A Particle Method for History-Dependent Materials

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Summary

• First paper of Material Point Method

Abstract

A broad class of engineering problems including penetration, impact and large rotations of solid bodies causes severe numerical problems. For these problems, the constitutive equations are history dependent so material points must be followed; this is difficult to implement in a Eulerian scheme. On the other hand, purely Lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement. Remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables, a process which can introduce errors. Proposed here is an extension of the particle-in-cell method in which particles are interpreted to be material points that are followed through the complete loading process. A fixed Eulerian grid provides the means for determining a spatial gradient. Because the grid can also be interpreted as an updated Lagrangian frame, the usual convection term in the acceleration associated with Eulerian formulations does not appear. With the use of maps between material points and the grid, the advantages of both Eulerian and Lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history. Example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced. Also, it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity.

Author

• D.Sulskya
• Z.Chenb
• H.L.Schreyer

Journal/Conference

Computer Methods in Applied Mechanics and Engineering 1994

Link

• Paper: https://prod-ng.sandia.gov/techlib-noauth/access-control.cgi/1993/937044.pdf
• ScienceDirect: https://www.sciencedirect.com/science/article/pii/0045782594901120

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Difference from conventional FEM

• The mass matrix varies with time and must be computed at each time step
• The gradient, stress and strain are evaluated at material points that can move from one element to another
• The use of the point mass representation results in the appearance of point masses in expressions for the internal and external force vectors
• The specific stress is used to provide a convenient form of the equation of motion