Keywords: finite element method, scaled boundary finite element method, cohesive crack model, discrete crack propagation, concrete beams, local arc-length method.

This study develops an innovative method, which couples the finite element method (FEM) and the scaled boundary finite element method (SBFEM), to fully-automatically model cohesive discrete crack growth in quasi-brittle materials. The linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone (FPZ), while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a "shadow subdomain", are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The following conclusions are drawn from this study:

1. The semi-analytical nature of the SBFEM offers various benefits to automatic cohesive crack propagation modelling. It allows accurate stress intensity factors to be extracted directly from the analytical stress solutions without refining crack-tip meshes or using singular elements as needed in FEM. This makes the very simple remeshing procedure developed in the previous and present studies possible. The remeshing procedure usually leads to a slight increase in degrees of freedom (DOFs). The semi-analytical nature of SBFEM solutions also makes mesh mapping of state variables after remeshing extremely simple but very accurate, which contributes greatly to the numerical stability and convergence in solving the highly nonlinear equation system. The semi-analytical nature also allows a structure to be modelled by a small number of DOFs. This not only simplifies remeshing and mesh mapping, but also has positive effects on computational efficiency and numerical stability.
2. The special cohesive interface elements used in this study are found effective in modelling energy dissipation in the FPZ and useful in analysing the evolution of the FPZ. The Petersson's bilinear softening curve is found to be sufficient for accurate prediction of structural responses in concrete beams.
3. The local arc-length solver proves very powerful in tracing complex equilibrium paths characterised by strong snapback and local instability.
4. The LEFM criteria are able to predict satisfactory crack growth direction in concrete beams. The K_I>=0 criterion is able to determine crack propagation timing with good objectivity with respect to the crack incremental length. The "shadow-domain" method developed in this study to calculate K_I when cohesive tractions exist along the crack is effective and efficient.

The developed method thus provides a competitive alternative to the FEM- and XFEM-based cohesive crack propagation modelling approaches. More work is needed to extend this method to fully-automatically model multiple cohesive crack propagation.

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