In this paper, analysis of the M-integral in plane elasticity is carried out. An infinite plate with any number of inclusions and cracks and with any applied forces and remote tractions is considered. To study the problem, the mutual work difference integral (abbreviated as MWDI) is introduced, which is defined by the difference of works done by each other stress field on a large circle. The concept of the derivative stress field is also introduced, which is a real elasticity solution and is derived from the physical stress field. It is found that the M-integral on a large circle is equal to a MWDI from the physical stress field and a derivative stress field. Finally, the expression for M-integral on a large circle is obtained. The variation for the M-integral with respect to the coordinate transformation is addressed. An illustrative example for the use of M-integral is presented.

1.
Rice
,
J. R.
,
1968
, “
A Path-Independent Integral and the Approximation Analysis of Strain Concentration by Notches and Cracks
,”
ASME J. Appl. Mech.
,
35
, pp.
379
386
.
2.
Knowles
,
J. K.
, and
Sternberg
,
E.
,
1972
, “
On a Class of Conservation Laws in Linearized and Finite Elastostatics
,”
Arch. Ration. Mech. Anal.
,
44
, pp.
187
211
.
3.
Budiansky
,
B.
, and
Rice
,
J. R.
,
1973
, “
Conservation Laws and Energy-Release Rates
,”
ASME J. Appl. Mech.
,
40
, pp.
201
203
.
4.
Fruend
,
L. B.
,
1978
, “
Stress Intensity Factor Calculation Based on a Conservation Integral
,”
Int. J. Solids Struct.
,
14
, pp.
241
250
.
5.
Herrmann
,
A. G.
, and
Hermann
,
G.
,
1981
, “
On Energy-Release Rates for a Plane Crack
,”
ASME J. Appl. Mech.
,
48
, pp.
525
528
.
6.
Bui
,
H. D.
,
1974
, “
Dual Path Independent Integrals in the Boundary-Value Problems of Cracks
,”
Eng. Fract. Mech.
,
6
, pp.
287
296
.
7.
Chen
,
Y. Z.
,
1985
, “
New Path Independent Integrals in Linear Elastic Fracture Mechanics
,”
Eng. Fract. Mech.
,
22
, pp.
673
686
.
8.
Cherepanov, G. P., 1979, Mechanics of Brittle Fracture, McGraw-Hill, New York.
9.
Kanninen, M. F., and Popelar, C. H., 1985, Advanced Fracture Mechanics, Oxford University Press, Oxford, UK.
10.
Eshelby
,
J. D.
,
1951
, “
The Force on an Elastic Singularity
,”
Philos. Trans. R. Soc. London, Ser. A
,
A244
, pp.
87
112
.
11.
Chen
,
Y. Z.
,
1985
, “
A Technique for Evaluating the Stress Intensity Factors by Means of the M-Integral
,”
Eng. Fract. Mech.
,
23
, pp.
777
780
.
12.
Suo
,
Z.
,
2000
, “
Zener’s Crack and the M-integral
,”
ASME J. Appl. Mech.
,
67
, pp.
417
418
.
13.
Chen
,
Y. H.
,
2001
, “
M-Integral Analysis for Two Dimensional Solids With Strongly Interacting Microcracks. Part I: In an Infinite Brittle Solid
,”
Int. J. Solids Struct.
,
38
, pp.
3193
3212
.
14.
Muskhelishvili, N. I., 1953, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Netherlands.
15.
Chen
,
Y. Z.
, and
Lee
,
K. Y.
,
2002
, “
Some Properties of J-Integral in Plane Elasticity
,”
ASME J. Appl. Mech.
,
69
, pp.
195
198
.
16.
Chen
,
Y. Z.
,
2003
, “
Analysis of L-Integral and Theory of the Derivative Stress Field in Plane Elasticity
,”
Int. J. Solids Struct.
,
40
, pp.
3589
3602
.
You do not currently have access to this content.