Abstract :
Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation
in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan,
Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York,
1968, pp. 498–586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990;
D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow.
We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described
by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the
displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position
derived from the energy balance law. We handle both equations separately, assuming at first that the
crack position is known. Then the weak and strong solvability of the wave equation will be studied and the
crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially.
Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an
ordinary differential equation for the motion of the crack tip.
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