MÉTODO A VALIDAR Resultado

The final problem is the most complicated where a tree-shaped crack with up to ten crack tips is included. The configuration of the problem is in Figure 4.20 where w = 6mm, h = w, a = b = 2c = 1mm, α = 45◦ _{and β = 90}◦_{, the same as in [258]. A biaxial}

tensile loading σ = 100MPa is applied at four edges of the plate, and an initial particle arrangement is set as in Figure 4.21 (a).

Adaptive particle arrangements are shown in Figures 4.21 (b-d), and the number of particles goes up to 848 from 262. Fine particles are generated around the four crack tips on the left and right sides while the requirement of refinements for the other six is low. Calculating the SIFs at four crack tips A, B, C and D marked in Figure 4.20, converged results can be obtained after adaptive steps, Figure 4.22, where the superscripts indicate the crack tip, and the results are consistent with the results from Ma et al. [258], although minor differences for KII at crack tip A are found. In this calculation, KII at crack tip

Figure 4.20: Configuration of the tree-shaped crack in a square plate.

Figure 4.21: Adaptive steps for the tree-shaped crack problem: (a) initial particles; (b) step 2; (c) step 4; (d) step 6.

and II SIFs at crack tips B and C are much smaller than at crack tips A and D, which explains why the refinement level at crack tips B and C is much lower than at crack tips A and D. The deformation of this problem is presented in Figure 4.23, from which it can be seen that some cracks are opening and some are closing. There are some non-physical results for the closing cracks as shown in Figure 4.23 (b), because the current algorithm

4.4. Numerical examples 103

Figure 4.22: Validation of calculated SIFs for the tree-shaped crack problem: (a) KA I ; (b)

0 2 4 6 8 10 12 0 2 4 6 8 10 12 14 (a) 3 4 5 6 7 8 9 4 5 6 7 8 9 (b)

Figure 4.23: Displacement enlarged by 50 times for the tree-shaped crack problem: (a) displacement; (b) enlarged view.

Figure 4.24: Configuration of tree-shaped crack in a double cantilever beam.

does not cover contacts between two surfaces.

It is a more challenging problem for the tree-shaped crack to start from the edge, e.g. in a double cantilever beam as in Figure 4.24. The configuration of the beam is w = 24mm and w/h = 3, and the crack pattern is kept the same as above. The beam is fixed at the right side and is loaded under a pair of tractions σ = 100MPa at the left edge.

Adaptive particle arrangements are presented in Figure 4.25, where fine particles are generated automatically around all ten crack tips while no refinement occurs on the right part of the beam. The final number of particles increases to 3359 from 283, which is about 11 times larger than the initial number. The final deformation is illustrated in Figure 4.26, including an enlarged view of the rectangular zone. A clear crack opening

4.4. Numerical examples 105

Figure 4.25: Adaptive steps for tree-shaped crack in a double cantilever beam: (a) initial particles; (b) step 2; (c) step 4; (d) step 6.

Figure 4.26: Crack opening for the tree-shaped crack in a double cantilever beam enlarged by 20 times: (a) overall deformation; (b) partial enlarged view.

can be seen at the three crack branches on the right; while for the two branched cracks, only one part is opened; and for two single cracks small crack openings are detected. Extracting the SIFs at crack tips A, B, C and D, converged values are obtained as given in Figure 4.27. It can be seen that KA

I and KID are almost three times larger that KIB,

while KC

I is much smaller, which explains why the crack opening is large at crack tips A

and D and why the deformation at crack tip C is negligible. KA

II is nearly zero, and KIIB

and KD

Figure 4.27: Calculated SIFs for the tree-shaped crack in a double cantilever beam: (a) K_IA; (b) K_IIA (c) K_IB; (d) K_IIB; (e) K_IC; (f) K_IIC; (g) K_ID; (h) K_IID

4.5. Summary 107

B and D are dominated by the mode I type fracture. No reference has been found till now to the author’s knowledge so the accuracy regarding the final SIFs cannot be fully assessed, however this example has shown the ability of the modified CPM to handle a complex crack problem.

It is notable that the proposed methodology is able to handle the simulation of crack coalescence, e.g. one crack propagating into and splitting another, by adding one particle at the intersection position and subsequently splitting it according to the crack patterns. But in this type of situation, there will be issues in the calculation of the J-integral when two crack tips are in close proximity, which is not covered in this chapter.

4.5

Summary

A multi-split particle method is proposed for the simulation of complex cracks and mul- tiple cracks in 2D. In this methodology, discontinuities at cracks are achieved by the modification of the influence domain of particles, and the displacement approximation involves no enrichment function and therefore is kept simple. The cracking particle at a crack branch point is multiply split into several subparticles, and the influence domain of this particle is also divided. The crack opening at the crack branch point is obtained by the relative displacements of these subparticles. Crack geometries are described by a set of crack segments rather than using level set functions, so the computational cost for modelling multiple cracks is controlled. An adaptivity strategy developed in Section 3.3 is introduced here to amend the density of particle distribution, by which the cal- culation efficiency is improved. Finally, several examples of multiple cracks including two independent cracks, a cross crack, a star-shaped crack and a tree-shaped crack are used to demonstrate the ability of the proposed methodology, where good agreement with previous results has been obtained.

Configurational-force-driven cracking

particle method for 2D problems

5.1

Introduction

In Chapters 3 and 4, crack propagation was determined by the maximum circumferential stress criterion, however there are other options developed for controlling the behaviour of crack growth. This chapter comprises a discussion and study of different criteria for crack propagation in the numerical modelling of fracture. The problem of crack initiation is not included, in that a new crack induces a severe topological change rarely supported by current numerical methods.

Accurate determination of the mechanism of crack propagation in mixed-mode sit- uations is of great importance in fracture mechanics, since the correctness of a crack propagation criterion is directly linked to the accuracy of crack growth prediction. A crack propagation criterion must provide answers to the following two questions: whether a crack is going to propagate and in which direction? These two questions are checked at each time of load step for a successive crack propagation process. Some crack prop- agation criteria are based on the local stress and displacement field at the crack tip, e.g. the maximum circumferential stress criterion [259] and the minimum strain energy density criterion [260], while others follow a global approach accounting for the energy distribution throughout the cracked part, e.g. the maximum strain energy release rate criterion [261]. In addition, the concept of the configurational force (CF) has been used