Energy minimising multi-crack growth in linear-elastic materials using the extended finite element method with application to Smart-CutTM silicon wafer splitting

Doctoral thesis (2016)

We investigate multiple crack evolution under quasi-static conditions in an isotropic linear-elastic solid based on the principle of minimum total energy, i.e. the sum of the potential and fracture ... [more ▼]

We investigate multiple crack evolution under quasi-static conditions in an isotropic linear-elastic solid based on the principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith’s theory of cracks. The technique, which has been implemented within the extended finite element method, enables minimisation of the total energy of the mechanical system with respect to the crack extension directions. This is achieved by finding the orientations of the discrete crack-tip extensions that yield vanishing rotational energy release rates about their roots. In addition, the proposed energy minimisation technique can be used to resolve competing crack growth problems. Comparisons of the fracture paths obtained by the maximum tension (hoop-stress) criterion and the energy minimisation approach via a multitude of numerical case studies show that both criteria converge to virtually the same fracture solutions albeit from opposite directions. In other words, it is found that the converged fracture path lies in between those obtained by each criterion on coarser numerical discretisations. Upon further investigation of the energy minimisation approach within the discrete framework, a modified crack growth direction criterion is proposed that assumes the average direction of the directions obtained by the maximum hoop stress and the minimum energy criteria. The numerical results show significant improvements in accuracy (especially on coarse discretisations) and convergence rates of the fracture paths. The XFEM implementation is subsequently applied to model an industry relevant problem of silicon wafer cutting based on the physical process of Smart-CutTM technology where wafer splitting is the result of the coalescence of multiple pressure-driven micro-crack growth within a narrow layer of the prevailing micro-crack distribution. A parametric study is carried out to assess the influence of some of the Smart-CutTM process parameters on the post-split fracture surface roughness. The parameters that have been investigated, include: mean depth of micro-crack distribution, distribution of micro-cracks about the mean depth, damage (isotropic) in the region of micro-crack distribution, and the influence of the depth of the buried-oxide layer (a layer of reduced stiffness) beneath the micro-crack distribution. Numerical results agree acceptably well with experimental observations. [less ▲]

Minimum energy multiple crack propagation. Part III: XFEM computer implementation and applications.

in Engineering Fracture Mechanics (n.d.)

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼]

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. The key contributions of Part-III of the three-part paper are as follows: (1) implementation of XFEM in Matlab with emphasis on the design of the code to enable fast and efficient computational times of fracture problems involving multiple cracks and arbitrary crack intersections, (2) verification of the minimum energy criterion and comparison with the maximum tension criterion via multiple benchmark studies, and (3) we propose a numerical improvement to the crack growth direction criterion that gives significant improvements in accuracy and convergence rates of the fracture paths, especially on coarse meshes. The comparisons of the fracture paths obtained by the maximum tension (or maximum hoop-stress) criterion and the energy minimisation approach via a multitude of numerical case studies show that both criteria converge to virtually the same fracture solutions albeit from opposite directions. In other words, it is found that the converged fracture path lies in between those obtained by each criterion on coarser meshes. Thus, a modified crack growth direction criterion is proposed that assumes the average direction of the directions obtained by the maximum tension and the minimum energy criteria. The numerical results show significant improvements in accuracy (especially on coarse discretisations) and convergence rates of the fracture paths. Finally, the open-source Matlab code, documentation, benchmarks and example cases are included as supplementary material. [less ▲]

Minimum energy multiple crack propagation. Part II: Discrete Solution with XFEM.

in Engineering Fracture Mechanics (n.d.)

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and ... [more ▼]

The three-part paper deals with energy-minimal multiple crack propagation in a linear elastic solid under quasi-static conditions. The principle of minimum total energy, i.e. the sum of the potential and fracture energies, which stems directly from the Griffith's theory of cracks, is applied to the problem of arbitrary crack growth in 2D. The proposed formulation enables minimisation of the total energy of the mechanical system with respect to the crack extension directions and crack extension lengths to solve for the evolution of the mechanical system over time. The three parts focus, in turn, on (I) the theory of multiple crack growth including competing cracks, (II) the discrete solution by the extended finite element method using the minimum-energy formulation, and (III) the aspects of computer implementation within the Matlab programming language. This Part-II of our three-part paper examines three discrete solution methods for solving fracture mechanics problems based on the principle of minimum total energy. The discrete solution approach is chosen based on the stability property of the fracture configuration at hand. The first method is based on external load-control. It is suitable for stable crack growth and stable fracture configurations. The second method is based on fractured area-control. This method is applicable to stable or unstable fracture growth but it is required that the fracture front be stable. The third solution method is based on a gradient-descent approach. This approach can be applied to arbitrary crack growth problems; however, the gradient-descent formulation cannot be guaranteed to yield the optimal solution in the case of competing crack growth and an unstable fracture front configuration. The main focus is on the gradient-descent solution approach within the framework of the extended finite element discretisation. Although a viable solution method is finally proposed for resolving competing crack growth in the case of an unstable fracture front configuration, the method is not implemented within the present XFEM code but rather exists as a separate proof-of-concept algorithm that is tested against several fabricated benchmark problems. The open-source Matlab code, documentation and example cases are included as supplementary material. [less ▲]