Questioning numerical integration methods for microsphere (and microplane) constitutive equations

•Cubature methods for microsphere and microplane constitutive equations are studied.•Three different formulations are compared by computing large strain invariants.•The classical Bažant and Oh 42 points algorithm is revealed ineffective.•For the two other methods, a very large number of integration...

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Bibliographic Details
Published in:Mechanics of materials Vol. 89; pp. 216 - 228
Main Author: Verron, Erwan
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.10.2015
Subjects:
Constitutive equations
Constitutive relationships
Mathematical analysis
Mathematical models
Microsphere models
Microspheres
Numerical integration
Numerical integration algorithms
Spherical harmonics
Stress-strain relationships
Microsphere models
Constitutive equations
Numerical integration algorithms
ISSN:0167-6636, 1872-7743
Online Access:Get full text
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Summary:•Cubature methods for microsphere and microplane constitutive equations are studied.•Three different formulations are compared by computing large strain invariants.•The classical Bažant and Oh 42 points algorithm is revealed ineffective.•For the two other methods, a very large number of integration points is required. In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated. Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.
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ISSN:0167-6636
1872-7743
DOI:10.1016/j.mechmat.2015.06.013