Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T07:52:34.035Z Has data issue: false hasContentIssue false

A mathematical model of damage accumulation taking into account microstructural effects

Published online by Cambridge University Press:  26 September 2008

G. I. Barenblatt
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23, ul. Krasikova, Moscow 117259, Russia
V. M. Prostokishin
Affiliation:
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, 23, ul. Krasikova, Moscow 117259, Russia

Abstract

It is shown that the influence of microstructure in the damage accumulation process leads to a nonlinear diffusion effect, with a strongly stress-dependent diffusion coefficient. A nonlinear parabolic equation with a source term is obtained for the damage parameter. This equation is relevant to blow-up and quenching problems well known to mathematicians with rupture corresponding to blow-up or quenching. However, the damage accumulation equation possesses an additional nonlinearity due to the non-healing of damage. Depending on the value of a dimensionless constant parameter (the ratio of a properly defined microstructural length-size to a characteristic length-size of the initial damage distribution), two essentially different types of damage accumulation process appear to be possible for a given initial damage distribution over the bar length. In processes of the first type, the damage accumulation remains non-homogeneous over the length of the bar, so that the lifetime for the whole specimen is determined by the maximal initial damage within the bar. For processes of the second type the damage distribution over the specimen at first becomes homogeneous (at least in a considerable part of specimen), and then the damage accumulation proceeds uniformly over all or part of the specimen. The lifetime for processes of the second type is essentially longer than the first. Results of a numerical experiment based on the proposed model are presented. In particular, the origin and development of damage propagation waves is demonstrated. Also, it is demonstrated that when there is substantial damage transfer, the ultimate value of the damage parameter in the life-time calculation is of no significance.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Kachanov, L. M. 1958 On the life-time under creep conditions. Izvesiiya (Bull.). USSR Ac. Sci., Division of Technical Sciences (8), pp. 2631 (in Russian).Google Scholar
[2]Kachanov, L. M. 1961 Rupture time under creep conditions. In Problems of Continuum Mechanics, Radok, J. R. M., ed., SIAM: Philadelphia, 202218.Google Scholar
[3]Odqvist, F. K. G. & Hult, J. 1966 Some aspects of creep rupture. Arkiv för Fysik 19 (26), 379382.Google Scholar
[4]Odqvist, F. K. G. 1966 Mathematical Theory of Creep and Creep Rupture. Clarendon: Oxford.Google Scholar
[5]Yokobori, T. 1965 The Strength, Fracture and Fatigue of Materials. Noordhoff.Google Scholar
[6]Tvergaard, V. 1991 Micromechanical model of creep rupture. Z. Angew Math. Mech. 70 (4), T23T32.Google Scholar
[7]Van der Giessen, E. & Tvergaard, V. 1991 A creep rupture model accounting for cavitation at sliding grain boundaries. Int. J. Fracture 48, 153178.CrossRefGoogle Scholar
[8]Samarsky, A. A. & Sobol', I. M. 1963 Examples of numerical calculation of thermal waves. J. Comput. Math, and Math. Physics 3 (4), 703719 (in Russian).Google Scholar
[9]Zmitrenko, N. V. & Kurdyumov, S. P. 1972 Self-similar compression regimes of a finite mass of a matter by a piston. Heat and Mass Transfer 8, 2023 (in Russian).Google Scholar
[10]Samarsky, A. A., Galaktionov, V. A., Kurdyumov, S. P. & Mikhailov, A. P. 1987 Blow-up regimes in problems for quasilinear parabolic equations. Nauka, Moscow (in Russian).Google Scholar
[11]Brezis, H. & Merle, F. 1991 Uniform estimates and blow-up behaviour for solutions of −λu = V(x)eu in two dimensions. Publication du Laboratoire d'Analyse Numerique, Université Pierre et Marie Curie, Paris.Google Scholar
[12]Giga, Y. & Kohn, R. V. 1985 Asymptotically self-similar blow-up of semilinear heat equations. Communications on Pure and Applied Mathematics 38, 297319.CrossRefGoogle Scholar
[13]Giga, Y. & Kohn, R. V. 1987 Characterizing blow-up using similarity variables. Indiana University Mathematics Journal 36 (1), 140.CrossRefGoogle Scholar
[14]Berger, M. & Kohn, R. V. 1988 A rescaling algorithm for numerical calculation of blowing-up solutions. Communications on Pure and Applied Mathematics 41, 841863.CrossRefGoogle Scholar
[15]Giga, Y. & Kohn, R. V. 1989 Nondegeneracy of blow-up for semilinear heat equations. Communications on Pure and Applied Mathematics 42, 845884.CrossRefGoogle Scholar
[16]Kawarada, H. 1976 On solutions of initial boundary value problem for ut = uxx + 1/(1 −u). RIMS Kyoto University 10, 729736.CrossRefGoogle Scholar
[17]Levine, H. A. 1989 Quenching, non-quenching, and beyond quenching for solutions of some parabolic equations. Annali di Matematica Pura ed Applicata (IV). CLV, 243260.CrossRefGoogle Scholar
[18]Langer, J. S. & Tang, C. 1991 Rupture propagation in a model of an earthquake fault. Phys. Rev. Letters 67 (8), 10431046.CrossRefGoogle Scholar