The $M$-Integral Description for a Brittle Plane Strip With Two Cracks Before and After Coalescence

In this paper we extend the $M$-integral concept (Eshelby, J. D., 1956, The Continuum Theory of Lattice Defects, Solid State Physics, F. Seitz and D. Turnbull, eds., Academic, New York, pp. 79–141; Eshelby, J. D., 1970, The Energy Momentum Tensor in Continuum Mechanics, Inelastic Behavior of Solids, M. F. Kanninen, ed., McGraw-Hill, New York, pp. 77–115; Eshelby, J. D., 1975, “The Elastic Energy-Momentum Tensor,” J. Elast., 5, pp. 321–335; 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; Budiansky, B., and Rice, J. R., 1973, “Conservation Laws and Energy Release Rates,” ASME J. Appl. Mech., 40, pp. 201–203; Freund, L. B., 1978, “Stress Intensity Factor Calculations Based on a Conservation Integral,” Int. J. Solids Struct., 14, pp. 241–250; Herrmann, G. A., and Herrmann, G., 1981, “On Energy Release Rates for a Plane Cracks,” ASME J. Appl. Mech., 48, pp. 525–530; King, R. B., and Herrmann, G., 1981, “Nondestructive Evaluation of the J- and M-Integrals,” ASME J. Appl. Mech., 48, pp. 83–87) to study the degradation of a brittle plan strip caused by irreversible evolution: the cracks coalescence under monotonically increasing loading. Attention is focused on the change of the $M$-integral before and after coalescence of two neighborly located cracks inclined each other. The influences of different orientations of the two cracks and different coalescence paths connecting the two cracks on the $M$-integral are studied in detail. Finite element analyses reveal that different orientations of the two cracks lead to different critical values of the $M$-integral at which the coalescence occurs. It is concluded that the $M$-integral does play an important role in the description of the damage extent and damage evolution. However, it only provides some outside variable features. This means that the complete failure mechanism due to damage evolution cannot be governed by a single parameter $MC$ as proposed by Chang and Peng, 2004, “Use of M integral for Rubbery Material Problems Containing Multiple Defects,” J. Eng. Mech., 130(5), pp. 589–598. It is found that there is an inherent relation between the $M$-integral and the reduction of the effective elastic moduli as the orientation of one crack varies, i.e., the larger the $M$-integral is, the larger the reduction is. Of great significance is that the $M$-integral is inherently related to the change of the total potential energy for a damaged brittle material regardless of the detailed damage features or damage evolution. Therefore, this provides a useful and convenient experimental technique to measure the values of $M$-integral for a damaged brittle material from initial damage to final failure without use of many stain gages (King, R. B., and Herrmann, G., 1981, “Nondestructive Evaluation of the J- and M-Integrals,” ASME J. Appl. Mech., 48, pp. 83–87).