Modal analysis of the dynamic crack growth and arrest in a DCB specimen

This paper discusses dynamic crack growth and arrest in an elastic double cantilever beam (DCB) specimen, simulated using the Bernoulli-Euler beam theory. The specimen is made from two different materials. The section of interest, where the dynamic crack growth takes place, is made from a material, the fracture energy of which will be denoted 2Γ₁. The initial crack grows slowly in a starter material with a fracture energy 2Γ₀ (Γ₀ ≥ Γ₁), while opposed displacements on both arms of the specimen are continuously increased. As the crack reaches the material interface at l = l_c, the loading displacement is instantly suspended, and the crack suddenly propagates through the test zone, until it stops at l = l_A. During this process, the energy 2Γ₁(l_A - l_c) is dissipated. The beam motion and the fracture process during the fast crack growth stage are investigated, based on the balance energy associated to the Griffith criterion. The motion equations are approximated using a modal decomposition up to order N of the beam deflection (the analysis has been performed up to N = 10 but in most cases N = 5 is sufficient to obtain an accurate solution). This process leads to a set of N second order differential equations whose unknowns are the mode amplitudes and their derivatives, and another equation the unknowns of which are the current crack length l(t), velocity l-.(t) and acceleration l-..(t). To demonstrate the accuracy of this method, it is first tested on a one dimensional peeling stretched film problem, with an insignificant bending energy. An exact solution exists, accurately approximated by the modal solution. The method is then applied to the DCB specimen described above. Despite the rather crude nature of the Bernoulli-Euler model, the results crack kinematics, and specially the arrest length, correspond well to those obtained by the combined use of finite elements and cohesive zone models, even for a few modes. Moreover, for the basic mode N = 0 (also referred to as Mott solution), even if the crack kinematics is not accurately reproduced, the prediction of the crack arrest length remains correct for moderate ratios. Some parametric studies about the beam geometry and the initial crack velocity are performed. The relative crack arrest l_A/l_c appears to be almost insensitive to these parameters, and is mainly governed by the ratio R =Γ₀/Γ₁ which is the key parameter to predict the crack arrest. (authors)