 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| V. I. Kondaurov, "A tensor model of damage and long-term strength of elastic bodies," Mech. Solids. 36 (5), 109-122 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
5 |
Pages |
109-122 |
| Title |
A tensor model of damage and long-term strength of elastic bodies |
| Author(s) |
V. I. Kondaurov (Moscow) |
| Abstract |
At the present time, damage mechanics [1] which phenomenologically describes the nucleation and evolution of numerous defects in a body, such as micropores and microcracks [2], becomes widely used in theoretical and applied investigations. The traditional approach to modeling damage [3-5] is based on the assumptions similar to those of the theory of plastic (viscoplastic) flow. The basic assumptions of this approach are as follows:
• in the space of strains (stresses), there exists a region of elastic (viscoelastic) behavior of the material, in which damage is not accumulated;
• the strain and strength characteristics of the material depend on the rate of loading;
• the velocity hardening takes place, depending on
the rate of loading;
• the kinetics of the damage accumulation depends on the stress and level of damage in the material;
• there exists a strength criterion characterizing the limit state of an element of the medium.
The strength criterion and damage evolution equation do not, as a rule, depend on the rheology of the material. At best, the dependence on the current temperature, strain rate, and some other parameters of the history of loading is taken into account.
In [6, 7], the model was suggested to describe the damage
accumulation in elastic bodies on the basis of the energy approach which is similar to the Griffith approach [8] used in the mechanics of an isolated crack. The basic idea of this approach is to take into account the local balance between the strain potential energy released during the formation (growth) of a defect due to the partial unloading of the material near this defect and the energy spent to form new surfaces of micropores and microcracks. Using only "basic principles" and some additional assumptions of general character, this model can describe qualitative effects typical of the crack accumulation, such as
• existence of the threshold stresses at which microdefects begin to develop,
• damage accumulation under both normal and tangential stresses,
• unloading in accordance with the elastic law,
• substantial difference between the threshold stresses under tension and compression,
• dilatancy effects, and
• existence of the descending part in the stress-strain diagram in the case of the developed damage.
The drawback of this model that it utilizes a scalar parameter of damage, which leads to only volume residual strains in all cases of the stress state. To eliminate this drawback, one can introduce a damage tensor of rank ≥2. In [9], it has been already pointed out that it is necessary to use
such a tensor to take into account the
specific features of the stress fields.
In the present study, which develops the ideas of [6, 7], a tensor model of damage of elastic bodies is considered. This model allows us to describe the above mentioned effects on the basis of the general thermodynamic assumptions. New forms of the elastic potential and density of effective surface energy of the medium are suggested. The evolution equation for the damage tensor, which thermodynamically correlates with the internal energy dissipation in the fracture process, is stated. The long-term strength criterion related to the rheological stability of the material is used. This criterion defines the conditions under which a weak discontinuity of the solution becomes a strong discontinuity, i.e., the strain localization occurs. The effect of "yield drop" is revealed for small strain rates. It is shown that the rate of loading considerably influences the strength properties of the material. |
| References |
| 1. | J. A. Lemaitre, Course on Damage Mechanics, Springer,
Berlin, 1992. |
| 2. | V. N. Kukudzhanov, "A micromechanical model of fracture of
an inelastic material and its application to
the analysis of strain localization," Izv. AN. MTT [Mechanics of
Solid], No. 5, pp. 72-87, 1999. |
| 3. | L. M. Kachanov, Fundamentals of Fracture Mechanics [in
Russian], Nauka, Moscow, 1974. |
| 4. | Yu. N. Rabotnov, Mechanics of Solids [in
Russian], Nauka, Moscow, 1979. |
| 5. | G. I. Kanel, S. V. Razorenov, A. V. Utkin, and
V. E. Fortov, Shock Wave Phenomena in Condensed Media [in Russian],
"Yanus-K", Moscow, 1996. |
| 6. | V. I. Kondaurov, "Damage of nonlinearly elastic bodies,"
PMM [Applied Mathematics and Mechanics], Vol. 52,
No. 2, pp. 302-310, 1988. |
| 7. | V. I. Kondaurov and L. V. Nikitin, "A model of damage of
viscoelastic bodies," Izv. AN SSSR. MTT [Mechanics of
Solids], No. 3, pp. 131-139, 1989. |
| 8. | A. A. Griffith, "The phenomena of rupture and flow in
solids," Phil. Trans. Soc. London. Ser. A, Vol. 221, pp. 163-198,
1920. |
| 9. | A. A. Il'yushin, "On a theory of long-term strength,"
Inzh. Zh. MTT, No. 3, pp. 21-35, 1967. |
| 10. | L. I. Sedov, Continuum Mechanics. Volume 2 [in Russian],
Nauka, Moscow, 1970. |
| 11. | V. I. Kondaurov, "On the specific features of the waves of fracture
in highly homogeneous brittle materials,"
PMM [Applied Mathematics and Mechanics], Vol. 62,
No. 4, pp. 704-714, 1998. |
| 12. | V. S. Nikiforovskii and E. I. Shemyakin, Dynamic Fracture
of Solids [in Russian], Nauka, Novosibirsk, 1979. |
| 13. | V. N. Kukudzhanov, Propagation of Elastoplastic Waves in
a Rod Taking into Account the Influence of the Deformation Rate [in
Russian], Izd-vo VTs AN SSSR, Moscow, 1967. |
| 14. | Yu. V. Suvorova and M. B. Akhundov, "Long-term fracture
of an isotropic medium in the conditions of complex stress state,"
Mashinovedenie, No. 4, pp. 40-46, 1986. |
| 15. | C. Truesdell, A First Course in Rational Continuum Mechanics
[Russian translation], Mir, Moscow, 1975. |
|
| Received |
23 May 2001 |
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