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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- Articles
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- artykuły w czasopismach
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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
- Bibliography: test
-
- 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
- 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
- Maugin, G.A.: Non-classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017) open in new tab
- 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
- 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
- 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
- Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385-414 (1962) open in new tab
- Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51-78 (1964) open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- Soubestre, J., Boutin, C.: Non-local dynamic behavior of linear fiber reinforced materials. Mech. Mater. 55, 16-32 (2012) open in new tab
- 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
- 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
- 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
- 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
- Sabik, A.: Direct shear stress vs strain relation for fiber reinforced composites. Compos. Part B: Eng. 139, 24-30 (2018) open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Oterkus, E., Madenci, E.: Peridynamic analysis of fiber-reinforced composite materials. J. Mech. Mater. Struct. 7(1), 45-84 (2012) open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Ciarlet, P.: Mathematical Elasticity. Theory of Plates, vol. II. Elsevier, Amsterdam (1997) open in new tab
- Ciarlet, P.: Mathematical Elasticity. Theory of Shells, vol. III. Elsevier, Amsterdam (2000) open in new tab
- Vorovich, I.I.: Nonliner Theory of Shallow Shells. Applied Mathematical Sciences, vol. 133. Springer, New York (1999) open in new tab
- Lebedev, L.P., Vorovich, I.I.: Functional Analysis in Mechanics. Springer, New York (2003) open in new tab
- Svetlitsky, V.A.: Statics of Rods. Springer, Berlin (2000) open in new tab
- 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
- 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
- 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
- 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
- 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
- Ciarlet, P.G.: Mathematical Elasticity. Three-Dimensional Elasticity, vol. I. North-Holland, Amsterdam (1988) open in new tab
- Eremeyev, V.A., Lebedev, L.P.: Existence of weak solutions in elasticity. Math. Mech. Solids 18(2), 204-217 (2013) open in new tab
- Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010) open in new tab
- Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Première partie: théorie du second gradient. J. Mécanique 12, 236-274 (1973)
- Fichera, G.: Linear Elliptic Differential Systems and Eigenvalue Problems. Lecture Notes in Mathematics, vol. 8. Springer, Berlin (1965) open in new tab
- 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
- 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
- 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
- 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
- Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 1. Wiley, New York (1978)
- Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 2. Wiley, New York (1979)
- Besov, O.V., II'in, V.P., Nikol'skii, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1996). (in Russian)
- Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics, vol. 100. Birkhäuser, Basel (2006) open in new tab
- 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
- Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) open in new tab
- 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
- 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
- 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
- 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
- Evans, L.C.: Partial Differential Equations. Graduate Series in Mathematics, vol. 19, 2nd edn. AMS Providence, Rhode Island (2010) open in new tab
- 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
- 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
- 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
- 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
- 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
- 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
- Romano, G., Barretta, R., Diaco, M.: Iterative methods for nonlocal elasticity problems. Contin. Mech. Thermodyn. 31(3), 669-689 (2019) open in new tab
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