Abstract
AbstractThis paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method. More precisely, we are interested in explicit algorithms. We prove that by using a specific lumping technique, the critical time step is exactly the same as if no crack were present. This somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack. The new lumping technique is obtained by using a lumping strategy initially developed to handle elements containing voids. To be precise, the results obtained are valid only when the crack is modelled by the Heaviside enrichment. Note also that the resulting lumped matrix is block diagonal (blocks of size 2 × 2). For constant strain elements (linear simplex elements) the critical time step is not modified when the element is cut. Thanks to the lumped mass matrix, the critical time step never tends to zero. Moreover, the lumping techniques conserve kinetic energy for rigid motions. In addition, tensile stress waves do not propagate through the discontinuity. Hence, the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack. Both these techniques will be used in a numerical experiment. Copyright © 2007 John Wiley & Sons, Ltd.
Highlights
The extended finite element method (X-FEM) allows one to introduce a crack within an existing mesh without the need to modify it
According to the original papers on the subject, the enrichment is composed of a tip enrichment [2, 3] and a Heaviside enrichment [4]
Normalized ∆tc X-FEM: diagonal mass Normalized ∆tc X-FEM: block-diagonal mass but it has to be noted that the use of the block-diagonal mass matrix allows one to obtain a bigger critical time step
Summary
The extended finite element method (X-FEM) allows one to introduce a crack within an existing mesh without the need to modify it. The use of an objected-oriented language such as C++ is quite convenient in dealing with the variable number of element degrees of freedom (in space and time as the crack is growing). The stability of X-FEM with implicit algorithms was studied in [12] and with explicit algorithms in [13, 14]. The latter pap√er [14] introduces a special lumping technique leading to a critical time step smaller (factor 1/ 2) than the one in the FEM case. In this paper we shall introduce a new lumping technique allowing one to use the same critical time step as in FEM (considering only Heaviside enrichment).
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