Two-Dimensional Analysis of Size Effects in Strain-Gradient Granular Solids with Damage-Induced Anisotropy Evolution
Publication: Journal of Engineering Mechanics
Volume 147, Issue 11
Abstract
We analyze in two dimensions the mechanical behavior of materials with granular microstructures modeled by means of a variationally formulated strain-gradient continuum approach based on micromechanics and show that it can capture microstructural-size-dependent effects. Tension-compression asymmetry of grain-assembly interactions, as well as microscale damage, is taken into account and the continuum scale is linked to the grain-scale mechanisms. Numerical results are provided for finite deformations and substantiate previous research. As expected, results show interesting size-dependent effects that are typical of strain-gradient modeling.
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Data Availability Statement
All data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
Acknowledgments
AM is partially supported by the United States National Science Foundation grant CMMI-1727433. LP is partially supported by the RESBA project (from Politecnico di Torino).
References
Abali, B., W. Müller, and F. dell’Isola. 2017. “Theory and computation of higher gradient elasticity theories based on action principles.” Arch. Appl. Mech. 87 (9): 1495–1510. https://doi.org/10.1007/s00419-017-1266-5.
Alibert, J.-J., P. Seppecher, and F. dell’Isola. 2003. “Truss modular beams with deformation energy depending on higher displacement gradients.” Math. Mech. Solids 8 (1): 51–73. https://doi.org/10.1177/1081286503008001658.
Andreaus, U., F. dell’Isola, I. Giorgio, L. Placidi, T. Lekszycki, and N. Rizzi. 2016. “Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity.” Int. J. Eng. Sci. 108 (Nov): 34–50. https://doi.org/10.1016/j.ijengsci.2016.08.003.
Auffray, N., F. dell’Isola, V. Eremeyev, A. Madeo, and G. Rosi. 2015a. “Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids.” Math. Mech. Solids 20 (4): 375–417. https://doi.org/10.1177/1081286513497616.
Auffray, N., J. Dirrenberger, and G. Rosi. 2015b. “A complete description of bi-dimensional anisotropic strain-gradient elasticity.” Int. J. Solids Struct. 69–70 (Sep): 195–206. https://doi.org/10.1016/j.ijsolstr.2015.04.036.
Bourdin, B., G. Francfort, and J.-J. Marigo. 2008. “The variational approach to fracture.” J. Elast. 91 (1): 5–148. https://doi.org/10.1007/s10659-007-9107-3.
Carcaterra, A., F. dell’Isola, R. Esposito, and M. Pulvirenti. 2015. “Macroscopic description of microscopically strongly inhomogeneous systems: A mathematical basis for the synthesis of higher gradients metamaterials.” Arch. Ration. Mech. Anal. 218 (3): 1239–1262. https://doi.org/10.1007/s00205-015-0879-5.
Cuomo, M., F. dell’Isola, and L. Greco. 2016. “Simplified analysis of a generalized bias test for fabrics with two families of inextensible fibres.” Zeitschrift für angewandte Mathematik und Physik 67 (3): 61. https://doi.org/10.1007/s00033-016-0653-z.
Cuomo, M., F. dell’Isola, L. Greco, and N. Rizzi. 2017. “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 115 (Apr): 423–448. https://doi.org/10.1016/j.compositesb.2016.08.043.
dell’Isola, F., I. Giorgio, M. Pawlikowski, and N. L. Rizzi. 2016. “Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium.” Proc. R. Soc. London, Ser. A: Math. Phys. Eng. Sci. 472 (2185): 20150790. https://doi.org/10.1098/rspa.2015.0790.
dell’Isola, F., A. Madeo, and P. Seppecher. 2009. “Boundary conditions at fluid-permeable interfaces in porous media: A variational approach.” Int. J. Solids Struct. 46 (17): 3150–3164. https://doi.org/10.1016/j.ijsolstr.2009.04.008.
dell’Isola, F., and L. Placidi. 2011. “Variational principles are a powerful tool also for formulating field theories.” In Variational models and methods in solid and fluid mechanics, 1–15. Vienna, Austria: Springer.
dell’Isola, F., P. Seppecher, and A. Della Corte. 2015. “The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: A review of existing results.” Proc. R. Soc. London, Ser. A 471 (2183): 20150415. https://doi.org/10.1098/rspa.2015.0415.
Del Piero, G., G. Lancioni, and R. March. 2007. “A variational model for fracture mechanics: Numerical experiments.” J. Mech. Phys. Solids 55 (12): 2513–2537. https://doi.org/10.1016/j.jmps.2007.04.011.
Eremeyev, V. A. 2018. “On the material symmetry group for micromorphic media with applications to granular materials.” Mech. Res. Commun. 94 (Dec): 8–12. https://doi.org/10.1016/j.mechrescom.2018.08.017.
Misra, A., P. Luca, and E. Turco. 2020a. “Variational methods for discrete models of granular materials.” In Encylopedia of continuum mechanics. Berlin: Springer.
Misra, A., N. Nejadsadeghi, M. De Angelo, and L. Placidi. 2020b. “Chiral metamaterial predicted by granular micromechanics: Verified with 1D example synthesized using additive manufacturing.” Continuum Mech. Thermodyn. 32 (5): 1497–1513. https://doi.org/10.1007/s00161-020-00862-8.
Misra, A., and P. Poorsolhjouy. 2020. “Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics.” Math. Mech. Solids 25 (10): 1778–1803. https://doi.org/10.1177/1081286515576821.
Misra, A., and V. Singh. 2013. “Micromechanical model for viscoelastic materials undergoing damage.” Continuum Mech. Thermodyn. 25 (2): 343–358. https://doi.org/10.1007/s00161-012-0262-9.
Misra, A., and V. Singh. 2015b. “Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model.” Contin. Mech. Thermodyn. 27 (4–5): 787. https://doi.org/10.1007/s00161-014-0360-y.
Nejadsadeghi, N., and A. Misra. 2020. “Extended granular micromechanics approach: A micromorphic theory of degree .” Math. Mech. Solids 25 (2): 407–429. https://doi.org/10.1177/1081286519879479.
Placidi, L. 2015. “A variational approach for a nonlinear 1-dimensional second gradient continuum damage model.” Continuum Mech. Thermodyn. 27 (4–5): 623. https://doi.org/10.1007/s00161-014-0338-9.
Placidi, L. 2016. “A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model.” Continuum Mech. Thermodyn. 28 (1–2): 119–137. https://doi.org/10.1007/s00161-014-0405-2.
Placidi, L., E. Barchiesi, and A. Misra. 2018a. “A strain gradient variational approach to damage: A comparison with damage gradient models and numerical results.” Math. Mech. Complex Syst. 6 (2): 77–100. https://doi.org/10.2140/memocs.2018.6.77.
Placidi, L., E. Barchiesi, A. Misra, and U. Andreaus. 2020. “Variational methods in continuum damage and fracture mechanics.” In Encylopedia of continuum mechanics. Berlin: Springer.
Placidi, L., R. Greve, H. Seddik, and S. Faria. 2010. “Continuum-mechanical, anisotropic flow model for polar ice masses, based on an anisotropic flow enhancement factor.” Continuum Mech. Thermodyn. 22 (3): 221–237. https://doi.org/10.1007/s00161-009-0126-0.
Placidi, L., A. Misra, and E. Barchiesi. 2018b. “Two-dimensional strain gradient damage modeling: A variational approach.” Zeitschrift für angewandte Mathematik und Physik 69 (3): 56. https://doi.org/10.1007/s00033-018-0947-4.
Placidi, L., A. Misra, and E. Barchiesi. 2019. “Simulation results for damage with evolving microstructure and growing strain gradient moduli.” Continuum Mech. Thermodyn. 31 (4): 1143–1163. https://doi.org/10.1007/s00161-018-0693-z.
Poorsolhjouy, P., and A. Misra. 2017. “Effect of intermediate principal stress and loading-path on failure of cementitious materials using granular micromechanics.” Int. J. Solids Struct. 108 (Mar): 139–152. https://doi.org/10.1016/j.ijsolstr.2016.12.005.
Rahali, Y., I. Giorgio, J. Ganghoffer, and F. dell’Isola. 2015. “Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices.” Int. J. Eng. Sci. 97 (Dec): 148–172. https://doi.org/10.1016/j.ijengsci.2015.10.003.
Seddik, H., R. Greve, L. Placidi, I. Hamann, and O. Gagliardini. 2008. “Application of a continuum-mechanical model for the flow of anisotropic polar ice to the EDML core, Antarctica.” J. Glaciol. 54 (187): 631–642. https://doi.org/10.3189/002214308786570755.
Simone, A., G. Wells, and L. Sluys. 2003. “From continuous to discontinuous failure in a gradient-enhanced continuum damage model.” Comput. Methods Appl. Mech. Eng. 192 (41): 4581–4607. https://doi.org/10.1016/S0045-7825(03)00428-6.
Steigmann, D., and F. dell’Isola. 2015. “Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching.” Acta Mech. Sin. 31 (3): 373–382. https://doi.org/10.1007/s10409-015-0413-x.
Timofeev, D., E. Barchiesi, A. Misra, and L. Placidi. 2021. “Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution.” Math. Mech. Solids 26 (5): 738–770. https://doi.org/10.1177/1081286520968149.
Turco, E., F. dell’Isola, A. Cazzani, and N. Rizzi. 2016. “Hencky-type discrete model for pantographic structures: Numerical comparison with second gradient continuum models.” Z. Angew. Math. Phys. 67 (85): 1–28. https://doi.org/10.1007/s00033-016-0681-8.
Wang, C., and X. Qian. 2018. “Heaviside projection based aggregation in stress constrained topology optimization.” Int. J. Numer. Methods Eng. 115 (7): 849–871. https://doi.org/10.1002/nme.5828.
Yang, Y., and A. Misra. 2010. “Higher-order stress-strain theory for damage modeling implemented in an element-free Galerkin formulation.” Comput. Model. Eng. Sci. 64 (1): 1–36. https://doi.org/10.3970/cmes.2010.064.001.
Yang, Y., and A. Misra. 2012. “Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity.” Int. J. Solids Struct. 49 (18): 2500–2514. https://doi.org/10.1016/j.ijsolstr.2012.05.024.
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Published In
Journal of Engineering Mechanics
Volume 147 • Issue 11 • November 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Feb 25, 2021
Accepted: Jul 23, 2021
Published online: Sep 10, 2021
Published in print: Nov 1, 2021
Discussion open until: Feb 10, 2022
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