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GeneralizedJ-integral and its applications I —Basic theory—

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Abstract

GeneralizedJ-integral is proposed by the author to express three-dimensional fracture phenomena. In this paper we will extend the concept of generalizedJ-integral and proposeJR-integral to be applicable for perturbations of singularities which occur in elliptic boundary value problems. It is shown in Sections 4 and 5 that global energy variation (energy release rate) and local energy variation are expressed byJR-integral.

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References

  1. C. Atkinson and C. J. Coleman, On some steady-state moving boundary problems in the linear theory of viscoelasticity. J. Inst. Math. Appl.,20 (1977), 85–106.

    Article  MATH  Google Scholar 

  2. B. Budiansky and J. R. Rice, Conservation laws and energy release rates. J. Appl. Mech.,40 (1973), 201–203.

    MATH  Google Scholar 

  3. F. Chen and R. T. Shield, Conservation laws in elasticity of theJ-integral type. J. Appl. Math. Phys.,28 (1977), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. P. Cherepanov, Crack propagation in continuous media. J. Appl. Math. Mech.,31 (1967), 503–512.

    Article  MATH  Google Scholar 

  5. H. G. deLorenzi, On the energy release rate and theJ-integral for 3-D crack configuration. Internal. J. Fracture,19 (1982), 183–193.

    Article  Google Scholar 

  6. P. Destuynder and M. Djaona. Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile. Math. Methods Appl. Sci.,3 (1981), 70–87.

    Article  MATH  MathSciNet  Google Scholar 

  7. G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin-Heidelberg-New York, 1976.

    MATH  Google Scholar 

  8. J. D. Eshelby, The continuum theory of lattice defects. Solid State Phys.,3 (1956), 79–144.

    Article  Google Scholar 

  9. A. C. Eringen, Theory of micropolar elasticity. “Fracture” Vol. 2 (ed. H. Liebowitz), Academic Press, New York. San Francisco. London, 1968, 621–729.

    Google Scholar 

  10. D. Fujiwara and S. Ozawa, The Hadamard variational formula for the Green functions of some normal elliptic boundary value problems. Proc. Japan Acad., Ser. A,54 (1978), 215–220.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Garabedian and M. Schiffer, Convexity of domain functionals. J. Anal. Math.,2 (1952/53), 281–368.

    Article  MathSciNet  Google Scholar 

  12. A. A. Griffith, The phenomena of rupture and flow in solids. Philos. Trans. Roy. Soc. Ser. A,221 (1920), 163–198.

    Article  Google Scholar 

  13. W. Günther, Über einige Randintegrale der Elastomechanik. Abh. Braunschweig. Wiss. Ges.,14 (1962), 53–72.

    MATH  Google Scholar 

  14. J. Hadamard, Mémoire sur le Problèm d’Analyse Relatif à l’Equilibre des Plaques Élastiques Encastrées. Oeuvres, 2, 1968, 515–631.

    Google Scholar 

  15. S. C. Hunter, The rolling contact of a rigid cylinder with a viscoelastic half space. J. Appl. Mech.,28 (1961), 611–617.

    MATH  MathSciNet  Google Scholar 

  16. J. Knowles and E. Sternberg, On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics. Internat. J. Solids and Structures,11 (1975), 1173–1201.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. Knowles and E. Sternberg, On a class of conservation laws in linearized and finite elastostatics. Arch. Rational Mech. Anal.,44 (1972), 187–211.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. D. Kupradze (ed.), Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam New York. Oxford, 1979.

    MATH  Google Scholar 

  19. D. A. Moran, Raising the differentiability class of a manifold in Euclidean space. J. Indian Math. Soc.,29 (1965), 119–133.

    MATH  MathSciNet  Google Scholar 

  20. J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies. An Introduction. Elsevier, Amsterdam-Oxford-New York, 1981.

    MATH  Google Scholar 

  21. E. Noether, Invariante Variationsprobleme. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl.,2 (1918), 235–257.

    Google Scholar 

  22. K. Ohtsuka,J-integral and two-dimensional fracture mechanics. Kôkyûroku, Res. Inst. Math. Sci., Kyoto Univ.,386, 1980, 231–248.

    Google Scholar 

  23. K. Ohtsuka, GeneralizedJ-integral and three dimensional fracture mechanics I. Hiroshima Math. J.,11 (1981), 21–52.

    MATH  MathSciNet  Google Scholar 

  24. K. Ohtsuka, GeneralizedJ-integral and its application—An interpretation of Hadamard’s variational formula—. Kôkyûroku, Res. Inst. Math. Sci., Kyoto Univ.,462, 1982, 149–165.

    Google Scholar 

  25. K. Ohtsuka, Remark on Griffith’s energy balance for three-dimensional fracture problems. Mem. Hiroshima-Denki Inst. Tech. and the Hiroshima Junior College Automotive Engrg.,16 (1983), 31–36.

    Google Scholar 

  26. J. P. Olver, Conservation laws in elasticity II. Linear homogeneous isotropic elasticities. Arch. Rational Mech. Anal.,85 (1984), 131–160.

    Article  MATH  MathSciNet  Google Scholar 

  27. P. J. Olver, Conservation laws in elasticity I. General results. Arch. Rational Mech. Anal.,85 (1984), 111–129.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Palmerio, Hadamard’s variational formula for a mixed problem and an application to a problem related to a Signorini-like variational inequality. Numer. Funct. Anal. Optim.,1 (1979), 113–144.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. Rayleigh, The Theory of Sound. Second Ed., Dover Publications, London, 1894/96.

    MATH  Google Scholar 

  30. J. R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech.,35 (1968), 379–386.

    Google Scholar 

  31. J. L. Sanders, On the Griffith-Irwin fracture theory, J. Appl. Mech.,27 (1960), 352.

    MATH  MathSciNet  Google Scholar 

  32. F. Treves, Topological Vector Spaces, Distributions and Kernels. Academic Press, New York. San Francisco. London, 1967.

    MATH  Google Scholar 

  33. C. Yatomi, The energy release rate and the work done by the surface traction in quasi-static elastic crack growth. Internat. J. Solids and Structures,19 (1983), 183–187.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Liberal Arts, Hiroshima-Denki Institute of Technology, 6-20-1 Nakano, Aki-ku, 739-03, Hiroshima, Japan

    Kohji Ohtsuka

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  1. Kohji Ohtsuka
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Ohtsuka, K. GeneralizedJ-integral and its applications I —Basic theory—. Japan J. Appl. Math. 2, 329–350 (1985). https://doi.org/10.1007/BF03167081

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  • DOI: https://doi.org/10.1007/BF03167081

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