Dual integrals in small strain elasticity with body forces

Vlado A. Lubarda; Xanthippi Markenscoff

doi:10.1016/j.crme.2007.11.009

Dual integrals in small strain elasticity with body forces
[Intégrales duales en élasticité infinitésimale avec forces de masse]

Vlado A. Lubarda ¹ ; Xanthippi Markenscoff ¹

¹ Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA

Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 190-202.

Résumé
Abstract

Des intégrales duales en élasticité infinitésimale sont obtenues et reliées aux taux de restitution d'énergie associés au mouvement d'un défaut en présence de forces de masse. On définit un tenseur d'énergie–impulsion qui inclut un terme de travail des forces de masse, et qui fournit des expressions simples des forces configurationnelles en fonction des intégrales $J_{k}$ , $L_{k}$ et M. Du fait que l'énergie potentielle complémentaire n'inclut pas explicitement les forces de masse, le tenseur d'énergie-impulsion complémentaire a la même structure qu'en élasticité sans forces de masse. Les expressions des intégrales non-conservées $J_{k}$ , $L_{k}$ et M et de leurs duales sont reliées à des intégrales de volume dont les intégrandes dépendent des forces de masse et de leurs gradients. Si les forces de masse sont spatialement uniformes, les lois de conservation $J_{k} = {\hat{J}}_{k} = 0$ s'appliquent aux problèmes tant 2D que 3D, de même que la loi $L_{3} = {\hat{L}}_{3} = 0$ aux problèmes antiplans. La loi de conservation $M = \hat{M} = 0$ s'applique en l'absence de forces de masse ou si ce sont des fonctions homogènes de degré particulier des coordonnées.

Dual integrals of small strain elasticity are derived and related to the energy release rates associated with a defect motion in the presence of body forces. A modified energy momentum tensor is used, which includes a work term due to body forces, and which yields simple expressions for the configurational forces in terms of the $J_{k}$ , $L_{k}$ , and M integrals. Since the complementary potential energy does not include body forces explicitly, the complementary energy momentum tensor has the same structure as in the elasticity without body forces. The expressions for the nonconserved $J_{k}$ , $L_{k}$ , and M integrals, and their duals, are related to the volume integrals whose integrands depend on body forces and their gradients. If body forces are spatially uniform, the conservation laws $J_{k} = {\hat{J}}_{k} = 0$ hold for both 2D and 3D problems, and $L_{3} = {\hat{L}}_{3} = 0$ for the antiplane strain problems. The conservation law $M = \hat{M} = 0$ holds if body forces are absent, or if they are homogeneous functions of particular degree in spatial coordinates.

Publié le : 2008-01-01

DOI : 10.1016/j.crme.2007.11.009

Keywords: Computational solid mechanics, Complementary energy, Body force, Configurational force, Dual integrals, Energy momentum tensor, Potential energy
Mot clés : Mécanique des solides numérique, Énergie complémentaire, Forces de masse, Forces configurationnelles, Intégrales duales, Tenseur d'énergie–impulsion, Énergie potentielle

Affiliations des auteurs :

Vlado A. Lubarda ¹ ; Xanthippi Markenscoff ¹

¹ Department of Mechanical and Aerospace Engineering, University of California, San Diego; La Jolla, CA 92093-0411, USA

@article{CRMECA_2008__336_1-2_190_0,
     author = {Vlado A. Lubarda and Xanthippi Markenscoff},
     title = {Dual integrals in small strain elasticity with body forces},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {190--202},
     publisher = {Elsevier},
     volume = {336},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crme.2007.11.009},
     language = {en},
}

TY  - JOUR
AU  - Vlado A. Lubarda
AU  - Xanthippi Markenscoff
TI  - Dual integrals in small strain elasticity with body forces
JO  - Comptes Rendus. Mécanique
PY  - 2008
SP  - 190
EP  - 202
VL  - 336
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crme.2007.11.009
LA  - en
ID  - CRMECA_2008__336_1-2_190_0
ER  -

%0 Journal Article
%A Vlado A. Lubarda
%A Xanthippi Markenscoff
%T Dual integrals in small strain elasticity with body forces
%J Comptes Rendus. Mécanique
%D 2008
%P 190-202
%V 336
%N 1-2
%I Elsevier
%R 10.1016/j.crme.2007.11.009
%G en
%F CRMECA_2008__336_1-2_190_0

Vlado A. Lubarda; Xanthippi Markenscoff. Dual integrals in small strain elasticity with body forces. Comptes Rendus. Mécanique, Volume 336 (2008) no. 1-2, pp. 190-202. doi : 10.1016/j.crme.2007.11.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.11.009/

Bibliographie
Cité par

[1] J.D. Eshelby The force on an elastic singularity, Philos. Trans. Roy. Soc. A, Volume 244 (1951), pp. 87-112

[2] J.D. Eshelby The continuum theory of lattice defects, Solid State Phys., Volume 3 (1956), pp. 79-144

[3] W. Günther Über einige Randintegrale der Elastomechanik, Abh. Braunschw. Wiss. Ges., Volume 14 (1962), pp. 53-72

[4] J.K. Knowles; E. Sternberg On a class of conservation laws in linearized and finite elastostatics, Arch. Ration. Mech. Anal., Volume 44 (1972), pp. 187-211

[5] B. Budiansky; J.R. Rice Conservation laws and energy-release rates, J. Appl. Mech., Volume 40 (1973), pp. 201-203

[6] H.D. Bui Dualité entre les intégrals indépendentes du contour dans la théorie des solides fissurés, C. R. Acad. Sci. Paris, Volume 276 (1973), pp. 1425-1428

[7] H.D. Bui Dual path independent integrals in the boundary-value problems of cracks, Eng. Fract. Mech., Volume 6 (1974), pp. 287-296

[8] J.R. Rice A path independent integral and approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., Volume 38 (1968), pp. 379-386

[9] A.J. Carlsson Path independent integrals in fracture mechanics and their relation to variational principles (G.C. Sih; H.C. van Elst; D. Broek, eds.), Prospects of Fracture Mechanics, Noordhoff, Leyden, The Netherlands, 1974, pp. 139-158

[10] S.-X. Sun Dual conservation laws in elastostatics, Int. J. Engrg. Sci., Volume 23 (1985), pp. 1179-1186

[11] X. Li Dual conservation laws in elastostatics, Eng. Fract. Mech., Volume 29 (1988), pp. 233-241

[12] V.A. Lubarda; X. Markenscoff Dual conservation integrals and energy release rates, Int. J. Solids Struct., Volume 44 (2007), pp. 4079-4091

[13] H.D. Bui Associated path-independent J-integrals for separating mixed modes, J. Mech. Phys. Solids, Volume 31 (1983), pp. 439-448

[14] H.D. Bui Inverse Problems in the Mechanics of Materials: An Introduction, CRC Press, Boca Raton, 1994

[15] B. Moran; C.F. Shih A general treatment of crack tip contour integrals, Int. J. Fract., Volume 35 (1987), pp. 295-310

[16] C. Trimarco; G.A. Maugin Bui's path-independent integral in finite elasticity, Meccanica, Volume 30 (1995), pp. 139-145

[17] S. Li On dual conservation laws in linear elasticity: stress function formalism, Nonlinear Dynamics, Volume 36 (2004), pp. 77-96

[18] S. Li; A. Gupta On dual configurational forces, J. Elasticity, Volume 84 (2006), pp. 13-31

[19] V.A. Lubarda; X. Markenscoff Complementary energy release rates and dual conservation integrals in micropolar elasticity, J. Mech. Phys. Solids, Volume 55 (2007), pp. 2055-2072

[20] M. Lazar; H.O.K. Kirchner The Eshelby stress tensor, angular momentum tensor and scaling flux in micropolar elasticity, Int. J. Solids Struct., Volume 44 (2007), pp. 4613-4620

[21] V.A. Lubarda; X. Markenscoff Conservation integrals in couple stress elasticity, J. Mech. Phys. Solids, Volume 48 (2000), pp. 553-564

[22] V.A. Lubarda; X. Markenscoff On Conservation integrals in micropolar elasticity, Phil. Mag. A, Volume 83 (2003), pp. 1365-1377

[23] J.D. Eshelby Energy relations and the energy–momentum tensor in continuum mechanics (M.F. Kanninen; W.F. Adler; A.R. Rosenfield; R.I. Jaffee, eds.), Inelastic Behavior of Solids, McGraw-Hill, New York, 1970, pp. 77-115

[24] G.P. Cherepanov Mechanics of Brittle Fracture, McGraw-Hill, New York, 1979

[25] K. Kishimoto; S. Aoki; M. Sakata On the path independent integral- $\hat{J}$ , Engrg. Frac. Mech., Volume 13 (1980), pp. 841-850

[26] S.N. Atluri Path-independent integrals in finite elasticity and inelasticity, with body forces, inertia, and arbitrary crack-face conditions, Engrg. Frac. Mech., Volume 16 (1982), pp. 341-364

[27] M.F. Kanninen; C.H. Popelar Advanced Fracture Mechanics, Oxford Univ. Press, New York, 1985

[28] E. Kröner Configurational and material forces in the theory of defects in ordered structures, Mater. Sci. Forum, Volume 123–125 (1993), pp. 447-454

[29] G.A. Maugin Material forces: concepts and applications, Appl. Mech. Rev., Volume 48 (1995), pp. 247-285

[30] T. Honein; G. Herrmann Conservation laws in nonhomogeneous plane elastostatics, J. Mech. Phys. Solids, Volume 45 (1997), pp. 789-805

[31] H.O.K. Kirchner The force on an elastic singularity in a homogeneous medium, J. Mech. Phys. Solids, Volume 47 (1999), pp. 993-998

[32] G. Herrmann; R. Kienzler Conservation laws and their application in configurational mechanics (R. Kienzler; G.A. Maugin, eds.), Configurational Mechanics of Materials, Springer-Verlag, Wien, 2001, pp. 1-53

[33] V.A. Lubarda, The energy momentum tensor in the presence of body forces and the Peach–Koehler force on a dislocation, Int. J. Solids Struct. (2008), in press

[34] A.C. Palmer; J.R. Rice The growth of slip surfaces in the progressive failure of over-consolidated clay, Proc. Roy. Soc. Lond. A, Volume 332 (1973), pp. 527-548

Cité par Sources :

Commentaires - Politique