Predicting the Structural Performance of Composite Structures Under Cyclic Loading

Abstract

The increased use of advanced composite materials on primary aircraft structure has brought back to the forefront the question of how such structures perform under repeated loading. In particular, when damage or other stress risers are present, tests have shown that the load to cause failure after a given number of cycles is a decreasing function of these cycles. This is a result of damage that was already present in the structure or was created during cyclic loading. In composites, multiple types of damage may be present in the structure at the same time such as matrix cracks, fiber kinks, delaminations, broken fibers, etc. These types of damage may interact and transition from one type to another and are, ultimately, responsible for structural failure. In trying to predict the number of cycles to failure of a composite structure it is, therefore, necessary to understand how damage is created, how it evolves and how different types of damage may interact or coalesce. A first step in that direction, using what is one of the simplest models that can be used, is the subject of this thesis. The number of cycles to failure is related to the residual strength of the structure for constant amplitude loading. A simple first-order model is postulated that determines the residual strength at any point during the fatigue life as a function of the residual strength at any earlier point in time. For constant amplitude loading, the resulting expression relates the maximum applied load, the number of cycles, the cycles to failure corresponding to the applied load, and the residual strength at the beginning of a test, to the residual strength at the end of the test. With the residual strength known as a function of cycles, a cycle-by-cycle probability of failure is introduced. It is shown that, if the static (or residual) strength follows a two-parameter Weibull distribution, the cycle-by- cycle probability of failure is constant and independent of the number of cycles. For the case of constant cycle-by-cycle probability of failure, the number of cycles to failure is determined as the value that maximizes the likelihood of failure. The resulting expression is in terms of the cycle-by-cycle probability of failure. If the residual strength distribution is known, the cycles to failure can be expressed in terms of parameters of this distribution. Simple closed-form expressions are obtained for two-parameter Weibull distributions. For other types of distributions (normal or lognormal for example) no closed form expressions were found. The effect of R ratio is incorporated using a simple proportional relation that accounts for the load excursion being different from that for R=0. The predictions of this approach for constant amplitude loading situations were compared to test results in the literature for a wide variety of laminates, materials, and loading conditions. While in some cases the agreement of test results with predictions was excellent, in others the discrepancy clearly suggested that the analytical model must be improved. The analytical model was also used to construct Goodman diagrams and determine omission levels for tests. Comparison of analytically predicted Goodman diagrams to test results showed good agreement in the tension-dominated portion of the diagram but some disagreement in the compression-dominated portion. This is attributed to the simplicity of the model which does not accurately capture interaction of failure modes when both tension and compression loads are present. The omission level is the load level below which no damage is created, no growth of existing damage is observed, and no failure occurs for a prescribed number of cycles. This allows shortening of test programs by eliminating cycles with loads below the omission level. Comparisons of predictions to test results showed very good agreement over a wide variety of tests, materials, R ratios, notches, and layups. The model, in its simplest form, was then extended to spectrum loading cases. This was done by creating an equivalence between different load levels and applied cycles by matching the residual strength at the end of each load level. For this approach to work, the failure mode and damage type dominating the fatigue life must be the same for the two (or more) load segments of interest. This then allows a single quantity, the residual strength, to accurately describe the damage state. Simple closed form expressions were obtained for the number of cycles or load segments to failure under spectrum loading. Comparisons with test results showed good agreement for tension-dominated spectra but major discrepancies for compression-dominated spectra again pointing to the need for improving the model to account for interaction of multiple failure modes and types of damage. The main reason for the discrepancies between test results and analytical predictions was the constant cycle-by-cycle probability of failure that resulted from the original assumptions in the model. If there is one dominant failure mode the cycle-by-cycle probability of failure is constant. However, when more than one types of damage or failure modes are present, their interaction and the resulting load redistribution in the structure changes the cycle-by-cycle probability of failure. The model was, therefore, modified by assuming that the probability of failure is constant over a limited number of cycles until another failure mode or damage type occurs and changes the residual strength and the cycle-by-cycle probability of failure. This can become quite complex even for the apparently simple case of a uni-directional laminate under tension where, during cyclic loading, weak fibers fail and their load is redistributed to adjacent fibers. The main difficulty is then in creating an analytical model that can accurately determine stresses throughout the structure as damage evolves and, on the basis of these stresses, predict the residual strength. The improved model was applied to two cases, a uni-directional laminate and a cross-ply laminate of the form [0m/90n]s under tension-tension fatigue. For the uni-directional laminate, the improved predictions for cycles to failure were in excellent agreement with test results. For the cross-ply laminate, the accuracy of the predictions ranged from excellent to poor depending on the ratio of the thickness of internal 90o plies to that of the surrounding 0o plies. The main issue in this case is that the analytical model developed for predicting stresses around matrix cracks and the associated load redistribution in the laminate are not very accurate as the crack density increases beyond a certain point. More accurate analytical modeling of this situation is expected to improve the predictions for cycles to failure. The analysis method proposed here is still in its infancy. In its simplest form, it is shown to work well in many cases but not well in others. What is important is that a framework for performing fatigue analysis of composites is presented, which relies on the residual strength and how that varies with cycles as damage is created and evolves. Essentially, what is proposed here is a wear-out model. Wear-out models have been proposed before. The main difference and potential improvement here is that there is no need for curve fitting test data or experimentally determined fatigue parameters. The equations governing the model are determined analytically and, in some cases, in closed form. While the model needs further improvements mainly in how the creation of different types of damage is predicted and how their interaction and evolution is accounted for, it is very promising because it provides a general and purely analytical methodology to predict cycles to failure under constant amplitude or spectrum loading.