In this work, we consider the diffusion-controlled axisymmetric fracture in an infinite space, and half-space. An important example of diffusion-controlled fracture growth is given by hydrogen induced cracking. In metals, hydrogen is typically dissolved in the proton form. When protons reach the crack surface, they recombine with electrons and form molecular hydrogen in the crack cavity. Then, the fracture can propagate even in the absence of any external loading, that is, only under the excessive pressure of gas hydrogen accumulated inside the crack.
Our results show that in the long-time asymptotic approximation (based on the quasi-static solution), the diffusion-controlled delamination propagates with constant velocity. We determine a maximum critical concentration that limits the use of the quasi-static solution. A transient solution, representing a short-time asymptotic approximation, is used when the concentration of gas exceeds the critical concentration. We then match these two end-member cases by using the method of Padé approximations and present closed-form solutions for both internal and near-surface diffusion-controlled crack propagation at different time scales.
]]>
En este trabajo, consideramos la fractura de difución controlada axisimétrica en un espacio infinito, y en el semiespacio. Un ejemplo importante del crecimiento de una fractura de difusión controlada es dado por el hidrogeno inducido en agrietamiento. En metales, el hidrógeno es típicamente disuelto en forma de protones. Cuando los protones alcanzan la superficie de la grieta, se recombinan con electrones y forman hidrógeno molecular en la cavidad de la grieta. Entonces, la fractura puede propagar aún en ausencia de cualquier carga externa, esto es, sólo bajo presión excesiva de gas hidrogeno acumulado dentro de la grieta.
Nuestros resultados muestran que en la aproximación asintótica a largo plazo (basada en la solución cuasiestática), la delaminación de difusión controlada propaga con velocidad constante. Nosotros de- terminamos una concentración crítica máxima que limita el uso de la solución cuasiestática. Una solución transitoria, que representa una aproximación asintótica de corto plazo, es usada cuando la concentración del gas excede la concentración crítica. Entonces apareamos estos dos casos usando el método de aproximaciones de Padé y presentamos soluciones en forma cerrada tanto para propagación de grietas de difusión controlada internas como cercanas a la superficie, en diferentes escalas de tiempo.
Palabras clave: difusión, propagación de grietas, análisis asintótico, aproximación de Padé.
[1] Balueva, A.V; Dashevski, I.D. (1995) “Qualitative estimates of gas filled crack growth”, Mechanics of Solids6: 122–128. [ Links ]
[2] Balueva, A.V.; Goldstein, R.V. (1992) “Kinetic crack propagation in a layer, with gas diffusion”, Mechanics of Solids2: 114–123. [ Links ]
[3] Carslaw, H.S.; Jaeger, J.C. (1992) Conditions of Heat in Solids, 2nd Edition. Clarendon Press, Oxford. [ Links ]
[4] Davis, J.R. (2000) Corrosion. Understanding the Basic. ASM International Material Park, Ohio 44073-0002. [ Links ]
[5] Eliaz, N., Banks-Sills,L., Ashkenazi,D., and Eliasi,R. (2004) “Modeling failure of metallic glasses due to hydrogen embrittlement”, Acta Materialia52: 93. [ Links ]
[6] Gapharov, N.A., Goncharov, A.A., Kushnarenko, V.M. (1998) Corrosion and Protection of Equipment for Hydrogen Sulfide Oil Deposits. Nedra Publishers, Moscow, 438 pp. (in Russian). [ Links ]
[7] Germanovich, L.N. (1986) “Temperature stresses in an elastic half space with heat sources”, Mechanics of Solids21(1): 77–88. [ Links ]
[8] Germanovich, L.N.; Kill’, I.D. (1985) “Convective heating of a half- space (nonsymmetric case)”, Journal of Engineering Physics 48(1): 113–114. [ Links ]
[9] Goldstein R.V.; Entov V.M.; Pavlovsky, B.R. (1977) “Model of development of hydrogen Cracks in metal”, Academiia nauk SSSR. Doklady 237(4): 828–831. [ Links ]
[10] Goldstein, R.V.; Zazovskii, A.F.; Pavlovsky, B.R. (1985) “Development of penny shaped layering in a metal sheet”, Fisiko Khimicheskaia Mekhanika Materialov21(3): 621–656. [ Links ]
[12] Hick, P.D.; Altstetter, C.J. (1992) “Hydrogen-enhanced cracking of superalloys”, Metallurgical Transactions A. Physical Metallurgy and Materials Science23(1): 237–249. [ Links ]
[13] Hirth, J.P. (1984) “Theories of hydrogen induced cracking of steels”, in: R. Gibala & .F. Hehemann (Eds.) Hydrogen Embrittlement and Stress Corrosion Cracking, American Society for Metals, Metal Park, Ohio: 29-41. [ Links ]
[14] Krom, A.H.; Koers, R.W.; Bakker, A. (1999) “Hydrogen transport near a blunting crack tip”, J. Mech. Phys. Solids47(4): 971–992. [ Links ]
[15] Van Leeuwen, H.P. (1974) “The kinetics of hydrogen embrittlement: a quantitative diffusion model”, Engineering Fracture Mechanics6: 141–161. [ Links ]
[16] Panasyuk, V.V.; Andreikiv, A.E.; Kharin, Y.S. (1987) “A model of crack growth in deformed metals under the action of hydrogen”, Soviet Materials Science23: 111–124. [ Links ]
[17] Polyakov, V.N. (1996) “Catastrophes of large diameter pipelines: the role of hydrogen fields”, Hydrogen Effects in Materials, Moran, WY (11-14 Sept. 1994): 991–1000. [ Links ]
[18] Polyanin, A.D.; Manzhirov, A.D. (1998) Handbook of Integral Equations. CRC Press, Boca Raton. [ Links ]
[19] Rice, J.R. (1968) “Mathematical analysis in the mechanics of fracture”, in: H. Liebowitz (Ed.) Fracture, an Advanced Treatise, Vol. II: Mathematical Fundamentals, Academic Press, New York: 191–311. [ Links ]
[20] Sneddon, I.N. (1972) The Use of Integral Transforms. McGraw-Hill, New York. [ Links ]
[21] Speidel, M.O. (1984) “Hydrogen embrittlement and stress corrosion cracking of aluminum alloys”, in: R. Gibala & R.F. Hehemann (Eds.) Hydrogen Embrittlement and Stress Corrosion Cracking, American Society for Metals, Metal Park, Ohio: 271–296. [ Links ]
[22] Timoshenko, S.P.; Goodier, J.N. (1970) Theory of Elasticity.McGraw-Hill, New York. [ Links ]
[23] Toribio, J.; Kharin, V. (1998) “The effect of history on hydrogen assisted cracking: 2. A revision of K-dominance”, Int. J. Fract.88: 247–258.
[24] Turnbull, A. (1993) “Modeling of environment assisted cracking”, Corrosion Science34(6): 921–960. [ Links ]
[25] Vehoff, H. (1997) “Hydrogen related material problems”, Hydrogen in Metals III, Topics in Applied Physics, Springer 73: 215–278. [ Links ]
[26] Zapffe, C.A.; Moore, G.A. (1943) “A micrographic study of the cleavage of hydrogenized ferrite”, Trans. Amer. Inst. Min. Met. Eng. 154: 335–359. [ Links ]
[27] Zhong, W.; Cai, Y.; Tomanek, D. (1993) “Computer simulation of hydrogen embrittlement in metals”, Nature 362: 435–437. [ Links ]
*Correspondencia a: Alla V. Balueva. Gainesville State College, Mathematics Department, P.O. Box 1358, Gainesville, Georgia, U.S.A. E-Mail: abalueva@gsc.edu Leonid N. Germanovich. Georgia Institute of Technology, Department of Civil Engineering, Atlanta, Georgia, U.S.A. E-Mail: leonid@ce.gatech.edu *Gainesville State College, Mathematics Department, P.O. Box 1358, Gainesville, Georgia, U.S.A. E-Mail: abalueva@gsc.edu †Georgia Institute of Technology, Department of Civil Engineering, Atlanta, Georgia, U.S.A. E-Mail: leonid@ce.gatech.edu
Received: 18 Feb 2010; Revised: 28 Jun 2012; Accepted: 29 Jun 2012