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Finite-volume homogenization and localization of nanoporous materials with cylindrical voids. II: New results. (English) Zbl 1406.74556
Summary: The finite-volume homogenization theory with surface-elasticity effects based on the Gurtin-Murdoch model developed in Part 1 [the authors, ibid. 70, 141–155 (2018; Zbl 1406.74557)] is employed to investigate little explored aspects of nanoporous materials’ response. New results illustrate the effects of pore array architecture, aspect ratio and mean radius of elliptical porosities on local stress fields and homogenized moduli in an admissible range of porosity volume fractions. These results highlight the importance of adjacent pore interactions neglected in the classical micromechanics models, quantified herein for the first time. The theory is also shown to capture the highly oscillatory stress fields associated with surface-elasticity induced solution instabilities in this class of materials with negative surface moduli without ill-conditioning problems. Differences and similarities between comparable finite-element and finite-volume solutions of the unit cell boundary-value problem are delineated, including identification of pore radii, and associated aspect ratios and volume fractions, at which instabilities initiate. Consistent with reported and herein generated finite-element based results, the solution instability is also shown to depend on finite-volume mesh refinement. Hence care is required to identify the admissible range of parameters in the calculation of homogenized moduli. The new theory provides an alternative and independent means of identifying stable solution ranges, and hence is a good tool in assessing the finite-element method’s predictive capability of generating stable solutions. Comparison with molecular dynamics simulations is included in further support of the theory’s potential to capture both the initial homogenized response and local stress fields that may lead to failure.

74Q05 Homogenization in equilibrium problems of solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74M25 Micromechanics of solids
74S10 Finite volume methods applied to problems in solid mechanics
Full Text: DOI
[1] Aktulga, H. M.; Fogarty, J. C.; Pandit, S. A.; Grama, A. Y., Parallel reactive molecular dynamics: numerical methods and algorithmic techniques, Parallel Comput., 38, 4-5, 245-259, (2012)
[2] Cavalcante, M. A.A.; Khatam, H.; Pindera, M.-J., Homogenization of elastic-plastic periodic materials by FVDAM and FEM approaches - an assessment, Composites Part B, 42, 1713-1730, (2011)
[3] Chatzigeorgiou, G.; Meraghni, F.; Javili, A., Generalized interfacial energy and size effects in composites, J. Mech. Phys. Solid., 106, 257-282, (2017)
[4] Chen, Q.; Wang, G.; Chen, X., Three-dimensional parametric finite-volume homogenization of periodic materials with multi-scale structural applications, Int. J. Appl. Mech., 10, 4, 1-30, (2018), 1850045
[5] Davydov, D.; Javili, A.; Steinmann, P., On molecular statics and surface-enhanced continuum modeling of nano-structures, Comp. Mater. Sci., 69, 510-519, (2013)
[6] Finnis, M. W.; Sinclair, J. E., A simple empirical N-body potential for transition metals, Philos. Mag. A, 50, 1, 45-55, (1984)
[7] Gao, W.; Yu, S. W.; Huang, G. Y., Finite element characterization of the size-dependent mechanical behaviour in nanosystems, Nanotechnology, 17, 1118-1122, (2006)
[8] Gurtin, M. E.; Murdoch, A. I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291-323, (1975) · Zbl 0326.73001
[9] Javili, A.; McBride, A.; Steinmann, P.; Reddy, B. D., Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies, Phil. Mag., 92, 28-30, 3540-3563, (2012)
[10] Javili, A.; McBride, A. T.; Mergheim, J.; Steinmann, P.; Schmidt, U., Micro-to-macro transitions for continua with surface structure at the microscale, Int. J. Solid Struct., 50, 2561-2572, (2013)
[11] Javili, A.; Chatzigeorgiou, G.; McBride, A. T.; Seinmann, P.; Linder, C., Computational homogenization of nano-materials accounting for size effects via surface elasticity, GAMM-Mitt., 38, 2, 285-312, (2015)
[12] Khatam, H.; Pindera, M.-J., Microstructural scale effects in the nonlinear elastic response of bio-inspired wavy multilayers undergoing finite deformation, Composites B, 43, 3, 2012, (2012)
[13] Kochmann, D. M., Stability criteria for continuous and discrete elastic composites and the influence of geometry on the stability of a negative-stiffness phase, Phys. Status Solidi B, 249, 7, 1399-1411, (2012)
[14] Kochmann, D. M.; Drugan, W. J., Analytical stability conditions for elastic composite materials with a non-positive-definite phase, Proc. R. Soc. A., 468, 2230-2254, (2012) · Zbl 1371.74013
[15] Mogilevskaya, S. G.; Crouch, S. L.; Stolarski, H. K., Multiple interacting circular nano-inhomogeneities with surface/interface effects, J. Mech. Phys. Solid., 56, 2298-2327, (2008) · Zbl 1171.74398
[16] Tian, L.; Rajapakse, R. K.N. D., Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity, Int. J. Solid Struct., 44, 7988-8005, (2007) · Zbl 1167.74525
[17] Tian, L.; Rajapakse, R. K.N. D., Finite element modelling of nanoscale inhomogeneities in an elastic matrix, Comput. Mater. Sci., 41, 44-53, (2007)
[18] Tu, W.; Pindera, M.-J., Cohesive zone-based damage evolution in periodic materials via finite-volume homogenization, J. Appl. Mech., 81, 10, 1-12, (2014), 01005
[19] Wang, G. F.; Feng, X. Q.; Yu, S. W., Surface buckling of a bending microbeam due to surface elasticity, EPL, 77, 1-4, (2007), 44002
[20] Winter, N.; Becton, M.; Zhang, L.; Wang, X., Effects of pore design on mechanical properties of nanoporous silicon, Acta Mater., 124, 127-136, (2017)
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