Signorini problem as a variational limit of obstacle problems in nonlinear elasticity. (English) Zbl 07926334
Summary: An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.
MSC:
| 74-XX | Mechanics of deformable solids |
| 49-XX | Calculus of variations and optimal control; optimization |
Keywords:
calculus of variations; Signorini problem; linear elasticity; nonlinear elasticity; finite elasticity; gamma-convergence; asymptotic analysis; unilateral constraintReferences:
| [1] | D. R. Adams, L. I. Hedberg, Function spaces and potential theory, Springer, 1999. |
| [2] | L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Academic, 2000. · Zbl 0957.49001 |
| [3] | V. Agostiniani, G. Dal Maso, A. De Simone, , Linear elasticity obtained from finite elasticity by \Gamma-convergence under weak coerciveness conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 715-735. https://doi.org/10.1016/j.anihpc.2012.04.001 doi: 10.1016/j.anihpc.2012.04.001 · Zbl 1250.74008 · doi:10.1016/j.anihpc.2012.04.001 |
| [4] | R. Alicandro, G. Dal Maso, G. Lazzaroni, M. Palombaro, Derivation of a linearised elasticity model from singularly perturbed multiwell energy functionals, Arch. Rational Mech. Anal., 230 (2018), 1-45. https://doi.org/10.1007/s00205-018-1240-6 doi: 10.1007/s00205-018-1240-6 · Zbl 1398.35229 · doi:10.1007/s00205-018-1240-6 |
| [5] | R. Alicandro, G. Lazzaroni, M. Palombaro, Derivation of linear elasticity for a general class of atomistic energies, SIAM J. Math. Anal., 53 (2021), 5060-5093. https://doi.org/10.1137/21M1397179 doi: 10.1137/21M1397179 · Zbl 1482.74011 · doi:10.1137/21M1397179 |
| [6] | G. Anzellotti, S. Baldo, D. Percivale, Dimension reduction in variational problems, asymptotic development in \Gamma-convergence and thin structures in elasticity, Asymptotic Anal., 9 (1994), 61-100. https://doi.org/10.3233/ASY-1994-9105 doi: 10.3233/ASY-1994-9105 · Zbl 0811.49020 · doi:10.3233/ASY-1994-9105 |
| [7] | H. Attouch, G. Buttazzo, G. Michaille, Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization, 2 Eds., Society for Industrial and Applied Mathematics, 2014. https://doi.org/10.1137/1.9781611973488 · Zbl 1311.49001 · doi:10.1137/1.9781611973488 |
| [8] | B. Audoly, Y. Pomeau, Elasticity and geometry, Oxford University Press, 2010. · Zbl 1223.74001 |
| [9] | C. Baiocchi, F. Gastaldi, F. Tomarelli, Inéquations variationnelles non coercives, C. R. Acad. Sci. Paris, 299 (1984), 647-650. |
| [10] | C. Baiocchi, F. Gastaldi, F. Tomarelli, Some existence results on noncoercive variational inequalities, Ann. Scuola Normale Sup. Pisa., Cl.Sci., 13 (1986), 617-659. · Zbl 0644.49004 |
| [11] | T. Bagby, Quasi topologies and rational approximation, J. Funct. Anal., 10 (1972), 259-268. https://doi.org/10.1016/0022-1236(72)90025-0 doi: 10.1016/0022-1236(72)90025-0 · Zbl 0266.30024 · doi:10.1016/0022-1236(72)90025-0 |
| [12] | C. Baiocchi, G. Buttazzo, F. Gastaldi, F. Tomarelli, General existence theorems for unilateral problems in continuum mechanics, Arch. Rational Mech. Anal., 100 (1988), 149-189. https://doi.org/10.1007/BF00282202 doi: 10.1007/BF00282202 · Zbl 0646.73011 · doi:10.1007/BF00282202 |
| [13] | P. Bella, R. V. Kohn, Wrinkles as the result of compressive stresses in an annular thin film, Commun. Pure Appl. Math., 67 (2014), 693-747. https://doi.org/10.1002/cpa.21471 doi: 10.1002/cpa.21471 · Zbl 1302.74105 · doi:10.1002/cpa.21471 |
| [14] | G. Buttazzo, F. Tomarelli, Compatibility conditions for nonlinear Neumann problems, Adv. Math., 89 (1991), 127-143. https://doi.org/10.1016/0001-8708(91)90076-J doi: 10.1016/0001-8708(91)90076-J · Zbl 0746.49012 · doi:10.1016/0001-8708(91)90076-J |
| [15] | M. Carriero, A. Leaci, F. Tomarelli, Strong solution for an elastic-plastic plate, Calc. Var. Partial Differ. Equ., 2 (1994), 219-240. https://doi.org/10.1007/BF01191343 doi: 10.1007/BF01191343 · Zbl 0806.49006 · doi:10.1007/BF01191343 |
| [16] | G. Dal Maso, An introduction to \Gamma-convergence, Boston: Birkhäuser Boston Inc., 1993. · Zbl 0816.49001 |
| [17] | G. Dal Maso, P. Longo, \Gamma-limits of obstacles, Ann. Mat. Pura Appl., 128 (1981), 1-50. https://doi.org/10.1007/BF01789466 doi: 10.1007/BF01789466 · Zbl 0467.49004 · doi:10.1007/BF01789466 |
| [18] | G. Dal Maso, M. Negri, D. Percivale, Linearized elasticity as \Gamma-limit of finite elasticity, Set-Valued Anal., 10 (2002), 165-183. https://doi.org/10.1023/A:1016577431636 doi: 10.1023/A:1016577431636 · Zbl 1009.74008 · doi:10.1023/A:1016577431636 |
| [19] | L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions, New York: Chapman and Hall/CRC, 2015. https://doi.org/10.1201/b18333 · Zbl 1310.28001 · doi:10.1201/b18333 |
| [20] | M. Egert, P. Tolksdorf, Characterizations of Sobolev functions that vanish on a part of the boundary, Discrete Cont. Dyn. Syst.-Ser. S, 10 (2016), 729-743. https://doi.org/10.3934/dcdss.2017037 doi: 10.3934/dcdss.2017037 · Zbl 1378.46024 · doi:10.3934/dcdss.2017037 |
| [21] | G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ. Ser., Rend., Cl. Sci. Fis. Mat. Nat., 8 (1963), 138-142. · Zbl 0128.18305 |
| [22] | G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti. Acad. Naz. Lincei. Mem. Cl. Sci. Nat., 8 (1964), 91-140. · Zbl 0146.21204 |
| [23] | G. Fichera, Boundary value problems of elasticity with unilateral constraints, In: C. Truesdell, Linear theories of elasticity and thermoelasticity, Springer, 1972,391-424. https://doi.org/10.1007/978-3-662-39776-3_4 · doi:10.1007/978-3-662-39776-3_4 |
| [24] | G. Frieseke, R. D. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Commun. Pure Appl. Math., 55 (2002), 1461-1506. https://doi.org/10.1002/cpa.10048 doi: 10.1002/cpa.10048 · Zbl 1021.74024 · doi:10.1002/cpa.10048 |
| [25] | G. Frieseke, R. D. James, S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Rational Mech. Anal., 180 (2006), 183-236. https://doi.org/10.1007/s00205-005-0400-7 doi: 10.1007/s00205-005-0400-7 · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7 |
| [26] | D. Grandi, M. Kru\check{\hbox{z}}ik, E. Mainini, U. Stefanelli, Equilibrium for multiphase solids with Eulerian interfaces, J. Elast., 142 (2020), 409-431. https://doi.org/10.1007/s10659-020-09800-w doi: 10.1007/s10659-020-09800-w · Zbl 1459.74151 · doi:10.1007/s10659-020-09800-w |
| [27] | M. E. Gurtin, An introduction to continuum mechanics, Academic Press, 1981. · Zbl 0559.73001 |
| [28] | V. P. Havin, Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR, 178 (1968), 1025-1028. · Zbl 0182.40201 |
| [29] | R. Haller-Dintelmann, J. Rehberg, M. Egert, Hardy’s inequality for functions vanishing on a part of the boundary, Potential Anal., 43 (2015), 49-78. https://doi.org/10.1007/s11118-015-9463-8 doi: 10.1007/s11118-015-9463-8 · Zbl 1331.26031 · doi:10.1007/s11118-015-9463-8 |
| [30] | T. Kilpelainen, A remark on the uniqueness of quasicontinuous functions, Ann. Acad. Sci. Fenn. Math., 23 (1998), 261-262. · Zbl 0919.31006 |
| [31] | D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, New York: Academic Press, 1980. · Zbl 0457.35001 |
| [32] | A. E. Love, A treatise on the mathematical theory of elasticity, Dover, 1944. · Zbl 0063.03651 |
| [33] | J. L. Lions, G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math., 20 (1967), 493-519. https://doi.org/10.1002/cpa.3160200302 doi: 10.1002/cpa.3160200302 · Zbl 0152.34601 · doi:10.1002/cpa.3160200302 |
| [34] | F. Maddalena, D. Percivale, Variational models for peeling problems, Interfaces Free Bound., 10 (2008), 503-516. https://doi.org/10.4171/ifb/199 doi: 10.4171/ifb/199 · Zbl 1160.35310 · doi:10.4171/ifb/199 |
| [35] | F. Maddalena, D. Percivale, G. Puglisi, L. Truskinowsky, Mechanics of reversible unzipping, Continuum Mech. Thermodyn., 21 (2009), 251-268. https://doi.org/10.1007/s00161-009-0108-2 doi: 10.1007/s00161-009-0108-2 · Zbl 1234.74013 · doi:10.1007/s00161-009-0108-2 |
| [36] | F. Maddalena, D. Percivale, F. Tomarelli, Adhesive flexible material structures, Discrete Cont. Dyn. Syst.-Ser. S, 17 (2012), 553-574. https://doi.org/10.3934/dcdsb.2012.17.553 doi: 10.3934/dcdsb.2012.17.553 · Zbl 1235.49038 · doi:10.3934/dcdsb.2012.17.553 |
| [37] | F. Maddalena, D. Percivale, F. Tomarelli, Elastic structures in adhesion interaction, In: G. Buttazzo, A. Frediani, Variational analysis and aerospace engineering: mathematical challenges for aerospace design, Springer, 66 (2012), 289-304. https://doi.org/10.1007/978-1-4614-2435-2_12 · Zbl 1243.76003 · doi:10.1007/978-1-4614-2435-2_12 |
| [38] | F. Maddalena, D. Percivale, F. Tomarelli, Local and nonlocal energies in adhesive interaction, IMA J. Appl. Math., 81 (2016), 1051-1075. https://doi.org/10.1093/imamat/hxw044 doi: 10.1093/imamat/hxw044 · Zbl 1406.35396 · doi:10.1093/imamat/hxw044 |
| [39] | F. Maddalena, D. Percivale, F. Tomarelli, Variational problems for Föppl-von Kármán plates, SIAM J. Math. Anal., 50 (2018), 251-282. https://doi.org/10.1137/17M1115502 doi: 10.1137/17M1115502 · Zbl 1394.49014 · doi:10.1137/17M1115502 |
| [40] | F. Maddalena, D. Percivale, F. Tomarelli, The gap between linear elasticity and the variational limit of finite elasticity in pure traction problems, Arch. Rational Mech. Anal., 234 (2019), 1091-1120. https://doi.org/10.1007/s00205-019-01408-2 doi: 10.1007/s00205-019-01408-2 · Zbl 1429.74022 · doi:10.1007/s00205-019-01408-2 |
| [41] | F. Maddalena, D. Percivale, F. Tomarelli, A new variational approach to linearization of traction problems in elasticity, J. Optim. Theory Appl., 182 (2019), 383-403, https://doi.org/10.1007/s10957-019-01533-8 doi: 10.1007/s10957-019-01533-8 · Zbl 1420.49014 · doi:10.1007/s10957-019-01533-8 |
| [42] | F. Maddalena, D. Percivale, F. Tomarelli, Elastic-brittle reinforcement of flexural structures, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 32 (2021), 691-724. https://doi.org/10.4171/RLM/954 doi: 10.4171/RLM/954 · Zbl 1486.49013 · doi:10.4171/RLM/954 |
| [43] | E. Mainini, R. Ognibene, D. Percivale, Asymptotic behavior of constrained local minimizers in finite elasticity, J. Elast., 152 (2022), 1-27. https://doi.org/10.1007/s10659-022-09946-9 doi: 10.1007/s10659-022-09946-9 · Zbl 1513.74124 · doi:10.1007/s10659-022-09946-9 |
| [44] | E. Mainini, D. Percivale, Variational linearization of pure traction problems in incompressible elasticity, Z. Angew. Math. Phys., 71 (2020), 146. https://doi.org/10.1007/s00033-020-01377-7 doi: 10.1007/s00033-020-01377-7 · Zbl 1447.74007 · doi:10.1007/s00033-020-01377-7 |
| [45] | E. Mainini, D. Percivale, Sharp conditions for the linearization of finite elasticity, Calc. Var. Partial Differ. Equ., 60 (2021), 164. https://doi.org/10.1007/s00526-021-02037-y doi: 10.1007/s00526-021-02037-y · Zbl 1486.49019 · doi:10.1007/s00526-021-02037-y |
| [46] | C. Maor, M. G. Mora, Reference configurations versus optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces, J. Nonlinear Sci., 31 (2021), 62. https://doi.org/10.1007/s00332-021-09716-2 doi: 10.1007/s00332-021-09716-2 · Zbl 1470.74013 · doi:10.1007/s00332-021-09716-2 |
| [47] | M. G. Mora, F. Riva, Pressure live loads and the variational derivation of linear elasticity, Proc. Roy. Soc. Edinb., 153 (2022), 1929-1964. https://doi.org/10.1017/prm.2022.79 doi: 10.1017/prm.2022.79 · Zbl 1539.74065 · doi:10.1017/prm.2022.79 |
| [48] | D. Percivale, F. Tomarelli, Scaled Korn-Poincaré inequality in BD and a model of elastic plastic cantilever, Asymptotic Anal., 23 (2000), 291-311. · Zbl 0978.74045 |
| [49] | D. Percivale, F. Tomarelli, From SBD to SBH: the elastic-plastic plate, Interfaces Free Bound., 4 (2002), 137-165. https://doi.org/10.4171/ifb/56 doi: 10.4171/ifb/56 · Zbl 0998.35057 · doi:10.4171/ifb/56 |
| [50] | D. Percivale, F. Tomarelli, A variational principle for plastic hinges in a beam, Math. Mod. Meth. Appl. Sci., 19 (2009), 2263-2297. https://doi.org/10.1142/S021820250900411X doi: 10.1142/S021820250900411X · Zbl 1187.49008 · doi:10.1142/S021820250900411X |
| [51] | D. Percivale, F. Tomarelli, Smooth and broken minimizers of some free discontinuity problems, In: P. Colli, A. Favini, E. Rocca, G. Schimperna, J. Sprekels, Solvability, regularity, and optimal control of boundary value problems for PDEs, Springer INdAM Series, 22 (2017), 431-468, https://doi.org/10.1007/978-3-319-64489-9_17 · doi:10.1007/978-3-319-64489-9_17 |
| [52] | P. Podio-Guidugli, On the validation of theories of thin elastic structures, Meccanica, 49 (2014), 1343-1352. https://doi.org/10.1007/s11012-014-9901-5 doi: 10.1007/s11012-014-9901-5 · Zbl 1375.74064 · doi:10.1007/s11012-014-9901-5 |
| [53] | B. D. Reddy, F. Tomarelli, The obstacle problem for an elastoplastic body, Appl. Math. Optim., 21 (1990), 89-110. https://doi.org/10.1007/BF01445159 doi: 10.1007/BF01445159 · Zbl 0709.73020 · doi:10.1007/BF01445159 |
| [54] | A. Signorini, Questioni di elasticit non linearizzata e semilinearizzata, Rend. Mat. Appl., 18 (1959), 95-139. · Zbl 0091.38006 |
| [55] | F. Tomarelli, Signorini problem in Hencky plasticity, Ann. Univ. Ferrara, 36 (1990), 73-84. https://doi.org/10.1007/BF02837208 doi: 10.1007/BF02837208 · Zbl 0772.73033 · doi:10.1007/BF02837208 |
| [56] | C. Truesdell, W. Noll, The non-linear field theories of mechanics, In: The non-linear field theories of mechanics/die nicht-linearen feldtheorien der mechanik, Springer, 1965, 1-541. https://doi.org/10.1007/978-3-642-46015-9_1 · Zbl 0137.19501 · doi:10.1007/978-3-642-46015-9_1 |
| [57] | W. P. Ziemer, Weakly differentiable functions, Springer, 1989. · Zbl 0692.46022 |
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