Journal
INTERNATIONAL JOURNAL OF DAMAGE MECHANICS
Volume 20, Issue 4, Pages 618-652Publisher
SAGE PUBLICATIONS LTD
DOI: 10.1177/1056789510386852
Keywords
fracture; energy methods; nonlocal damage; stability; variational inequalities; finite elements
Categories
Funding
- French research network MoMaS (Modelisations Mathematiques et Simulations numeriques liees aux problemes de gestion des dechets nucleaires)
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In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In this article, we first formulate the quasi-static evolution problem for a general class of such damage models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. We construct homogeneous as well as localized damage solutions in a closed form and illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress-displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant material parameters, such as limit stress and internal length. Finally, the 1D analytical results are compared with the numerical solution of the evolution problem in a 2D setting.
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