Finite-volume homogenization and localization of nanoporous materials with cylindrical voids. Part 2: New results

Abstract The finite-volume homogenization theory with surface-elasticity effects based on the Gurtin-Murdoch model developed in Part 1 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.

[1]  R.K.N.D. Rajapakse,et al.  Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity , 2007 .

[2]  S. L. Crouch,et al.  Multiple interacting circular nano-inhomogeneities with surface/interface effects , 2008 .

[3]  D. Kochmann.  Stability criteria for continuous and discrete elastic composites and the influence of geometry on the stability of a negative‐stiffness phase , 2012 .

[4]  Micro‐to‐macro transitions for continua with surface structure at the microscale , 2012 .

[5]  R. Rajapakse,et al.  Finite element modelling of nanoscale inhomogeneities in an elastic matrix , 2007 .

[6]  Gang Wang,et al.  Surface buckling of a bending microbeam due to surface elasticity , 2007 .

[7]  Yu Shouwen,et al.  Finite element characterization of the size-dependent mechanical behaviour in nanosystems , 2006, Nanotechnology.

[8]  Paul Steinmann,et al.  Computational homogenization of nano‐materials accounting for size effects via surface elasticity , 2015 .

[9]  Marek-Jerzy Pindera,et al.  Homogenization of elastic–plastic periodic materials by FVDAM and FEM approaches – An assessment , 2011 .

[10]  Paul Steinmann,et al.  On molecular statics and surface-enhanced continuum modeling of nano-structures , 2013 .

[11]  Wenqiong Tu,et al.  Cohesive Zone-Based Damage Evolution in Periodic Materials Via Finite-Volume Homogenization , 2014 .

[12]  Ananth Grama,et al.  Parallel reactive molecular dynamics: Numerical methods and algorithmic techniques , 2012, Parallel Comput..

[13]  Paul Steinmann,et al.  Relationships between the admissible range of surface material parameters and stability of linearly elastic bodies , 2012 .

[14]  Morton E. Gurtin,et al.  A continuum theory of elastic material surfaces , 1975 .

[15]  W. Drugan,et al.  Analytical stability conditions for elastic composite materials with a non-positive-definite phase , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[16]  M. Finnis,et al.  A simple empirical N-body potential for transition metals , 1984 .

[17]  Guannan Wang,et al.  Three-Dimensional Parametric Finite-Volume Homogenization of Periodic Materials with Multi-Scale Structural Applications , 2018 .

[18]  Ali Javili,et al.  Generalized interfacial energy and size effects in composites , 2017 .

[19]  Xianqiao Wang,et al.  Effects of pore design on mechanical properties of nanoporous silicon , 2017 .

[20]  M. Pindera,et al.  Microstructural scale effects in the nonlinear elastic response of bio-inspired wavy multilayers undergoing finite deformation , 2012 .