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1. Introduction
Fracture mechanics is a field of study that deals with the behavior of materials when subjected to stress or strain. The fundamental concept of fracture mechanics is that the failure of a material can be predicted by analyzing its internal stresses and strains. The three-dimensional fracture problem deals with the behavior of materials under complex loadings, where the stresses and strains are not constant in all directions. The aim of this paper is to review the various methods used to model the three-dimensional fracture problem with a special focus on the displacement discontinuity method.
2. Literature Review
There are various methods used to model the three-dimensional fracture problem, some of which include analytical, numerical and experimental methods. Analytical methods involve solving the governing equations of the problem using mathematical techniques, such as the boundary element method, finite element method, and integral equations. However, analytical methods are limited to simple geometries due to the complexity of the equations involved.
Numerical methods involve discretizing the domain into small elements and solving the governing equations on each element using numerical techniques, such as the finite element method, boundary element method, and meshless methods. While numerical methods can be used to simulate complex geometries, they require high computational resources and may suffer from numerical instabilities.
Experimental methods involve carrying out physical experiments on the material to observe its behavior under different loading conditions. However, experimental methods are expensive and time-consuming, and may not be feasible for some materials.
3. Displacement Discontinuity Method
The displacement discontinuity method is a numerical method used to model the three-dimensional fracture problem. The method involves separating the domain into two regions, namely the interior region and the fracture surface. The interior region is the region of the material that is not affected by the fracture, while the fracture surface is the region where the fracture occurs.
In the displacement discontinuity method, the displacement field is discontinuous across the fracture surface, which means that the displacement field on one side of the fracture is not the same as on the other side. The displacement field can be represented as a sum of two components, namely the continuous displacement field and the discontinuous displacement field.
The continuous displacement field represents the displacement field in the interior region, where the material is not affected by the fracture. It can be calculated using the finite element method or any other numerical technique. The discontinuous displacement field represents the displacement field across the fracture surface and can be calculated using the crack displacement method.
4. Crack Displacement Method
The crack displacement method is a numerical technique used to calculate the discontinuous displacement field across the fracture surface. The method involves assuming that the displacement field across the fracture surface can be represented by a set of displacement functions. The total displacement field can be expressed as a sum of these displacement functions multiplied by their respective coefficients.
The coefficient values can be obtained by enforcing the following two conditions:
(i) The displacement field must be continuous across the crack faces.
(ii) The stress field must be balanced at the crack faces.
The crack displacement method is a powerful tool for modeling the three-dimensional fracture problem as it can be used to simulate a wide range of fracture patterns, such as transverse, oblique and mixed modes of fracture.
5. Conclusion
The displacement discontinuity method is a powerful tool for modeling the three-dimensional fracture problem. The method involves separating the domain into two regions, namely the interior region and the fracture surface, and assuming that the displacement field is discontinuous across the fracture surface. The continuous displacement field can be calculated using the finite element method or any other numerical technique, while the discontinuous displacement field can be calculated using the crack displacement method. The crack displacement method is a powerful tool for modeling a wide range of fracture patterns and can be used to simulate complex geometries under different loading conditions.