Yu. Paramonov

A Probabilistic Model of the Fatigue Life of Composite Materials for Fatigue-Curve Approximations

A simple mathematical model for the fatigue life of composite materials in which the basic load is taken up by parallel rigid components (fiber strands) is offered. Based on the Markov chain theory, the model allows us to obtain an S-type curve for the growth of internal stresses and to calculate the relationship between the parameters of static strength distribution and the S-N fatigue curve (Wholer curve). The model is too simple and does not provide a numerical coincidence with experimental fatigue-test data, but, when used with additional parameters, it agrees with experiments and can be applied to the model of nonlinear regression in describing the fatigue curve and its variation as the parameters of static strength change.

Markov model of fatigue of a composite material with the poisson process of defect initiation

As a development of the model where only one weak microvolume (WMV) and only a pulsating cyclic loading are considered, in the current version of the model, we take into account the presence of several weak sites where fatigue damage can accumulate and a loading with an arbitrary (but positive) stress ratio. The Poisson process of initiation of WMVs is considered, whose rate depends on the size of a specimen. The cumulative distribution function (cdf) of the fatigue life of every individual WMV is calculated using the Markov model of fatigue. For the case where this function is approximated by a lognormal distribution, a formula for calculating the cdf of fatigue life of the specimen (modeled as a chain of WMVs) is obtained. Only a pulsating cyclic loading was considered in the previous version of the model. Now, using the modified energy method, a loading cycle with an arbitrary stress ratio is “transformed” into an equivalent cycle with some other stress ratio. In such a way, the entire probabilistic fatigue diagram for any stress ratio with a positive cycle stress can be obtained. Numerical examples are presented.

INSPECTION PROGRAM DEVELOPMENT BY THE USE OF APPROVAL TEST RESULTS

The use of p-set function and minimax approach is offered for inspection program development on a basis of lifetime approval test result processing. In this case, there is no need to resort to the dubious compromise choice of the required reliability and confidence probability.

Modelling the Strength and Fatigue Life of a Unidirectional Fibrous Composite by Using Daniels’ Sequence and Markov Chains

A review of the previous works of the authors dedicated to the use of Daniels’ sequence (DS) for analyzing the relation between the distribution of the static strength of components of a unidirectional fibrous composite (UFC) and the distribution of its fatigue life is presented. A generalization of the DS which can be used to analyze the association of distribution of the static strength of composite components with distribution of the static strength of the UFC itself is given. In analyzing the fatigue life of a UFC, unlike in Daniels’ model, the loading rate and randomness of the number of still workable components in the weak microvolume in which the destruction process takes place are taken into account. By analyzing the fatigue life, it is possible to explain the existence of the random fatigue strength and to calculate the maximum load at which the probability of absence of fatigue failure is great enough when the number of cycles of fatigue loading tends to infinity. Numerical examples of processing of experimental data are presented, and estimates for parameters of the corresponding nonlinear regression model, which can be interpreted as the strength parameters of UFC, are obtained.

Investigation of the reliability and longevity of parallel systems with defects using Daniels’ sequences

The Reliability and longevity of parallel systems is studied using the example of a fibrous composite material using Daniels sequences’ and the Markov chain theory. The connection between the strength distribution function of the separate longitudinal elements forming the parallel system and the longevity distribution function of this system under cyclic loads is analyzed. The obtained models do not give direct coincidence with the experimental data but can be used for their nonlinear regressive analysis. Numerical experimental data processing examples are given.

Estimating the parameters of fatigue curve of a composite material

By using the method of maximum likelihood, the parameters of two versions of a mathematical model for fatigue damage accumulation in a laminate are estimated. The models, which are founded on the Markov chain theory, are very simple: they do not take into account the specific structural features of a composite and therefore cannot provide numerical coincidence with experimental fatigue test data, but they can be used for a nonlinear regression analysis of fatigue curves. A simple method is offered for approximately estimating model parameters, some of which characterize the distribution of the local static strength. By using such models, we can predict the relative changes in fatigue curves from known relative variations in the parameters of static strength and also predict the distribution function of fatigue life in program fatigue tests.

Modeling the Residual Strength of a Fibrous Composite Using the Residual Daniels Function

The concept of a residual Daniels function (RDF) is introduced. Together with the concept of Daniels sequence, the RDF is used for estimating the residual (after some preliminary fatigue loading) static strength of a unidirectional fibrous composite (UFC) and its S–N curve on the bases of test data. Usually, the residual strength is analyzed on the basis of a known S–N curve. In our work, an inverse approach is used: the S–N curve is derived from an analysis of the residual strength. This approach gives a good qualitive description of the process of decreasing residual strength and explanes the existence of the fatigue limit. The estimates of parameters of the corresponding regression model can be interpreted as estimates of parameters of the local strength of components of the UFC. In order to approach the quantitative experimental estimates of the fatigue life, some ideas based on the mathematics of the semiMarkovian process are employed. Satisfactory results in processing experimental data on the fatigue life and residual strength of glass/epoxy laminates are obtained.

Decision procedures of automated system for inspection program development

Abstract Fatigue failure probability of airframe depends on unknown distribution parameters of fatigue crack growth model. The parameters should be estimated by processing of airframe full-scale fatigue test results. A definition of p-set function as specific statistical decision function is introduced. The use of the function, which defines some p-set for random vector, provides limitation of supremum of failure probability (as a function of unknown parameter) instead of dubious compromise combination of required reliability and confidence probability. The calculations are complex enough but it can be used in frame of automated system.