Conservation and balance laws in linear elasticity of grade three

Journal of Elasticity

Volumn 94, Issue 1, 2009, Pages 69-85

  Agiasofitou, E K ^a   Lazar, M ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Balance laws; Conservation laws; Eshelby stress tensor; Gradient elasticity

Indexed keywords

ELASTICITY; ELASTOHYDRODYNAMICS; TENSORS;

BALANCE LAWS; CONSERVATION LAWS; ESHELBY STRESS TENSOR; GRADIENT ELASTICITY; LINEAR ELASTICITIES; LINEAR ELASTODYNAMICS; MINDLIN; NOETHER'S THEOREMS; RECIPROCAL THEOREMS; STRAIN ENERGY DENSITIES; THIRD ORDERS;

LAWS AND LEGISLATION;

EID: 58149488793     PISSN: 03743535     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10659-008-9185-x     Document Type: Article

References (52)

1. Achenbach, J.D.: Reciprocity in Elastodynamics. Cambridge University Press, Cambridge (2003)
2. S.C. Anco G. Bluman 1997 Direct construction of conservation laws from field equations Phys. Rev. Lett. 78 2869 2873
3. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)
4. S. Deng J. Liu N. Liang 2007 Wedge and twist disclinations in second strain gradient elasticity Int. J. Solids Struct. 44 3646 3665
5. J.D. Eshelby 1975 The elastic energy-momentum tensor J. Elast. 5 321 335
6. D.C. Fletcher 1975 Conservation laws in linear elastodynamics Arch. Ration. Mech. Anal. 60 329 353
7. H.G. Georgiadis 2003 The mode III crack problem in microstructured solids governed by dipolar gradient elasticity ASME J. Appl. Mech. 70 517 530
8. H.G. Georgiadis D.S. Anagnostou 2008 Problems of the Flamant-Boussineasq and Kelvin type in dipolar gradient elasticity J. Elast. 90 71 98
9. H.G. Georgiadis C.G. Grentzelou 2006 Energy theorems and the J-integral in dipolar gradient elasticity Int. J. Solids Struct. 43 5690 5712
10. A.E. Giannakopoulos E. Amanatidou N. Aravas 2006 A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle Int. J. Solids Struct. 43 3875 3894
11. A.E. Green R.S. Rivlin 1964 Simple force and stress multipoles Arch. Ration. Mech. Anal. 16 325 353
12. A.E. Green R.S. Rivlin 1964 Multipolar continuum mechanics Arch. Ration. Mech. Anal. 17 113 147
13. M.Yu. Gutkin E.C. Aifantis 1999 Dislocations and disclinations in gradient elasticity Phys. Stat. Sol. (b) 214 245 284
14. W. Günther 1962 Über einige Randintegrale der Elastostatik Abh. Braunschweig. Wiss. Ges. 14 53 72
15. T. Honein G. Herrmann 1997 Conservation laws in nonhomogeneous plane elastostatics J. Mech. Phys. Solids 45 789 805
16. T. Honein N. Chien G. Herrmann 1991 On conservation laws for dissipative systems Phys. Lett. A 155 223 224
17. Ibragimov, N.H.: Lie Group Analysis of Differential Equations, Vol. 2, Applications in Engineering and Physical Sciences. CRC Press, Boca Raton (1995)
18. V.K. Kalpakides E.K. Agiasofitou 2002 On material equations in second gradient electroelasticity J. Elast. 67 205 227
19. V.K. Kalpakides G.A. Maugin 2004 Canonical formulation and conservation laws of thermoelasticity without dissipation Rep. Math. Phys. 53 371 391
20. Kienzler, R., Herrmann, G.: Mechanics in Material Space. Springer, Berlin (2000)
21. J.K. Knowles E. Sternberg 1972 On a class of conservation laws in linearized and finite elastostatics Arch. Ration. Mech. Anal. 44 187 211
22. R.W. Lardner 1971 Dislocations in materials with couple stress J. Inst. Math. Appl. 7 126 137
23. M. Lazar 2007 On conservation and balance laws in micromorphic elastodynamics J. Elast. 88 63 78
24. M. Lazar C. Anastassiadis 2006 Lie point symmetries and conservation laws in microstretch and micromorphic elasticity Int. J. Eng. Sci. 44 1571 1582
25. M. Lazar C. Anastassiadis 2007 Lie point symmetries, conservation and balance laws in linear gradient elastodynamics J. Elast. 88 5 25
26. M. Lazar H.O.K. Kirchner 2007 The Eshelby stress tensor, angular momentum tensor and dilatation flux in gradient elasticity Int. J. Solids Struct. 44 2477 2486
27. M. Lazar G.A. Maugin 2005 Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity Int. J. Eng. Sci. 43 1157 1184
28. M. Lazar G.A. Maugin 2006 A note on line forces in gradient elasticity Mech. Res. Commun. 33 674 680
29. M. Lazar G.A. Maugin E.C. Aifantis 2006 Dislocations in second strain gradient elasticity Int. J. Solids Struct. 43 1787 1817
30. S. Li I. Miskioglu B.S. Altan 2004 Solution to line loading of a semi-infinite solid in gradient elasticity Int J. Solids Struct. 41 3395 3410
31. V.A. Lubarda X. Markenscoff 2000 On conservation integrals in couple stress elasticity J. Mech. Phys. Solids 48 553 564
32. V.A. Lubarda X. Markenscoff 2003 On conservation integrals in micropolar elasticity Philos. Mag. 83 1365 1377
33. Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman and Hall, London (1993)
34. G.A. Maugin 1994 On the J-integral and energy-release rates in dynamical fracture Acta Mech. 105 33 47
35. G.A. Maugin C. Trimarco 1992 Pseudomomentum and material forces in nonlinear elasticity: variational formulations and application to brittle fracture Acta Mech. 94 1 28
36. R.D. Mindlin 1964 Micro-structure in linear elasticity Arch. Ration. Mech. Anal. 16 51 78
37. R.D. Mindlin 1965 Second gradient of strain and surface-tension in linear elasticity Int. J. Solids Struct. 1 417 438
38. R.D. Mindlin 1972 Elasticity, piezoelectricity and crystal lattice dynamics J. Elast. 2 217 282
39. R.D. Mindlin N.N. Eshel 1968 On first strain-gradient theories in linear elasticity Int. J. Solids Struct. 4 109 124
40. Morse, P.M., Feshbach, H.: Methods of Theoretical Physics I. McGraw-Hill, New York (1953)
41. Noether, E.: Invariante Variationsprobleme. Nachr. König. Gesell. Wissen. Göttingen, Math.-Phys. Kl, 235-257 (1918). (English translation: Invariant Variation Problems, 1971. Transport Theory and Stat. Phys. 1, 186-207)
42. P.J. Olver 1984 Conservation laws in elasticity. II. Linear homogeneous elastostatics Arch. Ration. Mech. Anal. 85 131 160 Errata in 102, 385-387 (1984)
43. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
44. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)
45. C. Polizzotto 2003 Gradient elasticity and nonstandard boundary conditions Int. J. Solids Struct. 40 7399 7423
46. E. Pucci G. Saccomandi 1990 Symmetries and conservation laws in micropolar elasticity Int. J. Eng. Sci. 28 557 562
47. C.Q. Ru E.C. Aifantis 1993 A simple approach to solve boundary-value problems in gradient elasticity Acta Mech. 101 59 68
48. K.O. Thielheim 1967 Note on classical fields of higher order Proc. Phys. Soc. 91 798 801
49. R.A. Toupin 1964 Theories of elasticity with couple-stress Arch. Ration. Mech. Anal. 17 85 112
50. J.R. Willis 1965 Dislocations and inclusions J. Mech. Phys. Solids 13 377 395
51. C.H. Wu 1992 Cohesive elasticity and surface phenomena Q. Appl. Math. L 1 73 103
52. X. Zhang K. Jiao P. Sharma B.I. Yakobson 2006 An atomistic and non-classical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene J. Mech. Phys. Solids 54 2304 2329