Numerical investigation of a particle system compared with first and second gradient continua: Deformation and fracture phenomena*

A discrete system constituted of particles interacting by means of a centroid-based law is numerically investigated. The elements of the system move in the plane, and the range of the interaction can be varied from a more local form (first-neighbours interaction) up to a generalized nth order interaction. The aim of the model is to reproduce the behaviour of deformable bodies with standard (Cauchy model) or generalized (second gradient) deformation energy density. The numerical results suggest that the considered discrete system can effectively reproduce the behaviour of first and second gradient continua. Moreover, a fracture algorithm is introduced and some comparison between first- and second-neighbour simulations are provided.

[1]  P. Seppecher,et al.  Edge Contact Forces and Quasi-Balanced Power , 1997, 1007.1450.

[2]  Antonio Cazzani,et al.  Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches , 2016 .

[3]  Ivan Giorgio,et al.  Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids , 2013 .

[4]  W. Wall,et al.  Isogeometric structural shape optimization , 2008 .

[5]  Francesco dell’Isola,et al.  Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response , 2015 .

[6]  Pierre Seppecher,et al.  Truss Modular Beams with Deformation Energy Depending on Higher Displacement Gradients , 2003 .

[7]  S Belouettar,et al.  A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization. , 2012, Journal of the mechanical behavior of biomedical materials.

[8]  Pierre Seppecher,et al.  Linear elastic trusses leading to continua with exotic mechanical interactions , 2011 .

[9]  Leopoldo Greco,et al.  B-Spline interpolation of Kirchhoff-Love space rods , 2013 .

[10]  Ching S. Chang,et al.  Application of uniform strain theory to heterogeneous granular solids , 1990 .

[11]  David J. Steigmann,et al.  Equilibrium of elastic nets , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[12]  Jean-François Ganghoffer,et al.  Construction of Micropolar Continua from the Homogenization of Repetitive Planar Lattices , 2011 .

[13]  Ugo Andreaus,et al.  Finite element analysis of the stress state produced by an orthodontic skeletal anchorage system based on miniscrews , 2013 .

[14]  Leopoldo Greco,et al.  An implicit G1 multi patch B-spline interpolation for Kirchhoff–Love space rod , 2014 .

[15]  A. N. Solov’ev,et al.  On the determination of eigenfrequencies for nanometer-size objects , 2006 .

[16]  Alessandro Della Corte,et al.  Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof , 2015 .

[17]  Nicola Luigi Rizzi,et al.  Heterogeneous elastic solids: a mixed homogenization-rigidification technique , 2001 .

[18]  Antonio Cazzani,et al.  Isogeometric analysis of plane-curved beams , 2016 .

[19]  F. dell'Isola,et al.  Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids , 2013, 1305.6744.

[20]  N. Morozov,et al.  Method of determining the eigenfrequencies of an ordered system of nanoobjects , 2007 .

[21]  David J. Steigmann,et al.  Two-dimensional models for the combined bending and stretching of plates and shells based on three-dimensional linear elasticity , 2008 .

[22]  A. Eringen.  Microcontinuum Field Theories , 2020, Advanced Continuum Theories and Finite Element Analyses.

[23]  Flavio Stochino,et al.  Constitutive models for strongly curved beams in the frame of isogeometric analysis , 2016 .

[24]  Ivan Giorgio,et al.  Modeling of the interaction between bone tissue and resorbable biomaterial as linear elastic materials with voids , 2015 .

[25]  Luca Placidi,et al.  Towards the Design of Metamaterials with Enhanced Damage Sensitivity: Second Gradient Porous Materials , 2014 .

[26]  P. Trovalusci,et al.  Masonry as structured continuum , 1995 .

[27]  Alessandro Della Corte,et al.  Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: Perspectives of continuum modeling via higher gradient continua , 2016 .

[28]  Francesco dell’Isola,et al.  Modeling Deformable Bodies Using Discrete Systems with Centroid-Based Propagating Interaction: Fracture and Crack Evolution , 2017 .

[29]  P. Boisse,et al.  Minimization of Shear Energy in Two Dimensional Continua with Two Orthogonal Families of Inextensible Fibers: The Case of Standard Bias Extension Test , 2016 .

[30]  L. Placidi,et al.  Application of a continuum-mechanical model for the flow of anisotropic polar ice to the EDML core, Antarctica , 2008 .

[31]  Francesco dell’Isola,et al.  Geometrically nonlinear higher-gradient elasticity with energetic boundaries , 2013 .

[32]  Ivan Giorgio,et al.  Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials , 2014 .

[33]  Francesco dell’Isola,et al.  The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power , 1995 .

[34]  Ivan Giorgio,et al.  Continuum and discrete models for structures including (quasi-) inextensible elasticae with a view to the design and modeling of composite reinforcements , 2015 .

[35]  V. Eremeyev,et al.  Wave processes in nanostructures formed by nanotube arrays or nanosize crystals , 2010 .

[36]  Luca Placidi,et al.  Thermodynamics of polycrystalline materials treated by the theory of mixtures with continuous diversity , 2006 .

[37]  Dionisio Del Vescovo,et al.  Dynamic problems for metamaterials: Review of existing models and ideas for further research , 2014 .

[38]  Leopoldo Greco,et al.  A variational model based on isogeometric interpolation for the analysis of cracked bodies , 2014 .

[39]  Giuseppe Piccardo,et al.  Linear instability mechanisms for coupled translational galloping , 2005 .

[40]  N. Morozov,et al.  The spectrum of natural oscillations of an array of micro- or nanospheres on an elastic substrate , 2007 .

[41]  F. Darve,et al.  A Continuum Model for Deformable, Second Gradient Porous Media Partially Saturated with Compressible Fluids , 2013 .

[42]  Francesco dell’Isola,et al.  Elastic pantographic 2 D lattices : a numerical analysis on the static response and wave propagation , 2015 .

[43]  J. Altenbach,et al.  On generalized Cosserat-type theories of plates and shells: a short review and bibliography , 2010 .

[44]  Ugo Andreaus,et al.  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 , 2013, 1310.5599.

[45]  M. Pulvirenti,et al.  Macroscopic Description of Microscopically Strongly Inhomogenous Systems: A Mathematical Basis for the Synthesis of Higher Gradients Metamaterials , 2015, 1504.08015.

[46]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[47]  I. Giorgio,et al.  A model for elastic flexoelectric materials including strain gradient effects , 2016 .

[48]  A. Della Corte,et al.  The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[49]  Ivan Giorgio,et al.  A micro‐structural model for dissipation phenomena in the concrete , 2015 .

[50]  Dionisio Del Vescovo,et al.  Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications , 2015 .

[51]  P. Germain,et al.  The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure , 1973 .

[52]  F. D’Annibale,et al.  On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday , 2015 .

[53]  M. Cuomo,et al.  Multi-Patch Isogeometric Analysis of Space Rods , 2012 .

[54]  Francesco dell’Isola,et al.  Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua , 2012 .

[55]  L. Placidi,et al.  Continuum-mechanical, Anisotropic Flow model for polar ice masses, based on an anisotropic Flow Enhancement factor , 2009, 0903.0688.

[56]  Francesco dell’Isola,et al.  How contact interactions may depend on the shape of Cauchy cuts in Nth gradient continua: approach “à la D’Alembert” , 2012 .

[57]  A. Luongo.  A unified perturbation approach to static/dynamic coupled instabilities of nonlinear structures , 2010 .

[58]  Victor A. Eremeyev,et al.  On vectorially parameterized natural strain measures of the non-linear Cosserat continuum , 2009 .

[59]  J. Ganghoffer,et al.  A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure , 2013, Biomechanics and Modeling in Mechanobiology.

[60]  On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets , 2004, Annals of Glaciology.

[61]  Ivan Giorgio,et al.  Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations , 2016 .

[62]  Francesco dell’Isola,et al.  Elastne kahemõõtmeline pantograafiline võre: Numbriline analüüs staatilisest tagasisidest ja lainelevist , 2015 .

[63]  Ivan Giorgio,et al.  Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: A review of presentations and discussions , 2017 .

[64]  S. Vidoli,et al.  Generalized Hooke's law for isotropic second gradient materials , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.