Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. (English) Zbl 1390.74028
Summary: We consider a bi-dimensional sheet consisting of two orthogonal families of inextensible fibres. Using the representation due to Rivlin and Pipkin for admissible placements, i.e. placements preserving the lengths of the inextensible fibres, we numerically simulate a standard bias extension test on the sheet, solving a non-linear constrained optimization problem. Several first and second gradient deformation energy models are considered, depending on the shear angle between the fibres and on its gradient, and the results obtained are compared. The proposed numerical simulations will be helpful in designing a systematic experimental campaign aimed at characterizing the internal energy for physical realizations of the ideal pantographic structure presented in this paper.
MSC:
| 74A60 | Micromechanical theories |
| 74A30 | Nonsimple materials |
| 74B20 | Nonlinear elasticity |
| 74K15 | Membranes |
Keywords:
bias extension test; constrained optimization; inextensible fibre network; pantographic sheet; second gradient theoryReferences:
| [1] | Nikopour H, Selvadurai APS (2014) Concentrated loading of a fibre-reinforced composite plate: experimental and computational modeling of boundary fixity. Composites B Eng 60:297-305 · doi:10.1016/j.compositesb.2013.12.034 |
| [2] | Selvadurai APS, Nikopour H (2012) Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: Experiments, theory and computations. Compos Struct 94(6):1973-1981 · doi:10.1016/j.compstruct.2012.01.019 |
| [3] | Hamila N, Boisse P (2013) Locking in simulation of composite reinforcement deformations. Analysis and treatment. Composites A Appl Sci Manuf 53:109-117 · doi:10.1016/j.compositesa.2013.06.001 |
| [4] | Boisse P (2011) Composite reinforcements for optimum performance. Elsevier, Amsterdam · doi:10.1533/9780857093714 |
| [5] | Nikopour H, Selvadurai A (2013) Torsion of a layered composite strip. Compos Struct 95:1-4 cited By 0 · doi:10.1016/j.compstruct.2012.08.027 |
| [6] | dell’Isola F, d’Agostino MV, Madeo A, Boisse P, Steigmann D (2016) Minimization of shear energy in two dimensional continua with two orthogonal families of inextensible fibers: the case of standard bias extension test. J Elast 122(2):131-155 · Zbl 1330.74024 |
| [7] | dell’Isola F, Steigmann DJ (2015) A two-dimensional gradient-elasticity theory for woven fabrics. J Elast 118(1):113-125 · Zbl 1305.74024 · doi:10.1007/s10659-014-9478-1 |
| [8] | Scerrato D, Giorgio I, Rizzi NL (2016) Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z Angew Math Phys 67:1-19 · Zbl 1464.74028 · doi:10.1007/s00033-016-0650-2 |
| [9] | Scerrato D, Zhurba Eremeeva IA, Lekszycki T, Rizzi NL (2016) On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. Z Angew Math Mech. doi:10.1002/zamm.201600066 · Zbl 07775122 · doi:10.1002/zamm.201600066 |
| [10] | dell’Isola F, Giorgio I, Andreaus U (2015) Elastic pantographic 2D lattices: a numerical analysis on the static response and wave propagation. Proc Estonian Acad Sc 64:219-225 · doi:10.3176/proc.2015.3.03 |
| [11] | dell’Isola F, Giorgio I, Pawlikowski M, Rizzi NL (2016) Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium. Proc Royal Soc Lond A: Math Phys Eng Sci 472(2185):1-23 · Zbl 1348.74271 |
| [12] | dell’Isola F, Della Corte A, Greco L, Luongo A (2015) Plane Bias Extension Test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange Multipliers and a Perturbation Solution. Submitted to: Int J Solids Struct |
| [13] | Laurent C, Durville D, Vaquette C, Rahouadj R, Ganghoffer J (2013) Computer-aided tissue engineering: application to the case of anterior cruciate ligament repair. Biomech Cells Tissues 9:1-44 · doi:10.1007/978-94-007-5890-2_1 |
| [14] | Del Vescovo D, Giorgio I (2014) Dynamic problems for metamaterials: review of existing models and ideas for further research. Int J Eng Sci 80:153-172 · Zbl 1423.74039 · doi:10.1016/j.ijengsci.2014.02.022 |
| [15] | dell’Isola F, Placidi L (2012) Variational principles are a powerful tool also for formulating field theories. In: dell’Isola F, Gavrilyuk S (eds) Variational models and methods in solid and fluid mechanics. Springer Science & Business Media, New York · Zbl 1247.70035 |
| [16] | Rivlin RS (1955) Plane strain of a net formed by inextensible cords. J Ration Mech Anal 4(6):951-974 · Zbl 0065.40202 |
| [17] | Rivlin R (1997) Plane strain of a net formed by inextensible cords. In: Collected Papers of RS Rivlin. Springer, New York, pp 511-534 |
| [18] | Pipkin AC (1980) Some developments in the theory of inextensible networks. Quart Appl Math 38(3):343-355 · Zbl 0484.73009 · doi:10.1090/qam/592201 |
| [19] | Pipkin AC (1981) Plane traction problems for inextensible networks. Quart J Mech Appl Math 34(4):415-429 · Zbl 0466.73037 · doi:10.1093/qjmam/34.4.415 |
| [20] | Wang W-B, Pipkin AC (1986) Inextensible networks with bending stiffness. Quart J Mech Appl Math 39(3):343-359 · Zbl 0594.73047 · doi:10.1093/qjmam/39.3.343 |
| [21] | Wang W-B, Pipkin AC (1987) Plane deformations of nets with bending stiffness. Acta Mech 65(1-4):263-279 · Zbl 0602.73034 · doi:10.1007/BF01176886 |
| [22] | Dos Reis F, Ganghoffer J (2012) Equivalent mechanical properties of auxetic lattices from discrete homogenization. Comput Mater Sci 51:314-321 · doi:10.1016/j.commatsci.2011.07.014 |
| [23] | Goda I, Assidi M, Belouettar S, Ganghoffer J (2012) A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. J Mech Behav Biomed Mater 16:87-108 · doi:10.1016/j.jmbbm.2012.07.012 |
| [24] | Alibert JJ, Della Corte A (2015) Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z Angew Math Phys 66:2855-2870 · Zbl 1327.74128 · doi:10.1007/s00033-015-0526-x |
| [25] | Forest S (1998) Mechanics of generalized continua: construction by homogenizaton. J Phys IV 8(PR4):Pr4-39 |
| [26] | Dos Reis F, Ganghoffer J (2012) Construction of micropolar models from lattice homogenization. Comput Struct 112—-113:354-363 · doi:10.1016/j.compstruc.2012.08.006 |
| [27] | Assidi M, Ben Boubaker B, Ganghoffer J (2011) Equivalent properties of monolayer fabric from mesoscopic modelling strategies. Int J Solid Struct 48(20):2920-2930 · doi:10.1016/j.ijsolstr.2011.06.010 |
| [28] | Goda I, Assidi M, Ganghoffer J (2013) Construction of micropolar models from lattice homogenization. J Mech Phys Solids 61(12):2537-2565 · doi:10.1016/j.jmps.2013.07.014 |
| [29] | Dos Reis F, Ganghoffer J (2014) Homogenized elastoplastic response of repetitive 2D lattice truss materials. Comput Mater Sci 84:145-155 · doi:10.1016/j.commatsci.2013.11.066 |
| [30] | Chaouachi F, Rahali Y, Ganghoffer J (2014) A micromechanical model of woven structures accounting for yarn-yarn contact based on Hertz theory and energy minimization. Comput Mater Sci 66:368-380 |
| [31] | Misra A, Poorsolhjouy P (2016) Based micromorphic model predicts frequency band gaps. Continuum Mech Thermodyn 28(1):215-234 · Zbl 1348.74271 · doi:10.1007/s00161-015-0420-y |
| [32] | Misra A, Poorsolhjouy P (2015) Identification of higher-order elastic constants for grain assemblies upon granular micromechanics. Math Mech Complex Syst 3(3):285-308 · Zbl 1329.74225 · doi:10.2140/memocs.2015.3.285 |
| [33] | Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51-78 · Zbl 0119.40302 · doi:10.1007/BF00248490 |
| [34] | Toupin RA (1964) Theories of elasticity with couple-stress. Arch Ration Mech Anal 17(2):85-112 · Zbl 0131.22001 · doi:10.1007/BF00253050 |
| [35] | Eringen AC (2012) Microcontinuum field theories: I. Foundations and solids. Springer, New York · Zbl 0953.74002 |
| [36] | Eringen AC (1965) Theory of micropolar fluids. Technical report, DTIC Document · Zbl 0436.76006 |
| [37] | Neff P, Forest S (2007) A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J Elast 87(2-3):239-276 · Zbl 1206.74019 · doi:10.1007/s10659-007-9106-4 |
| [38] | Neff P, Ghiba I-D, Madeo A, Placidi L, Rosi G (2014) A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech Thermodyn 26(5):639-681 · Zbl 1341.74135 · doi:10.1007/s00161-013-0322-9 |
| [39] | Altenbach H, Eremeyev VA, Lebedev LP, Rendón LA (2010) Acceleration waves and ellipticity in thermoelastic micropolar media. Arch Appl Mech 80(3):217-227 · Zbl 1271.74251 · doi:10.1007/s00419-009-0314-1 |
| [40] | Altenbach J, Altenbach H, Eremeyev VA (2010) On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch Appl Mech 80(1):73-92 · Zbl 1184.74042 · doi:10.1007/s00419-009-0365-3 |
| [41] | Eremeyev V (2005) Nonlinear micropolar shells: theory and applications. In: Shell structures: theory and applications (vol 2): proceedings of the 9th SSTA Conference, Jurata, Poland, pp.11-18 · Zbl 0920.73282 |
| [42] | Boutin C (1996) Microstructural effects in elastic composites. Int J Solids Struct 33(7):1023-1051 · Zbl 0920.73282 · doi:10.1016/0020-7683(95)00089-5 |
| [43] | Berezovski A, Giorgio I, Della Corte A (2015) Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math Mech Solids 21(1):37-51 · Zbl 1338.74060 · doi:10.1177/1081286515572244 |
| [44] | Giorgio I, Andreaus U, Madeo A (2014) The influence of different loads on the remodeling process of a bone and bio-resorbable material mixture with voids. Continuum Mech Thermodyn 28(1):21-40 · Zbl 1348.74242 |
| [45] | Madeo A, Neff P, Ghiba I-D, Placidi L, Rosi G (2015) Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps. Continuum Mech Thermodyn 28(1):1-20 · Zbl 1341.74085 |
| [46] | Federico S (2010) On the linear elasticity of porous materials. Int J Mech Sci 52(2):175-182 · doi:10.1016/j.ijmecsci.2009.09.006 |
| [47] | Misra A, Yang Y (2010) Micromechanical model for cohesive materials based upon pseudo-granular structure. Int J Solids Struct 47:2970-2981 · Zbl 1196.74161 · doi:10.1016/j.ijsolstr.2010.07.002 |
| [48] | Misra A, Singh V (2013) Micromechanical model for viscoelastic-materials undergoing damage. Continuum Mech Thermodyn 25:1-16 · Zbl 1343.74039 · doi:10.1007/s00161-012-0262-9 |
| [49] | Andreaus U, Giorgio I, Madeo A (2014) Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids. Z Angew Math Phys 66(1):209-237 · Zbl 1317.74064 · doi:10.1007/s00033-014-0403-z |
| [50] | Scerrato D, Giorgio I, Madeo A, Limam A, Darve F (2014) A simple non-linear model for internal friction in modified concrete. Int J Eng Sci 80:136-152 · Zbl 1423.74651 · doi:10.1016/j.ijengsci.2014.02.021 |
| [51] | Scerrato D, Giorgio I, Della Corte A, Madeo A, Limam A (2015) A micro-structural model for dissipation phenomena in the concrete. Int J Numer Anal Methods Geomec 39(18):2037-2052 · doi:10.1002/nag.2394 |
| [52] | Germain P (1973) The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J Appl Math 25(3):556-575 · Zbl 0273.73061 · doi:10.1137/0125053 |
| [53] | Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417-438 · doi:10.1016/0020-7683(65)90006-5 |
| [54] | Alibert J-J, Seppecher P, dell’Isola F (2003) Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids 8(1):51-73 · Zbl 1039.74028 · doi:10.1177/1081286503008001658 |
| [55] | Yang Y, Ching WY, Misra A (2011) Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film. J Nanomech Micromech 1(2):60-71 · doi:10.1061/(ASCE)NM.2153-5477.0000030 |
| [56] | Yang Y, Misra A (2010) Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation. Comput Model Eng Sci 64(1):1-36 · Zbl 1231.74023 |
| [57] | dell’Isola F, Andreaus U, Placidi L (2014) 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 · Zbl 1330.74006 · doi:10.1177/1081286513509811 |
| [58] | Rinaldi A, Placidi L (2014) A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices. ZAMM-J Appl Math Mech 94(10):862-877 · Zbl 1301.74042 · doi:10.1002/zamm.201300028 |
| [59] | Placidi L (2014) A variational approach for a nonlinear 1-dimensional second gradient continuum damage model. Continuum Mech Thermodyn 27(4-5):623-638 · Zbl 1341.74016 |
| [60] | Placidi L (2014) A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model. Continuum Mech Thermodyn 28(1):119-137 · Zbl 1348.74062 |
| [61] | Placidi L, Rosi G, Giorgio I, Madeo A (2014) Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math Mech Solids 19(5):555-578 · Zbl 1305.74047 · doi:10.1177/1081286512474016 |
| [62] | Andreaus U, Chiaia B, Placidi L (2013) Soft-impact dynamics of deformable bodies. Continuum Mech Thermodyn 25(2-4):375-398 · Zbl 1343.74037 · doi:10.1007/s00161-012-0266-5 |
| [63] | Selvadurai A (2009) On the surface displacement of an isotropic elastic halfspace containing an inextensible membrane reinforcement. Math Mech Solids 14(1-2):123-134 cited By 3 · Zbl 1257.74035 · doi:10.1177/1081286508092606 |
| [64] | Federico S, Grillo A, Imatani S (2014) The linear elasticity tensor of incompressible materials. Math Mech Solids 20(6):643-662 · Zbl 1327.74024 · doi:10.1177/1081286514550576 |
| [65] | Luongo A (2010) A unified perturbation approach to static/dynamic coupled instabilities of nonlinear structures. Thin-Wall Struct 48(10):744-751 · doi:10.1016/j.tws.2010.01.002 |
| [66] | Placidi L, Andreaus U, Giorgio I (2016) Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J Eng Math. doi:10.1007/s10665-016-9856-8 · Zbl 1390.74018 |
| [67] | Cuomo M, Contrafatto L, Greco L (2014) A varational model based on isogeometric interpolation for the analysis of cracked bodies. Int J Eng Sci 80:173-188 · Zbl 1423.74055 · doi:10.1016/j.ijengsci.2014.02.017 |
| [68] | Turco E, Caracciolo P (2000) Elasto-plastic analysis of Kirchhoff plates by high simplicity finite elements. Comput Methods Appl Mech Eng 190(5-7):691-706 · Zbl 1007.74080 · doi:10.1016/S0045-7825(99)00438-7 |
| [69] | Cazzani A, Malagù M, Turco E, Stochino F (2015) Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math Mech Solids 21(2):182-209 · Zbl 1333.74051 · doi:10.1177/1081286515577043 |
| [70] | Cazzani A, Malagù M, Turco E (2014) Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Continuum Mech Thermodyn 28:139-156 · Zbl 1348.74190 · doi:10.1007/s00161-014-0409-y |
| [71] | Cazzani A, Malagù M, Turco E (2016) Isogeometric analysis of plane-curved beams. Math Mech Solids 21(5):562-577 · Zbl 1333.74051 · doi:10.1177/1081286514531265 |
| [72] | Ciancio D, Carol I, Cuomo M (2006) On inter-element forces in the FEM-displacement formulation, and implication for stress recovery. Int J Numer Meth Eng 66(3):502-528 · Zbl 1110.74847 · doi:10.1002/nme.1564 |
| [73] | Ciancio D, Carol I, Cuomo M (2007) Crack opening at corner nodes in FE analysis with cracking along mesh lines. Eng Fract Mech 74(13):1963-1982 · doi:10.1016/j.engfracmech.2006.10.005 |
| [74] | dell’Isola F, Lekszycki T, Pawlikowski M, Grygoruk R, Greco L (2015) Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 66(6):3473-3498 · Zbl 1395.74002 · doi:10.1007/s00033-015-0556-4 |
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.
