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.

References

References
1.
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
.
2.
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
.
3.
Tsai
,
S. W.
, and
Wu
,
E. M.
,
1971
, “
A General Theory of Strength for Anisotropic Materials
,”
J. Compos. Mater.
,
5
(
1
), pp.
58
80
.
4.
Green
,
A.
, and
Mkrtichian
,
J.
,
1977
, “
Elastic Solids With Different Moduli in Tension and Compression
,”
J. Elasticity
,
7
(
4
), pp.
369
386
.
5.
Jones
,
R. M.
,
1977
, “
Stress-Strain Relations for Materials With Different Moduli in Tension and Compression
,”
AIAA J.
,
15
(
1
), pp.
16
23
.
6.
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
.
7.
Ortiz
,
M.
,
1985
, “
A Constitutive Theory for the Inelastic Behavior of Concrete
,”
Mech. Mater.
,
4
(
1
), pp.
67
93
.
8.
Del Piero
,
G.
,
1989
, “
Constitutive Equation and Compatibility of the External Loads for Linear Elastic Masonry-Like Materials
,”
Meccanica
,
24
(
3
), pp.
150
162
.
9.
Mazars
,
J.
,
Berthaud
,
Y.
, and
Ramtani
,
S.
,
1990
, “
The Unilateral Behaviour of Damaged Concrete
,”
Eng. Fract. Mech.
,
35
(
4–5
), pp.
629
635
.
10.
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
.
11.
Curnier
,
A.
,
He
,
Q.-C.
, and
Zysset
,
P.
,
1994
, “
Conewise Linear Elastic Materials
,”
J. Elasticity
,
37
(
1
), pp.
1
38
.
12.
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
.
13.
Bourdin
,
B.
,
Francfort
,
G. A.
, and
Marigo
,
J.-J.
,
2008
, “
The Variational Approach to Fracture
,”
J. Elasticity
,
91
(
1–3
), pp.
5
148
.
14.
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
.
15.
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
.
16.
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
.
17.
Freddi
,
F.
, and
Royer-Carfagni
,
G.
,
2010
, “
Regularized Variational Theories of Fracture: A Unified Approach
,”
J. Mech. Phys. Solids
,
58
(
8
), pp.
1154
1174
.
18.
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
.
19.
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
.
20.
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
.
21.
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
.
22.
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
.
23.
He
,
Q.-C.
,
2018
, “
Three-Dimensional Strain and Stress Orthogonal Decompositions Via an Elastic Energy Preserving Transformation
,” submitted.
24.
He
,
Q.-C.
, and
Zheng
,
Q.-S.
,
1996
, “
On the Symmetries of 2D Elastic and Hyperelastic Tensors
,”
J. Elasticity
,
43
(
3
), pp.
203
225
.
25.
Halmos
,
P. R.
,
1987
,
Finite-Dimensional Vector Spaces
, Springer, Berlin.
You do not currently have access to this content.