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).

Skip Nav Destination

yhchen2@mail.xjtu.edu.cn

Article navigation

November 2009

Research Papers

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

Hu Yi-Feng,

Hu Yi-Feng

School of Aerospace, SVL,

Xi’an Jiaotong University

, Xi’an, Shaanxi 710049, P. R. China
Search for other works by this author on:

Chen Yi-Heng

Chen Yi-Heng

School of Aerospace, SVL,

yhchen2@mail.xjtu.edu.cn
Xi’an Jiaotong University

, Xi’an, Shaanxi 710049, P. R. China
Search for other works by this author on:

Hu Yi-Feng

School of Aerospace, SVL,

Xi’an Jiaotong University

, Xi’an, Shaanxi 710049, P. R. China
Chen Yi-Heng

School of Aerospace, SVL,

Xi’an Jiaotong University

, Xi’an, Shaanxi 710049, P. R. Chinayhchen2@mail.xjtu.edu.cn

*J. Appl. Mech*. Nov 2009, 76(6): 061017 (10 pages)

**Published Online:**July 27, 2009

Article history

Received:

March 30, 2008

Revised:

October 30, 2008

Published:

July 27, 2009

Citation

Yi-Feng, H., and Yi-Heng, C. (July 27, 2009). "The $M$-Integral Description for a Brittle Plane Strip With Two Cracks Before and After Coalescence." ASME. *J. Appl. Mech*. November 2009; 76(6): 061017. https://doi.org/10.1115/1.3130818

Download citation file:

50
Views

0
Citations

### Get Email Alerts

### Cited By

Nonlinear Oscillations of Dielectric Elastomer Actuators With Stretch-Dependent Permittivity

J. Appl. Mech (November 2022)

Moiré Tuning of the Dynamic Behavior of a Twisted Bilayer van der Waals Material Resonator

J. Appl. Mech (December 2022)

### Related Articles

Fracture of Brittle Microbeams

J. Appl. Mech (May,2004)

Finite Element Analysis of Brittle Cracking due to Single Grit Rotating Scratch

J. Appl. Mech (January,2003)

Limit Load Analysis of Cracked Components Using the Reference Volume Method

J. Pressure Vessel Technol (August,2007)

Local Limit-Load Analysis Using the m β Method

J. Pressure Vessel Technol (May,2007)

### Related Proceedings Papers

### Related Chapters

Introduction and Definitions

Handbook on Stiffness & Damping in Mechanical Design

Data Tabulations

Structural Shear Joints: Analyses, Properties and Design for Repeat Loading

Approximate Analysis of Plates

Design of Plate and Shell Structures