., Long-range cohesive interactions of nonlocal continuum faced by fractional calculus, Int. J. Solids Struct., 45, 5642-5659 (2008) · Zbl 1273.74005
[20] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353.; Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015
[21] A. C. Eringen, Vistas of nonlocal continuum physics, Int. J. Eng. Sci. 30 (1992), 1551-1565.; Eringen, A. C., Vistas of nonlocal continuum physics, Int. J. Eng. Sci., 30, 1551-1565 (1992) · Zbl 0769.73004
[22] A. C. Eringen and D. G. B. Edelen, On nonlocal elasticity, Int. J. Eng. Sci. 10 (1972), 233-248.; Eringen, A. C.; Edelen, D. G. B., On nonlocal elasticity, Int. J. Eng. Sci., 10, 233-248 (1972) · Zbl 0247.73005
[23] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156-170.; Fiscella, A.; Valdinoci, E., A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94, 156-170 (2014) · Zbl 1283.35156
[24] M. G. D. Geers, R. de Borst and T. Peijs, Mixed numerical-experimental identification of non-local characteristics of random-fibre-reinforced composites, Compos. Sci. Technol. 59 (1999), 1569-1578.; Geers, M. G. D.; de Borst, R.; Peijs, T., Mixed numerical-experimental identification of non-local characteristics of random-fibre-reinforced composites, Compos. Sci. Technol., 59, 1569-1578 (1999)
[25] K. L. Hiebert, An evaluation of mathematical software that solves systems of nonlinear equations, ACM Trans. Math. Softw. 8 (1982), 5-20.; Hiebert, K. L., An evaluation of mathematical software that solves systems of nonlinear equations, ACM Trans. Math. Softw., 8, 5-20 (1982) · Zbl 0477.65037
[26] Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference-quadrature approach, SIAM J. Numer. Anal. 52 (2014), 3056-3084.; Huang, Y.; Oberman, A., Numerical methods for the fractional Laplacian: A finite difference-quadrature approach, SIAM J. Numer. Anal., 52, 3056-3084 (2014) · Zbl 1316.65071
[27] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (2014), 1-15.; Katugampola, U. N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6, 1-15 (2014) · Zbl 1317.26008
[28] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006.; Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006) · Zbl 1092.45003
[29] K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys. 317 (2013), 563-591.; Kirkpatrick, K.; Lenzmann, E.; Staffilani, G., On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317, 563-591 (2013) · Zbl 1258.35182
[30] P.-L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315-334.; Lions, P.-L., Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49, 315-334 (1982) · Zbl 0501.46032
[31] V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, 2nd ed., Grundlehren Math. Wiss. 342, Springer, Berlin, 2011.; Maz’ya, V., Sobolev Spaces with Applications to Elliptic Partial Differential Equations (2011) · Zbl 1217.46002
[32] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems. With a foreword by Jean Mawhin, Encyclopedia Math. Appl. 162, Cambridge University Press, Cambridge, 2016.; Molica Bisci, G.; Radulescu, V. D.; Servadei, R., Variational Methods for Nonlocal Fractional Problems. With a foreword by Jean Mawhin (2016) · Zbl 1356.49003
[33] J. J. Moré, B. S. Garbow and K. E. Hillstrom, User Guide for MINPACK-1, Technical Report ANL-80-74, Argonne National Laboratory, Argonne, 1980.; Moré, J. J.; Garbow, B. S.; Hillstrom, K. E., User Guide for MINPACK-1 (1980)
[34] S. Neukamm and I. Velcic, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Math. Models Methods Appl. Sci. 23 (2013), 2701-2748.; Neukamm, S.; Velcic, I., Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity, Math. Models Methods Appl. Sci., 23, 2701-2748 (2013) · Zbl 1282.35039
[35] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractionalp-Laplacian in \(R^N\), Calc. Var. Partial Differential Equations 54 (2015), 2785-2806.; Pucci, P.; Xiang, M.; Zhang, B., Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractionalp-Laplacian in \(R^N\), Calc. Var. Partial Differential Equations, 54, 2785-2806 (2015) · Zbl 1329.35338
[36] P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), 1529-1566.; Pucci, P.; Zhang, Q., Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations, 257, 1529-1566 (2014) · Zbl 1292.35135
[37] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975.; Reed, M.; Simon, B., Methods of Modern Mathematical Physics. II: Fourier Analysis, Self-Adjointness (1975) · Zbl 0308.47002
[38] A. Sapora, P. Cornetti and A. Carpinteri, Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 63-74.; Sapora, A.; Cornetti, P.; Carpinteri, A., Wave propagation in nonlocal elastic continua modelled by a fractional calculus approach, Commun. Nonlinear Sci. Numer. Simul., 18, 63-74 (2013) · Zbl 1253.35203
[39] S. A. Silling, Origin and Effect of Nonlocality in a Composite, Sandia Report SAND2013-8140, Sandia National Laboratories, Albuquerque, 2014.; Silling, S. A., Origin and Effect of Nonlocality in a Composite (2014)
[40] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971.; Stein, E. M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971) · Zbl 0232.42007
[41] V. E. Tarasov, Fractional gradient elasticity from spatial dispersion law, Condens. Matter Phys. 2014 (2014), Article ID 794097.; Tarasov, V. E., Fractional gradient elasticity from spatial dispersion law, Condens. Matter Phys. (2014)
[42] J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S 7 (2014), 857-885.; Vázquez, J. L., Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7, 857-885 (2014) · Zbl 1290.26010
[43] Q. Yang, F. Liu and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model. 34 (2010), 200-218.; Yang, Q.; Liu, F.; Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 34, 200-218 (2010) · Zbl 1185.65200
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