Some properties of the J-integral in plane elasticity are analyzed. An infinite plate with any number of inclusions, cracks, and any loading conditions is considered. In addition to the physical field, a derivative field is defined and introduced. Using the Betti’s reciprocal theorem for the physical and derivative fields, two new path-independent $D1$ and $D2$ are obtained. It is found that the values of $Jkk=1,2$ on a large circle are equal to the values of $Dkk=1,2$ on the same circle. Using this property and the complex variable function method, the values of $Jkk=1,2$ on a large circle is obtained. It is proved that the vector $Jkk=1,2$ is a gradient of a scalar function $Px,y.$

1.
Rice, J. R., 1968, Fracture: An Advanced Treatise, Vol. 2, H. Liebowitz, ed., Academic Press, San Diego, CA.
2.
Rice
,
J. R.
,
1968
, “
A Path-Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks
,”
ASME J. Appl. Mech.
,
35
, pp.
379
386
.
3.
Cherepanov, G. P., 1979, Mechanics of Brittle Fracture, McGraw-Hill, New York.
4.
Budiansky
,
B.
, and
Rice
,
J. R.
,
1973
, “
Conservation Laws and Energy-Release Rates
,”
ASME J. Appl. Mech.
,
40
, pp.
201
203
.
5.
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
.
6.
Freund
,
L. B.
,
1978
, “
Stress Intensity Factor Calculation Based on a Conservation Integral
,”
Int. J. Solids Struct.
,
14
, pp.
241
250
.
7.
Herrmann
,
A. G.
, and
Herrmann
,
G.
,
1981
, “
On Energy-Release Rates for a Plane Crack
,”
ASME J. Appl. Mech.
,
48
, pp.
525
530
.
8.
Chen
,
Y. Z.
,
1985
, “
New Path Independent Integrals in Linear Elastic Fracture Mechanics
,”
Eng. Fract. Mech.
,
22
, pp.
673
686
.
9.
Chen
,
Y. Z.
, and
Hasebe
,
Norio
,
1994
, “
Eigenfunction Expansion and Higher Order Weight Functions of Interface Cracks
,”
ASME J. Appl. Mech.
61
, pp.
843
849
.
10.
Chen
,
Y. H.
, and
Hasebe
,
Norio
,
1998
, “
A Consistency Check for Strongly Interaction Multiple Crack Problems in Isotropic, Bimaterial, and Orthotropic Bodies
,”
Int. J. Fract.
,
89
, pp.
333
353
.
11.
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
.
12.
Eshelby
,
J. D.
,
1951
, “
The Force on an Elastic Singularity
,”
Philos. Trans. R. Soc. London, Ser. A
,
A244
, pp.
87
112
.
13.
Muskhelishvili, N. I., 1953, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoof, The Netherlands.