On the stochastic fatigue crack growth problem
Abstract
The research focuses on continuous and discrete stochastic models for fatigue crack growth which are based on Markov process theory. These models account for the random nature of fatigue crack growth which is not adequately explained by a deterministic approach. A hybrid finite element/finite difference solution methodology is developed and shown to be highly effective in determining the solution of the backward Kolmogorov equation and the PontryaginVitt equation yielding the probabilistic description of the time to reach a critical crack size as a function of the initial crack size. Excellent comparisons are shown between this method, previous analytical studies, and experimental results. A significant reduction in computer processing time and storage is achieved with this approach. Alternatively, the forward FokkerPlanckKolmogorov equation is formulated, and a twodimensional initial boundary value problem developed, to determine the distribution of crack sizes as a function of time. A twodimensional finite element solution approach is used for problem solution. A major advantage of this problem formulation is that the entire probability density function is obtained as a function of cycle number. Studies of discrete Markov process models are also considered for the characterization of fatigue crack growth. A celltocell mapping approach, which has been effectively utilized for other twostate problems in stochastic dynamics, is developed for the stochastic fatigue crack growth problem. In this approach the transitional probability matrix for crack transition from cell i to any other cell is determined using simulation with a twostate lognormal random process model. Repeated matrix multiplication is then used to determine the distribution of crack lengths at other times for a given initial flow size distribution. The effect of varying the initial fatigue quality may be evaluated without repeating the simulation of the probability transition matrix. Modifications are considered for improved solution performance. Lastly, a random load model for stochastic fatigue crack growth is presented and a boundary value problem formulated for the determination of the random time for a crack to grow to a critical length.
 Publication:

Ph.D. Thesis
 Pub Date:
 1991
 Bibcode:
 1991PhDT........14E
 Keywords:

 Boundary Value Problems;
 Crack Propagation;
 Cracks;
 Fatigue (Materials);
 Finite Difference Theory;
 Finite Element Method;
 Kolmogoroff Theory;
 Markov Processes;
 Probability Theory;
 Entire Functions;
 Flow Distribution;
 Length;
 Matrices (Mathematics);
 Multiplication;
 Random Loads;
 Random Processes;
 Repetition;
 Simulation;
 Size Distribution;
 Time Dependence;
 Structural Mechanics