宏观结构的三参数三维断裂研究
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摘要
大型复杂机械(如:航空飞行器、压力容器和高速铁路等等)的安全和可靠性要求对国防和国民经济都具有无可比拟的重要性。而对材料和结构进行损伤容限耐久性设计在很大程度上充分发挥了材料和结构的承载能力,进而提高了结构设计应用的可靠性和经济性。然而,实际工程中采用的损伤容限耐久性设计仍然基于二维断裂理论中的框架(如:线弹性主要采用K-T方法,弹塑性主要采用J-Q方法等等),而建立在二维理论基础之上的传统断裂理论和损伤容限分析方法已经无法满足其现代结构设计的可靠性和经济性要求,这使得人们必须进一步发展三维损伤容限设计理论。目前,虽然许多学者在三维断裂理论及剩余寿命预测方面已经做了大量优秀的工作,如:Guo建立的基于三维约束因子的三维疲劳断裂理论、准则和规律能够较好地解决实际工程中关键科学和技术问题。然而考虑三维离面约束影响的研究工作还主要集中在穿透直裂纹的研究上,对各种复杂形式一般三维裂纹的剩余寿命预测方面还没有很好地解决,而一般裂纹体的三维弹塑性裂纹的剩余强度方面也没有系统的研究,这种局面阻碍着三维断裂力学的广泛应用和断裂力学学科的发展。由于准确描述线弹性三维裂纹尖端应力场对准确评估结构剩余寿命起着关键性作用,同时,对于弹塑性材料,裂尖弹塑性应力场对结构剩余强度的评定也起着非常重要的作用,本文针对这类关键问题,采用连续介质力学理论和系统的三维弹性和弹塑性有限元计算,对各种形式的三维裂纹体进行了系统研究,取得了以下进展:
1)由于一般的三维裂纹弹性端部场的解析解很难得到,针对目前工程实际中复杂三维结构广泛存在的各种典型的裂纹缺陷(包括:穿透直裂纹、内埋椭圆裂纹、半椭圆表面裂纹和1/4椭圆角裂纹),用三维弹性体单元及裂纹尖端的奇异单元建立各种形式裂纹的有限元模型,并通过对软件的二次开发分别求出了裂纹前沿的三个参数(应力强度因子K,T应力和离面约束因子T_z)的分布,同时考虑泊松比对三个参数的影响,从而建立起利用K-T-T_z完整描述各种裂纹弹性端部场的三维三参数方法,最后利用最小二乘法给出了描述三个参数的经验公式。这一系列成果为工程实际中有效地预测结构剩余寿命建立了坚实的理论桥梁。
2)针对I型弹塑性平面应变裂纹,通过引入弹塑性泊松比veq(此时T_z=v_(eq))并进行详细的理论分析,给出了能准确描述平面应变裂纹前沿应力应变场的准解析解,该解可以将HRR解只能描述T_z=0.5区域内的应力应变场拓展到能完整描述整个裂纹前沿塑性区域内的应力场。同时给出了当给定一某个点(r/(J/σ_0),θ),通过方程的迭代就可以确定该点处的T_z,从而最终确定各点处各应力参数及能量参数,进而完整确定了裂纹端部场了。基于平面应变的理论结果,进而提出了三维裂纹端部场的一种在给定某个点(r/(J/σ_0),θ)及相应的veq(此时T_z≠v_(eq)),再确定T_z的方法,进而最终确定整个三维裂纹的端部场的思想。
3)针对复杂弹塑性结构中各种典型的三维裂纹,利用三维弹塑性体单元建立了各种形式裂纹(内埋椭圆裂纹、半椭圆表面裂纹和1/4椭圆角裂纹)的有限元模型,并通过对软件的二次开发分别求出了裂纹前沿附近的J积分、Q应力和离面约束因子T_z的分布,再结合弹塑性断裂力学理论并引入弹塑性泊松比v_(eq),推导出利用已有弹性的T_z分布结果可以较好地描述弹塑性离面约束因子T_z分布的半解析公式,同时给出了能够准确描述等效应力σ_e和等效应变ε_e的近似公式,从而建立起利用三参数J-Q_T-T_z对三维裂纹前沿整个塑性区域端部场的完整描述的方法,这一系列成果为弹塑性结构的剩余强度评定提供了坚实的理论基础。
在以上工作基础上,采用应变能密度因子理论和三维约束理论,对弹性和弹塑性三维断裂准则进行了探讨,发现建立在应变能密度因子理论基础上的三维断裂准则无法显示T应力的影响,最后通过试验结果验证了弹塑性断裂准则的有效性。同时给出了预测各种一般三维裂纹在疲劳载荷作用下裂纹扩展的近似经验公式。从而为三维结构的剩余寿命预测和剩余强度评估建立起联系的桥梁。
此外,本文对珍珠贝的力学性能和基于BP神经网络的多裂纹柱体扭转问题进行了研究,相关结果在附件中给出。作为珍珠贝的力学性能研究的基础,其材料中方解石晶体的弹性参量几十年来有大量实验和理论研究,但数据分散。附录A给出了基于深入细致的第一原理计算得到的一组更客观的弹性参量,可为珍珠贝研究提供更好的基础。附录B给出了利用有限元模拟珍珠贝纳米压痕试验的初步结果。附录C提出一种基于BP神经网络的多裂纹柱体扭转问题的数据新处理方法,通过对BP神经网络的改进,给出了更精确、收敛速度更快的理想设计方案。
The safety and reliability of huge-scale mechanical systems are significant for national economy and national defence. With the extensive application of complex structures (e.g. airplane, pressure vessels, high-speed railway, etc.), the precision and reliability of them must be improved. The structure design based on the two-dimensional (2D) fracture theory and damage tolerance method (e.g. linear elastic K-T theory and elastic-plastic J-Q method etc.) has been difficult to meet the demand of reliability and economy. Therefore, people must develop the three-dimensional theory in order to grasp the mechanical behavior of structures under the complex stress state. Recently, though the three-dimensional (3D) fracture theory has obtained quite great progress (e.g. the 3D fatigue fracture theory and fracture criterion by Guo can be used to solve some key problem in engineering based on 3D out-of-plane constraint factor). However, a lot of research is focused on the through-thickness straight crack when the 3D constraint factor is considered. Many key problems have not been solved about many typical three-dimensional elastic and elastic-plastic cracks at home and abroad, and how to accomplish the object needs a lot of work from the lab to the engineering applications, which is becoming more and more severe drawback of the development and engineering application for fracture mechanics. Because the 3D stress field of linear elastic crack front and elastic-plastic crack front has an important role on predicting the residual life and evaluating the residual strength of 3D structures, various typical three-dimensional cracks have been investigated based on detailed continuum mechanics and finite element method (FEM) in the dissertation. The following main creativities are achieved:
1) The analytical solution of elastic stress field near 3D cracks front is hardly available. Based on the 3D elastic FEM, we study four typical 3D cracks (through-thickness straight crack, embedded elliptical crack, semi-elliptical crack and quarter elliptical crack). The 3D singular elements with four mid-side nodes at the quarter points are used around the crack front to simulate the inverse square foot singularity at the crack tip, and then the 3D finite element model for the various 3D crack are completed. Through secondary development of finite element software, the distributions of the three parameters (stress intensity factor K, T-stress and out-of-plane constraint factor T_z) are all obained, and the effect of Poisson’s ratio is also consided. By fitting the numerical results with the least squares method, empirical fromulae have been given for the convenience of engineering applications. Finally, the three-parameter K-T-T_z approach is provided, which can accurately describe the stress field around the crack front. The series of achievements provide the firm theory bridge against predicting the residual life of 3D structures effectively.
2) By introducing the elastic-plastic Poisson’s ratio v_(eq)(T_z=v_(eq)), the quasi-analytical solution for the mode I elastic-plastic plane strain crack tip fields is presented. The solution extends the dominating region from the region T_z=0.5 (HRR solution) to whole plastic zone. For a given point (r/(J/σ_0),θ), the value of v_(eq) at this point can be determined by an iteration procedure, and then various stress and energy parameters can be obtained. Based on the above results, the T_z can be determined by introducing veq (T_z≠veq) for 3D crack, and then the 3D crack tip fields can be described finally.
3) Based on the 3D elastic-plastic FEM, we study three typical 3D cracks (embedded elliptical crack, semi-elliptical crack and quarter elliptical crack). Through secondary development of finite element software, the distribution of the three parameters (J-integral, Q-stress and out-of-plane constraint factor T_z) are all obained, By the numerical computation and theoretical analysis, the semi-analytical fromulae of T_z,σ_e andε_e have been given for the convenience of engineering applications. Finally, the three-parameter J-Q_T-T_z approach is provided, which can accurately describe the stress field around the crack front. The series of achievements provide the firm theory foundation against evaluating the residual strength of 3D structures effectively.
From the above results, the 3D elastic and elastic-plastic fracture criteria are proposed based on the strain energy density factor theory and 3D constaint theory, and then they are used to the experimental results. The effects of T-stress effect are hardly displayed in the 3D fracture criteria. At the same time, the elastic-plastic fracture criterion has been proved to be effective by use of experimental data. Finally, the prediction of general 3D fatigue crack growth has been given by a series of experiential formulae. The achievements provide the firm theory bridge from the residual strength to the residual life of 3D structures.
Furthermore, the mechanical properties of nacre and the multi-cracked prismatic shaft torsion problem are both investigated based on the Back-propagation Neural Network theory. As a fundamental issue for the mechanical properties of nacre, the elastic properties single-crystal of calcite have long been made great efforts to study, while the large scatter of the available data are arise widely. In appendix A, the elastic constants of single-crystal calcite (CaCO_3) have been obtained by extensive first-principles calculations based on the density functional theory. In appendix B, the nanoindentation process of nacre has been simulatied based on 3D elastic-plastic FEM. In appendix C, a new processing method for the multi-cracked prismatic shaft torsion problem is presented based on the Back-propagation Neural Network. We provide an optimizing project of Back-propagation training and fast simulate the experimental results of the torsion rigidity. The examples prove that the project introduced in this paper is accurate and converge quickly.
1)由于一般的三维裂纹弹性端部场的解析解很难得到,针对目前工程实际中复杂三维结构广泛存在的各种典型的裂纹缺陷(包括:穿透直裂纹、内埋椭圆裂纹、半椭圆表面裂纹和1/4椭圆角裂纹),用三维弹性体单元及裂纹尖端的奇异单元建立各种形式裂纹的有限元模型,并通过对软件的二次开发分别求出了裂纹前沿的三个参数(应力强度因子K,T应力和离面约束因子T_z)的分布,同时考虑泊松比对三个参数的影响,从而建立起利用K-T-T_z完整描述各种裂纹弹性端部场的三维三参数方法,最后利用最小二乘法给出了描述三个参数的经验公式。这一系列成果为工程实际中有效地预测结构剩余寿命建立了坚实的理论桥梁。
2)针对I型弹塑性平面应变裂纹,通过引入弹塑性泊松比veq(此时T_z=v_(eq))并进行详细的理论分析,给出了能准确描述平面应变裂纹前沿应力应变场的准解析解,该解可以将HRR解只能描述T_z=0.5区域内的应力应变场拓展到能完整描述整个裂纹前沿塑性区域内的应力场。同时给出了当给定一某个点(r/(J/σ_0),θ),通过方程的迭代就可以确定该点处的T_z,从而最终确定各点处各应力参数及能量参数,进而完整确定了裂纹端部场了。基于平面应变的理论结果,进而提出了三维裂纹端部场的一种在给定某个点(r/(J/σ_0),θ)及相应的veq(此时T_z≠v_(eq)),再确定T_z的方法,进而最终确定整个三维裂纹的端部场的思想。
3)针对复杂弹塑性结构中各种典型的三维裂纹,利用三维弹塑性体单元建立了各种形式裂纹(内埋椭圆裂纹、半椭圆表面裂纹和1/4椭圆角裂纹)的有限元模型,并通过对软件的二次开发分别求出了裂纹前沿附近的J积分、Q应力和离面约束因子T_z的分布,再结合弹塑性断裂力学理论并引入弹塑性泊松比v_(eq),推导出利用已有弹性的T_z分布结果可以较好地描述弹塑性离面约束因子T_z分布的半解析公式,同时给出了能够准确描述等效应力σ_e和等效应变ε_e的近似公式,从而建立起利用三参数J-Q_T-T_z对三维裂纹前沿整个塑性区域端部场的完整描述的方法,这一系列成果为弹塑性结构的剩余强度评定提供了坚实的理论基础。
在以上工作基础上,采用应变能密度因子理论和三维约束理论,对弹性和弹塑性三维断裂准则进行了探讨,发现建立在应变能密度因子理论基础上的三维断裂准则无法显示T应力的影响,最后通过试验结果验证了弹塑性断裂准则的有效性。同时给出了预测各种一般三维裂纹在疲劳载荷作用下裂纹扩展的近似经验公式。从而为三维结构的剩余寿命预测和剩余强度评估建立起联系的桥梁。
此外,本文对珍珠贝的力学性能和基于BP神经网络的多裂纹柱体扭转问题进行了研究,相关结果在附件中给出。作为珍珠贝的力学性能研究的基础,其材料中方解石晶体的弹性参量几十年来有大量实验和理论研究,但数据分散。附录A给出了基于深入细致的第一原理计算得到的一组更客观的弹性参量,可为珍珠贝研究提供更好的基础。附录B给出了利用有限元模拟珍珠贝纳米压痕试验的初步结果。附录C提出一种基于BP神经网络的多裂纹柱体扭转问题的数据新处理方法,通过对BP神经网络的改进,给出了更精确、收敛速度更快的理想设计方案。
The safety and reliability of huge-scale mechanical systems are significant for national economy and national defence. With the extensive application of complex structures (e.g. airplane, pressure vessels, high-speed railway, etc.), the precision and reliability of them must be improved. The structure design based on the two-dimensional (2D) fracture theory and damage tolerance method (e.g. linear elastic K-T theory and elastic-plastic J-Q method etc.) has been difficult to meet the demand of reliability and economy. Therefore, people must develop the three-dimensional theory in order to grasp the mechanical behavior of structures under the complex stress state. Recently, though the three-dimensional (3D) fracture theory has obtained quite great progress (e.g. the 3D fatigue fracture theory and fracture criterion by Guo can be used to solve some key problem in engineering based on 3D out-of-plane constraint factor). However, a lot of research is focused on the through-thickness straight crack when the 3D constraint factor is considered. Many key problems have not been solved about many typical three-dimensional elastic and elastic-plastic cracks at home and abroad, and how to accomplish the object needs a lot of work from the lab to the engineering applications, which is becoming more and more severe drawback of the development and engineering application for fracture mechanics. Because the 3D stress field of linear elastic crack front and elastic-plastic crack front has an important role on predicting the residual life and evaluating the residual strength of 3D structures, various typical three-dimensional cracks have been investigated based on detailed continuum mechanics and finite element method (FEM) in the dissertation. The following main creativities are achieved:
1) The analytical solution of elastic stress field near 3D cracks front is hardly available. Based on the 3D elastic FEM, we study four typical 3D cracks (through-thickness straight crack, embedded elliptical crack, semi-elliptical crack and quarter elliptical crack). The 3D singular elements with four mid-side nodes at the quarter points are used around the crack front to simulate the inverse square foot singularity at the crack tip, and then the 3D finite element model for the various 3D crack are completed. Through secondary development of finite element software, the distributions of the three parameters (stress intensity factor K, T-stress and out-of-plane constraint factor T_z) are all obained, and the effect of Poisson’s ratio is also consided. By fitting the numerical results with the least squares method, empirical fromulae have been given for the convenience of engineering applications. Finally, the three-parameter K-T-T_z approach is provided, which can accurately describe the stress field around the crack front. The series of achievements provide the firm theory bridge against predicting the residual life of 3D structures effectively.
2) By introducing the elastic-plastic Poisson’s ratio v_(eq)(T_z=v_(eq)), the quasi-analytical solution for the mode I elastic-plastic plane strain crack tip fields is presented. The solution extends the dominating region from the region T_z=0.5 (HRR solution) to whole plastic zone. For a given point (r/(J/σ_0),θ), the value of v_(eq) at this point can be determined by an iteration procedure, and then various stress and energy parameters can be obtained. Based on the above results, the T_z can be determined by introducing veq (T_z≠veq) for 3D crack, and then the 3D crack tip fields can be described finally.
3) Based on the 3D elastic-plastic FEM, we study three typical 3D cracks (embedded elliptical crack, semi-elliptical crack and quarter elliptical crack). Through secondary development of finite element software, the distribution of the three parameters (J-integral, Q-stress and out-of-plane constraint factor T_z) are all obained, By the numerical computation and theoretical analysis, the semi-analytical fromulae of T_z,σ_e andε_e have been given for the convenience of engineering applications. Finally, the three-parameter J-Q_T-T_z approach is provided, which can accurately describe the stress field around the crack front. The series of achievements provide the firm theory foundation against evaluating the residual strength of 3D structures effectively.
From the above results, the 3D elastic and elastic-plastic fracture criteria are proposed based on the strain energy density factor theory and 3D constaint theory, and then they are used to the experimental results. The effects of T-stress effect are hardly displayed in the 3D fracture criteria. At the same time, the elastic-plastic fracture criterion has been proved to be effective by use of experimental data. Finally, the prediction of general 3D fatigue crack growth has been given by a series of experiential formulae. The achievements provide the firm theory bridge from the residual strength to the residual life of 3D structures.
Furthermore, the mechanical properties of nacre and the multi-cracked prismatic shaft torsion problem are both investigated based on the Back-propagation Neural Network theory. As a fundamental issue for the mechanical properties of nacre, the elastic properties single-crystal of calcite have long been made great efforts to study, while the large scatter of the available data are arise widely. In appendix A, the elastic constants of single-crystal calcite (CaCO_3) have been obtained by extensive first-principles calculations based on the density functional theory. In appendix B, the nanoindentation process of nacre has been simulatied based on 3D elastic-plastic FEM. In appendix C, a new processing method for the multi-cracked prismatic shaft torsion problem is presented based on the Back-propagation Neural Network. We provide an optimizing project of Back-propagation training and fast simulate the experimental results of the torsion rigidity. The examples prove that the project introduced in this paper is accurate and converge quickly.
引文
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