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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March 2019
Research-Article
Closed-Form Coordinate-Free Decompositions of the Two-Dimensional Strain and Stress for Modeling Tension–Compression Dissymmetry
Q.-C. He,
Q.-C. He
School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China;
Université Paris-Est,
Laboratoire Modélisation et Simulation
Multi Echelle,
MSME UMR 8208 CNRS,
5 bd Descartes,
Marne-la-Vallée Cedex 2 77454, France
e-mail: qi-chang.he@u-pem.fr
Southwest Jiaotong University,
Chengdu 610031, China;
Université Paris-Est,
Laboratoire Modélisation et Simulation
Multi Echelle,
MSME UMR 8208 CNRS,
5 bd Descartes,
Marne-la-Vallée Cedex 2 77454, France
e-mail: qi-chang.he@u-pem.fr
Search for other works by this author on:
Q. Shao
Q. Shao
School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China
Southwest Jiaotong University,
Chengdu 610031, China
Search for other works by this author on:
Q.-C. He
School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China;
Université Paris-Est,
Laboratoire Modélisation et Simulation
Multi Echelle,
MSME UMR 8208 CNRS,
5 bd Descartes,
Marne-la-Vallée Cedex 2 77454, France
e-mail: qi-chang.he@u-pem.fr
Southwest Jiaotong University,
Chengdu 610031, China;
Université Paris-Est,
Laboratoire Modélisation et Simulation
Multi Echelle,
MSME UMR 8208 CNRS,
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
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. Mar 2019, 86(3): 031007 (6 pages)
Published Online: January 8, 2019
Article history
Received:
October 28, 2018
Revised:
December 5, 2018
Citation
He, Q., and Shao, Q. (January 8, 2019). "Closed-Form Coordinate-Free Decompositions of the Two-Dimensional Strain and Stress for Modeling Tension–Compression Dissymmetry." ASME. J. Appl. Mech. March 2019; 86(3): 031007. https://doi.org/10.1115/1.4042217
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