Research Papers

Closed-Form Coordinate-Free Decompositions of the Two-Dimensional Strain and Stress for Modeling Tension–Compression Dissymmetry

[+] Author and Article Information
Q.-C. He

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China;
Université Paris-Est,
Laboratoire Modélisation et Simulation
Multi Echelle,
5 bd Descartes,
Marne-la-Vallée Cedex 2 77454, France
e-mail: qi-chang.he@u-pem.fr

Q. Shao

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 28, 2018; final manuscript received December 5, 2018; published online January 8, 2019. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 86(3), 031007 (Jan 08, 2019) (6 pages) Paper No: JAM-18-1612; doi: 10.1115/1.4042217 History: Received October 28, 2018; Revised December 05, 2018

The modeling of the different mechanical behaviors of brittle and quasi-brittle materials in tension and compression leads to partitioning of the strain (or stress) tensor into a positive part and a negative part. In this study, applying a recently proposed general method to the two-dimensional (2D) strain and stress tensors, closed-form coordinate-free expressions are obtained for their decompositions which are orthogonal in the sense of an inner product where the forth-order elastic stiffness or compliance acts as a metric. The orthogonal decompositions are given analytically and explicitly for all possible 2D elastic symmetries, i.e., isotropic, orthotropic, square, and totally anisotropic elastic materials. These results can be directly used, for example, in developing phase field methods for modeling and simulating the fracture of isotropic and anisotropic brittle and quasi-brittle materials.

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Grahic Jump Location
Fig. 1

Four elastic symmetry classes in the 2D case: (a) isotropy, (b) square symmetry, (c) orthotropy, and (d) total anisotropy



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