On the (Meshless Local Petrov-Galerkin) MLPG-Eshelby Method in Computational Finite Deformation Solid Mechanics - Part II

This paper presents a new method for the computational mechanics of large strain deformations of solids, as a fundamental departure from the currently popular finite element methods (FEM). The currently widely popular primal FEM: (1) uses element-based interpolations for displacements as the trial functions, and element-based interpolations of displacement-like quantities as the test functions; (2) uses the same type and class of trial & test functions, leading to a Galerkin approach; (3) uses the trial and test functions which are most often continuous at the inter-element boundaries; (4) leads to sparsely populated symmetric tangent stiffness matrices; (5) computes piecewise-linear predictor solutions based on the global weak-forms of the Newtonian Momentum Balance Laws for a Lagrangean Stress tensor, such as the symmetric Second Piola-Kirchhoff Stress tensor S [≡ JF−1 ·σ ·F−t , where σ is the Cauchy Stress tensor and F the deformation gradient] in the initial or any other known reference configuration; and (6) computes a corrector solution, using Newton-Raphson or other Jacobian-inversion-free iterations, based on the global weak-forms of the Newtonian Momentum Balance Laws for the symmetric Cauchy Stress tensor σ in the current configuration. In a radical departure, the present approach blends the Energy-Conservation Laws of Noether and Eshelby, and the Meshless Local Petrov Galerkin (MLPG) Methods of Atluri, and is designated herein as the MLPG-Eshelby Method. In the MLPGEshelby Method, we: (1) use meshless node-based functions δX, for configurational changes of the undeformed configuration, as the trial functions; (2) meshless node-based functions δx, for configurational changes of the deformed configuration, as the test functions; (3) the trial functions δX and the test functions δx are necessarily different and belong to different classes of functions, thus naturally 1 Livermore Software Technology Corporation, Livermore, CA, 94551, USA. 2 International Collaboratory for Fundamental Studies in Engineering and the Sciences, 4131 Engineering Gateway, University of California, Irvine, Irvine, CA, 92697, USA. 3 Fellow & Eminent Scholar, Texas Institute for Advanced Study, TAMU-3474, College Station, TX 77843, USA. 200 Copyright © 2014 Tech Science Press CMES, vol.97, no.3, pp.199-237, 2014 leading to a Petrov-Galerkin approach; (4) leads to sparsely populated unsymmetric tangent stiffness matrices; (5) the trial functions δX, as well as the test functions δx, may either be continuous or be discontinuous in their respective configurations; (6) generate piecewise-linear predictor solutions based on the local weak-forms of the Noether/Eshelby Energy Conservation Laws for the Lagrangean unsymmetric Eshelby Stress tensor T in the undeformed configuration [T = W I−P ·F; where P = JF−1 ·σ is the first Piola-Kirchhoff Stress tensor, and W is the stress-work density per unit initial volume of the solid] and (7) generate corrector solutions, based on Newton-Raphson or Jacobian-inversion-free iterations, using the local weak-forms of the Noether/Eshelby Energy Conservation Laws in the current configuration, for a newly introduced Eulerean symmetric Stress tensor S̃ [which is the counter part of T] in the current configuration [S̃ = (W/J)I−σ , often called by chemists as the Chemical Potential Tensor]. It is shown in the present paper that the present MLPG-Eshelby Method, based on the meshless local weak-forms of the Noether/Eshelby Energy Conservation Laws, converges much faster and leads to much better accuracies than the currently popular FEM based on the global weakforms of the Newtonian Momentum Balance Laws. The present paper is limited to hyperelasticity, while large strains of inelastic solids will be considered in our forthcoming papers.