A new deformation measure for micropolar plates subjected to in-plane loads
Tóm tắt
This work aims to analyze the effect of a new deformation measure in the response of 2D plates subjected to in-plane loads. The proposed formulation allows to clearly distinguish the energetic contribution of every involved deformation mechanism. The action functional is supposed to depend on a strain, a wryness and a new relative rotation tensor in the nonlinear hypothesis; therefore, the suggested approach seems to have a considerable potential to study granular materials. Some 2D samples characterized by complicated geometries are analyzed by means of the finite element method; parametric analyses are performed to test the role played by each material constant which appears in the new constitutive laws: both isotropic and orthotropic models are investigated.
Tài liệu tham khảo
Cosserat, E., Cosserat, F.: Theories of the Deformable Bodies. A. Herrmann et Fils, Paris (1909)
Nowacki, W.: The Linear Theory of Micropolar Elasticity. Micropolar Elasticity. Springer, New York (1974)
Maugin, G.A., Metrikine, A.V.: Mechanics of Generalized Continua: One Hundred Years after the Cosserats. Springer, New York (2010)
Altenbach, J., Altenbach, H., Eremeyev, V.A.: On generalized Cosserat-type theories of plates and shells: a short review and bibliography. Arch. Appl. Mech. 80, 73–92 (2010)
Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer, Berlin (2013)
Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus. Premiére partie. Théorie du second gradient. Journal de Mécanique 12, 235–274 (1973)
Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)
Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)
Mindlin, R.D., Tiersten, H.F.: Effects of couple stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)
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)
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics \(\grave{a}\) la Hamilton-Piola: least action principle for second gradient continua and capillary fluids. Math. Mech. Solids (2013). https://doi.org/10.1177/1081286513497616
dell’Isola, F., Steigmann, D.J.: Discrete and Continuum Models for Complex Metamaterials. Cambridge University Press, Cambridge (2020)
Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids 24(1), 212–234 (2019)
Giorgio, I., Spagnuolo, M., Andreaus, U., Scerrato, D., Bersani, A.M.: In-depth gaze at the astonishing mechanical behavior of bone: a review for designing bio-inspired hierarchical metamaterials. Math. Mech. Solids 26(7), 1074–1103 (2021)
dell’Isola, F., Seppecher, P., Alibert, J.J., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. 31, 851–884 (2019)
Rahali, Y., Giorgio, I., Ganghoffer, J.F., dell’Isola, F.: Homogenization \(\grave{a}\) la Piola produces second gradient continuum models for linear pantographic lattices. Int. J. Eng. Sci. 97, 148–172 (2015)
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)
Eremeyev, V.A., dell’Isola, F., Boutin, C., Steigmann, D.: Linear pantographic sheets: existence and uniqueness of weak solutions. ffhal-01466979f (2017)
Ciallella, A.: Research perspective on multiphysics and multiscale materials: a paradigmatic case. Contin. Mech. Thermodyn. 32, 527–539 (2020)
Duan, W.H., Challamel, N., Wang, C.M., Ding, Z.: Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams. Int. J. Appl. Phys. 114(10), 104312–104323 (2013)
Challamel, N., Lerbet, J., Darve, F., Nicot, F.: Buckling of granular systems with discrete and gradient elasticity Cosserat continua. Ann. Solid Struct. Mech. 12(3), 1–16 (2020)
Misra, A., Poorsolhjouy, P.: Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics. Math. Mech. Solids 25(10), 1778–1803 (2020)
Turco, E., dell’Isola, F., Misra, A.: A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations. Int. J. Numer. Anal. Methods Geomech. 43(5), 1051–1079 (2019)
Matsushima, T., Saomoto, H., Tsubokawa, Y., Yamada, Y.: Grain rotation versus continuum rotation during shear deformation of granular assembly. Soils Found. 43(4), 95–106 (2003)
Bourrier, F., Kneib, F., Chareyre, B., Fourcaud, T.: Discrete modeling of granular soils reinforcement by plant roots. Ecol. Eng. (2013). https://doi.org/10.1016/j.ecoleng.2013.05.002
Giorgio, I., dell’Isola, F., Misra, A.: Chirality in 2D Cosserat media related to stretch-micro-rotation coupling with links to granular micromechanics. Int. J. Solids Struct. 202, 28–38 (2020)
Jasiuk, I., Ostoja-Starzewski, M.: From Lattices and Composites to Micropolar Continua. Springer, Dordrecht (2004)
Ahmad, S., Ali, K., Bashir, H.: Interaction of micropolar fluid structure with the porous media in the flow due to a rotating cone. Alex. Eng. J. 60(1), 1249–1257 (2021)
Aganovic, I., Tutek, Z.: Nonstationary micropolar fluid flow through porous medium. Nonlinear Anal. Theory Methods Appl. 30(5), 3171–3178 (1997)
Misra, A., Placidi, L., dell’Isola, F., Barchiesi, E.: Identification of a geometrically nonlinear micromorphic continuum via granular micromechanics. Z. Angew. Math. Phys. 72, 1–21 (2021)
Placidi, L., Barchiesi, E., Misra, A., Timofeev, D.: Micromechanics-based elasto-plastic-damage energy formulation for strain gradient solids with granular microstructure. Contin. Mech. Thermodyn. 33, 2213–2241 (2021)
Barchiesi, E., Misra, A., Placidi, L., Turco, E.: Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations. Z. Angew. Math. Mech. (2021). https://doi.org/10.1002/zamm.202100059
Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)
Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. Z. Angew. Math. Mech. 89(4), 242–256 (2009)
Casolo, S.: Macroscopic modelling of structured materials: relationship between orthotropic Cosserat continuum and rigid elements. Int. J. Solids Struct. 43(3–4), 475–496 (2006)
Misra, A., Poorsolhjouy, P.: Elastic behavior of 2D grain packing modeled as micromorphic media based on granular micromechanics. J. Eng. Mech. 143(1), 1–12 (2016)
Giorgio, I., Angelo, M.D., Turco, E., Misra, A.: A Biot-Cosserat two-dimensional elastic nonlinear model for a micromorphic medium. Contin. Mech. Thermodyn. 32, 1357–1369 (2020)
Vasiliev, A.A., Miroshnichenko, A.E., Ruzzene, M.: A discrete model and analysis of one-dimensional deformations in a structural interface with micro-rotations. Mech. Res. Commun. 37(2), 225–229 (2010)
Massoumi, S., Challamel, N., Lerbet, J.: Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity Cosserat models (2021). https://doi.org/10.1016/j.jsv.2020.115839
Massoumi, S., Challamel, N., Lerbet, J.: Bending/Shear Wave Dispersion Analysis of Granular Chains—Discrete and Enriched Continuous Cosserat Modelling. Manuscript submitted for publication (2021)
Manevich, A.I.: Dynamics of Timoshenko beam on linear and nonlinear foundation: phase relations, significance of the second spectrum, stability. J. Sound Vib. 344, 209–220 (2015)
Rosi, G., Placidi, L., Nguyen, V.H., Naili, S.: Wave propagation across a finite heterogeneous interphase modeled as an interface with material properties. Mech. Res. Commun. 84, 43–48 (2017)
Placidi, L., Rosi, G., Giorgio, I., Madeo, A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second gradient materials. Math. Mech. Solids 19, 555–578 (2014)
dell’Isola, F., Madeo, A., Placidi, L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. Z. Angew. Math. Phys. 92, 52–71 (2012)
Nejadsadeghi, N., Placidi, L., Romeo, M., Misra, A.: Frequency band gaps in dielectric granular metamaterials modulated by electric field. Mech. Res. Commun. 95, 96–103 (2019)
Gauthier, R.D., Jahsman, W.E.: A quest for micropolar elastic constants. J. Appl. Mech. 42(2), 369–374 (1975)
Lakes, R.S.: Experimental methods for study of Cosserat elastic solids and other generalized continua. In: Muhlhaus, H. (ed.) Continuum Models for Materials with Micro-Structure, pp. 1–22. Wiley, New York (1995)
Mora, R., Waas, A.M.: Measurement of the Cosserat constant of circular-cell polycarbonate honeycomb. Philos. Mag. Phys. Condens. Matter Struct. Defects 80(7), 1699–1713 (2000)
Beveridge, A.J., Wheel, M.A., Nash, D.H.: The micropolar elastic behaviour of model macroscopically heterogeneous materials. Int. J. Solids Struct. 50(1), 246–255 (2013)
Lakes, R.S., Gorman, D., Bonfield, W.: Holographic screening method for microelastic solids. J. Mater. Sci. 20(8), 2882–2888 (1985)
Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3, 285–308 (2015)
Ould Ouali, M., Poorsolhjouy, P., Placidi, L., Misra, A.: Evaluation of the effects of stress concentrations on plates using granular micromechanics. Constr. Build. Mater. 290, 123227 (2021)
Timofeev, D., Barchiesi, E., Misra, A., Placidi, L.: Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math. Mech. Solids 26, 738–770 (2021)
Misra, A., Nejadsadeghi, N., De Angelo, M., Placidi, L.: Chiral metamaterial predicted by granular micromechanics: verified with 1D example synthesized using additive manufacturing. Contin. Mech. Thermodyn. 32, 1497–1513 (2020)
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 (2018). https://doi.org/10.1098/rspa.2018.0063
Cielecka, I., Wozniak, M., Wozniak, C.: Elastodynamic behaviour of honeycomb cellular media. J. Elast. 60, 1–17 (2000)
Larsson, R., Zhang, Y.: Homogenization of microsystem interconnects based on micropolar theory and discontinuous kinematics. J. Mech. Phys. Solids 55(4), 819–841 (2007)
Ostoja-Starzewski, M.: Lattice models in micromechanics. J. Appl. Mech. 55(1), 35–60 (2002)
Barchiesi, E., Misra, A., Placidi, L., Turco, E.: Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations. J. Appl. Math. Mech. (2021). https://doi.org/10.1002/zamm.202100059
Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Environ. Civ. Eng. 21(5), 509–554 (2017)
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. 89, 1333–1358 (2019)
Yang, H., Ganzosch, G., Giorgio, I., Abali, B.E.: Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Z. Angew. Math. Phys. 69(4), 105–116 (2018)
Javanbakht, M., Ghaedi, M.S., Barchiesi, E., Ciallella, A.: The effect of a pre-existing nanovoid on martensite formation and interface propagation: a phase field study. Math. Mech. Solids (2020). https://doi.org/10.1177/1081286520948118
Bojanczyk, W., Lutoborski, A.: Computation of the Euler angles of a symmetric \(3\times 3\) matrix. J. Matrix Anal. Appl. 12(1), 41–48 (1991)
dell’Isola, F., Gavrilyuk, S.L.: Variational Models and Methods in Solid and Fluid Mechanics. Springer, Vienna (2011)
dell’Isola, F., Auffray, N., Eremeyev, V.A., Madeo, A., Placidi, L., Rosi, G.: Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola’s point of view. In: dell’Isola F., Maier G., Perego U., Andreaus U., Esposito R., Forest S. (eds.) The Complete Works of Gabrio Piola: Volume I. Advanced Structured Materials, pp. 606–694. Springer, Cham (2014)
dell’Isola, F., Placidi, L.: 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, pp. 1–15. Springer, Vienna (2011)
dell’Isola, F., Guarascio, M., Hutter, K.: A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi’s effective stress principle. Arch. Appl. Mech. 70(5), 323–337 (2000)
dell’Isola, F., Di Cosmo, F.: Lagrange Multipliers in Infinite-Dimensional Systems. Encyclopedia of Continuum Mechanics. hal-02099527 (2018)
Bersani, A., dell’Isola, F., Seppecher, P.: Lagrange multipliers in infinite dimensional spaces, examples of application. hal-02269765 (2019)
Kafadar, C.B., Eringen, A.C.: Micropolar media—I. The classical theory. Int. J. Eng. Sci. 9(3), 271–305 (1971)
Zheng, Q.S.: Theory of representations for tensor functions-a unified invariant approach. Appl. Mech. Rev. 47, 545–587 (1994)