弹性和塑性V形切口应力奇异性分析与界面强度的扩展边界元法研究
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摘要
本文在调查和总结现有的分析线弹性和塑性v形切口／裂纹尖端附近区域的应力奇异性方法和断裂强度分析的基础上,研究了使用插值矩阵法分析线弹性、塑性v形应力奇异性和边界元法分析V形切口/裂纹结构的力学场问题。创立了一个新的分析途径—扩展边界元法(the extended boundary element method—XBEM),研制了相应的计算程序,有效和准确地求解了线弹性、塑性V形切口/裂纹应力奇异性指数和尖端附近区域的奇异应力场。全文主要研究工作及创新点如下：
1提出插值矩阵法分析固体结构切口尖端区域热流密度奇异性。基于在切口尖端附近区域温度场的渐近展开表达式,提出了计算切口/裂纹尖端处热流密度奇异性特征指数的新方法。将温度场的表达式引入稳态热传导微分方程,得到关于各向同性材料切口/裂纹奇异点处的一组非线性常微分方程的特征值问题,再采用变量代换法,将该非线性常微分方程组转化为一组线性常微分方程组。运用插值矩阵法求解,获得各向同性材料切口/裂纹尖端处多阶的热流密度奇异指数,同时获得其相应的特征角函数。
2提出插值矩阵法分析复合材料结构切口尖端区域应力奇异性。基于复合材料切口尖端位移场的渐近展开,将切口的反平面平衡控制方程转化为关于切口奇性指数的微分方程特征值问题,采用插值矩阵法计算该方程组的特征值,获取了切口尖端的应力奇性指数。研究了单相材料切口、双相材料切口以及止于异质界面切口的奇异性问题,算例表明本文方法可以一次性计算出多阶奇异性指数。对所取得的非奇异指数尽管切口不表现出奇性状态,但它们却是描述切口尖端完整应力场必不可少的参量。
3提出插值矩阵法分析三维柱向切口/裂纹尖端区域应力奇异性难题。在三维柱向切口根部区域引用位移渐近展开式,代入线弹性力学控制方程,导得切口/裂纹应力奇性指数的常微分方程组特征值问题。然后采用插值矩阵法,一次性地计算出三维柱向切口的各阶应力奇性指数,并可同时获取相应的特征角函数。算例结果表明本文方法是分析三维切口应力奇异指数的一个有效的路径,三维切口的前若干阶应力奇性指数解收敛于平面应变切口应力奇性指数理论值,但若直接用平面应变理论预测三维切口应力奇性指数将导致部分奇性指数缺失。本文方法的一个重要优点是以上求解的特征角函数和它们各阶导函数具有同阶精度,并且一次性地求出前若干阶特征对,插值矩阵法计算量小,易于和其他方法联合使用。这些优点在后续求解尖端区域完全应力场和温度梯度场非常优越。
4创立了扩展边界元法,用于分析线弹性平面V形切口/裂纹结构的位移场、应力场和裂纹扩展过程。对切口/裂纹尖端区域采用Williams渐近展开式表达,将其代入弹性力学基本方程中,尖端区域的应力奇异性指数及其对应的位移和应力角函数由插值矩阵法求解常微分方程组获得。由于在远离切口尖端的区域无应力奇异性,将尖端区域挖出后,其外围的剩余结构应力场无奇异性,由常规的边界元法分析。将尖端区域Williams渐近展开的特征分析法与边界积分方程结合,解出切口尖端附近应力奇异性区域的各应力场渐近展开项系数,获得平面切口／裂纹结构完整的位移和应力场,从而建立了扩展边界元法。①采用扩展边界元法研究了非奇异应力项对中央含V形切口试样的表观断裂韧度和临界荷载预测值的影响。结果表明：考虑非奇异应力项时,脆性断裂的表观断裂韧度和临界荷载的预测值要比忽略非奇异应力项时的预测值更接近实验值。②基于考虑非奇异应力项贡献的最大周向应力脆性断裂准则,运用扩展边界元法分析了边缘含V形切口/裂纹半圆形弯曲试样在荷载作用下的启裂方向,对切口/裂纹扩展过程给出了自动跟踪方法,通过算例证明了扩展边界元法的正确性和有效性。
5提出了分析幂硬化塑性材料V形切口和裂纹尖端区域的应力奇异性一个新途径。首先在切口和裂纹区域采用自尖端径向度量的渐近位移场假设,将其代入塑性全量理论的基本微分方程后,经过一系列推导,得出包含应力奇异指数和特征函数的非线性常微分方程特征值问题。然后采用插值矩阵法迭代求解导出的控制方程,一次性得到一般性塑性材料V形切口和裂纹的前若干阶应力奇异阶和相应的特征函数,本文获得的前3阶应力奇异指数有3～5位有效数字,并且同一阶的特征函数和其导函数的计算精度与对应的奇异指数计算精度同阶。目前关于平面塑性V形切口他人文献中鲜见有第2阶以上的可靠解。
6创立了扩展边界元法分析V形切口/裂纹尖端局部弹塑性奇异应力场。将含V形切口结构分成围绕切口尖端的塑性局部区域和外围的剩余结构两部分。基于切口尖端区域特征分析求出的多重塑性应力奇性指数和相应的位移、应力特征角函数,将尖端区域塑性变形的位移和应力表示成有限项奇性指数和特征角函数的线性组合,然后在挖去小区域后的剩余结构考虑为线弹性变形,由边界积分方程离散求解。两部分计算列式联立,由此精细地计算出V形切口尖端区域的塑性位移场、多重奇异应力场和应力强度因子。本文的扩展边界元法解符合切口尖端局部塑性奇异应力场的解析规律,为弹塑性V形切口/裂纹的疲劳和断裂扩展分析提供了一个有效新途径。
Based on the review of the analytic methods for the singularity orders of V-notches and interfacial strength of V-notched structures, the singularity analysis for the elastic and plastic V-notches by the interpolating matrix method and the mechanical field analysis at the V-notch tip of V-notched/cracked structures by the boundary element method are proposed in this thesis. A new approach named the extended boundary element method (XBEM) is established in order to effectively solve elastic or plastic singular stress fields in V-notch/crack tip region. The corresponding calculation program for the XBEM is developed. The main work and contribution of this thesis are given as follows:
1A new way of analyzing the heat flux singularity of the V-notched structures is proposed by the use of the interpolating matrix method. Based on the asymptotic expression of the temperature field near the notch tip, a characteristic analysis method for calculating the singularity orders of heat flux density at the V-notch/crack tip is proposed. After introducing the expression of temperature field into the differential equation of the steady-state heat conduction problem, the governing equations are transformed into a set of non-linear characteristic ordinary differential equations with the singularity orders. By adopting the variable substitution technique, the established non-linear characteristic equations are transformed into a set of linear ones. Then the interpolating matrix method is adopted to solve the established equations so that the leading heat flux singularity orders and the associated angular functions are obtained.
2The interpolating matrix method is proposed to analyze the stress singularity at the tip of composite V-notch. Based on the asymptotic extension of the displacement field at the composite notch tip, the governing equations for the notch subjected to the anti-plane loading are transformed into the characteristic differential equations with the notch singularity orders. A transformation is applied to converting these equations into a set of linear characteristic ordinary different equations. Then the interpolating matrix method is used to solve the established equations for obtaining the notch singularity orders. The single material notch, bi-material notch and the notch terminated at the bi-material interface are successively studied by the present method. The examples indicate that the present method can provide all the leading stress singularity orders synchronously. Although the eigenpairs corresponding to the non-singular orders in the asymptotic extension do not yield the singular stress components for the notched structures, they are the indispensable extension terms for evaluating the complete stress field in the notch tip region.
3The difficulty of the analysis of stress singularity at three-dimensional (3-D) V-notch notch tip is tackled by the interpolating matrix method. After the expression of displacement asymptotic expansion in the notched root zone of3-D structures with column notch is introduced into the linear elasticity governing equation, the eigenvalue problem of ordinary differential equations with the stress singularity orders for3-D V-notch root zone are established by a series of deduction. Then by applying the interpolating matrix method to solve the established equations, all the leading stress singularity orders and the associated displacement/stress angular eigenfunctions can be achieved simultaneously. The numerical results show that some of the singularity orders at the3-D notch problem are converging to the theoretic solutions of the plane strain V-notch problem. However, the number of the singularity orders for3-D V-notch is more than one of2-D plane strain V-notch. If the plane strain theory is used to predict the stress singularity orders of3-D V-notch, some important terms in the asymptotic expansion will be lost. One of the important advantages of the present method is that the computed results of the angular eigenfunctions and their derivative functions corresponding to each asymptotic expansion term have the same order of accuracy. Another advantage is that all the useful eigenpairs in the asymptotic expansion can be yielded at the same time. Moreover the interpolating matrix method takes a small amount of computation to solve the eigenvalue problems and is easily used. These advantages are very beneficial to solve the stress field and temperature gradient in V-notches and cracks tip region subsequently.
4A new numerical method named the extended boundary element method (XBEM) is established in the present thesis, which is used to analyze the displacement and stress field in the linear elastic plane notched/cracked structures and simulate the crack propagation beginning from the notch/crack tip. First of all, the displacement and stress fields in the characteristic radius region from a notch tip are expressed by the Williams asymptotic expansions. After the series expansion is substituted into the governing equation in elasticity, the stress singularity order and the associated angular functions can be obtained by solving the characteristic differential equations. Because there is no stress singularity in the remained region after the V-notch tip region is dug out, the conventional boundary element method (CBEM) can be used to analyze it. Hence all the leading unknown coefficients in the Williams series expansion and the complete stress field for the notched structures can be calculated by applying CBEM and combining with the results of the former eigenanalysis for the notch tip region. Here this is called eXtended Boundary Element Method (XBEM). The singular stress terms together with the non-singular stress terms in the tip region can be conveniently obtained. Then the effect of the non-singular stress on the fracture toughness and the critical loading for the centrally sharp V-notched specimen problem are discussed in detail. The numerical results show that the predicted critical loading and fracture toughness of V-notched structures when the non-singular stress is taken into consideration are more accurate than the predicted ones only considering the singular stress by comparing with the experimental results. Based on the consideration of the contribution of non-singular stress term and the maximum circumferential stress criterion of brittle fracture, the crack initiation extended direction from V-shaped notch/crack tip in a semicircular bending specimen can be determined by the XBEM. The strategy for XBEM tracking the crack propagation process is given. The numerical examples show that the XBEM is correct and effective for simulating the propagation process on plane crack.
5A new approach of analyzing the stress singularity of the plane V-notches and cracks in power law hardening materials is proposed. Firstly, the asymptotic displacement field in terms of radial coordinates in the notch tip region is adopted. As soon as the notch tip region appears the plastic deformation, the Von Mises yield criterion and total strain plastic theory are adopted. By introducing the displacement expressions into the governing differential equations of the plastic theory, a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions are proposed after a series of derivation. Then the interpolating matrix method is used to solve the eigenvalue problem by an iteration process. Several leading plastic stress singularity orders of plane V-notches and cracks can be obtained at a time. Simultaneously, the associated displacement and stress eigenvectors in the notch tip region have been determined as the same degree of accuracy with the corresponding singularity order. There are three to five significant figures in the computed values of the first three stress singularity orders obtained by the interpolating matrix method. Few of the published literatures gave out the effective and creditable second stress singularity order for the plane plastic V-notch.
6The extended boundary element method (XBEM) is proposed to determinate the singular stress field of the V-notched structures with local plastic deformation near notch tip region. Firstly, the elastic-plastic V-notched structure is divided into two parts, a small region around the V-notch tip and the remaining structure without the small region. Consider that the small region around the V-notch tip yields the plastic deformation due to the stress concentration. Secondly, based on the computed results of the multiple plastic stress singularity orders and the associated eigenfunctions in the singularity analysis for the small region, the displacement and stress components in the small region are expressed as the linear combinations of the finite terms of the series expansion with the leading singularity orders. The remaining structure is considered as linear elastic region, where the discrete boundary integral equations (BIE) can be established along the boundary of the remaining structures, included the connection border with the small region. Then the results of the eigen analysis for the stress singularity in the notch tip region are combined together with the discrete BIE, which is called XBEM. The XBEM can determine the elastic-plastic displacement and stress fields of the V-notched structures and the multiple plastic stress intensity factors at the notch tip.The stress solutions obtained by the XBEM is in agreement with analytical characteristics of local plastic singular stress field at the notch tip. Hence the XBEM provides an effective new way to investigate the fatigue fracture and crack propagation process of elastic-plastic V-notched and cracked structures.
1提出插值矩阵法分析固体结构切口尖端区域热流密度奇异性。基于在切口尖端附近区域温度场的渐近展开表达式,提出了计算切口/裂纹尖端处热流密度奇异性特征指数的新方法。将温度场的表达式引入稳态热传导微分方程,得到关于各向同性材料切口/裂纹奇异点处的一组非线性常微分方程的特征值问题,再采用变量代换法,将该非线性常微分方程组转化为一组线性常微分方程组。运用插值矩阵法求解,获得各向同性材料切口/裂纹尖端处多阶的热流密度奇异指数,同时获得其相应的特征角函数。
2提出插值矩阵法分析复合材料结构切口尖端区域应力奇异性。基于复合材料切口尖端位移场的渐近展开,将切口的反平面平衡控制方程转化为关于切口奇性指数的微分方程特征值问题,采用插值矩阵法计算该方程组的特征值,获取了切口尖端的应力奇性指数。研究了单相材料切口、双相材料切口以及止于异质界面切口的奇异性问题,算例表明本文方法可以一次性计算出多阶奇异性指数。对所取得的非奇异指数尽管切口不表现出奇性状态,但它们却是描述切口尖端完整应力场必不可少的参量。
3提出插值矩阵法分析三维柱向切口/裂纹尖端区域应力奇异性难题。在三维柱向切口根部区域引用位移渐近展开式,代入线弹性力学控制方程,导得切口/裂纹应力奇性指数的常微分方程组特征值问题。然后采用插值矩阵法,一次性地计算出三维柱向切口的各阶应力奇性指数,并可同时获取相应的特征角函数。算例结果表明本文方法是分析三维切口应力奇异指数的一个有效的路径,三维切口的前若干阶应力奇性指数解收敛于平面应变切口应力奇性指数理论值,但若直接用平面应变理论预测三维切口应力奇性指数将导致部分奇性指数缺失。本文方法的一个重要优点是以上求解的特征角函数和它们各阶导函数具有同阶精度,并且一次性地求出前若干阶特征对,插值矩阵法计算量小,易于和其他方法联合使用。这些优点在后续求解尖端区域完全应力场和温度梯度场非常优越。
4创立了扩展边界元法,用于分析线弹性平面V形切口/裂纹结构的位移场、应力场和裂纹扩展过程。对切口/裂纹尖端区域采用Williams渐近展开式表达,将其代入弹性力学基本方程中,尖端区域的应力奇异性指数及其对应的位移和应力角函数由插值矩阵法求解常微分方程组获得。由于在远离切口尖端的区域无应力奇异性,将尖端区域挖出后,其外围的剩余结构应力场无奇异性,由常规的边界元法分析。将尖端区域Williams渐近展开的特征分析法与边界积分方程结合,解出切口尖端附近应力奇异性区域的各应力场渐近展开项系数,获得平面切口／裂纹结构完整的位移和应力场,从而建立了扩展边界元法。①采用扩展边界元法研究了非奇异应力项对中央含V形切口试样的表观断裂韧度和临界荷载预测值的影响。结果表明：考虑非奇异应力项时,脆性断裂的表观断裂韧度和临界荷载的预测值要比忽略非奇异应力项时的预测值更接近实验值。②基于考虑非奇异应力项贡献的最大周向应力脆性断裂准则,运用扩展边界元法分析了边缘含V形切口/裂纹半圆形弯曲试样在荷载作用下的启裂方向,对切口/裂纹扩展过程给出了自动跟踪方法,通过算例证明了扩展边界元法的正确性和有效性。
5提出了分析幂硬化塑性材料V形切口和裂纹尖端区域的应力奇异性一个新途径。首先在切口和裂纹区域采用自尖端径向度量的渐近位移场假设,将其代入塑性全量理论的基本微分方程后,经过一系列推导,得出包含应力奇异指数和特征函数的非线性常微分方程特征值问题。然后采用插值矩阵法迭代求解导出的控制方程,一次性得到一般性塑性材料V形切口和裂纹的前若干阶应力奇异阶和相应的特征函数,本文获得的前3阶应力奇异指数有3～5位有效数字,并且同一阶的特征函数和其导函数的计算精度与对应的奇异指数计算精度同阶。目前关于平面塑性V形切口他人文献中鲜见有第2阶以上的可靠解。
6创立了扩展边界元法分析V形切口/裂纹尖端局部弹塑性奇异应力场。将含V形切口结构分成围绕切口尖端的塑性局部区域和外围的剩余结构两部分。基于切口尖端区域特征分析求出的多重塑性应力奇性指数和相应的位移、应力特征角函数,将尖端区域塑性变形的位移和应力表示成有限项奇性指数和特征角函数的线性组合,然后在挖去小区域后的剩余结构考虑为线弹性变形,由边界积分方程离散求解。两部分计算列式联立,由此精细地计算出V形切口尖端区域的塑性位移场、多重奇异应力场和应力强度因子。本文的扩展边界元法解符合切口尖端局部塑性奇异应力场的解析规律,为弹塑性V形切口/裂纹的疲劳和断裂扩展分析提供了一个有效新途径。
Based on the review of the analytic methods for the singularity orders of V-notches and interfacial strength of V-notched structures, the singularity analysis for the elastic and plastic V-notches by the interpolating matrix method and the mechanical field analysis at the V-notch tip of V-notched/cracked structures by the boundary element method are proposed in this thesis. A new approach named the extended boundary element method (XBEM) is established in order to effectively solve elastic or plastic singular stress fields in V-notch/crack tip region. The corresponding calculation program for the XBEM is developed. The main work and contribution of this thesis are given as follows:
1A new way of analyzing the heat flux singularity of the V-notched structures is proposed by the use of the interpolating matrix method. Based on the asymptotic expression of the temperature field near the notch tip, a characteristic analysis method for calculating the singularity orders of heat flux density at the V-notch/crack tip is proposed. After introducing the expression of temperature field into the differential equation of the steady-state heat conduction problem, the governing equations are transformed into a set of non-linear characteristic ordinary differential equations with the singularity orders. By adopting the variable substitution technique, the established non-linear characteristic equations are transformed into a set of linear ones. Then the interpolating matrix method is adopted to solve the established equations so that the leading heat flux singularity orders and the associated angular functions are obtained.
2The interpolating matrix method is proposed to analyze the stress singularity at the tip of composite V-notch. Based on the asymptotic extension of the displacement field at the composite notch tip, the governing equations for the notch subjected to the anti-plane loading are transformed into the characteristic differential equations with the notch singularity orders. A transformation is applied to converting these equations into a set of linear characteristic ordinary different equations. Then the interpolating matrix method is used to solve the established equations for obtaining the notch singularity orders. The single material notch, bi-material notch and the notch terminated at the bi-material interface are successively studied by the present method. The examples indicate that the present method can provide all the leading stress singularity orders synchronously. Although the eigenpairs corresponding to the non-singular orders in the asymptotic extension do not yield the singular stress components for the notched structures, they are the indispensable extension terms for evaluating the complete stress field in the notch tip region.
3The difficulty of the analysis of stress singularity at three-dimensional (3-D) V-notch notch tip is tackled by the interpolating matrix method. After the expression of displacement asymptotic expansion in the notched root zone of3-D structures with column notch is introduced into the linear elasticity governing equation, the eigenvalue problem of ordinary differential equations with the stress singularity orders for3-D V-notch root zone are established by a series of deduction. Then by applying the interpolating matrix method to solve the established equations, all the leading stress singularity orders and the associated displacement/stress angular eigenfunctions can be achieved simultaneously. The numerical results show that some of the singularity orders at the3-D notch problem are converging to the theoretic solutions of the plane strain V-notch problem. However, the number of the singularity orders for3-D V-notch is more than one of2-D plane strain V-notch. If the plane strain theory is used to predict the stress singularity orders of3-D V-notch, some important terms in the asymptotic expansion will be lost. One of the important advantages of the present method is that the computed results of the angular eigenfunctions and their derivative functions corresponding to each asymptotic expansion term have the same order of accuracy. Another advantage is that all the useful eigenpairs in the asymptotic expansion can be yielded at the same time. Moreover the interpolating matrix method takes a small amount of computation to solve the eigenvalue problems and is easily used. These advantages are very beneficial to solve the stress field and temperature gradient in V-notches and cracks tip region subsequently.
4A new numerical method named the extended boundary element method (XBEM) is established in the present thesis, which is used to analyze the displacement and stress field in the linear elastic plane notched/cracked structures and simulate the crack propagation beginning from the notch/crack tip. First of all, the displacement and stress fields in the characteristic radius region from a notch tip are expressed by the Williams asymptotic expansions. After the series expansion is substituted into the governing equation in elasticity, the stress singularity order and the associated angular functions can be obtained by solving the characteristic differential equations. Because there is no stress singularity in the remained region after the V-notch tip region is dug out, the conventional boundary element method (CBEM) can be used to analyze it. Hence all the leading unknown coefficients in the Williams series expansion and the complete stress field for the notched structures can be calculated by applying CBEM and combining with the results of the former eigenanalysis for the notch tip region. Here this is called eXtended Boundary Element Method (XBEM). The singular stress terms together with the non-singular stress terms in the tip region can be conveniently obtained. Then the effect of the non-singular stress on the fracture toughness and the critical loading for the centrally sharp V-notched specimen problem are discussed in detail. The numerical results show that the predicted critical loading and fracture toughness of V-notched structures when the non-singular stress is taken into consideration are more accurate than the predicted ones only considering the singular stress by comparing with the experimental results. Based on the consideration of the contribution of non-singular stress term and the maximum circumferential stress criterion of brittle fracture, the crack initiation extended direction from V-shaped notch/crack tip in a semicircular bending specimen can be determined by the XBEM. The strategy for XBEM tracking the crack propagation process is given. The numerical examples show that the XBEM is correct and effective for simulating the propagation process on plane crack.
5A new approach of analyzing the stress singularity of the plane V-notches and cracks in power law hardening materials is proposed. Firstly, the asymptotic displacement field in terms of radial coordinates in the notch tip region is adopted. As soon as the notch tip region appears the plastic deformation, the Von Mises yield criterion and total strain plastic theory are adopted. By introducing the displacement expressions into the governing differential equations of the plastic theory, a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions are proposed after a series of derivation. Then the interpolating matrix method is used to solve the eigenvalue problem by an iteration process. Several leading plastic stress singularity orders of plane V-notches and cracks can be obtained at a time. Simultaneously, the associated displacement and stress eigenvectors in the notch tip region have been determined as the same degree of accuracy with the corresponding singularity order. There are three to five significant figures in the computed values of the first three stress singularity orders obtained by the interpolating matrix method. Few of the published literatures gave out the effective and creditable second stress singularity order for the plane plastic V-notch.
6The extended boundary element method (XBEM) is proposed to determinate the singular stress field of the V-notched structures with local plastic deformation near notch tip region. Firstly, the elastic-plastic V-notched structure is divided into two parts, a small region around the V-notch tip and the remaining structure without the small region. Consider that the small region around the V-notch tip yields the plastic deformation due to the stress concentration. Secondly, based on the computed results of the multiple plastic stress singularity orders and the associated eigenfunctions in the singularity analysis for the small region, the displacement and stress components in the small region are expressed as the linear combinations of the finite terms of the series expansion with the leading singularity orders. The remaining structure is considered as linear elastic region, where the discrete boundary integral equations (BIE) can be established along the boundary of the remaining structures, included the connection border with the small region. Then the results of the eigen analysis for the stress singularity in the notch tip region are combined together with the discrete BIE, which is called XBEM. The XBEM can determine the elastic-plastic displacement and stress fields of the V-notched structures and the multiple plastic stress intensity factors at the notch tip.The stress solutions obtained by the XBEM is in agreement with analytical characteristics of local plastic singular stress field at the notch tip. Hence the XBEM provides an effective new way to investigate the fatigue fracture and crack propagation process of elastic-plastic V-notched and cracked structures.
引文
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