Mixed mode partition using numerical methods

2.3.3 Mixed mode partition using numerical methods

Virtual crack closure technique (VCCT) and cohesive interface element model are commonly used in the finite element analysis of delamination features. The two methods employ similar principle, and they simulate the crack propagation by the energy release rate criterion. The principle of VCCT is based on classical fracture mechanics, which investigates the behaviour of crack propagation when an initial crack is designated. While cohesive element is a numerical model based on damage mechanics, in which the stiffness of the element decrease when passing a critical stress, until a complete failure is reached, and then the bonding

22 element is eliminated to simulate the propagation of a fracture. Therefore, the entire process of crack initiation to propagation could be analysed by cohesive models.

The crack closure method is based on Irwin’s crack closure integral [35]. The method is based on the assumption that the energy released when the crack is extended by a is identical to the energy required to re-close the crack by that same distance. This method needs two complete FEM simulations (before and after crack extension). The VCCT is based on the same assumptions as the crack closure method. However, in addition, it is assumed that a crack extension ofadoes not significantly alter the state at the crack tip. As a result of this assumption, the ERR partition can be calculated with one FEM simulation only using the VCCT.

In recent years, Ronald Krueger [62-65] carried out series of the benchmark assessment of automated delamination propagation capability in finite element codes for standard software Abaqus, Ansys and MD Nastran and Marc. However, the assessments are mainly based on DCB, ENF and MMB beams, the feasibility of using these standard FEA software for specific problem, like buckling driven delamination, mixed mode partition for bi-material interface is not very clear. At least further validation works are required before using these standard FEA codes.

Rybicki and Kanninen [66] seems to be the first to evaluate of both mode I and mode II stress intensity factors from the results of a single analysis. The method does not use stresses, the conventional constant strain elements have been used, and a coarse grid near the crack tip was found to be sufficient. The axial and vertical forces at the crack tip were obtained by placing very stiff springs between adjacent points and evaluating the forces in these springs.

The better results are obtained by using four-node quadrilateral, non- singular elements.

It is convenient to maintain the same size for the elements, of which nodal force and displacement are used. If this is not the case, then a modification to handle this case is needed. At first, the VCCT was used to calculate energy release rate mode I and mode II.

Then mode I and mode II stress intensity factors are calculated from energy release rate.

The relationship between G and the stress intensity factor for an isotropic material in plane strain is established using Irwin’s crack closure integral [35],

2 2 2

2) (1 )

1 (

II

I K

K E

G E   (2.63)

23 The accuracy to predict strain energy release rate for bi-material interface depends on the mesh density applied to crack propagation area. From Raju’s study [61], the individual energy release rate did not show convergence as the delamination tip elements were made smaller. In contrast, the total strain energy release rate, G, converged and remained unchanged as the delamination tip elements were made smaller and agreed with the total G analytical calculated.

For two different anisotropic materials, the singularity is not the classical square root singularity but is of the form r^{ 21/ i}, where r is the radial distance measured from the delamination tip. The  depends on the material properties of the two materials. This imaginary power leads to the stress oscillations very close to the delamination tip. The oscillatory component of the singularity may cause the non-convergence of the individual G components. In the finite element analysis, this means that the computed mode I and mode II strain energy release rates will be dependent on the crack tip element size and do not show convergence as the crack tip elements are made smaller.

B. Dattaguru, et al. performed another convergence study for simulate mode I and mode II energy release rate for bi-material interface delamination [67]. The strain energy release rate components GI and GII in mode I and mode II at the tip of an interface crack in a bi-material plate under tension in a direction normal to the interface were evaluated using finite element analysis. The strain energy release rate components GI and GII are calculated at the crack tip.

The results show that GI and GII are likely to show an oscillatory trend at infinitesimally smalla /a. Based on the results obtained, it appears that such oscillations are likely to occur only when the virtual crack extension proposed a is less than the contact zone size r. Figure 2-4 shows the changes of GI, GII and total GT with reducing the mesh size as the mesh size at crack tip becomes smaller, aa/ 0, the GII is close to total GT.

24 Figure 2-4: ERR convergence of an edge cracked bi-material plate subjected to tension [67].

The difficulty with the convergence of the strain energy release rate components GI and GII as 0

 aa/  led Raju [61] to consider a finite thickness adhesive layer at the interface with the crack at the centre of the adhesive layer. The comparison energy release rates done by Raju [61] for the 'bare' interface laminate, i.e. one without the resin layer, and for the laminate with the resin showed that the 'bare' interface models are a very good approximation for the resin case if the delamination tip elements were one-quarter to one-half of the ply thickness.