缺口件疲劳寿命分布及参数敏度分析
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摘要
工程结构中存在很多缺口件，缺口件的疲劳寿命或强度控制着结构的疲劳寿命或强度，完善而准确的缺口件疲劳寿命分布信息是结构疲劳可靠性分析的基础。缺口件型式繁多、尺寸多样，通过试验获得其寿命分布的成本很大，而光滑件的疲劳寿命分布容易获得。因此通过光滑件疲劳寿命试验数据预测缺口件疲劳寿命分布具有非常重要的理论意义和工程实际价值。
缺口件疲劳寿命分散性有很多来源，本文主要考虑局部应力应变的分散性和材料微观结构的不均匀性，其中后者引起的寿命分散性可由光滑件疲劳寿命试验数据获得，局部应力应变的分散性是本文的主要研究内容。
首先，基于最弱环节理论并结合场强法的思想，提出了疲劳有效应力这一新的局部应力应变量，它能在概率意义上更好地反映缺口的疲劳严重程度，根据计算出的疲劳有效应力查取光滑件的疲劳寿命试验数据可以方便地得到缺口件疲劳寿命，该方法可以同时考虑到应力梯度和试件尺寸对缺口件疲劳寿命的影响。
其次，研究了基于随机有限元的局部应力应变分散性计算，包括四方面的内容。一是，利用弹性模量和硬度的联系，通过对两种金属材料的平板试件表面各点洛氏硬度值的测量，并以相关函数为工具对测量数据进行统计分析，完成了弹性模量随机场相关特性的测定。二是，研究了随机场离散的KL展开Galerkin法，并选择用单元形函数作为基函数来实现该方法，给出了随机场网格为四边形单元时的具体实现过程，编制了相应的算法程序。算例表明该程序离散精度很高，满足工程计算要求。三是，研究了基于变分原理的摄动随机有限元法，编制了二维平面应力问题的随机有限元计算程序，该程序的计算精度比较令人满意。四是，提出了一种求解缺口根部一点处应力概率分布的方法，该方法将影响应力概率分布的因素分为载荷和材料性能参数两部分，且材料性能参数部分可近似等效为一个正态随机变量，从而缺口根部一点处应力可表示为载荷随机变量和材料性能参数随机变量的线性组合，容易计算得到其分布函数，算例表明这种近似比较精确和有效。
最后，建立了基于随机有限元方法的缺口件疲劳寿命分布计算模型，该模型将由随机有限元方法计算得到的疲劳有效应力分散性同微观结构不均匀性引起的寿命分散性结合起来，最终计算出缺口件寿命分布。应用该模型进行了缺口件寿命分布对材料、载荷和几何特性三种不确定性设计变量的敏度分析，分析结果表明它们对缺口件寿命分布影响程度的大小顺序依次为载荷、几何和材料性能，并且大小在同一量级上。
There are numerous notched components in engineering structures. Their fatigue life or strengthdetermine that of structures. So complete and accurate data of fatigue life distribution of notchedspecimens is the basis for the fatigue reliability analysis of structures. Due to the variety of types anddimensions of notched specimens, the cost involved in getting their fatigue life distribution throughexperiments is very high. On the contrary, the cost for the fatigue life distribution of smoothspecimens is quite small. So the research in predicting the fatigue life distribution of notchedspecimens from fatigue life test data of smooth specimens has significant value in both theory andpractical engineering.
There are many factors that cause the randomness of fatigue life of notched specimens. Two ofthem are considered in this work. They are the dispersion of local stress and strain, and thenon-homogeneity of microstructure. The randomness of fatigue life induced by the latter can beobtained from the fatigue life test data of smooth specimens. So the research emphasis is placed onthe dispersion of local stress and strain.
In the beginning, based on the weakest-link theory and the stress field intensity approach, a newlocal stress and strain parameter, fatigue effective stress, is proposed. It can express the fatigueseverity of notch in the sense of probability. Employing this parameter to refer to the fatigue life testdata of smooth specimens would lead to the fatigue life of notched specimens. This model reflects theinfluence of both stress gradient and component size on the notched specimen fatigue life.
Then, the calculation for the dispersion of local stress and strain using the stochastic finite elementmethod(SFEM) is studied, including four parts. Firstly, based on the relation between the elasticmodulus and the hardness，the correlation characteristics of the elastic modulus random fields for twometals are determined, by experimental measurements of Rockwell hardness values at a series ofpoints on the specimen and a statistical analysis of the test result utilizing the correlation function.Secondly, the Galerkin-type procedure for Karhunen-Loève expansion of a random field isinvestigated. The shape function of the random field element is chosen to be the basis to implementthis procedure. Details regarding how to realize the procedure when the random field mesh iscomposed of quadrilaterals are given. The corresponding algorithmic routine is compiled. A exampleshows that the accuracy of this routine is high, meeting the requirement of engineering computation.Thirdly, the perturbation stochastic finite element method based on the variational principle is studied.According to the idea of this method, a stochastic finite element calculation program for two dimensional plane stress problem is developed. The precision of this program is moderately satisfying.Fourthly, A method for the probability distribution of the stress at a single point at notch root isproposed. In this method, the factors which affect the probability distribution of the stress areseparated into two parts, the load and the material. The contribution of the material is approximatelyequivalent to be a normal random variable. Then the stress can be expressed as a linear combinationof the random variable of the load and the random variable of the material. Thus the probabilitydistribution of the stress can be easily obtained. This treatment is accurate and effective.
Finally, the calculation model based on SFEM for fatigue life distribution of notched specimens isestablished. In this model, the dispersion of the fatigue effective stress worked out by SFEM isorganically combined with the randomness of fatigue life induced by the non-homogeneity ofmicrostructure to eventually obtain the fatigue life distribution of notched specimens. Making use ofthis calculation model, a sensitivity analysis for fatigue life distribution of notched specimens withregard to three types of design variables with uncertainty (material, load and geometry) is carried out.The result shows that the load has the largest influence on the fatigue life distribution of notchedspecimens, then the geometry, lastly the material. However, the values of sensitivity of these threefactors are at the same magnitude.
缺口件疲劳寿命分散性有很多来源，本文主要考虑局部应力应变的分散性和材料微观结构的不均匀性，其中后者引起的寿命分散性可由光滑件疲劳寿命试验数据获得，局部应力应变的分散性是本文的主要研究内容。
首先，基于最弱环节理论并结合场强法的思想，提出了疲劳有效应力这一新的局部应力应变量，它能在概率意义上更好地反映缺口的疲劳严重程度，根据计算出的疲劳有效应力查取光滑件的疲劳寿命试验数据可以方便地得到缺口件疲劳寿命，该方法可以同时考虑到应力梯度和试件尺寸对缺口件疲劳寿命的影响。
其次，研究了基于随机有限元的局部应力应变分散性计算，包括四方面的内容。一是，利用弹性模量和硬度的联系，通过对两种金属材料的平板试件表面各点洛氏硬度值的测量，并以相关函数为工具对测量数据进行统计分析，完成了弹性模量随机场相关特性的测定。二是，研究了随机场离散的KL展开Galerkin法，并选择用单元形函数作为基函数来实现该方法，给出了随机场网格为四边形单元时的具体实现过程，编制了相应的算法程序。算例表明该程序离散精度很高，满足工程计算要求。三是，研究了基于变分原理的摄动随机有限元法，编制了二维平面应力问题的随机有限元计算程序，该程序的计算精度比较令人满意。四是，提出了一种求解缺口根部一点处应力概率分布的方法，该方法将影响应力概率分布的因素分为载荷和材料性能参数两部分，且材料性能参数部分可近似等效为一个正态随机变量，从而缺口根部一点处应力可表示为载荷随机变量和材料性能参数随机变量的线性组合，容易计算得到其分布函数，算例表明这种近似比较精确和有效。
最后，建立了基于随机有限元方法的缺口件疲劳寿命分布计算模型，该模型将由随机有限元方法计算得到的疲劳有效应力分散性同微观结构不均匀性引起的寿命分散性结合起来，最终计算出缺口件寿命分布。应用该模型进行了缺口件寿命分布对材料、载荷和几何特性三种不确定性设计变量的敏度分析，分析结果表明它们对缺口件寿命分布影响程度的大小顺序依次为载荷、几何和材料性能，并且大小在同一量级上。
There are numerous notched components in engineering structures. Their fatigue life or strengthdetermine that of structures. So complete and accurate data of fatigue life distribution of notchedspecimens is the basis for the fatigue reliability analysis of structures. Due to the variety of types anddimensions of notched specimens, the cost involved in getting their fatigue life distribution throughexperiments is very high. On the contrary, the cost for the fatigue life distribution of smoothspecimens is quite small. So the research in predicting the fatigue life distribution of notchedspecimens from fatigue life test data of smooth specimens has significant value in both theory andpractical engineering.
There are many factors that cause the randomness of fatigue life of notched specimens. Two ofthem are considered in this work. They are the dispersion of local stress and strain, and thenon-homogeneity of microstructure. The randomness of fatigue life induced by the latter can beobtained from the fatigue life test data of smooth specimens. So the research emphasis is placed onthe dispersion of local stress and strain.
In the beginning, based on the weakest-link theory and the stress field intensity approach, a newlocal stress and strain parameter, fatigue effective stress, is proposed. It can express the fatigueseverity of notch in the sense of probability. Employing this parameter to refer to the fatigue life testdata of smooth specimens would lead to the fatigue life of notched specimens. This model reflects theinfluence of both stress gradient and component size on the notched specimen fatigue life.
Then, the calculation for the dispersion of local stress and strain using the stochastic finite elementmethod(SFEM) is studied, including four parts. Firstly, based on the relation between the elasticmodulus and the hardness，the correlation characteristics of the elastic modulus random fields for twometals are determined, by experimental measurements of Rockwell hardness values at a series ofpoints on the specimen and a statistical analysis of the test result utilizing the correlation function.Secondly, the Galerkin-type procedure for Karhunen-Loève expansion of a random field isinvestigated. The shape function of the random field element is chosen to be the basis to implementthis procedure. Details regarding how to realize the procedure when the random field mesh iscomposed of quadrilaterals are given. The corresponding algorithmic routine is compiled. A exampleshows that the accuracy of this routine is high, meeting the requirement of engineering computation.Thirdly, the perturbation stochastic finite element method based on the variational principle is studied.According to the idea of this method, a stochastic finite element calculation program for two dimensional plane stress problem is developed. The precision of this program is moderately satisfying.Fourthly, A method for the probability distribution of the stress at a single point at notch root isproposed. In this method, the factors which affect the probability distribution of the stress areseparated into two parts, the load and the material. The contribution of the material is approximatelyequivalent to be a normal random variable. Then the stress can be expressed as a linear combinationof the random variable of the load and the random variable of the material. Thus the probabilitydistribution of the stress can be easily obtained. This treatment is accurate and effective.
Finally, the calculation model based on SFEM for fatigue life distribution of notched specimens isestablished. In this model, the dispersion of the fatigue effective stress worked out by SFEM isorganically combined with the randomness of fatigue life induced by the non-homogeneity ofmicrostructure to eventually obtain the fatigue life distribution of notched specimens. Making use ofthis calculation model, a sensitivity analysis for fatigue life distribution of notched specimens withregard to three types of design variables with uncertainty (material, load and geometry) is carried out.The result shows that the load has the largest influence on the fatigue life distribution of notchedspecimens, then the geometry, lastly the material. However, the values of sensitivity of these threefactors are at the same magnitude.
引文
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