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A variational approach for the mixed problem in the elastostatics of bodies with dipolar structure. (English) Zbl 1404.70041

Summary: In this study, we address the mixed initial boundary value problem in the elastostatics of dipolar bodies. Using the equilibrium equations, we build the operator of dipolar elasticity and prove that this operator is positively defined even in the general case of an elastic inhomogeneous and anisotropic dipolar solid. This helps us to prove the existence of a generalized solution for first boundary value problem and also the uniqueness of the solution. Moreover, relying on this property of the operator of dipolar elasticity to be positively defined, we can apply the known variational method proposed by Mikhlin.

MSC:

70G75 Variational methods for problems in mechanics
53A45 Differential geometric aspects in vector and tensor analysis
58E30 Variational principles in infinite-dimensional spaces
74A20 Theory of constitutive functions in solid mechanics
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[1] Eringen, AC, Theory of thermo-microstretch elastic solids, Int. J. Engng. Sci., 28, 1291-1301, (1990) · Zbl 0718.73014 · doi:10.1016/0020-7225(90)90076-U
[2] Eringen, A.C.: Microcontinuum Field Theories. Springer, New York (1999) · Zbl 0953.74002 · doi:10.1007/978-1-4612-0555-5
[3] Iesan, D.; Pompei, A., Equilibrium theory of microstretch elastic solids, Int. J. Eng. Sci., 33, 399-410, (1995) · Zbl 0899.73461 · doi:10.1016/0020-7225(94)00067-T
[4] Iesan, D.; Quintanilla, R., Thermal stresses in microstretch bodies, Int. J. Eng. Sci., 43, 885-907, (2005) · Zbl 1211.74146 · doi:10.1016/j.ijengsci.2005.03.005
[5] Marin, M., Weak solutions in elasticity of dipolar porous materials, Math. Probl. Eng., 2008, 1-8, (2008) · Zbl 1149.74009 · doi:10.1155/2008/158908
[6] Marin, M., An approach of a heat-flux dependent theory for micropolar porous media, Meccanica, 51, 1127-1133, (2016) · Zbl 1381.74012 · doi:10.1007/s11012-015-0265-2
[7] Ciarletta, M., On the bending of microstretch elastic plates, Int. J. Eng. Sci., 37, 1309-1318, (1995) · Zbl 1210.74108 · doi:10.1016/S0020-7225(98)00123-2
[8] Marin, M.; Florea, O., On temporal behaviour of solutions in thermoelasticity of porous micropolar bodies, An. St. Univ. Ovidius Constanta-Seria Mathematics, 22, 169-188, (2014) · Zbl 1340.74023
[9] Marin, M.; Oechsner, A., An initial boundary value problem for modeling a piezoelectric dipolar body, Continuum Mech. Thermodyn., 30, 267-278, (2018) · Zbl 1392.74040 · doi:10.1007/s00161-017-0599-1
[10] Straughan, B.: Heat Waves. Applied Mathematical Sciences. Springer, New York (2011) · Zbl 1232.80001 · doi:10.1007/978-1-4614-0493-4
[11] Marin, M., Lagrange identity method for microstretch thermoelastic materials, J. Math. Anal. Appl., 363, 275-286, (2010) · Zbl 1267.74053 · doi:10.1016/j.jmaa.2009.08.045
[12] Mindlin, RD, Micro-structure in linear elasticity, Arch. Rational Mech. Anal., 16, 51-78, (1964) · Zbl 0119.40302 · doi:10.1007/BF00248490
[13] Green, AE; Rivlin, RS, Multipolar continuum mechanics, Arch. Rational Mech. Anal., 17, 113-147, (1964) · Zbl 0133.17604 · doi:10.1007/BF00253051
[14] Fried, E.; Gurtin, ME, Thermomechanics of the interface between a body and its environment, Contin. Mech. Thermodyn., 19, 253-271, (2007) · Zbl 1160.74303 · doi:10.1007/s00161-007-0053-x
[15] Brun, L., Méthodes énergetiques dans les systèmes linéaires, J. Mécanique, 8, 167-192, (1969) · Zbl 0194.26104
[16] Knops, RJ; Payne, LE, On uniqueness and continuous data dependence in dynamical problems of linear thermoelasticity, Int. J. Solids Struct., 6, 1173-1184, (1970) · Zbl 0209.56605 · doi:10.1016/0020-7683(70)90054-5
[17] Levine, HA, On a theorem of Knops and Payne in dynamical thermoelasticity, Arch. Rational Mech. Anal., 38, 290-307, (1970) · Zbl 0233.73024 · doi:10.1007/BF00281526
[18] Rionero, S.; Chirita, S., The Lagrange identity method in linear thermoelasticity, Int. J. Eng. Sci., 25, 935-947, (1987) · Zbl 0617.73007 · doi:10.1016/0020-7225(87)90126-1
[19] Wilkes, NS, Continuous dependence and instability in linear thermoelasticity, SIAM J. Appl. Math., 11, 292-299, (1980) · Zbl 0433.73013 · doi:10.1137/0511027
[20] Green, AE; Laws, N., On the entropy production inequality, Arch. Rational Mech. Anal., 45, 47-59, (1972) · doi:10.1007/BF00253395
[21] Mikhlin, S.G.: The Problem of the Minimum of a Quadratic Functional, Holden-Day Series in Mathematical Physics. Holden-Day Inc, San Francisco, Calif.-London-Amsterdam (1965)
[22] Iesan, D., Existence theorems in micropolar elastostatics, Int. J. Eng. Sci., 9, 59-78, (1971) · Zbl 0218.73003 · doi:10.1016/0020-7225(71)90013-9
[23] Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lectures Notes in Mathematics, Springer, Baltimore, MD (1965) · Zbl 0138.36104
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