摘要
断裂力学是固体力学近代发展的一个分支，它将裂纹作为损伤，建立起描述含裂纹构件的应力应变场的方法，给出了构件的断裂准则和裂纹扩展规律。上个世纪80年代，人们开始将断裂力学理论用于构件的寿命预测，使寿命预测理论得到了迅速的发展，但也遇到了许多有待解决的问题，例如，对缺陷和裂纹的定量描述，目前广泛采用的无损检测技术还不能提供很准确的信息；还没有一套完整的理论描述裂纹扩展行为的不确定性；人们对裂纹扩展行为认识的也不够完善等。
为了描述裂纹扩展行为的不确定性，人们开始将可靠性理论引入到断裂力学的研究中，用概率模型描述各参数和裂纹扩展行为的随机特性，并取得了一定的成果。然而，随着人们对断裂力学研究的不断深入，发现构件的断裂行为不仅存在随机性，而且还存在着广泛的模糊性，尤其是初始裂纹的确定更是如此，这就促进了模糊概率断裂力学的发展。用随机理论描述不确定性，用模糊理论描述由于人的参与而引起的不精确性，目前，这方面的研究还不多。木文基于现有这方面的研究成果，探讨了将模糊可靠性与断裂力学相结合进行寿命预测的可能性以及一般方法，其主要工作如下：
(1) 从总体拟合效果出发，利用数值模拟方法，定量的指出了传统确定数据分布函数方法的不足，提出了一种新的用于确定裂纹扩展数据分布类型的方法，这种方法将通过中位秩或平均秩计算得到的概率值模糊化，采用模糊线性凹归技术，比较了7中概率分布类型函数对样本数据的拟合效果，并以2024-T351材料的CT试样的裂纹扩展数据为例，验证了这种方法的有效性。结果表明，一定载荷循环次数下的裂纹扩展尺寸和一定应力强度因子变程下的裂纹扩展速率较好地服从对数正态分布。
(2) 在现有文献的基础上，深入探讨了加权函数法在确定应力强度因子中的应用。利用曲线拟合有限元分析结果获得了各种长宽比的有限尺寸平板2维直线边缘穿透裂纹和中心穿透裂纹的应力强度因子表达式，并将计算结果与有关文献比较，表明这种方法是行之有效的。
(3) 总结并发展了有关文献中关于确定初始裂纹尺寸和临界裂纹尺寸模糊随机特性的方法。给出了初始裂纹检测尺寸的隶属函数的确定方法。
(4) 提出了一种模糊随机加权回归办法，这种方法可以有效地消除试验数据异常点对直线回归的影响，并将其用于裂纹扩展速率曲线的确定中。
(5) 介绍了随机过程在进行裂纹扩展寿命可靠性分析中的应用，并探讨了模糊随机过程在这方面应用的可行性。
(6) 利用本文得到的概率裂纹扩展速率曲线和模糊概率裂纹扩展速率曲线，综合各种不确定性因素，通过模糊线性回归，得到模糊概率—裂纹扩展速率—应力强度因子变程曲线，并在此曲线的基础上，提出了一种裂纹扩展寿命模糊可靠性分析方法，并给出了计算公式。扩展了极大似然法在确定裂纹扩展分散特性中的应用。
模糊概率断裂力学的研究只是刚刚起步，本文仅从上述几个方面对其进行了研究。模糊概率断裂力学理论的发展和完善还需要更多学者的不断努力。
Fracture mechanics is an embranchment of solid meehanies in its recent development. It established a method to describe cracked bodies' fields of stress and strain regarding crack as damage, and it also proposed the fracture criteria of the component and the rules of crack growth. Since the 1980s, researchers have used fracture mechanics to predict component's life, which has made the theory of life prediction develop quickly. However, there are many problems to be solved in the process of using fracture mechanics to predict components' life. For example, the technology of the nondestructive inspection used widely to describe defects of crack quantitatively can not provide enough exact information. There isn't a set of integrated theories to describe stochastic properties of crack growth, and people have not known the crack growth fully, etc.
To describe the stochastic characters of the crack growth, researchers introduced the theory of the reliability to fracture mechanics and obtained lots of achievements. But, with the development of the research on fracture mechanics, people found that there are two uncertainties in the fracture of the component-both stochastic and fuzzy, especially in the inspection of the initial crack, which promoted the emergence and development of fuzzy probability fracture mechanics. Stochastic theory can be used to describe uncertainty, and fuzzy theory can be used to describe the imprecise property caused by the participation of the people. Currently, there is little research on this. This paper discusses the possibility and the general method of combining the theory of fuzzy reliability with fracture mechanics to predict the life of the component based on the current research achievements.
(1) From the purpose of fit on whole sample, defect of the traditional method for determining distribution function of data is pointed out through numerical simulation. With the help of the fuzzy linear regression method, a novel method for determining the distribution kinds is proposed, and this method is validated to be effective taking 2024-T351 CT sample's crack growth data as example. One can see from results that crack growth size under the certain load cycles and crack growth rate under the certain stress intensity factor range can be fitted with the log-normal distribution.
(2) Application of the weight function for determining the stress intensity factor based on the current literature is discussed in detail. Stress intensity factor expressions of 2D edge through crack and center through crack in the finite rectangle plate were obtained. This method are validated to be effective by comparing results with those proposed in other literatures.
(3) Method for determining the fuzzy stochastic property of the initial crack size and the critical crack size is summarized and developed.
(4) This paper proposes a novel linear regression with fuzzy weight method, which can eliminate the effect of the abnormal test data on the linear regression and is used to determine crack growth rate curve.
(5) Application of the stochastic process in the reliability analysis of crack growth life is introduced, as well as the application possibility of the fuzzy stochastic process in this aspect.
(6) Integrating various stochastic and fuzzy factors and using probability crack growth rate curve and
fuzzy probability crack growth rate curve proposed in this paper, a novel method and its formula for analyzing fuzzy reliability of crack life is proposed. The application of the maximum likelihood method for determining the stochastic and fuzzy properties of crack growth.
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