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On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Abstract

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradientincomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class ofmetamaterials whosemicrostructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that themathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. https://doi.org/10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

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Category:
Articles
Type:
artykuły w czasopismach
Published in:
CONTINUUM MECHANICS AND THERMODYNAMICS no. 31, pages 1843 - 1861,
ISSN: 0935-1175
Language:
English
Publication year:
2019
Bibliographic description:
Eremeev V., Alzahrani F. S., Cazzani A., Dell’isola F., Hayat T., Turco E., Konopińska-Zmysłowska V.: On existence and uniqueness of weak solutions for linear pantographic beam lattices models// CONTINUUM MECHANICS AND THERMODYNAMICS -Vol. 31,iss. 6 (2019), s.1843-1861
DOI:
Digital Object Identifier (open in new tab) 10.1007/s00161-019-00826-7
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  1. Maugin, G.A.: A historical perspective of generalized continuum mechanics. In: Altenbach, H., Erofeev, V.I., Maugin, G.A. (eds.) Mechanics of Generalized Continua. From the Micromechanical Basics to Engineering Applications, pp. 3-19. Springer, Berlin (2011) open in new tab
  2. Maugin, G.A.: Generalized continuum mechanics: various paths. In: Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective, Springer, Dordrecht, pp. 223-241 (2013) open in new tab
  3. Maugin, G.A.: Non-classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017) open in new tab
  4. 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(4), 852-872 (2017) open in new tab
  5. 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(4), 375-417 (2015) open in new tab
  6. dell'Isola, F., Eremeyev, V.A.: Some introductory and historical remarks on mechanics of microstructured materials. In: dell'Isola, F., Eremeyev, V.A., Porubov, A. (eds.) Advances in Mechanics of Microstructured Media and Structures, pp. 1-20. Springer, Cham (2018) open in new tab
  7. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385-414 (1962) open in new tab
  8. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51-78 (1964) open in new tab
  9. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109-124 (1968) open in new tab
  10. Eugster, S.R., dell'Isola, F.: Exegesis of the introduction and sect. I from "Fundamentals of the Mechanics of Continua"** by E. Hellinger. ZAMM 97(4), 477-506 (2017) open in new tab
  11. Eugster, S.R., dell'Isola, F.: Exegesis of Sect. II and III.A from "Fundamentals of the Mechanics of Continua" by E. Hellinger. ZAMM 98(1), 31-68 (2018) open in new tab
  12. Eugster, S.R., dell'Isola, F.: Exegesis of Sect. III.B from "Fundamentals of the Mechanics of Continua" by E. Hellinger. ZAMM 98(1), 69-105 (2018) open in new tab
  13. Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212-234 (2019) open in new tab
  14. di Cosmo, F., Laudato, M., Spagnuolo, M.: Acoustic metamaterials based on local resonances: homogenization, optimization and applications. In: Generalized Models and Non-classical Approaches in Complex Materials 1, pp. 247-274. Springer, New York (2018) open in new tab
  15. Soubestre, J., Boutin, C.: Non-local dynamic behavior of linear fiber reinforced materials. Mech. Mater. 55, 16-32 (2012) open in new tab
  16. Turco, E., dell'Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. ZAMP 67(4), 1-28 (2016) open in new tab
  17. Boisse, P., Colmars, J., Hamila, N., Naouar, N., Steer, Q.: Bending and wrinkling of composite fiber preforms and prepregs. A review and new developments in the draping simulations. Compos. Part B: Eng. 141, 234-249 (2018) open in new tab
  18. dell'Isola, F., Steigmann, D.: A two-dimensional gradient-elasticity theory for woven fabrics. J. Elast. 118(1), 113-125 (2015) open in new tab
  19. dell'Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev 67(6), 060804-1-21 (2016) open in new tab
  20. Sabik, A.: Direct shear stress vs strain relation for fiber reinforced composites. Compos. Part B: Eng. 139, 24-30 (2018) open in new tab
  21. Berrehili, Y., Marigo, J.-J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181-207 (2014) open in new tab
  22. dell'Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lat- tices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. Ser. A 472(2185), 20150790 (2016) open in new tab
  23. Boutin, C., dell'Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127-162 (2017) open in new tab
  24. Turco, E., Golaszewski, M., Giorgio, I., D'Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. Part B: Eng. 118, 1-14 (2017) open in new tab
  25. Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell'Isola, F.: Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148-172 (2015) open in new tab
  26. Misra, A., Placidi, L., Scerrato, D.: A review of presentations and discussions of the workshop Computational mechanics of generalized continua and applications to materials with microstructure that was held in Catania 29-31 October 2015. Math. Mech. Solids 22(9), 1891-1904 (2017) open in new tab
  27. Placidi, L., Giorgio, I., Della Corte, A., Scerrato, D.: Euromech 563 Cisterna di Latina 17-21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: a review of presentations and discussions. Math. Mech. Solids 22(2), 144-157 (2017) open in new tab
  28. Eremeyev, V.A., dell'Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions. J. Elast. 132, 175-196 (2018). https://doi.org/10.1007/s10659-017-9660-3 open in new tab
  29. Eremeyev, V.A., dell'Isola, F.: A note on reduced strain gradient elasticity. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds.) Generalized Models and Non-classical Approaches in Complex Materials 1, pp. 301-310. Springer, Cham (2018) open in new tab
  30. Nikol'skii, S.M.: On imbedding, continuation and approximation theorems for differentiable functions of several variables. Russian Math. Surv. 16(5), 55 (1961) open in new tab
  31. Kachala, V.V., Khemchyan, L.L., Kashin, A.S., Orlov, N.V., Grachev, A.A., Zalesskiy, S.S., Ananikov, V.P.: Target-oriented analysis of gaseous, liquid and solid chemical systems by mass spectrometry, nuclear magnetic resonance spectroscopy and electron microscopy. Russian Chem. Rev. 82(7), 648-85 (2013) open in new tab
  32. Kashin, A.S., Ananikov, V.P.: A SEM study of nanosized metal films and metal nanoparticles obtained by magnetron sputtering. Russian Chem, Bull. 60(12), 2602-2607 (2011) open in new tab
  33. Alibert, J.-J., Seppecher, P., dell'Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51-73 (2003) open in new tab
  34. Seppecher, P., Alibert, J.-J., dell'Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. J. Phys. Conf. Ser. 319(1), 012018 (2011) open in new tab
  35. Turco, E., Golaszewski, M., Cazzani, A., Rizzi, N.L.: Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete Lagrangian model. Mech. Res. Commun. 76, 51-56 (2016) open in new tab
  36. Eugster, S.R., Hesch, C., Betsch, P., Glocker, C.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng. 97(2), 111-129 (2014) open in new tab
  37. Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Contin. Mech. Thermodyn. 30, 1103-1123 (2018) open in new tab
  38. Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 103(1), 1-21 (2017) open in new tab
  39. Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67(4), 95 (2016) open in new tab
  40. Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1-2), 215-234 (2016) open in new tab
  41. Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285-308 (2015) open in new tab
  42. Misra, A., Poorsolhjouy, P.: Grain-and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model. Mech. Res. Commun. 81, 1-6 (2017) open in new tab
  43. Misra, A., Poorsolhjouy, P.: Elastic behavior of 2D grain packing modeled as micromorphic media based on granular micromechanics. J. Eng. Mech. 143(1), C4016005 (2016) open in new tab
  44. Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: Nonlinear FEM 2D failure onset prediction of composite shells based on 6-parameter shell theory. Thin-Walled Struct. 105, 207-219 (2016) open in new tab
  45. Balobanov, V., Niiranen, J.: Locking-free variational formulations and isogeometric analysis for the Timoshenko beam models of strain gradient and classical elasticity. Comput. Methods Appl. Mech. Eng. 339, 137-159 (2018) open in new tab
  46. Niiranen, J., Balobanov, V., Kiendl, J., Hosseini, S.B.: Variational formulations, model comparisons and numerical methods for Euler-Bernoulli micro-and nano-beam models. Math. Mech. Solids 24(1), 312-335 (2019) open in new tab
  47. Greco, L., Cuomo, M., Contrafatto, L., Gazzo, S.: An efficient blended mixed B-spline formulation for removing membrane locking in plane curved Kirchhoff rods. Comput. Methods Appl. Mech. Eng. 324, 476-511 (2017) open in new tab
  48. Chróścielewski, J., Schmidt, R., Eremeyev, V.A.: Nonlinear finite element modeling of vibration control of plane rod-type structural members with integrated piezoelectric patches. Contin. Mech. Thermodyn. 31(1), 147-188 (2019) open in new tab
  49. Chróścielewski, J., Sabik, A., Sobczyk, B., Witkowski, W.: 2-D constitutive equations for orthotropic Cosserat type laminated shells in finite element analysis. Compos. Part B: Eng. 165, 335-353 (2019) open in new tab
  50. Maurin, F., Greco, F., Desmet, W.: Isogeometric analysis for nonlinear planar pantographic lattice: discrete and continuum models. Contin. Mech. Thermodyn. 31(4), 1051-1064 (2019) open in new tab
  51. Alfano, G., De Angelis, F., Rosati, L.: General solution procedures in elasto/viscoplasticity. Comput. Methods Appl. Mech. Eng. 190(39), 5123-5147 (2001) open in new tab
  52. Palazzo, V., Rosati, L., Valoroso, N.: Solution procedures for j 3 plasticity and viscoplasticity. Comput. Methods Appl. Mech. Eng. 191(8-10), 903-939 (2001) open in new tab
  53. Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6(2), 77-100 (2018) open in new tab
  54. Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474(2210), 20170878 (2018) open in new tab
  55. Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3), 56 (2018) open in new tab
  56. Marmo, F., Toraldo, F., Rosati, A., Rosati, L.: Numerical solution of smooth and rough contact problems. Meccanica 53(6), 1415-1440 (2018) open in new tab
  57. Nadler, B., Steigmann, D.J.: A model for frictional slip in woven fabrics. Comptes Rendus Mecanique 331(12), 797-804 (2003) open in new tab
  58. Golaszewski, M., Grygoruk, R., Giorgio, I., Laudato, M., Di Cosmo, F.: Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions. Contin. Mech. Thermodyn. 31, 1015-1034 (2019) open in new tab
  59. Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn. 31(1), 33-45 (2019) open in new tab
  60. Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, pp. 239-258. Springer, Berlin (2017) open in new tab
  61. Turco, E., Misra, A., Sarikaya, R., Lekszycki, T.: Quantitative analysis of deformation mechanisms in pantographic sub- structures: experiments and modeling. Contin. Mech. Thermodyn. 31(1), 209-223 (2019) open in new tab
  62. Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Müller, W.H., dell'Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun. 89, 6-10 (2018) open in new tab
  63. dell'Isola, F., Seppecher, P., Alibert, J.J., Lekszycki, T., Grygoruk, R., Pawlikowski, M., Steigmann, D., Giorgio, I., Andreaus, U., Turco, E., Gołaszewski, M., Rizzi, N., Boutin, C., Eremeyev, V.A., Misra, A., Placidi, L., Barchiesi, E., Greco, L., Cuomo, M., Cazzani, A., Corte, A.D., Battista, A., Scerrato, D., Eremeeva, I.Z., Rahali, Y., Ganghoffer, J.-F., Müller, W., Ganzosch, G., Spagnuolo, M., Pfaff, A., Barcz, K., Hoschke, K., Neggers, J., Hild, F.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. 31(4), 851-884 (2019) open in new tab
  64. Carlen, E.A., Carvalho, M.C., Esposito, R., Lebowitz, J.L., Marra, R.: Droplet minimizers for the Gates-Lebowitz-Penrose free energy functional. Nonlinearity 22(12), 2919-2952 (2009) open in new tab
  65. Eremeyev, V.A., Pietraszkiewicz, W.: The non-linear theory of elastic shells with phase transitions. J. Elast. 74(1), 67-86 (2004) open in new tab
  66. Pietraszkiewicz, W., Eremeyev, V.A., Konopińska, V.: Extended non-linear relations of elastic shells undergoing phase transitions. J. Appl. Math. Mech.-ZAMM 87(2), 150-159 (2007) open in new tab
  67. De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Potts models in the continuum. Uniqueness and exponential decay in the restricted ensembles. J. Stat. Phys. 133(2), 281-345 (2008) open in new tab
  68. De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in Potts models in the continuum. J. Stat. Phys. 134(2), 243-306 (2009) open in new tab
  69. Atai, A.A., Steigmann, D.J.: On the nonlinear mechanics of discrete networks. Arch. Appl. Mech. 67(5), 303-319 (1997) open in new tab
  70. Luo, C., Steigmann, D.J.: Bending and twisting effects in the three-dimensional finite deformations of an inextensible network. In: Advances in the Mechanics of Plates and Shells, pp. 213-228. Springer, Berlin (2001) open in new tab
  71. Steigmann, D.J.: Continuum theory for elastic sheets formed by inextensible crossed elasticae. Int. J. Non-Linear Mech. 106, 324-329 (2018) open in new tab
  72. Gao, Y., Oterkus, S.: Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems. Contin. Mech. Thermodyn. 31, 907-937 (2019) open in new tab
  73. Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7(1), 45-84 (2012) open in new tab
  74. Oterkus, E., Madenci, E.: Peridynamic theory for damage initiation and growth in composite laminate. Key Eng. Mater. 488, 355-358 (2012) open in new tab
  75. Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152-168 (2015) open in new tab
  76. Diyaroglu, C., Oterkus, E., Oterkus, S.: An Euler-Bernoulli beam formulation in an ordinary state-based peridynamic framework. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517728424 open in new tab
  77. dell'Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S. (Eds.): The complete works of Gabrio Piola: Volume I, vol. 38 of Advanced Structured Materials, Springer, Cham (2014) open in new tab
  78. dell'Isola, F., Maier, G., Perego, U., Andreaus, U., Esposito, R., Forest, S. (Eds.), The complete works of Gabrio Piola: Volume II, vol. 97 of Advanced Structured Materials, Springer, Cham (2018) open in new tab
  79. 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(8), 887-928 (2015) open in new tab
  80. Giorgio, I., Harrison, P., dell'Isola, F., Alsayednoor, J., Turco, E.: Wrinkling in engineering fabrics: a comparison between two different comprehensive modelling approaches. Proc. R. Soc. A 474(2216), 20180063 (2018) open in new tab
  81. Boisse, P., Hamila, N., Vidal-Sallé, E., Dumont, F.: Simulation of wrinkling during textile composite reinforcement forming. Influence of tensile, in-plane shear and bending stiffnesses. Compos. Sci. Technol. 71(5), 683-692 (2011) open in new tab
  82. Buet-Gautier, K., Boisse, P.: Experimental analysis and modeling of biaxial mechanical behavior of woven composite reinforcements. Exp. Mech. 41(3), 260-269 (2001) open in new tab
  83. Gelin, J.C., Cherouat, A., Boisse, P., Sabhi, H.: Manufacture of thin composite structures by the RTM process: numerical simulation of the shaping operation. Compos. Sci. Technol. 56(7), 711-718 (1996) open in new tab
  84. Ciarlet, P.: Mathematical Elasticity. Theory of Plates, vol. II. Elsevier, Amsterdam (1997) open in new tab
  85. Ciarlet, P.: Mathematical Elasticity. Theory of Shells, vol. III. Elsevier, Amsterdam (2000) open in new tab
  86. Vorovich, I.I.: Nonliner Theory of Shallow Shells. Applied Mathematical Sciences, vol. 133. Springer, New York (1999) open in new tab
  87. Lebedev, L.P., Vorovich, I.I.: Functional Analysis in Mechanics. Springer, New York (2003) open in new tab
  88. Svetlitsky, V.A.: Statics of Rods. Springer, Berlin (2000) open in new tab
  89. Bîrsan, M., Altenbach, H., Sadowski, T., Eremeyev, V.A., Pietras, D.: Deformation analysis of functionally graded beams by the direct approach. Compos. Part B: Eng. 43(3), 1315-1328 (2012) open in new tab
  90. Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM 96(11), 1268-1279 (2016) open in new tab
  91. Spagnuolo, M., Barcz, K., Pfaff, A., dell'Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47-52 (2017) open in new tab
  92. Steigmann, D.J., dell'Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin. 31(3), 373-382 (2015) open in new tab
  93. Fichera, G.: Existence theorems in elasticity. In: Flügge, S. (ed.) Handbuch der Physik, vol. VIa/2, pp. 347-389. Springer, Berlin (1972) open in new tab
  94. Ciarlet, P.G.: Mathematical Elasticity. Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988) open in new tab
  95. Eremeyev, V.A., Lebedev, L.P.: Existence of weak solutions in elasticity. Math. Mech. Solids 18(2), 204-217 (2013) open in new tab
  96. Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010) open in new tab
  97. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient. J. Mécanique 12, 236-274 (1973)
  98. Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lecture Notes in Mathematics, vol. 8. Springer, Berlin (1965) open in new tab
  99. Egorov, Y.V., Shubin, M.A.: Foundations of the Classical Theory of Partial Differential Equations. Encyclopaedia of Mathematical Sciences 30, vol. 30, 1st edn. Springer, Berlin (1998) open in new tab
  100. Agranovich, M.: Elliptic boundary problems. In: Agranovich, M., Egorov, Y., Shubin, M. (eds.) Partial Differential Equations IX: Elliptic Boundary Problems. Encyclopaedia of Mathematical Sciences, vol. 79, pp. 1-144. Springer, Berlin (1997) open in new tab
  101. Hörmander, L.: The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients. A Series of Comprehensive Studies in Mathematics, vol. 257. Springer, Berlin (1983) open in new tab
  102. Palamodov, V.P.: Systems of linear differential equations. In: Gamkrelidze, R.V. (ed.) Mathematical Analysis. Progress in Mathematics, pp. 1-35. Springer, Boston (1971) open in new tab
  103. Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 1. Wiley, New York (1978)
  104. Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 2. Wiley, New York (1979)
  105. Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1996). (in Russian)
  106. Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006) open in new tab
  107. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, Ams- terdam (2003) open in new tab
  108. Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) open in new tab
  109. Lopatinskii, Y.B.: On a method of reducing boundary problems for a system of differential equations of elliptic type to a regular integral equation (in Russian. Ukrain. Math. Zhurnal. 5, 123-151 (1953) open in new tab
  110. Shapiro, Z.Y.: On general boundary problems for equations of elliptic type (in Russian). Izv. Akad. Nauk SSSR. Ser. Math. 17, 539-562 (1953) open in new tab
  111. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12(4), 623-727 (1959) open in new tab
  112. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17(1), 35-92 (1964) open in new tab
  113. Evans, L.C.: Partial Differential Equations. Graduate Series in Mathematics, vol. 19, 2nd edn. AMS Providence, Rhode Island (2010) open in new tab
  114. Polyanin, A.D., Nazaikinskii, V.E.: Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd edn. Chapman and Hall/CRC, Boca Raton (2016) open in new tab
  115. Laudato, M., Manzari, L., Barchiesi, E., Cosmo, F.D., Göransson, P.: First experimental observation of the dynamical behavior of a pantographic metamaterial. Mech. Res. Commun. 94, 125-127 (2018) open in new tab
  116. Eremeyev, V.A., Lebedev, L.P.: Existence theorems in the linear theory of micropolar shells. ZAMM 91(6), 468-476 (2011) open in new tab
  117. Gharahi, A., Schiavone, P.: Uniqueness of solution for plane deformations of a micropolar elastic solid with surface effects. Contin. Mech. Thermodyn. (2019). https://doi.org/10.1007/s00161-019-00779-x open in new tab
  118. Marin, M., Öchsner, A.: An initial boundary value problem for modeling a piezoelectric dipolar body. Contin. Mech. Thermodyn. 30(2), 267-278 (2018) open in new tab
  119. Marin, M., Öchsner, A., Taus, D.: On structural stability for an elastic body with voids having dipolar structure. Contin. Mech. Thermodyn. (2019). https://doi.org/10.1007/s00161-019-00793-z open in new tab
  120. Romano, G., Barretta, R., Diaco, M.: Iterative methods for nonlocal elasticity problems. Contin. Mech. Thermodyn. 31(3), 669-689 (2019) open in new tab
  121. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. open in new tab
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