Lie point symmetries, conservation and balance laws in linear gradient elastodynamics

Journal of Elasticity

Volumn 88, Issue 1, 2007, Pages 5-25

  Lazar, Markus ^a   Anastassiadis, Charalampos ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Conservation laws; Gradient elasticity; Path independent integrals; Symmetries

Indexed keywords

ELASTICITY; GRADIENT METHODS; THEOREM PROVING; VECTORS; VELOCITY MEASUREMENT;

CONSERVATION LAWS; GRADIENT ELASTICITY; PATH INDEPENDENT INTEGRALS; SYMMETRIES;

ELASTOHYDRODYNAMICS;

EID: 34547377302     PISSN: 03743535     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10659-007-9105-5     Document Type: Article

References (34)

1. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer (1989)
2. Budiansky, B., Rice, J.R.: Conservation laws and energy-release rate. J. Appl. Math. 40, 201-203 (1973)
3. Chen, J.Y., Wei, Y., Huang, Y., Hutchinson, J.W., Hwang, K.C.: The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses. Eng. Fract. Mech. 64, 625-648 (1999)
4. Eshelby, J.D.: The elastic energy-momentum tensor. J. Elast. 5, 321-335 (1975)
5. Eshelby, J.D.: The calculation of energy release rates. In: Sih, G.C. (ed.) Prospects of Fracture Mechanics, pp. 69-84. Noorhoff, Leydenpp (1975)
6. Fletcher, D.C.: Conservation laws in linear elastodynamics. Arch. Ration. Mech. Anal. 60, 329-353 (1975)
7. Georgiadis, H.G., Vardoulakis, I.: Anti-plane shear Lamb's problem treated by gradient elasticity with surface energy. Wave Motion 28, 353-366 (1998)
8. Georgiadis, H.G.: The mode III crack problem in microstructured solids governed by dipolar gradient elasticity. J. Appl. Mech. 70, 517-530 (2003)
9. Georgiadis, H.G., Vardoulakis, I., Velgaki, E.G.: Dispersive Rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity. J. Elast. 74, 17-45 (2004)
10. Georgiadis, H.G., Grentzelou, C.G.: Energy theorems and the J-integral in dipolar gradient elasticity. Int. J. Solids Struct. 43, 5690-5712 (2006)
11. Günther, W.: Über einige Randintegrale der Elastostatik Abh. Braunschweig. Wiss. Ges. 14, 53-72 (1962)
12. Huang, Y.-N., Batra, R.C.: Energy-momentum tensors in nonsimple elastic dielectrics. J. Elast. 42, 275-281 (1996)
13. Ibragimov, N.H.: Transformation Group Applied to Mathematical Physics. Dordrecht, Reidel (1985)
14. Kalpakides, V.K., Agiasofitou, E.K.: On material equations in second gradient electroelasticity. J. Elast. 67, 205-227 (2002)
15. Kienzler, R., Herrmann, G.: Mechanics in Material Space. Berlin, Springer (2000)
16. J.K. Knowles, Sternberg, E.: On a class of conservation laws in linearized and finite elastostatics, Arch. Ration. Mech. Anal. 44, 187-211 (1972)
17. M. Lazar, C. Anastassiadis, Lie point symmetries and conservation laws in microstretch and micromorphic elasticity. Int. J. Eng. Sci. 44, 1571-1582 (2006)
18. Lazar, M., Kirchner, H.O.K.: The Eshelby stress tensor, angular momentum tensor and dilatation flux in gradient elasticity. Int. J. Solids Struct. 44, 2477-2486 (2007)
19. Lazar, M. On conservation and balance laws in micromorphic elastodynamics, J. Elast. (in press) (2007)
20. Lubarda, V.A., Markenscoff, X.: On conservation integrals in micropolar elasticity. Philos. Mag. 83, 1365-1377 (2003)
21. Maugin, G.A., Trimarco, C.: Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture. Acta Mech. 94, (1992) 1-28
22. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman and Hall, London (1993)
23. Maugin, G.A.: Nonlinear Waves in Elastic Crystals. Oxford University Press (1999)
24. Maugin, G.A., Christov, C.I.: Nonlinear Waves and Conservation laws. Nonlinear Duality between Elastic Waves and Quasi-Particles. In: Christov, C.J., Guran, A. (eds.) Selected Topics in Nonlinear Wave Mechanics, pp. 116-160. Boston, MA: Birkhäuser (2002)
25. Meletlidou, E., Pouget, J., Maugin, G.A., Aifantis, E.C.: Invariant relations in Boussinesq-type equations. Chaos Solitons and Fractals 22, 613-625 (2004)
26. Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51-78 (1964)
27. Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109-124 (1968)
28. Mindlin, R.D.: Elasticity, piezoelectricity and crystal lattice dynamics. J. Elast. 2, 217-282 (1972)
29. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics I, McGraw-Hill, New York, (1953)
30. Olver, P.J.: Conservation laws in elasticity. II. Linear homogeneous elastostatics, Arch. Ration. Mech. Anal. 85, 131-160 (1984). Errata in 102, 385-387 (1988)
31. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York, (1986)
32. Podolsky, B., Kikuchi, C.: A generalized electrodynamics. Phys. Rev. 65, 228-234 (1944)
33. Pucci, E., Saccomandi, G.: Symmetries and conservation laws in micropolar elasticity, Int. J. Eng. Sci. 28, 557-562 (1990)
34. Thielheim, K.O.: Note on classical fields of higher order. Proc. Phys. Soc. 91, 798-801 (1967)