Weak solutions within the gradient-incomplete strain-gradient elasticity. (English) Zbl 1453.74012
Summary: In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.
MSC:
| 74B99 | Elastic materials |
| 74G22 | Existence of solutions of equilibrium problems in solid mechanics |
| 74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
References:
| [1] | F. dell’Isola, P. Seppecher, J. J. Alibert, T. Lekszycki, R. Grygoruk, M. Pawlikowski, D. Steigmann, I. Giorgio, U. Andreaus, E. Turco, M. Gołaszewski, N. Rizzi, C. Boutin, V. A. Eremeyev, A. Misra, et al., ‘‘Pantographic metamaterials: An example of mathematically driven design and of its technological challenges,’’ Continuum Mech. Thermodyn. 31, 851-884 (2019). |
| [2] | dell’Isola, F.; Seppecher, P.; Spagnuolo, M.; Barchiesi, E.; Hild, F.; Lekszycki, T.; Giorgio, I.; Placidi, L.; Andreaus, U.; Cuomo, M.; Eugster, S. R.; Pfaff, A.; Hoschke, K.; Langkemper, R.; Turco, E., Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses, Continuum Mech. Thermodyn., 31, 1231-1282 (2019) · doi:10.1007/s00161-019-00806-x |
| [3] | Mindlin, R. D., Micro-structure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302 · doi:10.1007/BF00248490 |
| [4] | Mindlin, R. D.; Eshel, N. N., On first strain-gradient theories in linear elasticity, Int. J. Solids Struct., 4, 109-124 (1968) · Zbl 0166.20601 · doi:10.1016/0020-7683(68)90036-X |
| [5] | Toupin, R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414 (1962) · Zbl 0112.16805 · doi:10.1007/BF00253945 |
| [6] | Maugin, G. A., Continuum Mechanics Through the Twentieth Century: A Concise Historical (2013), Dordrecht: Perspective Springer, Dordrecht · Zbl 1277.82007 · doi:10.1007/978-94-007-6353-1 |
| [7] | Maugin, G. A., Non-Classical Continuum Mechanics: A Dictionary (2017), Singapore: Springer, Singapore · Zbl 1346.76001 · doi:10.1007/978-981-10-2434-4 |
| [8] | Auffray, N.; dell’Isola, F.; Eremeyev, V. A.; Madeo, A.; Rosi, G., Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids, Math. Mech. Solids, 20, 375-417 (2015) · Zbl 1327.76008 · doi:10.1177/1081286513497616 |
| [9] | dell’Isola, F.; Andreaus, U.; Placidi, L., At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola, Math. Mech. Solids, 20, 887-928 (2015) · Zbl 1330.74006 · doi:10.1177/1081286513509811 |
| [10] | dell’Isola, F.; Della Corte, A.; Giorgio, I., Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives, Math. Mech. Solids, 22, 852-872 (2016) · Zbl 1371.74024 · doi:10.1177/1081286515616034 |
| [11] | J. G. Simmonds, A Brief on Tensor Analysis, 2nd ed. (Springer, New Yourk, 1994). · Zbl 0790.53014 |
| [12] | Eremeyev, V. A.; Cloud, M. J.; Lebedev, L. P., Applications of Tensor Analysis in Continuum Mechanics (2018), New Jersey: World Scientific, New Jersey · Zbl 1471.74001 · doi:10.1142/10959 |
| [13] | Healey, T. J.; Krömer, S., “Injective weak solutions in second-gradient nonlinear elasticity,” ESAIM: Control, Optim. Calculus Variat., 15, 863-871 (2009) · Zbl 1175.74017 · doi:10.1051/cocv:2008050 |
| [14] | Mareno, A.; Healey, T. J., Global continuation in second-gradient nonlinear elasticity, SIAM J. Math. Anal., 38, 103-115 (2006) · Zbl 1302.74023 · doi:10.1137/050626065 |
| [15] | Eremeyev, V. A.; dell’Isola, F.; Boutin, C.; Steigmann, D., Linear pantographic sheets: Existence and uniqueness of weak solutions, J. Elasticity, 132, 175-196 (2018) · Zbl 1398.74011 · doi:10.1007/s10659-017-9660-3 |
| [16] | Eremeyev, V. A.; Alzahrani, F. S.; Cazzani, A.; dell’Isola, F.; Hayat, T.; Turco, E.; Konopińska-Zmys, V., On existence and uniqueness of weak solutions for linear pantographic beam lattices models, Continuum Mech. Thermodyn., 31, 1843-1861 (2019) · doi:10.1007/s00161-019-00826-7 |
| [17] | Nikol’skii, S. M., On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russ. Math. Surv., 16, 55-104 (1961) · Zbl 0117.29101 · doi:10.1070/RM1961v016n05ABEH004113 |
| [18] | Besov, O. V., V. P. II’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (1978), New York: Wiley, New York · Zbl 0392.46022 |
| [19] | Besov, O. V., V. P. II’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (1979), New York: Wiley, New York · Zbl 0392.46023 |
| [20] | H. Triebel, Theory of Function Spaces III, Vol. 100 of Monographs in Mathematics (Birkhäuser, Basel, 2006). · Zbl 1104.46001 |
| [21] | Boutin, C.; dell’Isola, F.; Giorgio, I.; Placidi, L., Linear pantographic sheets: Asymptotic micro-macro models identification, Math. Mech. Complex Syst., 5, 127-162 (2017) · Zbl 1457.74171 · doi:10.2140/memocs.2017.5.127 |
| [22] | Placidi, L.; Andreaus, U.; Della Corte, A.; Lekszycki, T., Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients, Zeitschr. Angew. Math. Phys., 66, 3699-3725 (2015) · Zbl 1386.74018 · doi:10.1007/s00033-015-0588-9 |
| [23] | Placidi, L.; Greco, L.; Bucci, S.; Turco, E.; Rizzi, N. L., A second gradient formulation for a 2D fabric sheet with inextensible fibres, Zeitschr. Angew. Math. Phys., 67, 114 (2016) · Zbl 1432.74034 · doi:10.1007/s00033-016-0701-8 |
| [24] | Chandrasekhar, S., Liquid Crystals (1977), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0127.41403 |
| [25] | de Gennes, P. G.; Prost, J., The Physics of Liquid Crystals (1993), Oxford: Clarendon, Oxford |
| [26] | Oswald, P.; Pieranski, P., Smectic and Columnar Liquid Crystals, The Liquid Crystals Book Series (2006), Boca Raton: CRC, Boca Raton |
| [27] | V. A. Eremeyev and F. dell’Isola, ‘‘A note on reduced strain gradient elasticity,’’ in Generalized Models and Non-classical Approaches in Complex Materials 1, Ed. by H. Altenbach, J. Pouget, M. Rousseau, B. Collet, and T. Michelitsch, Vol. 89 of Advanced Structured Materials (Springer, Cham, 2018), pp. 301-310. · Zbl 1412.74004 |
| [28] | G. Fichera, ‘‘Existence theorems in elasticity,’’ in Handbuch der Physik, Ed. by S. Flügge (Springer, Berlin, 1972), Vol. VIa/2, pp. 347-389. |
| [29] | Ciarlet, P. G., Mathematical Elasticity, Three-Dimensional Elasticity (1988), Amsterdam: North-Holland, Amsterdam · Zbl 0648.73014 |
| [30] | Eremeyev, V. A.; Lebedev, L. P., Existence of weak solutions in elasticity, Math. Mech. Solids, 18, 204-217 (2013) · Zbl 07280041 · doi:10.1177/1081286512462187 |
| [31] | Adams, R. A.; Fournier, J. J. F., Sobolev Spaces (2003), Amsterdam: Academic, Amsterdam · Zbl 1098.46001 |
| [32] | Lebedev, L. P.; Vorovich, I. I., Functional Analysis in Mechanics (2003), New York: Springer, New York · Zbl 1014.46002 · doi:10.1007/b98851 |
| [33] | V. Maz’ya, Sobolev Spaces: With Applications to Elliptic Partial Differential Equations, Vol. 342 of Grundlehren der mathematischen Wissenschaften, 2nd ed. (Springer, Berlin, 2011). · Zbl 1217.46002 |
| [34] | Abali, B. E.; Müller, W. H.; dell’Isola, F., Theory and computation of higher gradient elasticity theories based on action principles, Arch. Appl. Mech., 87, 1495-1510 (2017) · doi:10.1007/s00419-017-1266-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
