On dual conservation laws in linear elasticity: Stress function formalism

Nonlinear Dynamics

Volumn 36, Issue 1, 2004, Pages 77-96

  Li, Shaofan ^a  

^a 109 McLaughlin Hall   (United States)

Author keywords

Conservation laws; Elasticity; J integral; Lie group; Lie B cklund transformation

Indexed keywords

FRACTURE MECHANICS; LINEAR EQUATIONS; MATHEMATICAL TRANSFORMATIONS; NAVIER STOKES EQUATIONS; STRESS ANALYSIS; THEOREM PROVING; VARIATIONAL TECHNIQUES;

CONSERVATION LAWS; J-INTEGRAL; LIE GROUP; LIE-BACKLUND TRANSFORMATION;

ELASTICITY;

EID: 3242782663     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/B:NODY.0000034648.08181.c5     Document Type: Article

References (30)

1. Eshelby, J. D., 'The force on an elastic singularity,' Philosophical Transactions of the Royal Society 87, 1951, 12-111.
2. Eshelby, J. D., 'The continuum theory of lattice defects,' in Solid State Physics, Vol. 3, F. Seitz and D. Turnbull (eds.), Academic Press. New York, 1956, pp. 79-144.
3. Rice, J. R., 'A path-independent integral and the approximate analysis of strain concentrations by notches and cracks,' Journal of Applied Mechanics 35, 1968, 379-386.
4. Knowles, J. K. and Stemberg, E., 'On a class of conservation laws in linear and finite elastostatics,' Archive for Rational Mechanics and Analysis. 44, 1972, 187-211.
5. Budiansky, B. and Rice, J. R., 'Conservation laws and energy-release rate,' ASME Journal of Applied Mechanics 40, 1985, 201-203.
6. Eshelby, J. D., 'The elastic energy-momentum tensor,' Journal of Elasticity 5, 1975, 321-335.
7. Fletcher, D. C., 'Conservation laws in linear elastodynamics,' Archive for Rational Mechanics and Analysis 60, 1976, 329-353.
8. Olver, P. J., 'Conservation laws in elasticity. I. General results,' Archive for Rational Mechanics and Analysis 85, 1984, 119-129.
9. Olver, P. J., 'Conservation laws in elasticity. II. Linear homogeneous elastostatics,' Archive for Rational Mechanics and Analysis 85, 1984, 131-160.
10. Olver, P. J., 'Conservation laws in elasticity. III. Planar linear anisotropic elastostatics,' Archive for Rational Mechanics and Analysis 102, 1988, 167-181.
11. Bui, H. D., 'Dual path independent integrals in the boundary-value problems of cracks,' Engineering Fracture Mechanics 6, 1974, 287-296.
12. Sun, S.-X., 'Dual conservation laws in elastostatics,' International Journal of Engineering Science 23, 1985, 1179-1186.
13. Li, X., 'Dual conservation laws in elasticity,' Engineering Fracture Mechanics 29, 1988, 233-241.
14. Christiansen, S. and Hougaard, P., 'An investigation of a pair of integral equations for the biharmonic problem,' Journal of Institute Mathematics Applications 22, 1978, 15-27.
15. Christiansen, S., 'On the elastostatic significance of four boundary integrals involving biharmonic functions,' Acta Mechanica 126, 1998, 37-43.
16. Horgan, C. O., 'On Saint-Venant's principle in plane anisotropic elasticity,' Journal of Elasticity 2, 1972, 169-180.
17. Morgan, C. O., 'Decay estimates for the biharmonic equation with applications to Saint-Venant principles in plane elasticity and Stokes flows,' Quarterly of Applied Mathematics 47, 1989, 147-157.
18. Miller, K. L. and Horgan, C. O., 'Conservation properties for plane deformations of isotropic and anisotropic linear elastic strips,' Journal of Elasticity 33, 1993, 311-318.
19. Flavin, J. N., 'Some convexity considerations for a two-dimensional traction problem,' Journal of Applied Mathematics and Physics (ZAMP) 39, 1988, 166-176.
20. Gurtin, M. E., 'The linear theory of elasticity,' in Handbuch der Physik, Vol. VIa/2, S. Flügge (ed.), (ed. by C. Truesdell), Springer-Verlag, Berlin, 1972, pp. 1-295.
21. Ibragimov, N. H., Transformation Group Applied to Mathematical Physics, Reidel, Boston, 1985.
22. Ibragimov, N. H., Lie Group Analysis of Differential Equations, Vols. I-III, CRC Handbook, CRC Press, Boca Raton, Florida, 1994.
23. Olver, P. J., Applications of Lie Group to Differential Equations, Springer, New York, 1986.
24. Ibragimov, N. H. and Andersen, R. L., 'Lie-Bäcklund tangent transformations,' Journal of Mathematics and Aanalysis and Applications 59, 1977, 145.
25. Bluman, G. W. and Gregory, R. D., 'On transformation of the biharmonic equations,' Mathematika 32, 1985, 118-130.
26. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, 1953.
27. Eshelby, J. D., 'Energy relations and the energy-momentum tensor in continuum mechanics,' in Inelastic Behavior of Solids, M. F. Kanninen et al. (eds.), McGraw Hill, New York, 1970, pp. 77-115.
28. Günther, W., 'Über einige Randintegrale der Elastomechanik,' Braunschweiger Wissenschaftliche Gesellschaft 14, 1962, 53.
29. Noether, E., 'Invariante Variationsprobleme,' Göttinger Nachrichten (Mathematisch-physikalische Klasse) 2, 1918, 235-257 [translation Transport Theory and Statistical Physics 1, 1971, 186-207].
30. Noether, E., 'Invariante Variationsprobleme,' Göttinger Nachrichten (Mathematisch-physikalische Klasse) 2, 1918, 235-257 [translation Transport Theory and Statistical Physics 1, 1971, 186-207].