疲劳裂纹扩展数值模拟方法及其在核压力管道LBB分析中的应用
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摘要
本文针对核压力管道疲劳裂纹扩展的特点,提出一种可以直接施加本质边界条件的耦合有限元/无网格Galerkin算法(FE/EFG)。将整个计算模型划分成两种类型的子域,FE子域和EFG子域。在裂纹前沿附近区域设置EFG子域,其余区域设置FE子域,充分利用有限元法计算效率高,EFG方法计算精度高的特点,减少计算规模,提高计算精度。此外,在疲劳裂纹扩展模拟过程中,采用FE/EFG耦合算法,数值模型能够方便、快捷地得到更新。
根据有限元方法与无网格方法之间的内在联系,提出整个数值模型区域采用有限单元进行离散。将这些单元按其使用性质和位置分为A、B、C三类。A类单元为常规有限单元;B类单元为与耦合面相连的单元;C类单元为处于EFG子域中的单元且仅作为背景积分网格使用,为了提高计算精度,可对其进一步细分。通过这种单元分类方式,FE/EFG耦合算法的计算程序流程更加清晰明了,同时提高了程序的通用性。
本文针对当前三维裂纹应力强度因子计算方法中存在的困难,提出一种基于EFG方法的虚拟裂纹闭合法。在裂纹前沿附近区域设置一辅助有限元区域,用以计算该区域的刚度矩阵,将该刚度矩阵与辅助有限元节点位移相乘即可得到裂纹尖端的节点力,最终得到裂纹前沿的应变能释放率。通过应变能释放率与应力强度因子之间的关系确定应力强度因子。数值算例表明该方法计算精度高,即适合线性材料也适合非线性材料。
在疲劳裂纹扩展模拟中,为了利用EFG方法高精度、无网格性的优点,同时避免EFG子域过大而带来计算量增大的不足,提出一种动态子域划分法。将整个数值模型区域采用有限元网格进行离散。计算中,裂纹附近区域的有限元网格仅作为EFG方法背景积分网格使用,通过一定的尺寸参数控制裂纹前沿附近EFG子域的规模,且EFG子域可随着裂纹前沿动态移动。这样,计算中,EFG子域和FE子域都可动态调整,从而减少了计算规模,提高了计算效率。
采用Paris疲劳裂纹扩展公式对裂纹扩展过程进行数值模拟。根据压力管道破前漏(Leak before break,LBB)评估技术,只考虑I型疲劳裂纹扩展情况,这样裂纹扩展方向可由裂纹前沿的法线确定。采用本论文提出的数值方法模拟了中心裂纹板疲劳扩展过程并与实验结果作了比较,数值算例表明,两者结果一致;同时模拟了中心半椭圆形表面裂纹扩展的过程,计算得到的应力强度因子与文献中的结果一致。此外,数值模拟结果表明,半椭圆形表面裂纹扩展贯穿前裂纹沿深度方向扩展较快,贯穿后裂纹前沿逐渐向平直线发展,这与试验中观察到的现象一致。
根据压力管道破前漏分析技术(LBB),在裂纹贯穿管壁后,需计算裂纹张开面积(COA)进而确定管内介质泄漏量是否达到了泄漏监测系统能够监测到的最小泄漏量,因此裂纹张开面积是一个重要的计算参数。针对目前各种工程方法在计算裂纹张开面积中的不足,给出一种能够有效计算压力管道裂纹张开面积的数值算法,并用中心裂纹板裂纹张开面积的计算证明了方法的正确性,通过与工程算法的比较,证明了数值算法的有效性。
采用本论文提出的数值方法,模拟研究了典型核反应堆压力管道轴向、环向半椭圆形表面裂纹的疲劳扩展。数值模拟结果表明,内表面裂纹的扩展速率要稍快于外表面裂纹的扩展速率,两种类型的表面裂纹在贯穿前沿深度方向发展较快,贯穿后,裂纹前沿逐渐发展到平直状态。
最后,阐述了常规LBB分析方法的基本步骤,并以中国实验快堆余热排放系统中间回路中压力管道的LBB分析为例来说明本文所提压力管道疲劳裂纹扩展数值模拟方法在核压力管道评估中的应用。算例结果表明,采用本文数值模拟方法对核压力管道进行LBB评估是可行和有效的。
In this paper, according to the characteristics of fatigue crack propagation of nuclear pressure pipe, a coupled finite element/meshless Galerkin algorithm (FE/EFG) is presented, by which the essential boundary conditions can be imposed directly. In order to take advantage of the merits of FE method and EFG method, the whole numerical model is discretized into two types of sub-domains, FE sub-domain and EFG sub-domain. A discretize scheme which can ruduce computing scale and improve calculation accuracy is taken to determine the two types of sub-domans by which EFG sub-domain is used on the part closed to the crack front, the remain part FE sub-domain.
Based on the instrinc relation betwin the FE method and EFG method, the whole computational model is discretized using finite elements which are catigried into three types: A, conventional finite element, B, element linked to interface, C, element in EFG sub-domain, according to their utilize characteristics and location. C elements are only used as background integral grids and can be furtherly divided into several small sub-elements in order to improve computational precise. Using this kind of catigary scheme for elements, the apparentment and convention of computational programme can be improved evidently.
In view of the deficiency of current methods to calculate stress intensity factor, a virtual crack closure method based on EFG is proposed by which an auxiliary finite element domain is set up to calculate the stiffness matrix of local area around the crack tip. Nodal force of crack tip is determined by multiplying the matrix and nodal displacement vector of the auxiliary finite element. Finally, strain energy release rate of crack front is determined, which can be eventually changed into stress intensity factor. It is showed by several numerical examples that the current method has high computational precision and is not only suitable for the linear material but also for nolinear material.
In the simulation of fatigue crack propogation, in order to take advantage of the merit of EFG method and advoid the calculation increase due to the expand of EFG sub-domain, a kind of dynamic sub-domain partition method is proposed. Using the method, finite element grids are used to discretize the whole domain and the grids closed to crack front are only used as background integral grid for EFG method. EFG sub-domain can dynamically move together with crack front using a certain size parameter controlling the scale of EFG sub-domain near to crack front. Thus, EFG sub-domain and FE sub-domain can be adjusted dynamically, which greatly reduces the computing scale and increase calculation efficiency.
Paris law is used to determine the stepinterval of crack growth. According to the leak before break technique (LBB), only mode I crack is considered in the study. Thus, the normal of crack front can be used as the crack propagation direction. The current method is used to simulate the extension of centre through crack and semic-elliptic surface crack. It is showed that the result including stress intensity factor obtained from current method is consistent to those from literature. Before penetration, the crack grow faster in depth than in other direction. After penetration, the shape of the crack front grow toward the shape of centre-penetrate crack, which is consistent to phenomenon observed in experiment.
According to the LBB method, after the penetration, the crack opening area (COA) should be determined to estimate whether the medium leakage in pressure pipe is enough to be detected by monitor system of refrigerant. Therefore, crack opening area is a very important parameter for LBB technology. Noting the drawback of engineering algorithm for COA calculation, an numerical method is proposed to determine the COA of cracked pressure pipe and it’s efficiency is proved by means of the COA calculation of centre-crack panel and pressure pipe.
The method presented in this paper is also used to simulate the propagation of surface crack of pressure pipe including axial surface semic-elliptic crack and circumferential surface semic-elliptic crack. It was found that the growth rate of inner surface semic-elliptic crack is faster than those of outer urface semic-elliptic crack and both two types of crack grow faster in depth direction than the other direction before penetration. After penetration, the shape of the crack front develop toward the shape of centre-penetrate crack, which is consistent to phenomenon observed in experiment.
Finally, the basic steps of conventional LBB method are presented and the application of numerical method in the method is introduced by means of a typical example of LBB evaluation. The result from the example indicates that it is feasible to adopt numerical method in this paper to analyse the fracture of nuclear pressure pipe.
根据有限元方法与无网格方法之间的内在联系,提出整个数值模型区域采用有限单元进行离散。将这些单元按其使用性质和位置分为A、B、C三类。A类单元为常规有限单元;B类单元为与耦合面相连的单元;C类单元为处于EFG子域中的单元且仅作为背景积分网格使用,为了提高计算精度,可对其进一步细分。通过这种单元分类方式,FE/EFG耦合算法的计算程序流程更加清晰明了,同时提高了程序的通用性。
本文针对当前三维裂纹应力强度因子计算方法中存在的困难,提出一种基于EFG方法的虚拟裂纹闭合法。在裂纹前沿附近区域设置一辅助有限元区域,用以计算该区域的刚度矩阵,将该刚度矩阵与辅助有限元节点位移相乘即可得到裂纹尖端的节点力,最终得到裂纹前沿的应变能释放率。通过应变能释放率与应力强度因子之间的关系确定应力强度因子。数值算例表明该方法计算精度高,即适合线性材料也适合非线性材料。
在疲劳裂纹扩展模拟中,为了利用EFG方法高精度、无网格性的优点,同时避免EFG子域过大而带来计算量增大的不足,提出一种动态子域划分法。将整个数值模型区域采用有限元网格进行离散。计算中,裂纹附近区域的有限元网格仅作为EFG方法背景积分网格使用,通过一定的尺寸参数控制裂纹前沿附近EFG子域的规模,且EFG子域可随着裂纹前沿动态移动。这样,计算中,EFG子域和FE子域都可动态调整,从而减少了计算规模,提高了计算效率。
采用Paris疲劳裂纹扩展公式对裂纹扩展过程进行数值模拟。根据压力管道破前漏(Leak before break,LBB)评估技术,只考虑I型疲劳裂纹扩展情况,这样裂纹扩展方向可由裂纹前沿的法线确定。采用本论文提出的数值方法模拟了中心裂纹板疲劳扩展过程并与实验结果作了比较,数值算例表明,两者结果一致;同时模拟了中心半椭圆形表面裂纹扩展的过程,计算得到的应力强度因子与文献中的结果一致。此外,数值模拟结果表明,半椭圆形表面裂纹扩展贯穿前裂纹沿深度方向扩展较快,贯穿后裂纹前沿逐渐向平直线发展,这与试验中观察到的现象一致。
根据压力管道破前漏分析技术(LBB),在裂纹贯穿管壁后,需计算裂纹张开面积(COA)进而确定管内介质泄漏量是否达到了泄漏监测系统能够监测到的最小泄漏量,因此裂纹张开面积是一个重要的计算参数。针对目前各种工程方法在计算裂纹张开面积中的不足,给出一种能够有效计算压力管道裂纹张开面积的数值算法,并用中心裂纹板裂纹张开面积的计算证明了方法的正确性,通过与工程算法的比较,证明了数值算法的有效性。
采用本论文提出的数值方法,模拟研究了典型核反应堆压力管道轴向、环向半椭圆形表面裂纹的疲劳扩展。数值模拟结果表明,内表面裂纹的扩展速率要稍快于外表面裂纹的扩展速率,两种类型的表面裂纹在贯穿前沿深度方向发展较快,贯穿后,裂纹前沿逐渐发展到平直状态。
最后,阐述了常规LBB分析方法的基本步骤,并以中国实验快堆余热排放系统中间回路中压力管道的LBB分析为例来说明本文所提压力管道疲劳裂纹扩展数值模拟方法在核压力管道评估中的应用。算例结果表明,采用本文数值模拟方法对核压力管道进行LBB评估是可行和有效的。
In this paper, according to the characteristics of fatigue crack propagation of nuclear pressure pipe, a coupled finite element/meshless Galerkin algorithm (FE/EFG) is presented, by which the essential boundary conditions can be imposed directly. In order to take advantage of the merits of FE method and EFG method, the whole numerical model is discretized into two types of sub-domains, FE sub-domain and EFG sub-domain. A discretize scheme which can ruduce computing scale and improve calculation accuracy is taken to determine the two types of sub-domans by which EFG sub-domain is used on the part closed to the crack front, the remain part FE sub-domain.
Based on the instrinc relation betwin the FE method and EFG method, the whole computational model is discretized using finite elements which are catigried into three types: A, conventional finite element, B, element linked to interface, C, element in EFG sub-domain, according to their utilize characteristics and location. C elements are only used as background integral grids and can be furtherly divided into several small sub-elements in order to improve computational precise. Using this kind of catigary scheme for elements, the apparentment and convention of computational programme can be improved evidently.
In view of the deficiency of current methods to calculate stress intensity factor, a virtual crack closure method based on EFG is proposed by which an auxiliary finite element domain is set up to calculate the stiffness matrix of local area around the crack tip. Nodal force of crack tip is determined by multiplying the matrix and nodal displacement vector of the auxiliary finite element. Finally, strain energy release rate of crack front is determined, which can be eventually changed into stress intensity factor. It is showed by several numerical examples that the current method has high computational precision and is not only suitable for the linear material but also for nolinear material.
In the simulation of fatigue crack propogation, in order to take advantage of the merit of EFG method and advoid the calculation increase due to the expand of EFG sub-domain, a kind of dynamic sub-domain partition method is proposed. Using the method, finite element grids are used to discretize the whole domain and the grids closed to crack front are only used as background integral grid for EFG method. EFG sub-domain can dynamically move together with crack front using a certain size parameter controlling the scale of EFG sub-domain near to crack front. Thus, EFG sub-domain and FE sub-domain can be adjusted dynamically, which greatly reduces the computing scale and increase calculation efficiency.
Paris law is used to determine the stepinterval of crack growth. According to the leak before break technique (LBB), only mode I crack is considered in the study. Thus, the normal of crack front can be used as the crack propagation direction. The current method is used to simulate the extension of centre through crack and semic-elliptic surface crack. It is showed that the result including stress intensity factor obtained from current method is consistent to those from literature. Before penetration, the crack grow faster in depth than in other direction. After penetration, the shape of the crack front grow toward the shape of centre-penetrate crack, which is consistent to phenomenon observed in experiment.
According to the LBB method, after the penetration, the crack opening area (COA) should be determined to estimate whether the medium leakage in pressure pipe is enough to be detected by monitor system of refrigerant. Therefore, crack opening area is a very important parameter for LBB technology. Noting the drawback of engineering algorithm for COA calculation, an numerical method is proposed to determine the COA of cracked pressure pipe and it’s efficiency is proved by means of the COA calculation of centre-crack panel and pressure pipe.
The method presented in this paper is also used to simulate the propagation of surface crack of pressure pipe including axial surface semic-elliptic crack and circumferential surface semic-elliptic crack. It was found that the growth rate of inner surface semic-elliptic crack is faster than those of outer urface semic-elliptic crack and both two types of crack grow faster in depth direction than the other direction before penetration. After penetration, the shape of the crack front develop toward the shape of centre-penetrate crack, which is consistent to phenomenon observed in experiment.
Finally, the basic steps of conventional LBB method are presented and the application of numerical method in the method is introduced by means of a typical example of LBB evaluation. The result from the example indicates that it is feasible to adopt numerical method in this paper to analyse the fracture of nuclear pressure pipe.
引文
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