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Crack dynamics via Lagrange's equations and generalized coordinates

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Cracks that are subjected to a sudden, sufficiently large, increase in stress respond with a combination of opening and unstable growth. This behavior is idealized here by representing the opening and growth as two generalized coordinates and determining the corresponding Lagrangean. Then the equation of motion are established for an arbitrary time-dependent driving pressure. Lagrange's equations are especially appropriate for systems involving dissipation, nonholonomic constraints, and other such processes that can not be accounted for in a Hamiltonian formulation. It is shown that the standard results for crack opening and stability are consequences of the Lagrange formulation, as is an algebraic expression for crack speed. Of course, the solution for the crack dynamics is approximate, but solving the full PDEs for the unstable cracks in a brittle structure (∼10,000 cracks/cc) is not practical, nor would it be appropriate since the details of the cracks vary between samples. Thus, the Lagrange formulation makes it possible to analyze brittle materials containing an ensemble of cracks in an efficient manner. In an example the ODEs are integrated numerically and it is shown that the results are consistent with analytic results. In that work the surface energy is considered constant, but when the stresses are modest, crack growth is governed by creep processes. It is shown that this slow growth can be accounted for by letting the surface energy depend on the stress-intensity factor in a manner that is based on crack-speed data.

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  1. J. K. Dienes

    Present address: Theoretical Division, Los Alamos National Laboratory, 87545, Los Alamos, NM, U.S.A

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    Dienes, J.K. Crack dynamics via Lagrange's equations and generalized coordinates. Acta Mechanica 148, 79–92 (2001). https://doi.org/10.1007/BF01183670

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