Analysis of L-integral and theory of the derivative stress field in plane elasticity

In this paper, analysis of the L-integral in plane elasticity is present. An infinite plate with any number of inclusions and cracks and with any remote tractions is assumed in analysis. Arbitrary forces are applied on the cracks, inclusions or at a point of the infinite medium. To study the problem, the concept of the derivative stress field is introduced, which is derived from a physical stress field. The mutual work difference integral (MWDI) is also introduced, which is defined as a difference of mutual works done by each other from the physical stress field and the derivative field. It is proved that the L(CR) (L-integral on a large circle) is equal to a particular MWDI. General expression for the L(CR) is obtained. For a given stress field, the variation of the L(CR) is studied when the coordinates have a translation or rotation. It is found that the L(CR) is an invariant with respect to the rotation of coordinates, and it has a variation when the coordinates have a translation.