# M- and \(\hbox {M}_{\text {int}}\)-integrals for cracks normal to the interface of anisotropic bimaterials

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## Abstract

A contour integral termed \(\hbox {M}_{\mathrm{int}}\) is presented for describing the fracture behavior of cracks passing through or terminating normally at a bimaterial interface. The \(\hbox {M}_{\mathrm{int}}\)-integral is defined by performing the conventional M-integral along a contour enclosing the cracks, with the coordinate system properly originated for measure of the integration points. The presented formulation is considered to be feasible for problems with generally anisotropic elastic materials. Physically, the energy parameter \(\hbox {M}_{\mathrm{int}}\) is shown to be equivalent to twice the released potential energy required for creation of the cracks. Such relation is exactly valid when the crack length is small compared with the size of the specimen, and approximately satisfied for problems containing relatively large cracks. Also, due to path-independence, the \(\hbox {M}_{\mathrm{int}}\)-integral can be performed along an arbitrary outer contour, which is chosen to be far from the crack tips. With this property, the complicated singular stress field in the near-tip areas can be avoided so that a complicated finite element model around the crack tips is not required in the calculation.

## Keywords

Crack normal to bimaterial interface M-integral \(\hbox {M}_{\mathrm{int}}\)-integral Path-independence Origin-dependence Released potential energy## Notes

### Acknowledgments

This work has been partially supported by National Science Council Grant No. NSC102-2221-E-008-060 to National Central University.

## References

- Abdi R, Valentin G (1989) Isoparametric elements for a crack normal to the interface between two bonded layers. Comp Struct 33:241–248CrossRefGoogle Scholar
- Bogy DB (1971) On the plane elastostatic problem of a loaded crack terminating at a material interface. J Appl Mech 38:911–918CrossRefGoogle Scholar
- Budiansky B, Rice JR (1973) Conservation laws and energy-release rates. ASME J Appl Mech 40:201–203Google Scholar
- Chang JH, Chien AJ (2002) Evaluation of M-integral for anisotropic elastic media with multiple defects. Int J Fract 114(3):267–289Google Scholar
- Chang JH, Wu DJ (2003) Calculation of mixed-mode stress intensity factors for a crack normal to a bimaterial Interface using contour integrals. Eng Fract Mech 70:1675–1695CrossRefGoogle Scholar
- Chang JH, Wu WH (2011) Using M-integral for multi-cracked problems subjected to nonconservative and nonuniform crack surface tractions. Int J Sol Struct 48:2605–2613Google Scholar
- Chen WH, Chen KT, Chiang CR (1988) On the applicability of \(\text{ J }_{{\rm Ro}}\) integral for perpendicular interface crack. Eng Fract Mech 30:13–19CrossRefGoogle Scholar
- Chen YH (2001) M-integral analysis for two-dimensional solids with strongly interacting microcracks. Part I: in an infinite brittle solid. Int J Sol Struct 38:3193–3212CrossRefGoogle Scholar
- Chen YH, Lu TJ (2003) Recent developments and applications of invariant integrals. Appl Mech Rev 56:515–552CrossRefGoogle Scholar
- Eischen JW, Herrmann G (1987) Energy release rates and related balance laws in linear elastic defect mechanics. J Appl Mech 54:388–392CrossRefGoogle Scholar
- Herrmann AG, Herrmann G (1981) On energy release rates for a plane crack. J Appl Mech 48:525–528CrossRefGoogle Scholar
- Kaddouri K, Belhouari M, Bachir B, Bouiadjra B (2006) Finite element analysis of crack perpendicular to bi-material interface: case of couple ceramic–metal. Ser Comp Mat Sci 35:53–60CrossRefGoogle Scholar
- Knowles JK, Sternberg E (1972) On a class of conservation laws in linearized and finite elastostatics. Arch Ration Mech Anal 44:187–343CrossRefGoogle Scholar
- Marsavina L, Sadowski T, Faur N (2009) Asymptotic stress field for a crack normal to a ceramic–metal interface. Key Eng Mater 417:489–492CrossRefGoogle Scholar
- Profant T, Sevecek O, Kotoul M (2008) Calculation of K-factor and T-stress for cracks in anisotropic bimaterials. Eng Fract Mech 75:3707–3726CrossRefGoogle Scholar
- Rogowski B (2012) Antiplane shear crack normal to and terminating at the Interface of two bonded piezo-electro-magneto-elastic materials. ISRN Mat Sci (2012) Article ID 659352:19 pagesGoogle Scholar
- Tracey DM, Cook TS (1979) Analysis of power type singularities using finite elements. Int J Numer Methods Eng 11:1225–1233CrossRefGoogle Scholar
- Wang DF, Chen YH (2002) M-integral analysis for a two-dimensional metal/ceramic bimaterial solid with extending subinterface microcracks. Arch Appl Mech 72:588–598CrossRefGoogle Scholar
- Yavari A, Sarkani S, Moyer ET (1999) On quadratic isoparametric transition elements or a crack normal to a bimaterial interface. Int J Numer Methods Eng 46:457–469CrossRefGoogle Scholar
- Zhong XC, Li XF, Kang YL (2009) Analysis of a mode-I crack perpendicular to an imperfect interface. Int J Sol Struct 46:1456–1463CrossRefGoogle Scholar