A Lagrangian meshfree method applied to linear and nonlinear elasticity

RESEARCH ARTICLE A Lagrangian meshfree method applied to linear and nonlinear elasticity Wade A. Walker* Independent Researcher, Austin, Texas, United States of America * Abstract a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Walker WA (2017) A Lagrangian meshfree method applied to linear and nonlinear elasticity. PLoS ONE 12(10): e0186345. https://doi. org/10.1371/journal.pone.0186345 The repeated replacement method (RRM) is a Lagrangian meshfree method which we have previously applied to the Euler equations for compressible fluid flow. In this paper we present new enhancements to RRM, and we apply the enhanced method to both linear and nonlinear elasticity. We compare the results of ten test problems to those of analytic solvers, to demonstrate that RRM can successfully simulate these elastic systems without many of the requirements of traditional numerical methods such as numerical derivatives, equation system solvers, or Riemann solvers. We also show the relationship between error and computational effort for RRM on these systems, and compare RRM to other methods to highlight its strengths and weaknesses. And to further explain the two elastic equations used in the paper, we demonstrate the mathematical procedure used to create Riemann and SedovTaylor solvers for them, and detail the numerical techniques needed to embody those solvers in code. Editor: Krishna Garikipati, University of Michigan, UNITED STATES Received: July 1, 2017 Introduction Accepted: October 1, 2017 The repeated replacement method (RRM) [1] is a Lagrangian meshfree method for the simulation of time-dependent systems of conservation laws. We previously used RRM to simulate the one-dimensional Euler equations for compressible fluid flow to establish the basic functionality of the method, and we compared the results to an exact Riemann solver for the Euler equations given by Toro [2]. In this paper, we enhance RRM to increase its speed and accuracy, and apply it to the more challenging problems of one-dimensional linear and nonlinear elasticity. This allows us to demonstrate that RRM works for a range of constitutive equations, while maintaining good accuracy and error scaling behavior. In this paper, we first motivate and derive our linear and nonlinear elastic constitutive equations, to define the terms and symbols used in subsequent sections. Second, we give anoverview of the repeated replacement method with a detailed comparison to earlier work, highlighting how the strengths and weaknesses of RRM differ from those of previous methods. Third, we explain the improvements made to RRM in the course of adapting it to handle elastic systems. Fourth, we present the derivation and explanation of our Riemann and SedovTaylor solvers. Then we show the results of RRM for eleven test problems, validating the Published: October 18, 2017 Copyright: © 2017 Wade A. Walker. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper and its Supporting Information files. Funding: The author received no specific funding for this work. Competing interests: The author has declared that no competing interests exist. PLOS ONE | https://doi.org/10.1371/journal.pone.0186345 October 18, 2017 1 / 49 A Lagrangian meshfree method applied to linear and nonlinear elasticity results against Riemann and Sedov-Taylor solvers, and show that RRM’s error convergence behavior is somewhat super-linear. Finally, we present a summary and conclusion. Elastic equations For both linear and nonlinear cases, we assume a homogeneous, one-dimensional, elastic continuum material that is subject to finite strain, which is a strain large enough that we cannot simplify our mathematical treatment by assuming it to be infinitesimal. The material’s reference (or undeformed) density is ρ0, and its spatial (or deformed) density at each point x is ρ(x). We will express the deformation in terms of the stretch λ, which is written in terms of density as λ(x) = ρ0/ρ(x). For a finite-sized piece of material, stretch is defined as λ  l/l0, where l0 and l are the undeformed and deformed lengths of the piece, respectively. Strain is defined as   (l − l0)/l0 = λ − 1. We assume that the internal energy density of the material due to its state of strain and temperature, per unit of reference length, can be expressed as some function C(λ, T) = Cs(λ) + Ct(λ, T), where Cs is the strain energy density, and Ct is the thermal energy density. The Cauchy stress in such a material has both an elastic and a thermal component, and can be derived from the energy density as sðl; TÞ ¼ se ðlÞ þ st ðl; TÞ ¼ @Cs ðlÞ @Ct ðl; TÞ þ @l @l ð1Þ A material whose stress is derived from an energy density function in this way is called a Green-elastic or hyperelastic material. Linear elasticity Linear elasticity is defined by the direct proportionality of stress to strain. The elastic part of the stress is se ðlÞ ¼ E ¼ Eðl 1Þ ð2Þ where E is the elastic modulus, which has the units (kg
m)/s2 in one dimension. To obtain the strain energy density, we note that the integral of stress over distance is strain energy, and strain energy density with respect to the reference length is simply strain energy/l0. So, making use of the fact that λ = l/l0, the strain energy density is Z 1 l E 2 Cs ðlÞ ¼ sðlÞ dl ¼ ðl 1Þ ð3Þ l0 l0 2 To complete this simple model, we choose a thermal energy density Ct(T) = Cv T where Cv is the heat capacity in J/(m
K). This Ct has no dependence on λ, since it is difficult to form such an expression that has the correct signs for both energy and stress equations without making the stress nonlinear in λ. This results in a total stress of sðlÞ ¼ @Cs ðlÞ @Ct ðTÞ þ ¼ Eðl @l @l 1Þ ð4Þ Note that this stress does not model thermal expansion, and also allows the material to be compressed down to zero length with finite work Cs(λ)|λ = 0 = E/2, so we have sacrificed physicality for the sake of linearity in this simple model. PLOS ONE | https://doi.org/10.1371/journal.pone.0186345 October 18, 2017 2 / 49 A Lagrangian meshfree method applied to linear and nonlinear elasticity The total energy density in spatial coordinates, including kinetic energy, is Ctot ðl; u; TÞ ¼ ru2 Cðl; TÞ ru2 E ¼ þ þ ðl l 2l 2 2 2 1Þ þ Cv T l ð5Þ where u is velocity, and the division of the C term by λ is needed to convert the energy density from reference to spatial coordinates. Finally, making use of the fact that λ = ρ0/ρ, the speed of sound in the material in terms of the density is rffiffiffiffiffi pffiffiffiffiffiffiffi Er0 @s ð6Þ aðrÞ ¼ ¼ @r r Nonlinear elasticity There are many existing models of nonlinear elasticity. Some, such as that of Ogden [3], are complex enough to form an (...truncated)