Abstract
The explicit definition of the L-integral is reviewed by the curl operation of the Lagrangian energy density moment in plane elasticity. The physical interpretation of the configurational stress tensor associated with the integrand of the L-integral is explored, and it is identified as the change of potential energy due to the rotation of one infinitesimal material element with respect to the fixed point. Further, the path-independence of the L-integral is analyzed by an explicit form. It is demonstrated that the L-integral shows the path-independent properties for an isotropic elasticity or the isotropic plane in a transversely isotropic material while the path-dependence is found for the anisotropic elasticity. Finally, it is explicitly concluded that the L-integral will be independent of the coordinate system attributing to the conservation laws of J 1 and J 2 integrals.
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References
Eshelby J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 244, 87–112 (1951)
Eshelby, J.D.: The continuum theory of lattice defects. In: Seitz, F., Turnbull, D. (eds.) Solid State Physics, vol. 3, pp. 79–141. Academic Press, New York (1956)
Rice J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)
Cherepanov G.P.: Crack propagation in continuous media. J. Appl. Math. Mech. 31, 476–488 (1967)
Knowles J.K., Steinberg E.: Large deformations near a tip of all interface and finite elastic. Arch. Ration. Mech. Anal. 1, 187–211 (1972)
Budiansky B., Rice J.R.: Conservation laws and energy-release rates. J. Appl. Mech. 40, 201–203 (1973)
Eshelby J.D.: The elastic energy–momentum tensor. J. Elast. 5, 321–335 (1975)
Herrmann A.G., Herrmann G.: On energy-release rates for a plane crack. J. Appl. Mech. 48, 525–528 (1981)
Eischen J.W., Herrmann G.: Energy release rates and related balance laws in linear elastic defect mechanics. J. Appl. Mech. 54, 388–392 (1987)
Kienzler R., Herrmann G.: Mechanics in Material Space with Application to Defect and Fracture Mechanics. Springer, New York (2000)
Chen Y.H.: Advances in Conservation Laws and Energy Release Rates. Kluwer, Dordrecht (2002)
Shi J.P., Liu X.H., Li J.M.: On the relation between the L-integral and the Bueckner work-conjugate integral. J. Appl. Mech. 67(4), 828–829 (2000)
Shi J.P., Chen Y.H.: The explicit formulations for the L-integral. Acta Mech. Solida Sin. 20(2), 104–112 (1999)
Kienzler R., Herrmann G.: On the properties of the Eshelly tensor. Acta Mech. 125, 73–91 (1997)
Li, Q., Hu, Y.F., Chen, Y.H.: On the physical interpretation of the M-integral in nonlinear elastic defect mechanics. Int. J. Damage Mech. (2012). doi:10.1177/1056789512456860
Freund L.B.: Stress intensity factor calculations based on a conservation integral. Int. J. Solids Struct. 14, 241–250 (1978)
Chang J.H., Chien A.J.: Evaluation of M-integral for anisotropic elastic media with multiple defects. Int. J. Fract. 114, 267–289 (2002)
Hu Y.F., Chen Y.H.: The M-integral description for a brittle plane strip with two holes before and after coalescence. Acta Mech. 204, 109–123 (2009)
Chen Y.Z.: Analysis of L-integral and theory of the derivative stress field in plane elasticity. Int. J. Solids Struct. 40, 3589–3602 (2003)
Noether E.: Invariante variations probleme. Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. 2, 235–257 (1918)
Noether E.: Invariant variational problems. Transp. Theory Stat. Phys. 1, 183–207 (1971)
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Guo, YL., Li, Q. On some fundamental properties of the L-integral in plane elasticity. Acta Mech 226, 137–148 (2015). https://doi.org/10.1007/s00707-014-1152-y
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DOI: https://doi.org/10.1007/s00707-014-1152-y