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摘要: 断裂力学是工程材料和结构的疲劳与断裂分析、损伤容限设计和结构完整性评定的理论基础. 应力强度因子作为线弹性裂纹尖端奇异场的单一表征参量和裂纹扩展驱动力, 在裂纹体的断裂力学分析中发挥着关键作用. 权函数法为复杂受载裂纹体的应力强度因子求解计算提供了强有力的解析工具, 不但具有远高于各类数值解法的计算效率, 而且精度可靠, 使用方便. 本文结合笔者团队在权函数法方面的长期研究工作, 对该方法自20世纪70年代初提出至今半个世纪以来, 国际断裂界在二维和三维权函数理论与应用方面的主要研究进展作了回顾和评述, 并对其未来发展提出了展望. 主要内容涵盖: 当前国际断裂界广泛应用的3种二维裂纹解析权函数法简介和以格林函数为基准的验证评价; 三维裂纹问题的片条合成权函数法和点载荷权函数法; 权函数法在复杂受载裂纹体的应力强度因子和裂纹张开位移等关键力学参量计算、内聚力/桥连等裂纹模型分析、共线多裂纹权函数理论及其在剩余强度预测等方面的应用, 以及复杂裂纹几何的工程化权函数分析和权函数法的反向应用问题.Abstract: Fracture mechanics is the theoretical foundation for fatigue and fracture analyses of engineering materials and structures, damage tolerance design, and structural integrity assessment. Being the single characterizing parameter of the linear elastic crack tip singular stress/strain field and the crack deriving force, the stress intensity factor (SIF) plays a vital role in fracture mechanics analysis. The weight function method (WFM) is a powerful method for the determination of SIFs for cracks under complex load conditions, with remarkable computational efficiency and reliable solution accuracy, and is easy to use. Combined with the authors’ teamwork on WFM research in the past several decades, this article presents a comprehensive review of the historical developments of various WFMs over the past 50 years and also a brief outlook. The main topics include: a brief introduction of three types of analytical weight function approaches for 2D crack problems and accuracy verification based on Green’s functions; the slice synthesis weight function method and the point weight function method for 3D crack problems; various practical applications of WFMs, including determination of the key fracture mechanics parameters of SIFs and crack opening displacements under complex loadings, cohesive/bridging model analyses, WFM for multiple collinear cracks and residual strength prediction of panels containing multiple site damage, engineering weight function approaches to complex crack configurations, and inverse application of WFM for the determination of stress distributions in un-cracked bodies.
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图 1 推动断裂力学和损伤容限设计准则发展的“里程碑”事故. (a)斯克内克塔迪自由轮在平静的港湾中整体断裂为两段(Tetelman & McEvily 1967), 该事故引起人们对裂纹尖端场的关注与研究; (b) 锻造缺陷引起的裂纹导致美国空军F-111战斗轰炸机机翼断裂坠毁, 该事故推动了损伤容限设计准则和规范的建立(Wanhill 2003); (c) 多位置损伤MSD导致阿罗哈航空公司波音737-200机身压力舱上部蒙皮被撕脱,该事故改变了大型运输机的适航规范 (Wanhill 2003, FAA 2010); (d) 疲劳裂纹导致CF6-6D航空发动机钛合金一级风扇盘破裂, DC-10客机坠毁(McEvily 2002), 它推动了发动机损伤容限设计准则和规范的建立.
图 3 三种裂纹几何的COD权函数解与精确解及有限元解比较. (a) 无限板周期性共线裂纹受裂纹面均匀应力, (b) 圆盘径向边缘裂纹受裂纹面均匀应力, (c) 无限板孔边径向双裂纹受远方拉伸
图 20 有限宽板单边缘裂纹4种格林函数与WCTSE的比较. (a) Bueckner与WCTSE, (b) Wu-Carlsson与WCTSE, (c) Fett-Munz与WCTSE, (d) Glinka-Shen与WCTSE
图 22 圆盘中心和边缘裂纹的三种II型格林函数结果比较. (a) 中心裂纹, (b) 边缘裂纹 (Xu et al. 2020a)
图 23 正交各向异性(ρ = 6.395)有限板单边裂纹的格林函数以及与各向同性板的比较. (a) 规范化解析权函数法和有限元法获得的格林函数比较, (b) 各向异性与各向同性有限板单边裂纹格林函数比较
图 27 三种权函数法对有限宽板单边缘裂纹参考解精度的敏感性比较(吴学仁等2019). (a) 远方拉伸的两个无量纲应力强度因子曲线, (b) Wu-Carlsson, (c) Fett-Munz, (d) Glinka-Shen
图 28 由Glinka-Shen通用权函数法得到的有限板单边缘裂纹格林函数(采用几何条件m´ = 0). 参考载荷: (a) 拉伸+弯曲, (b) 拉伸+集中力 (吴学仁等2019)
图 31 有限板圆孔边四分之一椭圆角裂纹的片条分析模型. (a) 三维裂纹体, (b) 片条裂纹长度及椭圆参量角, (c) 基本片条(a片条), (d) 弹簧片条(c片条).
图 32 (a) 无限板圆孔边沿x和y两个方向变化的三维应力场, (b) 用SSWFM结合二维和三维有限元分析的应力场得到的孔边角裂纹无量纲应力强度因子. 实线: 用2D应力解, 虚线: 用3D应力解 (Zhao et al. 1997b)
图 33 (a) 孔边表面裂纹远方拉伸下的应力强度因子: SSWFM结果与有限元解的比较(Newman et al. 1994, Wu et al. 1998), (b) 孔边对称双角裂纹远方拉伸下的应力强度因子: SSWFM与各种数值解法的比较(Bakuckas 2001)
图 34 远方拉伸SENT半圆缺口试样三维应力强度因子: SSWFM与有限元结果以及拟合方程比较. (a) 表面裂纹, (b) 角裂纹 (Newman et al. 1994; Wu et al. 1998)
图 35 9种方法求得的远方拉伸无限板孔边角裂纹(r/t = 1.0, a/c = 1.0, a/t = 0.2)三维应力强度因子比较(ERSI-USA). (a) r/W = 0.125, (b) r/W = 0.4167 (Newman & Wu 2021)
图 36 无限板孔边两条非对称角裂纹(r/t = 1.0)在远方拉伸和面外弯曲载荷下的三维应力强度因子: SSWFM与有限元软件Franc3D结果的比较. 裂纹面应力分布σ(x, y)采用了3D有限元分析结果(Zhang et al. 2022)
图 39 有限板半椭圆表面裂纹在3种双变应力作用下的无量纲应力强度因子3种解法的结果比较. 点载荷权函数法, 边界元数值法- FADD3D, 有限元法- FEACrack (McClung et al. 2013)
图 40 点载荷权函数法(PWFM)与边界元法(FADD3D)计算的孔边角裂纹无量纲应力强度因子比较. (a) 拉伸, (b) 弯曲 (McClung et al. 2013)
图 41 平板三维裂纹权函数的参数定义. (a) 半椭圆表面裂纹, (b) 四分之一椭圆角裂纹 (Zheng et al. 1996)
图 43 含单/双边缘裂纹圆盘在一对径向集中力P作用下的II型无量纲应力强度因子: 规范化解析权函数解与有限元解的比较. (a) 单裂纹, (b) 双裂纹 (Xu et al. 2020a; Wu & Xu 2022)
图 44 典型热应力和残余应力分布. (a) 圆柱体热冲击应力, (b) 冷挤压孔边残余应力, (c) 飞机机翼铝合金锻件的简化残余应力, (d) 圆管焊缝轴向残余应力 (吴学仁等 2019)
图 46 两种残余应力场作用下的无量纲应力强度因子. (a) 有限宽板激光冲击残余应力和三种方法计算的边缘裂纹K结果的比较 (Ribeiro & Hill 2016), (b) 无限板冷挤压孔的残余应力(虚线)和不同挤压量孔边裂纹的K
图 47 权函数/格林函数法在洛−马公司联合攻击机含残余应力的机翼梁锻件损伤容限设计中的应用成效. (a) 最终设计许用应力与基线设计许用应力的比值, (b) 因许用应力提高导致的翼梁970个控制部位的局部厚度相对变化(Ball 2008)
图 48 (a) 内聚力水力压裂模型(刘曰武等 2019, Chen 2012), (b) 孔边裂纹水力压裂分析模型(Dong et al. 2018)
图 49 混凝土重力大坝高压水劈裂分析模型. (a) 扩展有限元模型(XFEM), (b) 裂纹面水压分布 (Wang et al. 2015)
图 53 在激光冲击导致的剧烈变化残余应力作用下, 有限宽板边缘裂纹张开位移的解析权函数解与有限元结果比较(a/W = 0.25, 0.6). 图中左下方的插图是残余应力分布 (Ribeiro & Hill 2016)
图 57 用权函数法得到的桥连应力结果验证 (Buchanan et al. 1997). (a) 预设的3种桥连应力分布, (b) 采用权函数法结合最小二乘法获得的桥连应力与有限元结果的对比
图 58 用权函数法结合桥连应力离散化方法求得的无限板中心裂纹的内聚力模型解(Xu & Waas 2017). (a) 裂纹尖端过程区的大小及应力分布, (b) 裂纹尖端过程区内的张开位移
图 59 用权函数法求得的几种裂纹几何的裂纹张开应力以及与有限元等数值解的比较. (a) 半无限大板边缘裂纹的裂纹张开应力(R = 0), (b) 缺口和孔边裂纹的张开应力与裂纹长度与缺口/孔半径的关系(平面应力) (吴学仁等 2019)
图 60 三条等长裂纹受集中载荷及其无量纲应力强度因子解. (a) 各裂纹中心线受一对对称集中力, (b) 裂纹尖端A的无量纲应力强度因子, (c) 裂纹尖端B的无量纲应力强度因子, (d) 裂纹尖端C的无量纲应力强度因子 (Xu & Wu 2012)
图 61 (a) 加筋板三条对称共线裂纹受远端均匀应力, (b) 中心裂纹应力强度因子与文献结果比较(Zhang et al. 2020)
图 62 (a) 两条等长裂纹, (b) 三条对称裂纹, (c) 两条等长裂纹, 内侧裂尖A的无量纲应力强度因子, (d) 三条对称裂纹, 内侧裂尖A的无量纲应力强度因子 (Xu et al. 2011, Xu & Wu 2012,徐武 2012)
图 63 裂纹尖端A的塑性区尺寸与外载荷σ/σs关系, r0 = a0[sec(0.5πσ/σs). (a) 两条等长裂纹, (b) 三条等长裂纹(Xu et al. 2011; Xu & Wu 2012)
图 65 连续分布应力作用的分析模型. (a) 部分裂纹面受连续分布应力σ(x)作用, (b)一系列离散的部分裂纹面受均匀应力作用以模拟情况(a), (c) 单一中心裂纹部分裂纹面[xi, xi+1]受均匀载荷作用
图 66 裂纹尖端正则化塑性区尺寸和张开位移与外载荷关系r0 = a[sec(0.5πσ/σs) − 1], δ0 = 8aσs/(πE)ln[sec(0.5πσ/σs)], (2a + b)/w = 0.5. (a) 裂纹尖端A的塑性区尺寸, (b) 裂纹张开位移, a = b = 1/6 (徐武2012, Xu et al. 2014)
图 67 基于单裂纹权函数的“统一”分析方法和有限元法预测的共线多裂纹板剩余强度比较 (Xu et al. 2014, 吴学仁2019)
图 69 (a) 无限大板中心裂纹受焊接残余应力作用; (b) 用逆向权函数法求得的无限大板中心裂纹所受残余应力σ(ξ)/σ0, 以及与已知精确解的比较 (吴学仁等 2019)
图 70 利用替代几何计算复杂裂纹体的应力强度因子(Zerbst et al. 2007)
图 71 替代几何示例. (a) T形焊接板和平直板, (b) 远方拉伸T形板焊趾根部的无裂纹应力分布, (c) T形板和平直板的格林函数比较, (d) 4种方法计算的应力强度因子及与有限元结果比较 (吴学仁等 2019)
图 72 (a) 权函数合成原理(weight function composition principle), (b) 用合成权函数法得到的有限板半圆缺口边缘裂纹受纯弯曲的应力强度因子以及与Wu-Carlsson (1991)权函数结果比较(Brennan & Teh 2004)
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