| Title:
|
On the change of energy caused by crack propagation in 3-dimensional anisotropic solids (English) |
| Author:
|
Steigemann, Martin |
| Author:
|
Specovius-Neugebauer, Maria |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
401-416 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Crack propagation in anisotropic materials is a persistent problem. A general concept to predict crack growth is the energy principle: A crack can only grow, if energy is released. We study the change of potential energy caused by a propagating crack in a fully three-dimensional solid consisting of an anisotropic material. Based on methods of asymptotic analysis (method of matched asymptotic expansions) we give a formula for the decrease in potential energy if a smooth inner crack grows along a small crack extension. (English) |
| Keyword:
|
crack propagation |
| Keyword:
|
energy principle |
| Keyword:
|
stress intensity factor |
| MSC:
|
35Q74 |
| MSC:
|
41A60 |
| MSC:
|
74G10 |
| MSC:
|
74G70 |
| MSC:
|
74R10 |
| idZBL:
|
Zbl 06362269 |
| idMR:
|
MR3238850 |
| DOI:
|
10.21136/MB.2014.143865 |
| . |
| Date available:
|
2014-07-14T08:48:11Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143865 |
| . |
| Reference:
|
[1] Argatov, I. I., Nazarov, S. A.: Energy release caused by the kinking of a crack in a plane anisotropic solid. Translated from the Russian.J. Appl. Math. Mech. 66 (2002), 491-503. MR 1937462, 10.1016/S0021-8928(02)00059-X |
| Reference:
|
[2] Bach, M., Nazarov, S. A., Wendland, W. L.: Stable propagation of a mode-1 planar crack in an isotropic elastic space. Comparison of the Irwin and the Griffth approaches.Problemi attuali dell'analisi e della fisica matematica P. E. Ricci Dipartimento di Matematica, Univ. di Roma (2000), 167-189. MR 1809025 |
| Reference:
|
[3] Bourdin, B., Francfort, G. A., Marigo, J.-J.: Numerical experiments in revisited brittle fracture.J. Mech. Phys. Solids 48 (2000), 797-826. Zbl 0995.74057, MR 1745759, 10.1016/S0022-5096(99)00028-9 |
| Reference:
|
[4] Ciarlet, P. G.: An introduction to differential geometry with applications to elasticity.J. Elasticity 78-79 (2005), 1-215. Zbl 1086.74001, MR 2196098 |
| Reference:
|
[5] Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems I.Proc. R. Soc. Edinb., Sect. A 123 (1993), 109-155. Zbl 0791.35032, MR 1204855, 10.1017/S0308210500021272 |
| Reference:
|
[6] Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems II.Proc. R. Soc. Edinb., Sect. A 123 (1993), 157-184. Zbl 0791.35033, MR 1204855, 10.1017/S0308210500021284 |
| Reference:
|
[7] Costabel, M., Dauge, M.: Crack singularities for general elliptic systems.Math. Nachr. 235 (2002), 29-49. Zbl 1094.35038, MR 1889276, 10.1002/1522-2616(200202)235:1<29::AID-MANA29>3.0.CO;2-6 |
| Reference:
|
[8] Costabel, M., Dauge, M., Yosibash, Z.: A quasi-dual function method for extracting edge stress intensity functions.SIAM J. Math. Anal. 35 (2004), 1177-1202. Zbl 1141.35363, MR 2050197, 10.1137/S0036141002404863 |
| Reference:
|
[9] Dauge, M.: Elliptic Boundary Value Problems on Corner Domains.Lecture Notes in Mathematics 1341 Springer, Berlin (1988). Zbl 0668.35001, MR 0961439 |
| Reference:
|
[10] Lorenzi, H. G. de: On the energy release rate and the $J$-integral for 3-D crack configurations.Int. J. Fract. 19 (1982), 183-193. 10.1007/BF00017129 |
| Reference:
|
[11] Favier, E., Lazarus, V., Leblond, J.-B.: Coplanar propagation paths of 3D cracks in infinite bodies loaded in shear.Int. J. Solids Struct. 43 (2006), 2091-2109. Zbl 1121.74449, 10.1016/j.ijsolstr.2005.06.041 |
| Reference:
|
[12] Griffith, A. A.: The phenomena of rupture and flow in solids.Philos. Trans. Roy. Soc. London 221 (1921), 163-198. 10.1098/rsta.1921.0006 |
| Reference:
|
[13] Hartranft, R. J., Sih, G. C.: Stress singularity for a crack with an arbitrarily curved front.Engineering Fracture Mechanics 9 (1977), 705-718. 10.1016/0013-7944(77)90083-2 |
| Reference:
|
[14] Il'in, A. M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems.Translated from the Russian. Translations of Mathematical Monographs 102 American Mathematical Society, Providence (1992). Zbl 0754.34002, MR 1182791, 10.1090/mmono/102 |
| Reference:
|
[15] Irwin, G.: Fracture.Handbuch der Physik. Bd. 6: Elastizität und Plastizität S. Flügge Springer, Berlin 551-590 (1958). MR 0094946 |
| Reference:
|
[16] Kondrat'ev, V. A.: Boundary value problems for elliptic equations in domains with conical or angular points.Trans. Mosc. Math. Soc. 16 (1967), 227-313. MR 0226187 |
| Reference:
|
[17] Kozlov, V. A., Maz'ya, V. G., Rossmann, J.: Elliptic Boundary Value Problems in Domains with Point Singularities.Mathematical Surveys and Monographs 52 American Mathematical Society, Providence (1997). Zbl 0947.35004, MR 1469972 |
| Reference:
|
[18] Kühnel, W.: Differential Geometry. Curves--Surfaces--Manifolds. Translated from the German.Student Mathematical Library 16 American Mathematical Society, Providence (2002). MR 1882174 |
| Reference:
|
[19] Lazarus, V.: Brittle fracture and fatigue propagation paths of 3D plane cracks under uniform remote tensile loading.Int. J. Fract. 122 (2003), 23-46. 10.1023/B:FRAC.0000005373.73286.5d |
| Reference:
|
[20] Leblond, J.-B., Torlai, O.: The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body.J. Elasticity 29 (1992), 97-131. Zbl 0777.73054, 10.1007/BF00044514 |
| Reference:
|
[21] Maz'ya, V. G., Plamenevsky, B. A.: The coefficients in the asymptotic of the solutions of elliptic boundary-value problems in domains with conical points.Russian Math. Nachr. 76 (1977), 29-60. |
| Reference:
|
[22] Maz'ya, V. G., Rossmann, J.: Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten.German Math. Nachr. 138 (1988), 27-53. Zbl 0672.35020, MR 0975198, 10.1002/mana.19881380103 |
| Reference:
|
[23] Nazarov, S. A.: Stress intensity factors and crack deviation conditions in a brittle anisotropic solid.J. Appl. Mech. Techn. Phys. 46 (2005), 386-394. Zbl 1088.74025, MR 2144814, 10.1007/s10808-005-0088-3 |
| Reference:
|
[24] Nazarov, S. A., Plamenevsky, B. A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries.De Gruyter Expositions in Mathematics 13 Walter de Gruyter, Berlin (1994). Zbl 0806.35001, MR 1283387 |
| Reference:
|
[25] Nazarov, S. A., Polyakova, O. R.: Rupture criteria, asymptotic conditions at crack tips, and selfadjoint extensions of the Lamé operator.Russian Tr. Mosk. Mat. Obs. 57 (1996), 16-74. MR 1468975 |
| Reference:
|
[26] Parks, D. M.: A stiffness derivative finite element technique for determination of crack tip stress intensity factors.Int. J. Fract. 10 (1974), 487-502. 10.1007/BF00155252 |
| Reference:
|
[27] Sih, G. C., Paris, P. C., Irwin, G. R.: On cracks in rectilinearly anisotropic bodies.Int. J. Fract. 1 (1965), 189-203. |
| Reference:
|
[28] Steigemann, M.: Verallgemeinerte Eigenfunktionen und lokale Integralcharakteristiken bei quasi-statischer Rissausbreitung in anisotropen Materialien.German Berichte aus der Mathematik Shaker, Aachen (2009). Zbl 1181.35286 |
| Reference:
|
[29] Williams, M. L.: Stress singularities resulting from various boundary conditions in angular corners of plates in extension.J. Appl. Mech. 19 (1952), 526-528. |
| . |
