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Research Papers

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

[+] Author and Article Information
Hu Yi-Feng

School of Aerospace, SVL, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. China

Chen Yi-Heng1

School of Aerospace, SVL, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. Chinayhchen2@mail.xjtu.edu.cn

1

Corresponding author.

J. Appl. Mech 76(6), 061017 (Jul 27, 2009) (10 pages) doi:10.1115/1.3130818 History: Received March 30, 2008; Revised October 30, 2008; Published July 27, 2009

## Abstract

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).

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## Figures

Figure 2

A brittle strip with two neighborly located cracks before coalescence

Figure 3

Detailed configuration for the strip with the two cracks after coalescence

Figure 6

Finite element analyses for the maximum von-Mises stress along the two cracks (red points) before coalescence for the four orientation angles (a) 11.25 deg, (b) 22.5 deg, (c) 33.75 deg, and (d) 45 deg

Figure 7

Finite element analyses for the maximum von-Mises stress along the crack (red points) after coalescence for the four orientation angles (a) 11.25 deg, (b) 22.5 deg, (c) 33.75 deg, and (d) 45 deg

Figure 1

The picture after crack coalescence, provided by Professor Chen Zemao (Vice Dean, School of Aerospace, Xi’an Jiaotong University)

Figure 4

Finite element meshes before coalescence of the two cracks for the four orientation angles (a) 11.25 deg, (b) 22.5 deg, (c) 33.75 deg, and (d) 45 deg

Figure 5

Finite element meshes after coalescence of the two cracks for the four orientation angles (a) 11.25 deg, (b) 22.5 deg, (c) 33.75 deg, and (d) 45 deg

Figure 8

Four typical closed paths for calculating the M-integral before or after coalescence between the two cracks: (a) paths 1 and 2 are within the PMLA material and (b) paths 3 and 4 are in an imaginary material with infinite small Young’s modulus outside the PMLA material

Figure 9

The calculated curves of the M-integral and CTPE denoted by 2U for the angle 11.25 deg of the two cracks before and after coalescence

Figure 10

The calculated curves of the M-integral and CTPE denoted by 2U for the angle 22.5 deg of the two cracks before and after coalescence

Figure 11

The calculated curves of the M-integral and CTPE denoted by 2U for the angle 33.75 deg of the two cracks before and after coalescence

Figure 12

The calculated curves of the M-integral and CTPE denoted by 2U for the angle 45 deg of the two cracks before and after coalescence

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