Solving quasi‐static equations with the material‐point method

Summary The material-point method models continua by following a set of unconnected material points throughout the deformation of a body. This set of points provides a Lagrangian description of the material and geometry. Information from the material points is projected onto a background grid where equations of motion are solved. The grid solution is then used to update the material points. This paper describes how to use this method to solve quasi-static problems. The resulting discrete equations are a coupled set of nonlinear equations that are then solved with a Jacobian-free, Newton–Krylov algorithm. The technique is illustrated by examining two problems. The first problem simulates a compact tension test and includes a model of material failure. The second problem computes effective, macroscopic properties of a polycrystalline thin film. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[2]  P. E. Peterson,et al.  Fracture energy of concrete: Method of determination , 1980 .

[3]  Pieter A. Vermeer,et al.  Solution of quasi‐static large‐strain problems by the material point method , 2010 .

[4]  Deborah Sulsky,et al.  An unconditionally stable, energy–momentum consistent implementation of the material-point method , 2006 .

[5]  Jason Sanchez,et al.  A Critical Evaluation of Computational Fracture Using a Smeared Crack Approach in MPM , 2011 .

[6]  Helmut Schaeben,et al.  A novel pole figure inversion method: specification of the MTEX algorithm , 2008 .

[7]  Deborah Sulsky,et al.  Implicit dynamics in the material-point method , 2004 .

[8]  Helmut Schaeben,et al.  Calculating anisotropic physical properties from texture data using the MTEX open-source package , 2011 .

[9]  D. Sulsky,et al.  MPM simulation of dynamic material failure with a decohesion constitutive model , 2004 .

[10]  James E. Guilkey,et al.  Implicit time integration for the material point method: Quantitative and algorithmic comparisons with the finite element method , 2003 .

[11]  Hirozo Mihashi,et al.  Size effect on fracture energy of concrete , 1990 .

[12]  K. Peterson.  Modeling Arctic sea ice using the material-point method and an elastic-decohesive rheology , 2009 .

[13]  Howard L. Schreyer,et al.  Modelling surface orientation and stress at failure of concrete and geological materials , 2007 .

[14]  H. Wenk,et al.  Rietveld texture analysis from diffraction images , 2007 .

[15]  Deborah Sulsky,et al.  An energy‐consistent material‐point method for dynamic finite deformation plasticity , 2006 .

[16]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[17]  Howard L. Schreyer,et al.  Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems , 1996 .

[18]  D. Peroulis,et al.  Estimating the In-Plane Young's Modulus of Polycrystalline Films in MEMS , 2012, Journal of microelectromechanical systems.

[19]  Z. Bažant,et al.  Crack band theory for fracture of concrete , 1983 .

[20]  D. Sulsky,et al.  A particle method for history-dependent materials , 1993 .

[21]  L. Delannay,et al.  A texture discretization technique adapted to polycrystalline aggregates with non-uniform grain size , 2006 .

[22]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[23]  D. Sulsky.  Erratum: Application of a particle-in-cell method to solid mechanics , 1995 .