It is a quite challenging subject to efficiently perform fracture and fatigue analyses for complex structures with cracks in engineering. To precisely and efficiently study crack problems in practical engineering, an iterative method is developed in this study. The overall structure which contains no crack is analyzed by the traditional finite element method (FEM), and the crack itself is analyzed using analytical solution or other numerical solutions which are effective and efficient for solving crack problems. An iteration is carried out between the two abovementioned solutions, and the original crack problem could be solved based on the superposition principle. Several typical crack problems are studied using the present method, showing very high precision and efficiency of this method when making fracture and fatigue analyses of structures.
The stress intensity factors (SIFs) characterize the singularity of the area near the crack tip, which are the most significant parameters for fracture mechanics. However, the lack of precise solutions of stress intensity factors has hindered the progress and application of fracture mechanics to the fracture and fatigue analyses for various kinds of structures. Once the SIFs are precisely and efficiently computed, the fatigue crack propagation rate of a cracked structure under cyclic load can be determined, and its fatigue life estimation can then be rationally made. So, the calculation of fracture mechanics parameters for arbitrary surface and embedded cracks in complex civil, mechanical, and aerospace structures remains an important task for the structural integrity assessment and damage tolerance analyses [
Since the traditional finite element method (FEM) uses simple polynomial interpolations in numerical analysis, it is unsuitable and unwise to simulate cracks and their fatigue growth with FEM, which is largely attributed to its high inefficiency and labor costing of approximating stress and strain singularities using polynomial FEM shape functions. Then, embedded singularity elements [
To overcome the inherent difficulty of FEM when solving crack problems, many other numerical methods have been proposed and studied during the past several decades. The traditional boundary element methods (BEM) and dual boundary element methods were developed in fracture analysis [
In this paper, an iterative method is employed to obtain the SIFs of structures with different crack configurations and examine the fatigue crack growth subjected to cyclic loading. Crack growth rates are determined by the Paris fatigue law. For damage tolerance evaluation, the crack growth paths and number of loading cycles are predicted. The validations of the current iterative method are illustrated by the comparison of numerical results with available analytical solutions as well as empirical observations.
The finite element method is generally unsuitable for modeling cracks, because polynomial shape functions are inefficient for approximating singularities, and the shaped function-based method involves complicated remeshing procedures. On the other hand, the iterative method is a highly accurate and efficient method for solving fracture mechanics problems. The basic idea is to model the uncracked global structure using finite elements with very crude meshes, and model the crack with more suitable methods, such as the analytical methods, integral equations, and SGBEMs. By iterating between the solution for a crack in an infinite domain (or in a finite subdomain) and the solution by finite element for the uncracked global structure, the iterative methods obtain solution for the original problem. Tractions at the crack’s surface are obtained by FEM, and the stresses at field boundaries are corrected through the iteration process. No remeshing is involved, which is a significant advantage compared to the traditional FEM-based methods.
For linear elastic materials, stress intensity factors are additive like stress, strain, and displacements, as long as the mode of loading is consistent, which can be expressed as follows:
So, tractions acting on the crack face can replace tractions that act on the boundary, as long as these two configurations result in the same SIFs, as shown in Figure
Schematic diagram of superposition principle in linear elastic fracture mechanics.
Consider a finite body containing initial cracks. The crack surface with no traction is denoted as
The iterative method involves two different solutions for solving the original problem. The first is denoted as
The basic principle of the iterative method for crack analysis.
For the problem of
For the given crack surface load
Subtract the solution for
Substituting Equations (
Collecting the terms containing
This equation can be used to solve for
Also, substituting Equation (
Rearrange these two equations:
These two equations can be rewritten as follows:
Equation (
Similarly, the following linear system can be obtained for the traction
To directly solve these equations involves computation of
The solution is as follows:
Since
Since this fixed-point iteration usually converges very quickly, the iterative method is efficient in solving crack-related problems for practical engineering.
If the crack is subjected to constant amplitude cyclic loading, a plastic zone will form at the crack tip. Usually, the plastic zone is quite small and it is buried in an elastic singularity zone, so the conditions at the crack tip are defined by the value of stress intensity factors. The following equation is often used to express the functional relationship for fatigue crack growth:
Figure
Typical fatigue crack growth behavior in metals.
The linear region in Figure
Based on Equation (
Based on the Paris fatigue law, the number of loading cycles required to make a crack grow from an initial length
Indeed, plasticity-induced crack closure may account for the effects of plate thickness on crack growth rates. Here, the simple Paris law is used to predict the fatigue growth rate. However, other models which account for the effect of plate thickness can also be incorporated in the framework of the current iterative method, which will also be our future study. Moreover, some 3D effects of fracture mechanics are also neglected in this study, such as 3D corner singularity and mode II and III coupling, which are expected to have an insignificant effect for the damage tolerance of the cracked structures.
In this section, several typical numerical examples will be studied by the iterative method. For all of these examples, the material properties are Young’s elastic modulus
where
A finite plate with an edge crack subjected to a shear load is studied in this example. The size of the plate is
An edge crack in a finite plate under shear load.
After meshing the model in Patran, exporting it into a file and running the computational code, the computed stress intensity factors are listed as follows:
STRESS INTENSITY FACTORS EMBEDDED CRACK NO. 1 CRACK TIP A (-.4000E+01, 0.0000E+00) K_I=0.74935E+02 K_II=-0.76000E+01 CRACK TIP B (0.0000E+00, 0.0000E+00) K_I=0.33674E+02 K_II=0.13913E+01 Operation time: 0.2496016 seconds
The result is
This example studies the problem of three parallel cracks in a finite plate aligned normally to the tensile direction, which is shown in Figure
A finite plate with three parallel cracks subjected to tensile loading. (a) The geometric model. (b) The numerical model.
The computed SIF for
A finite plate with a slanted crack is studied in this example. The dimension and load are shown in Figure
A finite plate with a slanted crack subjected to tension load. (a) The geometric model. (b) The numerical model.
Here, fatigue crack growth is simulated by employing the Paris fatigue law, and the parameters for Equation (
Plot of the fatigue crack growth. (a) The overall plot. (b) The magnified plot, after 457178 fatigue load cycles in 10 steps.
Figure
In this example, a cantilever beam with the size of
Numerical model of the cantilever beam with an eccentric crack.
The shape of the eccentric crack after fatigue growth.
In this example, a circular fastener hole with two edge cracks at its left and right side is considered. As shown in Figure
A finite plate with edge cracks emanating from a circular fastener hole. (a) The geometric model. (b) The numerical model.
The shape of the cracks growing from a fastener hole after fatigue growth of 4939 loading cycles.
Figure
Assumed initial configurations of cracks emanating from loaded fastener holes.
The fatigue-related parameters are
Mesh of the plate with the two holes.
The propagation of the cracks is shown in Figure
Crack propagation for 80272 cycles in 50 steps. (a) The overall view. (b) The magnified view.
Stress intensity factors for crack mode I and mode II.
In this paper, based on superposition principle, the iterative method was developed and used to obtain the stress intensity factors and simulate fatigue crack propagation of various kinds of structures with different initial cracks. Several typical numerical examples were illustrated in the paper. The following conclusions can be drawn by carefully comparing the numerical results with analytical solutions as well as empirical observations available in the open literature.
The iterative method requires independent and very coarse meshes for both the uncracked global structure and the crack; it only requires very little computational burden and human labor cost for modeling the fatigue propagation of various kinds of cracks. The computed stress intensity factors for cracks using this method are in good agreement with analytical solutions or empirical solutions. The whole crack growth path up to failure of the structures can be easily and efficiently simulated using this method.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The research described in this paper was financially supported by the National Natural Science Foundation of China (grant number 51808114), the Natural Science Foundation of Jiangsu Province (grant number BK20170670), and the Fundamental Research Funds for the Central Universities (grant number 2242018K40143).