Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

1.
Betten
J.
,
1983
, “
Damage Tensors in Continuum Mechanics
,”
Journal of Me´canique The´orique et Applique´e
, Vol.
2
, No.
1
, pp.
13
32
.
2.
Chaboche, J. L., 1979, “Le concept de contrainte effective applique´ a l’ e´lasticite´ et a la viscoplasticite´ en pre´sence d’un endommagement anisotrope,” Colooque Euromech 115, Villard de Lans, June.
3.
Chaboche
J. L.
,
1981
, “
Continuum Damage Mechanics—A Tool to Describe Phenomena Before Crack Initiation
,”
Nuclear Engineering and Design
, Vol.
64
, pp.
233
247
.
4.
Chaboche
J. L.
,
1988
a, “
Continuum Damage Mechanics: Part I—General Concepts
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
55
, pp.
59
64
.
5.
Chaboche
J. L.
,
1988
b, “
Continuum Damage Mechanics: Part II—Damage Growth, Crack Initiation, and Crack Growth
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
55
, pp.
65
72
.
6.
Cordebois, J. P., and Sidoroff, F., 1982, “Anisotropic Damage in Elasticity and Plasticity,” Journal de Me´canique The´orique et Applique´e, Nume´ro Spe´cial, pp. 45–60.
7.
Ju
J. W.
,
1990
, “
Isotropic and Anisotropic Damage Variables in Continuum Damage Mechanics
,”
Journal of Engineering Mechanics
, Vol.
116
, No.
12
, pp.
2764
2770
.
8.
Kachanov
L. M.
,
1958
, “
On the Creep Fracture Time
,”
, Vol.
8
, pp.
26
31
.
9.
Krajcinovic
D.
,
1983
, “
Constitutive Equations for Damaging Materials
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
50
, pp.
335
360
.
10.
Krajcinovic
D.
,
1985
, “
Continuous Damage Mechanics Revisited: Basic Concepts and Definitions
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
52
, pp.
829
834
.
11.
Krajcinovic
D.
, and
Foneska
G. U.
,
1981
, “
The Continuum Damage Theory for Brittle Materials
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
48
, pp.
809
834
.
12.
Lee, H., Li, G., and Lee, S., 1986, “The Influence of Anisotropic Damage on the Elastic Behavior of Materials,” International Seminar on Local Approach of Fracture, Moret-sur-Loing, France, June, pp. 79–90.
13.
Lemaitre
J.
,
1985
, “
A Continuous Damage Mechaics Model of Ductile Fracture
,”
ASME Journal of Engineering Materials and Technology
, Vol.
107
, No.
42
, pp.
83
89
.
14.
Lemaitre
J.
,
1986
, “
Local Approach of Fracture
,”
Engineering Fracture Mechanics
, Vol.
25
, No.
5–6
, pp.
523
537
.
15.
Lu
T. J.
, and
Chow
C. L.
,
1990
, “
On Constitutive Equations of Inelastic Solids With Anisotropic Damage
,”
Theoretical and Applied Fracture Mechanics
, Vol.
14
, pp.
187
218
.
16.
Murakami
S.
,
1983
, “
Notion of Continuum Damage Mechanics and Its Application to Anisotropic Creep Damage Theory
,”
Journal of Engineering Materials and Technology
, Vol.
105
, pp.
99
105
.
17.
Murakami
S.
,
1988
, “
Mechanical Modeling of Material Damage
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
55
, pp.
280
286
.
18.
Murakami, S., and Ohno, N., 1980, “A Continuum Theory of Creep and Creep Damage,” Creep in Structures, A. R. S. Ponter and D. R. Hayhurst, eds., IUTAM 3rd Symposium, Leichester, U.K., Sept. 8–12, Springer-Verlag, New York, pp. 422–443.
19.
Sidoroff, F., 1979, “Description of Anisotropic Damage Application to Elasticity,” Mechanical Behavior of Anisotropic Solids (No. 295 Comportement Mechanique Des Solides Anisotropes), J.-P. Boehler, ed., Martinus Nijhoff, Dordrecht, The Netherlands.
20.
Sidoroff, F., 1980, “Description of Anisotropic Damage Application to Elasticity,” Physical Non-Linearities in Structural Analysis (IUTAM Series), J. Hult and J. Lemaitre, eds., Springer-Verlag, New York, pp. 237–244.
21.
G. Z.
, and
Kattan
P. I.
,
1992
, “
A Plasticity-Damage Theory for Large Deformation of Solids, Part I: Theoretical Formulation
,”
International Journal of Engineering Science
, Vol.
30
, No.
9
, pp.
1089
1108
.
22.
G. Z.
, and
Park
T.
,
1995
, “
Local and Interfacial Damage Analysis of Metal Matrix Composites
,”
International Journal of Engineering Science
, Vol.
33
, No.
11
, pp.
1595
1621
.
23.
G. Z.
, and
Venson
A. R.
,
1995
, “
Experimental Damage Investigation of a SiC-Ti Aluminide Metal Matrix Composite
,”
International Journal of Damage Mechanics
, Vol.
4
. No.
4
, Oct., pp.
338
361
.
This content is only available via PDF.