×

Mechanical metamaterials: a state of the art. (English) Zbl 1425.74036

Summary: In this paper, we give a review of the state of the art in the study of mechanical metamaterials. The very attractive property of having a microstructure capable of determining exotic and specifically tailored macroscopic behaviour makes the study of metamaterials a field that is actually in expansion, from both a theoretical and a technological point of view. This work is divided into two sections, describing the phenomenological and theoretical aspects of metamaterials. We first give an overview of some existing metamaterials, such as pentamode materials, auxetic materials, materials with negative mechanical constitutive coefficients and materials with enhanced mechanical properties. We also focus on some emerging areas, such as origami. Then, we present some theoretical studies in the field of mechanical metamaterials, such as those related to first- and second-gradient theories.

MSC:

74A40 Random materials and composite materials
74A60 Micromechanical theories
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [1] Lee, J, Singer, J, Thomas, E. Micro-/nanostructured mechanical metamaterials. Adv Mater 2012; 24(36): 4782-4810.
[2] [2] Wang, K, Chang, Y, Chen, Y. Designable dual-material auxetic metamaterials using three-dimensional printing. Mater Des 2015; 67: 159-164.
[3] [3] Rumpf, R, Pazos, J, Garcia, C. 3D printed lattices with spatially variant self-collimation. Prog Electromagn Res 2013; 139: 1-14.
[4] [4] Xing, J, Zheng, M, Duan, X. Two-photon polymerization microfabrication of hydrogels: an advanced 3D printing technology for tissue engineering and drug delivery. Chem Soc Rev 2015; 44(15): 5031-5039.
[5] [5] Ok, J, Youn, H, Kwak, M. Continuous and scalable fabrication of flexible metamaterial films via roll-to-roll nanoimprint process for broadband plasmonic infrared filters. Appl Phys Lett 2012; 101(22): 223102.
[6] [6] Morse, J . Nanofabrication technologies for roll-to-roll processing. In: NIST-NNN Workshop, 2011.
[7] [7] Lin, Y. Electrospinning polymer fibers for design and fabrication of new materials. PhD Thesis, University of Akron, 2011.
[8] [8] Lei, T, Lu, X, Yang, F. Fabrication of various micro/nano structures by modified near-field electrospinning. AIP Adv 2015; 5(4): 041301.
[9] [9] Bakshi, V. EUV lithography. Bellingham, WA: SPIE Press, 2009.
[10] [10] Madou, M. Manufacturing techniques for microfabrication and nanotechnology. Boca Raton, FL: CRC Press, 2011.
[11] [11] Henzie, J, Grünwald, M, Widmer-Cooper, A. Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices. Nat Mater 2012; 11(2): 131-137.
[12] [12] Kadic, M, Bückmann, T, Stenger, N. On the practicability of pentamode mechanical metamaterials. Appl Phys Lett 2012; 100(19): 191901.
[13] [13] Lakes, R. Foam structures with a negative Poisson’s ratio. Science 1987; 235: 1038-1041.
[14] [14] Bertoldi, K, Reis, P, Willshaw, S. Negative Poisson’s ratio behavior induced by an elastic instability. Adv Mater 2010; 22(3): 361-366.
[15] [15] Babaee, S, Shim, J, Weaver, J. 3D soft metamaterials with negative Poisson’s ratio. Adv Mater 2013; 25(36): 5044-5049.
[16] [16] Milton, G. Composite materials with Poisson’s ratios close to −1. J Mech Phys Solids 1992; 40(5): 1105-1137. · Zbl 0780.73047
[17] [17] Silverberg, J, Evans, A, McLeod, L. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 2014; 345(6197): 647-650.
[18] [18] Liu, Q. Literature review: materials with negative Poisson’s ratios and potential applications to aerospace and defence. Technical report, DTIC Document, 2006.
[19] [19] Bückmann, T, Schittny, R, Thiel, M. On three-dimensional dilational elastic metamaterials. New J Phys 2014; 16(3): 033032.
[20] [20] Milton, G. Complete characterization of the macroscopic deformations of periodic unimode metamaterials of rigid bars and pivots. J Mech Phys Solids 2013; 61(7): 1543-1560.
[21] [21] Nicolaou, Z, Motter, A. Mechanical metamaterials with negative compressibility transitions. Nat Mater 2012; 11(7): 608-613.
[22] [22] dell’Isola, F, Giorgio, I, Andreaus, U. Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation. Proc Est Acad Sci 2015; 64: 219-225.
[23] [23] dell’Isola, F, Della Corte, A, Giorgio, I. Pantographic 2D sheets: discussion of some numerical investigations and potential applications. Int J Non-Linear Mech 2016; 80: 200-208.
[24] [24] Madeo, A, Della Corte, A, Greco, L. Wave propagation in pantographic 2D lattices with internal discontinuities. Proc Est Acad Sci 2015; 64(3S): 325-330. · Zbl 1330.35440
[25] [25] Paulose, J, Meeussen, A, Vitelli, V. Selective buckling via states of self-stress in topological metamaterials. Proc Natl Acad Sci USA 2015; 112(25): 7639-7644.
[26] [26] Paulose, J, Chen, B, Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat Phys 2015; 11: 153-156.
[27] [27] Zheng, X, Lee, H, Weisgraber, T. Ultralight, ultrastiff mechanical metamaterials. Science 2014; 344(6190): 1373-1377.
[28] [28] Zheng, X, Smith, W, Jackson, J. Multiscale metallic metamaterials. Nat Mater 2016; 15: 1100-1106.
[29] [29] dell’Isola, F, Lekszycki, T, Pawlikowski, M. Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66: 3473-3498. · Zbl 1395.74002
[30] [30] Florijn, B, Coulais, C, van Hecke, M. Programmable mechanical metamaterials. Phys Rev Lett 2014; 113(17): 175503.
[31] [31] Rafsanjani, A, Akbarzadeh, A, Pasini, D. Snapping mechanical metamaterials under tension. Adv Mater 2015; 27(39): 5931-5935.
[32] [32] Liu, X, Hu, G, Huang, G. An elastic metamaterial with simultaneously negative mass density and bulk modulus. Appl Phys Lett 2011; 98(25): 251907.
[33] [33] Del Vescovo, D, Giorgio, I. Dynamic problems for metamaterials: review of existing models and ideas for further research. Int J Eng Sci 2014; 80: 153-172. · Zbl 1423.74039
[34] [34] dell’Isola, F, Steigmann, D, Della Corte, A. Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl Mech Rev 2015; 67(6): 060804.
[35] [35] Milton, G, Cherkaev, A. Which elasticity tensors are realizable? J Eng Mater Technol 1995; 117(4): 483-493.
[36] [36] Zadpoor, A. Mechanical meta-materials. Mater Horiz 2016; 3(5): 371-381.
[37] [37] Greaves, G, Greer, A, Lakes, R. Poisson’s ratio and modern materials. Nat Mater 2011; 10(11): 823-837.
[38] [38] Kadic, M, Bückmann, T, Schittny, R. On anisotropic versions of three-dimensional pentamode metamaterials. New J Phys 2013; 15(2): 023029.
[39] [39] Méjica, G, Lantada, A. Comparative study of potential pentamodal metamaterials inspired by Bravais lattices. Smart Mater Struct 2013; 22(11): 115013.
[40] [40] Evans, KE, Nkansah, MA, Hutchinson, IJ. Molecular network design. Nature 1991; 353: 124.
[41] [41] Yang, W, Li, Z, Shi, W. Review on auxetic materials. J Mater Sci 2004; 39(10): 3269-3279.
[42] [42] Alderson, A, Alderson, K. Auxetic materials. Proc Inst Mech Eng G J Aerosp Eng 2007; 221(4): 565-575.
[43] [43] Carneiro, VH, Meireles, J, Puga, H. Auxetic materials – a review. Mater Sci Poland 2013; 31(4): 561-571.
[44] [44] Mir, M, Ali, M, Sami, J. Review of mechanics and applications of auxetic structures. Adv Mater Sci Eng 2014; 2014: 753496.
[45] [45] Lim, T. Auxetic materials and structures. Singapore: Springer, 2014.
[46] [46] Coluci, V, Hall, L, Kozlov, M. Modeling the auxetic transition for carbon nanotube sheets. Phys Rev B: Condens Matter 2008; 78(11): 115408.
[47] [47] Cabras, L, Brun, M. Auxetic two-dimensional lattices with Poisson’s ratio arbitrarily close to −1 . Proc R Soc London, Ser A 2014; 470(2172). · Zbl 1371.74074
[48] [48] Cabras, L, Brun, M. A class of auxetic three-dimensional lattices. J Mech Phys Solids 2016; 91: 56-72. · Zbl 1482.74136
[49] [49] Milton, G. New examples of three-dimensional dilational materials. Phys Status Solidi B 2015; 252(7): 1426-1430.
[50] [50] Liu, Y, Hu, H. A review on auxetic structures and polymeric materials. Sci Res Essays 2010; 5(10): 1052-1063.
[51] [51] Gibson, L, Ashby, M, Schajer, G. The mechanics of two-dimensional cellular materials. Proc R Soc London, Ser A 1982; 382: 25-42.
[52] [52] Gibson, L. Biomechanics of cellular solids. J Biomech 2005; 38(3): 377-399.
[53] [53] Grima, J, Gatt, R, Ravirala, N. Negative Poisson’s ratios in cellular foam materials. Mater Sci Eng, A 2006; 423(1): 214-218.
[54] [54] Lorato, A, Innocenti, P, Scarpa, F. The transverse elastic properties of chiral honeycombs. Compos Sci Technol 2010; 70(7): 1057-1063.
[55] [55] Wojciechowski, K. Non-chiral, molecular model of negative Poisson ratio in two dimensions. J Phys A: Math Gen 2003; 36(47): 11765. · Zbl 1047.81581
[56] [56] Chan, N, Evans, K. Fabrication methods for auxetic foams. J Mater Sci 1997; 32(22): 5945-5953.
[57] [57] Friis, E, Lakes, R, Park, J. Negative Poisson’s ratio polymeric and metallic foams. J Mater Sci 1988; 23(12): 4406-4414.
[58] [58] Attard, D, Grima, J. Modelling of hexagonal honeycombs exhibiting zero Poisson’s ratio. Phys Status Solidi B 2011; 248(1): 52-59.
[59] [59] Grima, J, Alderson, A, Evans, K. Auxetic behaviour from rotating rigid units. Phys Status Solidi B 2005; 242(3): 561-575.
[60] [60] Grima, J, Evans, K. Auxetic behavior from rotating squares. J Mater Sci Lett 2000; 19(17): 1563-1565.
[61] [61] Grima, J, Alderson, A, Evans, K. Negative Poisson’s ratios from rotating rectangles. Comput Methods Sci Technol 2004; 10: 137-145.
[62] [62] Grima, J, Evans, K. Auxetic behavior from rotating triangles. J Mater Sci 2006; 41(10): 3193-3196.
[63] [63] Attard, D, Grima, J. Auxetic behaviour from rotating rhombi. Phys Status Solidi B 2008; 245(11): 2395-2404.
[64] [64] Grima, J, Zammit, V, Gatt, R. Auxetic behaviour from rotating semi-rigid units. Phys Status Solidi B 2007; 244(3): 866-882.
[65] [65] Chetcuti, E, Ellul, B, Manicaro, E. Modeling auxetic foams through semi-rigid rotating triangles. Phys Status Solidi B 2014; 251(2): 297-306.
[66] [66] Alderson, A, Evans, K. Rotation and dilation deformation mechanisms for auxetic behaviour in the α-cristobalite tetrahedral framework structure. Phys Chem Miner 2001; 28(10): 711-718.
[67] [67] Attard, D, Grima, J. A three-dimensional rotating rigid units network exhibiting negative Poisson’s ratios. Phys Status Solidi B 2012; 249(7): 1330-1338.
[68] [68] Prall, D, Lakes, R. Properties of a chiral honeycomb with a Poisson’s ratio of −1 . Int J Mech Sci 1997; 39(3): 305-314. · Zbl 0894.73018
[69] [69] Theocaris, P, Stavroulakis, G, Panagiotopoulos, P. Negative Poisson’s ratios in composites with star-shaped inclusions: a numerical homogenization approach. Arch Appl Mech 1997; 67(4): 274-286. · Zbl 0894.73087
[70] [70] Larsen, U, Signund, O, Bouwsta, S. Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio. J Microelectromec Syst 1997; 6(2): 99-106.
[71] [71] Lakes, R, Wojciechowski, K. Negative compressibility, negative Poisson’s ratio, and stability. Phys Status Solidi B 2008; 245(3): 545-551.
[72] [72] Gatt, R, Grima, J. Negative compressibility. Phys Status Solidi RRL 2008; 2(5): 236-238.
[73] [73] Panovko, Y, Gubanova, II, Larrick, CV. Stability and oscillations of elastic systems. J Appl Mech 1966; 33: 479.
[74] [74] Eremeyev, VA, Lebedev, LP. On the loss of stability of von Mises truss with the effect of pseudo-elasticity. Matemáticas: Enseanza Universitaria 2006; 14: 111-118. · Zbl 1149.74024
[75] [75] Wang, Y, Lakes, R. Extreme stiffness systems due to negative stiffness elements. Am J Phys 2004; 72(1): 40-50.
[76] [76] Lakes, R, Lee, T, Bersie, A. Extreme damping in composite materials with negative-stiffness inclusions. Nature 2001; 410(6828): 565-567.
[77] [77] Correa, DM, Klatt, T, Cortes, S. Negative stiffness honeycombs for recoverable shock isolation. Rapid Prototyping J 2015; 21(2): 193-200.
[78] [78] Lakes, R. Extreme damping in composite materials with a negative stiffness phase. Phys Rev Lett 2001; 86(13): 2897.
[79] [79] Wang, Y, Lakes, R. Composites with inclusions of negative bulk modulus: extreme damping and negative Poisson’s ratio. J Compos Mater 2005; 39(18): 1645-1657.
[80] [80] Drugan, W. Elastic composite materials having a negative stiffness phase can be stable. Phys Rev Lett 2007; 98(5): 055502.
[81] [81] Antoniadis, I, Chronopoulos, D, Spitas, V. Hyper-damping properties of a stiff and stable linear oscillator with a negative stiffness element. J Sound Vib 2015; 346: 37-52.
[82] [82] Lee, C, Goverdovskiy, V, Temnikov, A. Design of springs with “negative” stiffness to improve vehicle driver vibration isolation. J Sound Vib 2007; 302(4): 865-874.
[83] [83] Sarlis, A, Pasala, D, Constantinou, M. Negative stiffness device for seismic protection of structures. J Struct Eng 2012; 139(7): 1124-1133.
[84] [84] Baughman, R, Stafström, S, Cui, C. Materials with negative compressibilities in one or more dimensions. Science 1998; 279(5356): 1522-1524.
[85] [85] Grima, J, Caruana-Gauci, R. Mechanical metamaterials: materials that push back. Nat Mater 2012; 11(7): 565-566.
[86] [86] Fortes, A, Suard, E, Knight, K. Negative linear compressibility and massive anisotropic thermal expansion in methanol monohydrate. Science 2011; 331(6018): 742-746.
[87] [87] Grima, J, Attard, D, Caruana-Gauci, R. Negative linear compressibility of hexagonal honeycombs and related systems. Scr Mater 2011; 65(7): 565-568.
[88] [88] Grima, J, Caruana-Gauci, R, Attard, D. Three-dimensional cellular structures with negative Poisson’s ratio and negative compressibility properties. Proc R Soc London, Ser A 2012; 468: 3121-3138.
[89] [89] Barnes, D, Miller, W, Evans, K. Modelling negative linear compressibility in tetragonal beam structures. Mech Mater 2012; 46: 123-128.
[90] [90] Grima, J, Attard, D, Gatt, R. Truss-type systems exhibiting negative compressibility. Phys Status Solidi B 2008; 245(11): 2405-2414.
[91] [91] Maruszewski, T, Wojciechowski, K. Anomalous deformation of constrained auxetic square. Rev Adv Mater Sci 2010; 23: 169-174.
[92] [92] Giorgio, I, Andreaus, U, Lekszycki, T. The influence of different geometries of matrix/scaffold on the remodeling process of a bone and bioresorbable material mixture with voids. Math Mech Solids 2017; 22(5): 969-987. · Zbl 1371.74209
[93] [93] Grillo, A, Carfagna, M, Federico, S. Non-Darcian flow in fibre-reinforced biological tissues. Meccanica. Epub ahead of print 17May2017. DOI: 10.1007/s11012-017-0679-0. · Zbl 1394.76161
[94] [94] Dudte, LH, Vouga, E, Tachi, T. Programming curvature using origami tessellations. Nat Mater 2016; 15: 583-588.
[95] [95] Miura, K. Method of packaging and deployment of large membranes in space. The Institute of Space and Astronautical Science Report 1985; 618: 1.
[96] [96] Miura, K . Map fold a la Miura style, its physical characteristics and application to the space science. In: Takaki, R. (ed.) Research of pattern formation. Tokyo: KTK Scientific Publishers, 1994, 77-90.
[97] [97] Miura, K . A note on intrinsic geometry of origami. In: Takaki, R. (ed.) Research of pattern formation. Tokyo: KTK Scientific Publishers, 1994, 91-102.
[98] [98] Kawasaki, T. On the relation between mountain-creases and valley-creases of a flat origami. In: Proceedings of the First International Meeting of Origami Science and Technology, 1991.
[99] [99] Nishiyama, Y. Miura folding: applying origami to space exploration. Int J Pure Appl Math 2012; 79(2): 269-279. · Zbl 1254.51009
[100] [100] Nassar, H, Lebée, A, Monasse, L. Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern. Proc R Soc London, Ser A 2017; 473: 20160705. · Zbl 1404.53010
[101] [101] Feynman, R. There’s plenty of room at the bottom. Eng Sci 1960; 23(5): 22-36.
[102] [102] Milton, GW, Kohn, RV. Variational bounds on the effective moduli of anisotropic composites. J Mech Phys Solids 1988; 36(6): 597-629. · Zbl 0672.73012
[103] [103] Sigmund, O. Design of materials structures using topology optimization. PhD Thesis, Technical University of Denmark, 1994.
[104] [104] Bückmann, T, Kadic, M, Stenger, N. On the feasibility of pentamode mechanical metamaterials. In: The Sixth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics, 2012.
[105] [105] Cosserat, E, Cosserat, F. Théorie des corps déformables. Paris: A. Hermann et fils, 1909. · JFM 40.0862.02
[106] [106] Toupin, RA. Elastic materials with couple-stresses. Arch Ration Mech Anal 1962; 11(1): 385-414. · Zbl 0112.16805
[107] [107] Mindlin, RD. Micro-structure in linear elasticity. Arch Ration Mech Anal 1964; 16(1): 51-78. · Zbl 0119.40302
[108] [108] Germain, P. The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J Appl Math 1973; 25(3): 556-575. · Zbl 0273.73061
[109] [109] Eugster, SR, dell’Isola, F. Exegesis of the introduction and Sect. I from ‘Fundamentals of the mechanics of continua’ by E Hellinger. Zeitschrift Angew Math Mech 2017; 97(4): 477-506.
[110] [110] dell’Isola, F, Maier, G, Perego, U. The complete works of Gabrio Piola: volume I. Cham, Switzerland: Springer, 2014. · Zbl 1305.74003
[111] [111] Auffray, N, dell’Isola, F, Eremeyev, V. Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math Mech Solids 2015; 20(4): 375-417. · Zbl 1327.76008
[112] [112] 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 2015; 20(8): 887-928. · Zbl 1330.74006
[113] [113] 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 2017; 22(4): 852-872. · Zbl 1371.74024
[114] [114] Seppecher, P. Second-gradient theory: application to Cahn-Hilliard fluids. In: Maugin, GA, Drouot, R, Sidoroff, F (eds.) Continuum thermomechanics (Solid Mechanics and its Applications, vol. 76). Dordrecht: Springer, 2000. pp. 379-388.
[115] [115] dell’Isola, F, Sciarra, G, Vidoli, S. Generalized Hooke’s law for isotropic second gradient materials. Proc R Soc London, Ser A 2009: 2177-2196. · Zbl 1186.74019
[116] [116] Abali, BE, Müller, WH, dell’Isola, F. Theory and computation of higher gradient elasticity theories based on action principles. Arch Appl Mech 2017; 87: 1495-1510.
[117] [117] Koiter, WT . A consistent first approximation in the general theory of thin elastic shells. In: Proceedings IUTAM Symposium on the Theory of Thin Elastic Shells, 1959.
[118] [118] Rivlin, R . Plane strain of a net formed by inextensible cords. In: Barenblatt, GI, Joseph, DD (eds.) Collected papers of RS Rivlin. New York: Springer, 1997, 511-534.
[119] [119] Hilgers, M, Pipkin, A. Energy-minimizing deformations of elastic sheets with bending stiffness. J Elast 1993; 31(2): 125-139. · Zbl 0773.73051
[120] [120] Wang, WB, Pipkin, A. Plane deformations of nets with bending stiffness. Acta Mech 1987; 65(1-4): 263-279. · Zbl 0602.73034
[121] [121] Steigmann, DJ, Pipkin, A. Equilibrium of elastic nets. Philos Trans R Soc London, Ser A 1991; 335(1639): 419-454. · Zbl 0734.73098
[122] [122] Steigmann, DJ, dell’Isola, F. Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech Sin 2015; 31(3): 373-382. · Zbl 1346.74128
[123] [123] dell’Isola, F, Della Corte, A, Greco, L. Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange multipliers and a perturbation solution. Int J Solids Struct 2016; 81: 1-12.
[124] [124] dell’Isola, F, Giorgio, I, Pawlikowski, M. Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc London, Ser A 2016; 472: 20150790.
[125] [125] Giorgio, I. Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Z Angew Math Phys 2016; 67(4): 95. · Zbl 1359.74018
[126] [126] Carcaterra, A, dell’Isola, F, Esposito, R. Macroscopic description of microscopically strongly inhomogeneous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch Ration Mech Anal 2015; 218(3): 1239-1262. · Zbl 1352.37193
[127] [127] Seppecher, P, Alibert, JJ, dell’Isola, F. Linear elastic trusses leading to continua with exotic mechanical interactions. J Phys Conf Ser 2011; 319: 012018.
[128] [128] Giorgio, I, Della Corte, A, dell’Isola, F. Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn 2017; 88(1): 21-31.
[129] [129] Greco, L., Giorgio, I., Battista, A. (2017). In plane shear and bending for first gradient inextensible [sic] pantographic sheets: numerical study of deformed shapes and global constraint reactions. Mathematics and Mechanics of Solids, 22(10), 1950-1975. · Zbl 1386.74041
[130] [130] Cuomo, M, dell’Isola, F, Greco, L. First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Composites Part B 2017; 115: 423-448.
[131] [131] Turco, E, Golaszewski, M, Giorgio, I. Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Composites Part B 2017; 118: 1-14.
[132] [132] Scerrato, D, Giorgio, I, Rizzi, NL. Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z Angew Math Phys 2016; 67(3): 53. · Zbl 1464.74028
[133] [133] Giorgio, I, Della Corte, A, dell’Isola, F. Buckling modes in pantographic lattices. CR Mec 2016; 344(7): 487-501.
[134] [134] Turco, E, dell’Isola, F, Cazzani, A. Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1-28. · Zbl 1432.74158
[135] [135] Bourdin, B, Francfort, GA, Marigo, JJ. The variational approach to fracture. J Elast 2008; 91(1): 5-148. · Zbl 1176.74018
[136] [136] Francfort, GA, Marigo, JJ. Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 1998; 46(8): 1319-1342. · Zbl 0966.74060
[137] [137] Marigo, J. Formulation d’une loi d’endommagement d’un matériau élastique. CR Acad Sci, Ser II 1981; 292: 1309-1312. · Zbl 0485.73087
[138] [138] Marigo, JJ, Truskinovsky, L. Initiation and propagation of fracture in the models of Griffith and Barenblatt. Continuum Mech Thermodyn 2004; 16(4): 391-409. · Zbl 1066.74007
[139] [139] Forest, S. Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J Eng Mech 2009; 135(3): 117-131.
[140] [140] Placidi, L. A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Continuum Mech Thermodyn 2015; 27(4-5): 623. · Zbl 1341.74016
[141] [141] Placidi, L. A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continuum Mech Thermodyn 2016; 28(1-2): 119-137. · Zbl 1348.74062
[142] [142] Yang, Y, Misra, A. Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation. Comput Model Eng Sci 2010; 64(1): 1-36. · Zbl 1231.74023
[143] [143] Spagnuolo, M, Barcz, K, Pfaff, A. Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech Res Commun 2017; 83: 47-52.
[144] [144] Turco, E, dell’Isola, F, Rizzi, NL. Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech Res Commun 2016; 76: 86-90.
[145] [145] Turco, E, Rizzi, NL. Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech Res Commun 2016; 77: 65-69.
[146] [146] Popper, KR. Logik der Forschung: zur Erkenntnistheorie der moderner Naturwissenschaft. Vienna: Springer, 1935. · JFM 61.0977.04
[147] [147] Russo, L The forgotten revolution: how science was born in 300 BC and why it had to be reborn. Berlin: Springer, 2013.
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.