Minimizers of nonlocal polyconvex energies in nonlocal hyperelasticity

References

[1] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Suche 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/9780198502456.001.0001Suche 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/BF00279992Suche 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.1982.0095Suche 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/CBO9781107360198.002Suche 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_1Suche 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)90085-9Suche 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/s00205-017-1088-1Suche 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/s10659-020-09782-9Suche 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.2020.02.006Suche 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/s00526-020-01868-5Suche 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). 10.1515/anona-2022-0316Suche in Google Scholar

[13] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Suche 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.2019.03.011Suche 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/2096Suche 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. Suche 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. 10.1007/s13540-023-00196-7Suche in Google Scholar

[18] B. Dacorogna, Direct Methods in the Calculus of Variations, 2nd ed., Appl. Math. Sci. 78, Springer, New York, 2008. Suche 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/s00205-004-0351-4Suche 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/S0218202512500546Suche 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. Suche in Google Scholar

[22] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: L p Spaces, Springer Monogr. Math., Springer, New York, 2007. Suche in Google Scholar

[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)00034-9Suche 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. 10.1007/978-3-662-06218-0Suche 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/s00205-009-0271-4Suche 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/s00205-010-0395-6Suche 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.2011.016Suche 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.2021.112625Suche in Google Scholar

[29] G. Leoni, A First Course in Sobolev Spaces, Grad. Stud. Math. 105, American Mathematical Society, Providence, 2009. 10.1090/gsm/105Suche 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/s00526-014-0711-3Suche 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)30193-7Suche 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/BF00386070Suche 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-2014-0009Suche 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-2016-0056Suche 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/s00161-019-00797-9Suche 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)00029-0Suche in Google Scholar

[37] E. M. Stein, Singular Integrals and Differentiability Properties of functions, Princeton Math. Ser. 30, Princeton University, Princeton, 1970. 10.1515/9781400883882Suche 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/BF00282200Suche in Google Scholar