Abstract
We develop a theory of existence of minimizers of energy functionals in vectorial problems based on a nonlocal gradient under Dirichlet boundary conditions. The model shares many features with the peridynamics model and is also applicable to nonlocal solid mechanics, especially nonlinear elasticity. This nonlocal gradient was introduced in an earlier work, inspired by Riesz’ fractional gradient, but suitable for bounded domains. The main assumption on the integrand of the energy is polyconvexity. Thus, we adapt the corresponding results of the classical case to this nonlocal context, notably, Piola’s identity, the integration by parts of the determinant and the weak continuity of the determinant. The proof exploits the fact that every nonlocal gradient is a classical gradient.
Funding source: Agencia Estatal de Investigación
Award Identifier / Grant number: PID2020-116207GB-I00
Award Identifier / Grant number: PID2021-124195NB-C32
Award Identifier / Grant number: CEX2019-000904-S
Funding source: Junta de Comunidades de Castilla-La Mancha
Award Identifier / Grant number: SBPLY/19/180501/000110
Funding source: European Regional Development Fund
Award Identifier / Grant number: 2018/11744
Funding source: European Research Council
Award Identifier / Grant number: 834728
Funding statement: This work has been supported by the Agencia Estatal de Investigación of the Spanish Ministry of Research and Innovation, through projects PID2020-116207GB-I00 (J.C.B. and J.C.), PID2021-124195NB-C32 and the Severo Ochoa Programme for Centres of Excellence in R&D CEX2019-000904-S (C.M.-C.), by Junta de Comunidades de Castilla-La Mancha through project SBPLY/19/180501/000110 and European Regional Development Fund 2018/11744 (J.C.B. and J.C.), by the Madrid Government (Comunidad de Madrid, Spain) under the multiannual Agreement with UAM in the line for the Excellence of the University Research Staff in the context of the V PRICIT (Regional Programme of Research and Technological Innovation) (C.M.-C.), by the ERC Advanced Grant 834728 (C.M.-C.), and by Fundación Ramón Areces (J.C.).
References
[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Oxford University, New York, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar
[3] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1976/77), no. 4, 337–403. 10.1007/BF00279992Search in Google Scholar
[4] J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. Trans. Roy. Soc. A 306 (1982), no. 1496, 557–611. 10.1098/rsta.1982.0095Search in Google Scholar
[5] J. M. Ball, Singularities and computation of minimizers for variational problems, Foundations of Computational Mathematics (Oxford 1999), London Math. Soc. Lecture Note Ser. 284, Cambridge University, Cambridge (2001), 1–20. 10.1017/CBO9781107360198.002Search in Google Scholar
[6] J. M. Ball, Some open problems in elasticity, Geometry, Mechanics, and Dynamics, Springer, New York (2002), 3–59. 10.1007/0-387-21791-6_1Search in Google Scholar
[7] J. M. Ball, J. C. Currie and P. J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), no. 2, 135–174. 10.1016/0022-1236(81)90085-9Search in Google Scholar
[8] M. Barchiesi, D. Henao and C. Mora-Corral, Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 743–816. 10.1007/s00205-017-1088-1Search in Google Scholar
[9] J. C. Bellido, J. Cueto and C. Mora-Corral, Bond-based peridynamics does not converge to hyperelasticity as the horizon goes to zero, J. Elasticity 141 (2020), no. 2, 273–289. 10.1007/s10659-020-09782-9Search in Google Scholar
[10] J. C. Bellido, J. Cueto and C. Mora-Corral, Fractional Piola identity and polyconvexity in fractional spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), no. 4, 955–981. 10.1016/j.anihpc.2020.02.006Search in Google Scholar
[11] J. C. Bellido, J. Cueto and C. Mora-Corral, Γ-convergence of polyconvex functionals involving s-fractional gradients to their local counterparts, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 7. 10.1007/s00526-020-01868-5Search in Google Scholar
[12] J. C. Bellido, J. Cueto and C. Mora-Corral, Nonlocal gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings, Advances in Nonlinear Analysis, to appear (2023). Search in Google Scholar
[13] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar
[14] G. E. Comi and G. Stefani, A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up, J. Funct. Anal. 277 (2019), no. 10, 3373–3435. 10.1016/j.jfa.2019.03.011Search in Google Scholar
[15] G. E. Comi and G. Stefani, Failure of the local chain rule for the fractional variation, Port. Math. 80 (2023), no. 1/2, 1–25. 10.4171/PM/2096Search in Google Scholar
[16] J. Cueto, Mathematical analysis of fractional and nonlocal models from nonlinear solid mechanics, PhD thesis, Universidad de Castilla-La Mancha, 2021. Search in Google Scholar
[17] J. Cueto, C. Kreisbeck and H. Schönberger, A variational theory for integral functionals involving finite-horizon fractional gradients, preprint (2023), https://arxiv.org/abs/2302.05569. Search in Google Scholar
[18] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, New York, 2008. Search in Google Scholar
[19] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176 (2005), no. 2, 165–225. 10.1007/s00205-004-0351-4Search in Google Scholar
[20] Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci. 23 (2013), no. 3, 493–540. 10.1142/S0218202512500546Search in Google Scholar
[21] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar
[22]
I. Fonseca and G. Leoni,
Modern Methods in the Calculus of Variations:
[23] G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998), no. 8, 1319–1342. 10.1016/S0022-5096(98)00034-9Search in Google Scholar
[24] M. Giaquinta, G. Modica and J. Souček, Cartesian Currents in the Calculus of Variations. I, Ergeb. Math. Grenzgeb. (3) 37, Springer, Berlin, 1998. 10.1007/978-3-662-06218-0Search in Google Scholar
[25] D. Henao and C. Mora-Corral, Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity, Arch. Ration. Mech. Anal. 197 (2010), no. 2, 619–655. 10.1007/s00205-009-0271-4Search in Google Scholar
[26] D. Henao and C. Mora-Corral, Fracture surfaces and the regularity of inverses for BV deformations, Arch. Ration. Mech. Anal. 201 (2011), no. 2, 575–629. 10.1007/s00205-010-0395-6Search in Google Scholar
[27] D. Henao and C. Mora-Corral, Lusin’s condition and the distributional determinant for deformations with finite energy, Adv. Calc. Var. 5 (2012), no. 4, 355–409. 10.1515/acv.2011.016Search in Google Scholar
[28] C. Kreisbeck and H. Schönberger, Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation, Nonlinear Anal. 215 (2022), Paper No. 112625. 10.1016/j.na.2021.112625Search in Google Scholar
[29] G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math. 105, American Mathematical Society, Providence, 2009. 10.1090/gsm/105Search in Google Scholar
[30] T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies, Calc. Var. Partial Differential Equations 52 (2015), no. 1–2, 253–279. 10.1007/s00526-014-0711-3Search in Google Scholar
[31] S. Müller, T. Qi and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 2, 217–243. 10.1016/s0294-1449(16)30193-7Search in Google Scholar
[32] S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal. 131 (1995), no. 1, 1–66. 10.1007/BF00386070Search in Google Scholar
[33] T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations, Adv. Calc. Var. 8 (2015), no. 4, 321–336. 10.1515/acv-2014-0009Search in Google Scholar
[34] T.-T. Shieh and D. E. Spector, On a new class of fractional partial differential equations II, Adv. Calc. Var. 11 (2018), no. 3, 289–307. 10.1515/acv-2016-0056Search in Google Scholar
[35] M. Šilhavý, Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Contin. Mech. Thermodyn. 32 (2020), no. 1, 207–228. 10.1007/s00161-019-00797-9Search in Google Scholar
[36] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), no. 1, 175–209. 10.1016/S0022-5096(99)00029-0Search in Google Scholar
[37] E. M. Stein, Singular Integrals and Differentiability Properties of functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Search in Google Scholar
[38] V. Šverák, Regularity properties of deformations with finite energy, Arch. Ration. Mech. Anal. 100 (1988), no. 2, 105–127. 10.1007/BF00282200Search in Google Scholar
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