13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Hamiltonian analysis applied to the dynamic crack growth and arrest in a Double Cantilever Beam. Gilles DEBRUYNE,1,*, Radhi ABDELMOULA2 1 LaMSID-EDF-CEA, 1 avenue du Gal de Gaulle 92141 Clamart, France 2 Paris XIII University, LSPM ,Avenue J.B. Clément 93430 Villetaneuse, France * Corresponding author: gilles.debruyne@edf.fr Abstract The paper deals with the dynamic crack growth and arrest in an elastic DCB specimen, idealized by a Bernoulli-Euler beam. The surface energy of the material is 1 2Γ and the initial crack is blunted with the energy 0 1 2 2 Γ > Γ . The crack is then pushed ahead while the loading is frozen, the crack velocity and arrest length depending on the ratio 0 1 / R=Γ Γ . The analytical approach used to investigate the beam behaviour and the crack growth, is based on the Hamilton’s principle of stationary action, considering an approximate equation of motion, based on a N modes decomposition of the beam deflection . This process leads to a set of N second order differential equations where the unknowns are the mode amplitudes and their derivatives, coupled to a single equation exhibiting the current crack length ( )t l , velocity ( )t l& and acceleration ( )t&l . The results concerning the crack kinematics, particularly the arrest length, are in good accordance with those obtained by a Finite Element Model associated to a cohesive zone model. The method is then applied to a material with heterogeneous fracture properties, in particular with a distribution of small brittle flaws perturbating the crack kinematics. This method allows a large range of configurations with a low computational time. Keywords Crack arrest, Hamiltonian, Lagrange equations, Double Cantilever Beam, pops-in 1. Introduction The double cantilever beam (DCB) specimen is widely used for crack arrest investigations, and its geometry is suitable for one-dimensional analysis models. For dynamic crack growth in a finite solid, analytical solutions are uncommon. The first quantitative prediction for the kinematics of a fast propagating crack, using an energy balance, is due to Mott [1]. Whereas crack propagation problems with transient motion are often achieved only with Finite Element Models, some analytical procedures are available for one dimensional models as DCB specimens. We focus here on a bimaterial DCB specimen, with a high toughness part where the initial crack is blunted, welded to a low toughness section where the crack enters with a high velocity. Transient kinetic analyses may be also achieved with some analytical methods, such hyperbolic equations solved by a finite difference scheme applied to characteristic lines [2,3]. An original analytical way, based on solving Lagrange’s equation of a DCB specimen motion, considered as a Bernoulli-Euler beam, has been initiated by Burns and Webb [4], assuming a special form of the kinetic energy corresponding to a specific constant loading rate. Freund [5] has extended this approach for a general loading,
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