PSI - Issue 2_A

Timothy Crump et al. / Procedia Structural Integrity 2 (2016) 381–388 Crump / Structural Integrity Procedia 409 (2016) 000–000

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effects with a phenomenological opening-rate-dependent cohesive law, introduced on the main crack path only. This is rather than directly introducing cohesive surfaces to element boundaries at the meso-scale as with previous approaches; allowing the crack to cross elements, Xu and Needleman (2004). This paper presents a meso-scale approach to dynamic crack propagation, which incorporates a phenomenological rate-dependant cohesive zone model into the eXtended Finite Element Method for crack representation (XCZM). The approach combines the benefits of mesh independency of XFEM with a phenomenological cohesive law, which represents the velocity toughening nature of dynamic fracture. To resolve and integrate a propagating crack’s influence on a global model a quasi-explicit solver is used; treating the crack propagation implicitly while taking advantage of the efficiency of an explicit approach to resolve highly non-linear global dynamics due to the crack propagation, Doyen (2013). This approach is applied to a well-known experiment on a Double Cantilever Beam (DCB) made of glass (Homolite-100) and for which an analytical solution exists in 2D, Freund (1977) Kobayashi (1985) and Sampson (1982). 2. Modelling Approach The presentation of a crack via XFEM is independent of the mesh through the exploitation of Partition of Unity concept using level sets to define the crack tip and lips, as in Fig.2(a): ���� � ∑ � � �� � ��� � � ∑ � � �� � �������� � (4) The crack tip is representated by the interface between a fully adherent cohezive zone and the crack lips using the phenomenological rate-dependant cohesive law in Fig.2(b). The intially rigid cohezive law also has a critical coheizve length, l c , for which the the constitutive relation plays a role and is set by the elastic material properties of the solid: � � ��� � �� � � � � � � � � � ���� � � � (5) with a critical time step for stability being: �� � ���� � �� � ��� � � � ��� � � (6) where h min is smallest element size and c d is the dilation wave speed of the material.

(b)

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Fig. 2. (a) XFEM crack representation: triangles are enriched with discontinuous Heaviside function; (b) initially rigid phenomenological cohesive law.