Data-Driven multiscale modeling in mechanics

Abstract We present a Data-Driven framework for multiscale mechanical analysis of materials. The proposed framework relies on the Data-Driven formulation in mechanics (Kirchdoerfer and Ortiz, 2016), with the material data being directly extracted from lower-scale computations. Particular emphasis is placed on two key elements: the parametrization of material history, and the optimal sampling of the mechanical state space. We demonstrate an application of the framework in the prediction of the behavior of sand, a prototypical complex history-dependent material. In particular, the model is able to predict the material response under complex nonmonotonic loading paths, and compares well against plane strain and triaxial compression shear banding experiments.

[1]  M. Ortiz,et al.  A variational Cam-clay theory of plasticity , 2004 .

[2]  Xuxin Tu,et al.  Multiscale framework for behavior prediction in granular media , 2009 .

[3]  Qizhi He,et al.  A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Database , 2019, Computer Methods in Applied Mechanics and Engineering.

[4]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[5]  Trenton Kirchdoerfer,et al.  Data-driven computational mechanics , 2015, 1510.04232.

[6]  Timothy W. Simpson,et al.  Metamodels for Computer-based Engineering Design: Survey and recommendations , 2001, Engineering with Computers.

[7]  Thomas J. Santner,et al.  Design and analysis of computer experiments , 1998 .

[8]  S. Nemat-Nasser,et al.  A Micromechanical Description of Granular Material Behavior , 1981 .

[9]  F. Radjai,et al.  Fabric evolution and accessible geometrical states in granular materials , 2012 .

[10]  P. Alam,et al.  H , 1887, High Explosives, Propellants, Pyrotechnics.

[11]  Hans-Peter Kriegel,et al.  A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise , 1996, KDD.

[12]  T. Majmudar,et al.  Jamming transition in granular systems. , 2006, Physical review letters.

[13]  R. Regueiro,et al.  Concurrent Multiscale Computational Modeling for Dense Dry Granular Materials Interfacing Deformable Solid Bodies , 2011 .

[14]  A. Schofield,et al.  On The Yielding of Soils , 1958 .

[15]  David L. McDowell,et al.  Non-local separation constitutive laws for interfaces and their relation to nanoscale simulations , 2004 .

[16]  J. Andrade,et al.  All you need is shape: Predicting shear banding in sand with LS-DEM , 2018 .

[17]  WaiChing Sun,et al.  Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity , 2020, ArXiv.

[18]  José E. Andrade,et al.  Capturing strain localization in dense sands with random density , 2006 .

[19]  Giacomo Fiorin,et al.  Using collective variables to drive molecular dynamics simulations , 2013 .

[20]  P. Alam.  ‘W’ , 2021, Composites Engineering.

[21]  H. Fischmeister,et al.  Crack propagation in b.c.c. crystals studied with a combined finite-element and atomistic model , 1991 .

[22]  Majid T. Manzari,et al.  SIMPLE PLASTICITY SAND MODEL ACCOUNTING FOR FABRIC CHANGE EFFECTS , 2004 .

[23]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[24]  J. Andrade,et al.  Level set discrete element method for three-dimensional computations with triaxial case study , 2016 .

[25]  V. Bulatov,et al.  Connecting atomistic and mesoscale simulations of crystal plasticity , 1998, Nature.

[26]  Gaël Combe,et al.  Two-scale modeling of granular materials: a DEM-FEM approach , 2011 .

[27]  S. Hall,et al.  Grain-scale experimental investigation of localised deformation in sand: a discrete particle tracking approach , 2012 .

[28]  J. Andrade,et al.  Investigating the incremental behavior of granular materials with the level-set discrete element method , 2020, Journal of the Mechanics and Physics of Solids.

[29]  M. F. Bransby,et al.  Fault Rupture Propagation through Sand: Finite-Element Analysis and Validation through Centrifuge Experiments , 2007 .

[30]  D. C. Drucker,et al.  Soil mechanics and plastic analysis or limit design , 1952 .

[31]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[32]  G. Grest,et al.  Flow-Arrest Transitions in Frictional Granular Matter. , 2018, Physical review letters.

[33]  Francisco Chinesta,et al.  A Manifold Learning Approach to Data-Driven Computational Elasticity and Inelasticity , 2016, Archives of Computational Methods in Engineering.

[34]  J. Tinsley Oden,et al.  Hierarchical modeling of heterogeneous solids , 1996 .

[35]  Richard J. Bathurst,et al.  Observations on stress-force-fabric relationships in idealized granular materials , 1990 .

[36]  P. Alam.  ‘E’ , 2021, Composites Engineering: An A–Z Guide.

[37]  Laurent Stainier,et al.  Model-Free Data-Driven Inelasticity , 2018, Computer Methods in Applied Mechanics and Engineering.

[38]  David González,et al.  Thermodynamically consistent data-driven computational mechanics , 2018, Continuum Mechanics and Thermodynamics.

[39]  R. Chambon,et al.  Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography , 1996 .

[40]  Trenton Kirchdoerfer,et al.  Data‐driven computing in dynamics , 2017, 1706.04061.

[41]  J. Q. Broughton,et al.  Concurrent coupling of length scales: Methodology and application , 1999 .

[42]  R. Bathurst,et al.  Analytical study of induced anisotropy in idealized granular materials , 1989 .

[43]  A. Schofield,et al.  Critical State Soil Mechanics , 1968 .

[44]  Katalin Bagi,et al.  Analysis of microstructural strain tensors for granular assemblies , 2006 .

[45]  J. Hutchinson,et al.  PLASTICITY THEORY , 2008, How to Love Everyone and Almost Get Away with It.

[46]  R Madec,et al.  From dislocation junctions to forest hardening. , 2002, Physical review letters.

[47]  David Lifka,et al.  Instability dynamics in three-dimensional fracture: An atomistic simulation , 1997 .

[48]  J. Andrade,et al.  A novel experimental device for investigating the multiscale behavior of granular materials under shear , 2017 .

[49]  Reese E. Jones,et al.  Machine Learning Models of Plastic Flow Based on Representation Theory , 2018, Computer Modeling in Engineering & Sciences.

[50]  Jean-Paul Watson,et al.  Algorithmic dimensionality reduction for molecular structure analysis. , 2008, The Journal of chemical physics.

[51]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[52]  M. Satake,et al.  Fabric tensor in granular materials , 1982 .

[53]  S. Luding,et al.  On the fabric tensor of polydisperse granular materials in 2D , 2004 .

[54]  Reduced Gravity Effects on the Strength of Granular Matter: DEM Simulations versus Experiments , 2020 .

[55]  G. Midi,et al.  On dense granular flows , 2003, The European physical journal. E, Soft matter.

[56]  Trenton Kirchdoerfer,et al.  Data Driven Computing with Noisy Material Data Sets , 2017, 1702.01574.

[57]  I. Vardoulakis,et al.  The thickness of shear bands in granular materials , 1987 .

[58]  James H. Garrett,et al.  Knowledge-Based Modeling of Material Behavior with Neural Networks , 1992 .

[59]  Richard LeSar,et al.  Discrete dislocation dynamics simulations of plasticity at small scales , 2010 .

[60]  Yannis F. Dafalias,et al.  Anisotropic Critical State Theory: Role of Fabric , 2012 .

[61]  Gregory J. Wagner,et al.  Coupling of atomistic and continuum simulations using a bridging scale decomposition , 2003 .

[62]  Guy T. Houlsby,et al.  Potential particles: a method for modelling non-circular particles in DEM , 2009 .

[63]  Kun Wang,et al.  A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning , 2018, Computer Methods in Applied Mechanics and Engineering.

[64]  M. Gurtin,et al.  Thermodynamics with Internal State Variables , 1967 .

[65]  M Mozaffar,et al.  Deep learning predicts path-dependent plasticity , 2019, Proceedings of the National Academy of Sciences.

[66]  F. Radjai,et al.  Modeling Granular Materials: Century-Long Research across Scales , 2017 .

[67]  Ken Been,et al.  A STATE PARAMETER FOR SANDS , 1985 .

[68]  M. Ortiz,et al.  Quasicontinuum analysis of defects in solids , 1996 .

[69]  Rodney Hill,et al.  The essential structure of constitutive laws for metal composites and polycrystals , 1967 .

[70]  Laurent Stainier,et al.  Data-based derivation of material response , 2018 .

[71]  C. O’Sullivan.  Particulate Discrete Element Modelling: A Geomechanics Perspective , 2011 .

[72]  Kanatani Ken-Ichi.  DISTRIBUTION OF DIRECTIONAL DATA AND FABRIC TENSORS , 1984 .

[73]  T. Bieler,et al.  Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications , 2010 .