工程科学与技术   2018, Vol. 50 Issue (5): 239-243

Study on Residual Life of Pressure Vessels with Immerged Cracks Based on Defect Security Attenuation Path Simulation
LONG Wei, YAN Jiabing, LI Yanyan, LIU Huoguo, HUANG Jimin
School of Manufacturing Sci. and Eng., Sichuan Univ., Chengdu 610065, China
Abstract: The study of crack propagation is the key and foundation of pressure vessels’ residual life prediction. In view of security assessment for pressure vessels with immerged cracks, based on the security attenuation path simulation, another calculation method is put forward to predict the residual life. First of all, the iterative calculation based on Paris fatigue crack growth formula is conducted to build the model of geometric correlation change in crack length and depth. Next, this correlation model is used to simulate the security attenuation path track of immersed crack defects in R6 evaluation diagram. The immersed crack change represented by path track can be divided into immersed crack phase and surface crack phase. Then the numerical integration method is adopted to get the residual track segment of security attenuation path in different phases, and the calculation model of immersed crack safety margin is built. This calculation model can show related dynamic safety margins for different sizes of crack defect, and reflect the safety degree of pressure vessels. After building the mechanical model of pressure vessels’ crack defect, the stress intensity factor of special crack point in finite element method is calculated. And weight function method is adopted to get the function formula of the stress intensity factor range K of crack defect. The function formula of fatigue circulation time N is calculate by Paris formula integral. Lastly, according to the model of geometric correlation change in crack length and depth, the fatigue circulation times of immersed cracks is calculated by means of integration.
Key words: pressure vessel    buried crack    residual life    safety margin    weight function

 图1 射线法和平行线法评估缺陷安全性的示例 Fig. 1 Example of defect security assessment with radial method and parallel method

1 疲劳裂纹扩展理论

 $\frac{{{\rm d}a}}{{{\rm d}N}} = C{(\Delta K)^m}$ (1)

Newman等[9]假设表面裂纹在成长过程中一直保持半椭圆形状，且只采用一对Paris疲劳裂纹扩展公式进行计算以确定半椭圆形状。基于这个假设，埋藏裂纹在成长过程中一直保持椭圆形状，其从原点可剖分成两个半椭圆形状，类比于表面裂纹形式，同样只采用一对Paris疲劳裂纹扩展公式进行计算以确定埋藏裂纹的椭圆形状。即表面裂纹和埋藏裂纹可采用同一对Paris疲劳裂纹扩展公式表示其裂纹深度尖端和长度尖端与疲劳应力循环次数的关系，其数学表达式为：

 $\frac{{{\rm d}a}}{{{\rm d}N}} = {C_{\rm A}}{(\Delta {K_{\rm A}})^m}$ (2)
 $\frac{{{\rm d}c}}{{{\rm d}N}} = {C_{\rm C}}{(\Delta {K_{\rm C}})^m}$ (3)

 $\frac{{{\rm d}a}}{{{\rm d}c}} = {\left(\frac{{\Delta {K_{\rm A}}}}{{\Delta {K_{\rm C}}}}\right)^m}$ (4)

 \left\{ {\begin{aligned}& {\Delta {K_{\rm A}} = Y\Delta {\sigma _{\rm A}}\sqrt {{\text{π}} a} } \;,\\ & {\Delta {K_{\rm C}} = Y\Delta {\sigma _{\rm C}}\sqrt {{\text{π}} c} } \end{aligned}} \right. (5)

 $\Delta c/\Delta a = {\left( {{c_i}/{a_i}} \right)^{\textstyle\frac{m}{2}}}$ (6)

2 安全衰减路径与安全裕度模型

 图2 裂纹安全衰减路径仿真图 Fig. 2 Simulation chart of crack safety decay path

 S = \left\{ {\begin{aligned}& {{S\!\!_1}},\\ & {{S\!\!_2}}\end{aligned}} \right. (7)

 $A = 1 - \frac{{f(x)}}{{{f_{{\text{总}}}}}}$ (8)

3 剩余寿命的计算 3.1 权函数法

 $K = \int_0^a {\sigma (x)} m(a,x){\rm d}x$ (9)
 $m(a,x) = \frac{H}{K}\frac{{\partial U(a,x)}}{{\partial a}}$ (10)

 $\sigma (x) = {\sigma _0}\sum\limits_{n = 0}^n {{S\!\!_n}} {x^n}$ (11)

 $m(a,x) = \frac{2}{{\sqrt {2{\text{π}} (a - x)} }}\sum\limits_{j = 1}^3 {{C_j}} {(1 - \frac{x}{a})^{j/2}}$ (12)

3.2 剩余寿命计算

 $N = \int_0^N {{\rm d}N = \int_{{a_0}}^{{a_N}} {\frac{{{\rm d}a}}{{C{{(\Delta K)}^m}}}} }$ (13)

1）根据式（6），计算裂纹从初始尺寸 ${c_0}$ ${a_0}$ 扩展到下一尺寸 ${c_1}$ ${a_1}$ ，并用权函数方法求取当前时刻应力强度因子，再求出当前时刻的应力强度因子幅 $\Delta {K_1}$

2）利用Paris积分公式，求取裂纹从初始尺寸 ${c_0}$ ${a_0}$ 扩展到下一尺寸 ${c_1}$ ${a_1}$ 所经历的循环次数 ${N_1}$

3）若裂纹的尺寸没有达到裂纹的失效尺寸，则重复第1）、2）步骤，依次求取下一尺寸的循环次数 ${N_2}$ ${N_3}$ ${N_4}\cdots,{N_s}$ ，直至裂纹失效；

4）将前面求取的循环次数相加，即为裂纹的最终剩余寿命 $N$

 $N = \sum\limits_1^s {{N_i}}$ (14)

4 结　论

1）利用Paris公式的迭代处理，当长度和深度变化量足够小时，可得到关于埋藏裂纹的长度和深度几何关联变化模型，并通过其变化模型在R6评定图上建立埋藏裂纹的安全衰减路径仿真图，能够准确、有效地反应埋藏裂纹的扩展情况。基于裂纹的安全衰减路径轨迹可以在理论上给出裂纹缺陷在任何尺寸时对应的动态安全裕度，反应压力容器的安全程度。

2）运用裂纹扩展规律和安全衰减路径轨迹，通过权函数和Paris分步积分的方法，建立起含埋藏裂纹压力容器的剩余寿命模型。

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