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Relaxation in $$BV$$ versus quasiconvexification in $$W^{1,p}$$; a model for the interaction between fracture and damage. (English) Zbl 0847.73077
For the study of the connection between damage and fracture of a material sample, the authors discuss some improvements which could be made to the hypotheses used in the problem. They construct a mathematical model in the BV-space of the deformation fields, under the following assumptions: (a) the phenomenon is studied in a close vicinity of the fracture; (b) the material is only allowed to brutally drop from its healthy states to its damaged state; (c) the configurational forces need break atomic bonds, and the crack propagation is not affected by the damaging process; (d) the quasistatic evolution of both damage and fracture is governed by a yield criterion, in accordance with the brittleness of the material; (e) the above criterion is energetic; (f) both damage and fracture processes are irreversible. The adopted model is variationally studied. The potential energy having to be minimized has the form $\int_{\text{body}} W(\nabla u)dx + \lambda H^{N-1} (S(u))- \int_{\text{body}} f\cdot u dx,$ where $$u$$ is the deformation field, $$W(\xi)$$ is an “elastic-type” energy density, $$\lambda$$ is a dissipation rate, $$f$$ are body loadings, and $$S(u)$$ is the jump set of $$u$$.
As the solution of the above problem is not unique, the authors propose a selective criterion for the solution choice: the global stability. The quasistatic evolution is investigated at discretized times. A special attention is given to the stable damage and fracture evolution in a brittle elastic continuum.
Reviewer: S.Zanfir (Craiova)

MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74P10 Optimization of other properties in solid mechanics 74R99 Fracture and damage
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