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Three-point bending test of pantographic blocks: numerical and experimental investigation. (English) Zbl 07259266

Summary: The equilibrium forms of pantographic blocks in a three-point bending test are investigated via both experiments and numerical simulations. In the computational part, the corresponding minimization problem is solved with a deformation energy derived by homogenization within a class of admissible solutions. To evaluate the numerical simulations, series of measurements have been carried out with a suitable experimental setup guided by the acquired theoretical knowledge. The observed experimental issues have been resolved to give a robust comparison between the numerical and experimental results. Promising agreement between theoretical predictions and experimental results is demonstrated for the planar deformation of pantographic blocks.

MSC:

74-XX Mechanics of deformable solids
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[1] Kadic, M, Bückmann, T, Stenger, N, et al. On the practicability of pentamode mechanical metamaterials. Appl Phys Lett 2012; 100(19): 049902.
[2] Lee, JH, Singer, JP, Thomas, EL. Micro-/nanostructured mechanical metamaterials. Adv Mater 2012; 24(36): 4782-4810.
[3] Turco, E, Giorgio, I, Misra, A, et al. King post truss as a motif for internal structure of (meta)material with controlled elastic properties. R Soc Open Sci 2017; 4: 171153.
[4] Vangelatos, Z, Komvopoulos, K, Grigoropoulos, C. Vacancies for controlling the behavior of microstructured three-dimensional mechanical metamaterials. Math Mech Solids 2019; 24(2): 511-524. · Zbl 1447.74034
[5] Vangelatos, Z, Melissinaki, V, Farsari, M, et al. Intertwined microlattices greatly enhance the performance of mechanical metamaterials. Math Mech Solids 2019; 24(8): 2636-2648.
[6] Vangelatos, Z, Gu, GX, Grigoropoulos, CP. Architected metamaterials with tailored 3D buckling mechanisms at the microscale. Extreme Mech Lett 2019; 33: 100580.
[7] Eugster, S, dell’Isola, F, Steigmann, D. Continuum theory for mechanical metamaterials with a cubic lattice substructure. Math Mech Complex Syst 2019; 7(1): 75-98. · Zbl 1428.74004
[8] Nejadsadeghi, N, Placidi, L, Romeo, M, et al. Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech Res Commun 2019; 95: 96-103.
[9] Barchiesi, E, Spagnuolo, M, Placidi, L. Mechanical metamaterials: A state of the art. Math Mech Solids 2019; 24(1): 212-234. · Zbl 1425.74036
[10] Yu, X, Zhou, J, Liang, H, et al. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review. Prog Mater Sci 2018; 94: 114-173.
[11] dell’Isola, F, Seppecher, P, Alibert, JJ, et al. Pantographic metamaterials: An example of mathematically driven design and of its technological challenges. Continuum Mech Thermodyn 2019; 31(4): 851-884.
[12] dell’Isola, F, Seppecher, P, Spagnuolo, M, et al. Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses. Continuum Mech Thermodyn 2019; 31(4): 1231-1282.
[13] Alibert, JJ, Seppecher, P, dell’Isola, F. Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids 2003; 8(1): 51-73. · Zbl 1039.74028
[14] dell’Isola, F, Andreaus, U, Cazzani, A, et al. The complete works of Gabrio Piola: Volume II commented English translation. Cham: Springer, 2018.
[15] 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
[16] dell’Isola, F, Della Corte, A, Esposito, R, et al. Some cases of unrecognized transmission of scientific knowledge: From antiquity to Gabrio Piola’s peridynamics and generalized continuum theories. In: Altenbach, H, Forest, S (eds.) Generalized continua as models for classical and advanced materials (Advanced Structured Materials, vol. 42). Cham: Springer, 2016, 77-128.
[17] Auffray, N, dell’Isola, F, Eremeyev, VA, et al. 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
[18] Rahali, Y, Giorgio, I, Ganghoffer, J, et al. Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices. Int J Eng Sci 2015; 97: 148-172. · Zbl 1423.74794
[19] Pideri, C, Seppecher, P. A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech Thermodyn 1997; 9(5): 241-257. · Zbl 0893.73006
[20] Abdoul-Anziz, H, Seppecher, P. Strain gradient and generalized continua obtained by homogenizing frame lattices. Math Mech Complex Syst 2018; 6(3): 213-250. · Zbl 1403.35028
[21] Abdoul-Anziz, H, Seppecher, P, Bellis, C. Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms. Math Mech Solids 2019; 24(12): 3976-3999.
[22] dell’Isola, F, Giorgio, I, Pawlikowski, M, et al. 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(2185): 20150790.
[23] Cuomo, M, dell’Isola, F, Greco, L. Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres. Z Angew Math Phys 2016; 67(3): 61. · Zbl 1442.74056
[24] dell’Isola, F, Cuomo, M, Greco, L, et al. Bias extension test for pantographic sheets: Numerical simulations based on second gradient shear energies. J Eng Math 2017; 103(1): 127-157. · Zbl 1390.74028
[25] Placidi, L, Greco, L, Bucci, S, et al. A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z Angew Math Phys 2016; 67(5): 114. · Zbl 1432.74034
[26] Spagnuolo, M, Barcz, K, Pfaff, A, et al. Qualitative pivot damage analysis in aluminum printed pantographic sheets: Numerics and experiments. Mech Res Commun 2017; 83: 47-52.
[27] Andreaus, U, Spagnuolo, M, Lekszycki, T, et al. A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams. Continuum Mech Thermodyn 2018; 30(5): 1103-1123. · Zbl 1396.74070
[28] De Angelo, M, Spagnuolo, M, D’Annibale, F, et al. The macroscopic behavior of pantographic sheets depends mainly on their microstructure: Experimental evidence and qualitative analysis of damage in metallic specimens. Continuum Mech Thermodyn 2019; 31: 1181-1203.
[29] 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
[30] Placidi, L, Andreaus, U, Giorgio, I. Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J Eng Math 2017; 103(1): 1-21. · Zbl 1390.74018
[31] Yang, H, Ganzosch, G, Giorgio, I, et al. Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Z Angew Math Phys 2018; 69(4): 105. · Zbl 1401.74016
[32] De Angelo, M, Barchiesi, E, Giorgio, I, et al. Numerical identification of constitutive parameters in reduced-order bi-dimensional models for pantographic structures: Application to out-of-plane buckling. Arch Appl Mech 2019; 89(7): 1333-1358.
[33] Nejadsadeghi, N, De Angelo, M, Drobnicki, R, et al. Parametric experimentation on pantographic unit cells reveals local extremum configuration. Exp Mech 2019; 59(6): 927-939.
[34] Solyaev, Y, Lurie, S, Barchiesi, E, et al. On the dependence of standard and gradient elastic material constants on a field of defects. Math Mech Solids 2020; 25(1): 35-45.
[35] Turco, E . How the properties of pantographic elementary lattices determine the properties of pantographic metamaterials. In: Abali, B, Altenbach, H, dell’Isola, F, et al. (eds.) New achievements in continuum mechanics and thermodynamics (Advanced Structured Materials, vol. 108). Cham: Springer, 2019, 489-506. · Zbl 1425.74395
[36] 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
[37] Della Giorgio, I, Corte, A, dell’Isola, F, et al. Buckling modes in pantographic lattices. CR Mec 2016; 344(7): 487-501.
[38] Giorgio, I, Rizzi, NL, Turco, E. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc R Soc London, Ser A 2017; 473(2207): 20170636. · Zbl 1404.74064
[39] 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
[40] Niiranen, J, Niemi, AH. Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates. Eur J Mech A Solids 2017; 61: 164-179. · Zbl 1406.74446
[41] Khakalo, S, Balobanov, V, Niiranen, J. Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: Applications to sandwich beams and auxetics. Int J Eng Sci 2018; 127: 33-52. · Zbl 1423.74343
[42] Khakalo, S, Niiranen, J. Lattice structures as thermoelastic strain gradient metamaterials: Evidence from full-field simulations and applications to functionally step-wise-graded beams. Composites Part B 2019; 177: 107224.
[43] Barchiesi, E, Khakalo, S. Variational asymptotic homogenization of beam-like square lattice structures. Math Mech Solids 2019; 24(10): 3295-3318.
[44] Eremeyev, V . Nonlinear micropolar shells: Theory and applications. In: Pietraszkiewicz, W, Szymczak, C (eds.) Shell Structures: Theory and Applications. London: Taylor & Francis, 2005, 11-18.
[45] Eremeyev, VA, Lebedev, LP, Altenbach, H. Foundations of micropolar mechanics. Berlin: Springer Science & Business Media, 2012. · Zbl 1257.74002
[46] Eremeyev, V, Altenbach, H. Basics of mechanics of micropolar shells. In: Altenbach, H, Eremeyev, V (eds.) Shell-like structures. Cham: Springer, 2017, 63-111.
[47] Altenbach, H, Eremeyev, VA. On the theories of plates based on the Cosserat approach. In: Maugin, G, Metrikine, A (eds.) Mechanics of generalized continua (Advances in Mechanics and Mathematics, vol 21). New York: Springer, 2010, 27-35. · Zbl 1396.74072
[48] Altenbach, H, Eremeyev, VA. On the linear theory of micropolar plates. J Appl Math Mech 2009; 89(4): 242-256. · Zbl 1160.74030
[49] Altenbach, H, Eremeyev, VA. Cosserat-type shells. In: Altenbach, H, Eremeyev, VA (eds.) Generalized continua from the theory to engineering applications (CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 541). Vienna: Springer, 2013, 131-178. · Zbl 1279.74023
[50] Altenbach, H, Eremeyev, VA, Lebedev, LP. Micropolar shells as two-dimensional generalized continua models. In: Altenbach, H, Maugin, G, Erofeev, V (eds.) Mechanics of generalized continua (Advanced Structured Materials, vol. 7). Berlin: Springer, 2011, 23-55.
[51] Misra, A, Poorsolhjouy, P. Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math Mech Complex Syst 2015; 3(3): 285-308. · Zbl 1329.74225
[52] Placidi, L, Barchiesi, E, Turco, E, et al. A review on 2D models for the description of pantographic fabrics. Z Angew Math Phys 2016; 67(5): 121. · Zbl 1359.74019
[53] Turco, E, Golaszewski, M, Cazzani, A, et al. Large deformations induced in planar pantographic sheets by loads applied on fibers: Experimental validation of a discrete Lagrangian model. Mech Res Commun 2016; 76: 51-56.
[54] Turco, E, Barcz, K, Pawlikowski, M, et al. Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: Numerical simulations. Z Angew Math Phys 2016; 67(5): 122. · Zbl 1432.74156
[55] Turco, E, Golaszewski, M, Giorgio, I, et al. Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations. Composites Part B 2017; 118: 1-14.
[56] Turco, E, Misra, A, Sarikaya, R, et al. Quantitative analysis of deformation mechanisms in pantographic substructures: Experiments and modeling. Continuum Mech Thermodyn 2019; 31(1): 209-223.
[57] Battista, A, Rosa, L, dell’Erba, R, et al. Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena. Math Mech Solids 2017; 22(11): 2120-2134. · Zbl 1395.74005
[58] Greco, L, Giorgio, I, Battista, A. In plane shear and bending for first gradient inextensible pantographic sheets: Numerical study of deformed shapes and global constraint reactions. Math Mech Solids 2017; 22(10): 1950-1975. · Zbl 1386.74041
[59] Franciosi, P, Spagnuolo, M, Salman, OU. Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates. Continuum Mech Thermodyn 2019; 31(1): 101-132.
[60] Chiaia, B, Barchiesi, E, De Biagi, V, et al. A novel worst-case-based structural resilience index: Definition, computation and applications to portal frame structures. Mech Res Commun 2019; 99: 52-57.
[61] Barchiesi, E, Eugster, SR, Placidi, L, et al. Pantographic beam: A complete second gradient 1D-continuum in plane. Z Angew Math Phys 2019; 70(5): 135. · Zbl 1425.74387
[62] Spagnuolo, M, Andreaus, U. A targeted review on large deformations of planar elastic beams: Extensibility, distributed loads, buckling and post-buckling. Math Mech Solids 2019; 24(1): 258-280. · Zbl 1425.74267
[63] Green, A. Micro-materials and multipolar continuum mechanics. Int J Eng Sc 1965; 3(5): 533-537.
[64] Mindlin, RD. Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1965; 1(4): 417-438.
[65] Sedov, LI. Mathematical methods for constructing new models of continuous media. Russ Math Surv 1965; 20(5): 123. · Zbl 0151.37003
[66] Toupin, RA. Theories of elasticity with couple-stress. Arch Ration Mech Anal 1964; 17(2): 85-112. · Zbl 0131.22001
[67] 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
[68] 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.
[69] Misra, A, Lekszycki, T, Giorgio, I, et al. Pantographic metamaterials show atypical Poynting effect reversal. Mech Res Commun 2018; 89: 6-10.
[70] Javanbakht, M, Barati, E. Martensitic phase transformations in shape memory alloy: Phase field modeling with surface tension effect. Comput Mater Sci 2016; 115: 137-144.
[71] Mirzakhani, S, Javanbakht, M. Phase field-elasticity analysis of austenite-martensite phase transformation at the nanoscale: Finite element modeling. Comput Mater Sci 2018; 154: 41-52.
[72] dell’Isola, F, Lekszycki, T, Pawlikowski, M, et al. Designing a light fabric metamaterial being highly macroscopically tough under directional extension: First experimental evidence. Z Angew Math Phys 2015; 66(6): 3473-3498. · Zbl 1395.74002
[73] della Corte, A, Giorgio, I, Scerrato, D. Pantographic 2D sheets: Discussion of some numerical investigations and potential applications. Int J Non Linear Mech 2016; 80: 200-208.
[74] Fischer, P, Klassen, M, Mergheim, J, et al. Isogeometric analysis of 2D gradient elasticity. Comput Mech 2011; 47(3): 325-334. · Zbl 1398.74329
[75] Cuomo, M, Contrafatto, L, Greco, L. A variational model based on isogeometric interpolation for the analysis of cracked bodies. Int J Eng Sci 2014; 80: 173-188. · Zbl 1423.74055
[76] Cazzani, A, Malagù, M, Turco, E. Isogeometric analysis: A powerful numerical tool for the elastic analysis of historical masonry arches. Continuum Mech Thermodyn 2016; 28(1-2): 139-156. · Zbl 1348.74190
[77] Cazzani, A, Malagù, M, Turco, E, et al. Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math Mech Solids 2016; 21(2): 182-209. · Zbl 1333.74051
[78] Cazzani, A, Malagù, M, Turco, E. Isogeometric analysis of plane-curved beams. Math Mech Solids 2016; 21(5): 562-577. · Zbl 1370.74084
[79] Cazzani, A, Stochino, F, Turco, E. An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. J Appl Math Mech 2016; 96(10): 1220-1244. · Zbl 1338.74068
[80] Niiranen, J, Khakalo, S, Balobanov, V, et al. Variational formulation and isogeometric analysis for fourth-order boundary value problems of gradient-elastic bar and plane strain/stress problems. Comput Methods Appl Mech Eng 2016; 308: 182-211. · Zbl 1439.74036
[81] Niiranen, J, Kiendl, J, Niemi, AH, et al. Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates. Comput Methods Appl Mech Eng 2017; 316: 328-348. · Zbl 1439.74037
[82] Niiranen, J, Balobanov, V, Kiendl, J, et al. Variational formulations, model comparisons and numerical methods for Euler-Bernoulli micro-and nano-beam models. Math Mech Solids 2019; 24(1): 312-335. · Zbl 1425.74264
[83] 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 2018; 339: 137-159. · Zbl 1440.74179
[84] Khakalo, S, Niiranen, J. Isogeometric analysis of higher-order gradient elasticity by user elements of a commercial finite element software. Comput-Aided Des 2017; 82: 154-169.
[85] Khakalo, S, Niiranen, J. Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis. J Mech Phys Solids 2020; 134: 103728.
[86] Yaghoubi, ST, Balobanov, V, Mousavi, SM, et al. Variational formulations and isogeometric analysis for the dynamics of anisotropic gradient-elastic Euler-Bernoulli and shear-deformable beams. Eur J Mech A Solids 2018; 69: 113-123. · Zbl 1406.74409
[87] Yildizdag, ME, Demirtas, M, Ergin, A. Multipatch discontinuous Galerkin isogeometric analysis of composite laminates. Continuum Mech Thermodyn 2020; 32: 607-620.
[88] Yildizdag, ME, Ardic, IT, Demirtas, M, et al. Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach. Ocean Eng 2019; 172: 316-329.
[89] Capobianco, G, Eugster, SR, Winandy, T. Modeling planar pantographic sheets using a nonlinear Euler-Bernoulli beam element based on B-spline functions. Proc Appl Math Mech 2018; 18: e201800220.
[90] Greco, L, Cuomo, M, Contrafatto, L. A reconstructed local \(\overline{B}\) formulation for isogeometric Kirchhoff-Love shells. Comput Methods Appl Mech Eng 2018; 332: 462-487. · Zbl 1440.74395
[91] Balobanov, V, Kiendl, J, Khakalo, S, et al. Kirchhoff-Love shells within strain gradient elasticity: Weak and strong formulations and an H3-conforming isogeometric implementation. Comput Methods Appl Mech Eng 2019; 344: 837-857. · Zbl 1440.74057
[92] Eremeyev, VA, dell’Isola, F, Boutin, C, et al. Linear pantographic sheets: Existence and uniqueness of weak solutions. J Elast 2018; 132(2): 175-196. · Zbl 1398.74011
[93] Love, AEH. A treatise on the mathematical theory of elasticity. Cambridge: Cambridge University Press, 2013.
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