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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,
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

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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References

Ambartsumyan, S. , and Khachatryan, A. , 1966, “ Basic Equations in the Theory of Elasticity for Materials With Different Stiffness in Tension and Compression,” Mech. Solids, 1(2), pp. 29–34.
Ambartsumyan, S. , and Khachartryan, A. , 1969, “ The Basic Equations and Relations of the Different-Modulus Theory of Elasticity of an Anisotropic Body,” Mech. Solids, 4(3), pp. 48–56.
Tsai, S. W. , and Wu, E. M. , 1971, “ A General Theory of Strength for Anisotropic Materials,” J. Compos. Mater., 5(1), pp. 58–80. [CrossRef]
Green, A. , and Mkrtichian, J. , 1977, “ Elastic Solids With Different Moduli in Tension and Compression,” J. Elasticity, 7(4), pp. 369–386. [CrossRef]
Jones, R. M. , 1977, “ Stress-Strain Relations for Materials With Different Moduli in Tension and Compression,” AIAA J., 15(1), pp. 16–23. [CrossRef]
Medri, G. , 1982, “ A Nonlinear Elastic Model for Isotropic Materials With Different Behavior in Tension and Compression,” ASME J. Eng. Mater. Technol., 104(1), pp. 26–28. [CrossRef]
Ortiz, M. , 1985, “ A Constitutive Theory for the Inelastic Behavior of Concrete,” Mech. Mater., 4(1), pp. 67–93. [CrossRef]
Del Piero, G. , 1989, “ Constitutive Equation and Compatibility of the External Loads for Linear Elastic Masonry-Like Materials,” Meccanica, 24(3), pp. 150–162. [CrossRef]
Mazars, J. , Berthaud, Y. , and Ramtani, S. , 1990, “ The Unilateral Behaviour of Damaged Concrete,” Eng. Fract. Mech., 35(4–5), pp. 629–635. [CrossRef]
Mattos, H. C. , Fremond, M. , and Mamiya, E. , 1992, “ A Simple Model of the Mechanical Behavior of Ceramic-Like Materials,” Int. J. Solids Struct., 29(24), pp. 3185–3200. [CrossRef]
Curnier, A. , He, Q.-C. , and Zysset, P. , 1994, “ Conewise Linear Elastic Materials,” J. Elasticity, 37(1), pp. 1–38. [CrossRef]
Francfort, G. A. , and Marigo, J.-J. , 1998, “ Revisiting Brittle Fracture as an Energy Minimization Problem,” J. Mech. Phys. Solids, 46(8), pp. 1319–1342. [CrossRef]
Bourdin, B. , Francfort, G. A. , and Marigo, J.-J. , 2008, “ The Variational Approach to Fracture,” J. Elasticity, 91(1–3), pp. 5–148. [CrossRef]
Amor, H. , Marigo, J.-J. , and Maurini, C. , 2009, “ Regularized Formulation of the Variational Brittle Fracture With Unilateral Contact: Numerical Experiments,” J. Mech. Phys. Solids, 57(8), pp. 1209–1229. [CrossRef]
Borden, M. J. , Verhoosel, C. V. , Scott, M. A. , Hughes, T. J. R. , and Landis, C. M. , 2012, “ A Phase-Field Description of Dynamic Brittle Fracture,” Comput. Methods Appl. Mech. Eng., 217–220(1), pp. 77–95. [CrossRef]
Lancioni, G. , and Royer-Carfagni, G. , 2009, “ The Variational Approach to Fracture Mechanics: A Practical Application to the French Panthéon in Paris,” J. Elasticity, 95(1–2), pp. 1–30. [CrossRef]
Freddi, F. , and Royer-Carfagni, G. , 2010, “ Regularized Variational Theories of Fracture: A Unified Approach,” J. Mech. Phys. Solids, 58(8), pp. 1154–1174. [CrossRef]
Miehe, C. , Hofacker, M. , and Welschinger, F. , 2010, “ A Phase Field Model for Rate-Independent Crack Propagation: Robust Algorithmic Implementation Based on Operator Splits,” Comput. Methods Appl. Mech. Eng., 199(45–48), pp. 2765–2778. [CrossRef]
Miehe, C. , Welschinger, F. , and Hofacker, M. , 2010, “ Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field fe Implementations,” Int. J. Numer. Methods Eng., 83(10), pp. 1273–1311. [CrossRef]
Nguyen, T. T. , Yvonnet, J. , Zhu, Q.-Z. , Bornert, M. , and Chateau, C. , 2015, “ A Phase Field Method to Simulate Crack Nucleation and Propagation in Strongly Heterogeneous Materials From Direct Imaging of Their Microstructure,” Eng. Fract. Mech., 139, pp. 18–39. [CrossRef]
Li, T. , Marigo, J.-J. , Guilbaud, D. , and Potapov, S. , 2016, “ Gradient Damage Modeling of Brittle Fracture in an Explicit Dynamics Context,” Int. J. Numer. Methods Eng., 108(11), pp. 1381–1405. [CrossRef]
Wu, J.-Y. , and Nguyen, V. P. , 2018, “ A Length Scale Insensitive Phase-Field Damage Model for Brittle Fracture,” J. Mech. Phys. Solids., 119, pp. 20–42. [CrossRef]
He, Q.-C. , 2018, “ Three-Dimensional Strain and Stress Orthogonal Decompositions Via an Elastic Energy Preserving Transformation,” submitted.
He, Q.-C. , and Zheng, Q.-S. , 1996, “ On the Symmetries of 2D Elastic and Hyperelastic Tensors,” J. Elasticity, 43(3), pp. 203–225. [CrossRef]
Halmos, P. R. , 1987, Finite-Dimensional Vector Spaces, Springer, Berlin.

Figures

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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