力学学报 2018 , 50 (2): 307-314https://doi.org/10.6052/0459-1879-17-412

Orginal Article

热对流作用下筒壁涂层的边裂行为

彭中伏 ¹ ,陈学军 ¹ ^, ² ^,

1北京科技大学应用力学系,北京 100083
2.北京科技大学磁光电复合材料与界面科学北京市重点实验室,北京 100083

EDGE CRACKING BEHAVIOR OF A COATED HOLLOW CYLINDER DUE TO THERMAL CONVECTION

Peng Zhongfu ¹ ,Chen Xuejun ¹ ^, ² ^,

1.Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China
2.Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, Beijing 100083, China

中图分类号:O346.1

文献标识码:A

收稿日期:2017-12-11

接受日期:2017-12-11

网络出版日期:2018-04-17

基金资助: 国家自然科学基金资助项目(51171026).

作者简介:

作者简介：陈学军，副教授，主要研究方向：涂层/薄膜力学. E-mail:chenxuejun@ustb.edu.cn

展开

摘要

边裂(边缘开裂)是涂层热致损伤的主要模式之一. 边缘裂纹穿透涂层后,常导致界面脱粘从而驱使涂层与基体剥离,最终丧失对基体的保护作用. 本文以热应力强度因子表征边缘裂纹的扩展驱动力,研究筒壁涂层在热对流作用下的边裂行为. 首先,利用拉普拉斯变换法,得到了瞬态温度场及热应力场的封闭解. 其次,运用Fett等的三参数法确定了筒壁涂层边缘裂纹的权函数. 最后,基于叠加原理和权函数方法计算了边缘裂纹的热应力强度因子. 探讨了无量纲时间、边缘裂纹深度、基体/涂层厚度比、热对流强度等参数对热应力强度因子的影响规律. 结果表明：热应力强度因子的峰值既非发生在热载荷初始时刻,也非发生在热稳态时刻,而出现在时间历程的中间时刻;增大热对流强度不仅可提高热应力强度因子的峰值,而且使峰值提前出现;其他条件相同时,热应力强度因子随着边缘裂纹长度的增大而降低;增大涂层厚度或减小基体厚度可增强涂层抵抗瞬态热载荷的能力.

关键词： 热对流 ; 涂层 ; 边裂 ; 权函数 ; 应力强度因子

Abstract

Edge cracking is one of major damage modes for coatings subjected to thermal transients. After penetrating across coating thickness, edge cracks usually cause interfacial decohesion and hence result in the detachment of coating from substrate, which leads to the ultimate loss of the protective effect on the substrate. The edge cracking behavior due to thermal convection is studied in this paper for a coated hollow cylinder, where the thermal stress intensity factor is used to characterize the crack driving force. Firstly, by using the Laplace transform technique, closed-form solutions are obtained for the transient temperature as well as thermal stresses. Secondly, the weight function for an edge crack in a coated hollow cylinder is determined by using the three-parameter method proposed by Fett et al. Finally, the thermal stress intensity factor at the edge crack tip is evaluated based on the principle of superposition and the derived weight function. The dependence of the normalized thermal stress intensity factor is examined on the normalized time, edge crack depth, substrate/coating thickness ratio as well as thermal convection severity. It is shown that the peak thermal stress intensity factor occurs neither at the very beginning nor at the thermal steady state of a thermal transient, but at an intermediate instant. The severer thermal convection generates a peak thermal stress intensity factor not only higher in magnitude but also earlier in time. Should other conditions remain invariant, the thermal stress intensity factor is a decreasing function of the edge crack depth; a thicker coating or a thinner substrate may enhance the thermal transient resistance of a coating.

Keywords： thermal convection ; coating ; edge cracking ; weight function ; stress intensity factor

PDF (12728KB)元数据多维度评价相关文章收藏文章

本文引用格式导出EndNote Ris Bibtex

彭中伏 , 陈学军 . 热对流作用下筒壁涂层的边裂行为 [J]. 力学学报 , 2018, 50(2): 307-314https://doi.org/10.6052/0459-1879-17-412

Peng Zhongfu , Chen Xuejun . EDGE CRACKING BEHAVIOR OF A COATED HOLLOW CYLINDER DUE TO THERMAL CONVECTION [J]. Chinese Journal of Theoretical and Applied Mechanics , 2018, 50(2): 307-314https://doi.org/10.6052/0459-1879-17-412

引言

涂层技术在航空发动机、枪炮身管、核反应堆管道等国防及民用工业的关键热端部件中得到了广泛应用. 由于防护涂层通常具有优良的特殊性能(例如,耐高温、强热障、抗烧蚀等),能有效保护常为失效源发地的构件表面,相关构件的使用寿命得以显著延长. 但随着服役工况要求的不断提高,其寿命问题依然日益突出. 大量实验证据证实^[1,2,3,4,5],这是由于防护涂层过早地出现开裂与剥落现象,导致其优良性能难以充分发挥所致. 因此,使涂层在服役期间保持完整并附着于基体以延长其服役期限,是提高上述构件寿命的关键.

实际上,热端部件之防护涂层主要承受由温度梯度引起的热应力. 一旦该应力超过涂层材料的强度极限,涂层表面将萌生边缘裂纹(edge crack). 在周期热载荷的反复作用下,边缘裂纹穿透涂层至界面,导致界面脱粘从而驱使涂层与基体剥离,最终丧失对基体的保护作用^[1,2,3,4,5]. 对枪炮身管而言,贯穿涂层后的边缘裂纹即使未能导致涂层剥落,也将作为腐蚀气体的流动通道,造成界面及基体损伤^[3,6].

国内外相关文献表明^{[1,2,3,4,5,6,7,8,9,10]}：由于涂层与基体材料的热、物理性能不同,两者间产生失配应变 (mismatch strain),促使涂层开裂与剥落. 因此,人们一方面从改善涂层/基体的失配应变角度出发,设计出多层涂层、功能梯度涂层、纳米结构涂层等^[7,8,9]. 另一方面,为增强涂层的剥落抗力,研究者通过增强基体与涂层之间的结合性能来抑制界面裂纹的形成与扩展^[10,11].

基于断裂力学的观点,也可通过降低边缘裂纹的热致扩展驱动力来延缓涂层开裂,这就要求计算裂纹尖端的应力强度因子. 此外,为准确预测边缘裂纹的扩展速率,也需确定热应力强度因子的主要影响因素及相关影响规律. 就热载荷而言,考虑其瞬态特征比仅限于热稳态分析更符合实际工况^[1,2,3]. 有关涂层边裂纹的应力强度因子的计算方法很多^[12,13,14],如积分变换法、有限元法、奇异积分法等. 由于瞬态热应力在涂层厚度方向分布极度不均匀,宜采用权函数方法. 众所周知,权函数法^{[15,16,17,18,19,20,21,22]}是计算应力强度因子的简单、高效方法,它只依赖于裂纹体的几何构形,而与受载情况无关. 一旦已知裂纹体的权函数,可通过简单的线积分确定任意分布应力下的应力强度因子.

本文考虑温度场的瞬态特征,研究筒壁涂层在热对流作用下的边裂行为. 必须指出：在大多数情况下,实际涂层具有多条边缘裂纹. 为简单起见,本文仅考虑单条边缘裂纹,运用Fett等^[21,22]的三参数法确定边缘裂纹的权函数,基于叠加原理和权函数法计算其热应力强度因子. 着重探讨无量纲时间、边缘裂纹深度、基体/涂层厚度比、热对流强度等参数对热应力强度因子的影响规律. 研究结果将为筒壁涂层在热环境下的安全服役提供有益建议.

1 温度场及热应力场

考虑筒壁涂层/基体系统,其几何模型如图1所示,其中#1代表涂层,#2代表基体. 基体圆筒的内、外半径分别为 $a$ , $b$ ,涂层厚度为 $h$ (值为 $a - a_{0})$ ,边缘裂纹长度为 $a_{1}$ .假设涂层和基体材料都是均匀、各向同性的线弹性材料,所有力学性能均与温度无关, 且忽略热--弹耦合效应和惯性效应.

显示原图|下载原图ZIP|生成PPT

图1 筒壁涂层边缘裂纹简图

Fig. 1 Schematic of a coated hollow cylinder with an edge crack

设涂层/基体系统的初始温度为 $T_{0}$ ,圆筒外表面绝热. 在 $t$ =0时刻,涂层内表面( $r = a_{0}$ )受到对流冷却作用, 其中对流介质温度为 $T_{g}$ ,对流换热系数为 $H_{g}$ .由于裂纹面沿径向,不影响瞬态温度场的分布. 定义温度变化： $θ_{i} (r, t) = T_{i} (r, t) - T_{0}$ , $θ_{0} = T_{g} - T_{0}$ ,其中 $T_{i} (r, t)$ ( $i = 1$ , 2)分别表示涂层和基体的温度.

采用柱坐标系,涂层和基体的温度场满足如下热扩散方程^[23]

$\frac{\partial^{2} θ_{i} (r, t)}{\partial r^{2}} + \frac{1}{r} \frac{\partial θ_{i} (r, t)}{\partial r} = \frac{1}{D_{i}} \frac{\partial θ_{i} (r, t)}{\partial t} (i = 1,2)$ (1)

初始条件为

$θ_{1} (r, 0) = 0 (a_{0} < r < a)$ （2）

$θ_{2} (r, 0) = 0 (a < r < b)$ （3）

边界条件为

$k_{1} \frac{\partial}{\partial r} θ_{1} (a_{0}, t) = H_{g} [θ_{1} (a_{0}, t) - θ_{0}]$ （4）

$\frac{\partial θ_{2} (b, t)}{\partial r} = 0$ （5）

假设涂层/基体界面为理想结合,则温度和热流密度在界面处( $r = a)$ 连续, 即

$θ_{1} (a, t) = θ_{2} (a, t)$ （6）

$k_{1} \frac{\partial}{\partial r} θ_{1} (a, t) = k_{2} \frac{\partial}{\partial r} θ_{2} (a, t)$ （7）

以上各式中, $D = k / ρ c$ 为热扩散系数, $k$ , $ρ$ , $c$ 分别为热传导系数,密度和比热容. 定义下列无量纲参数^[24]

$\begin{matrix} \begin{matrix} {\overset{̅}{θ}}_{1} = \frac{θ_{1}}{θ_{0}}, {\overset{̅}{θ}}_{2} = \frac{θ_{2}}{θ_{0}}, Fo = \frac{D_{1} t}{h^{2}} \\ \overset{̅}{D} = \frac{D_{2}}{D_{1}}, Bi = \frac{H_{g} h}{k_{1}}, \overset{̅}{r} = \frac{r}{h} \end{matrix} \\ \begin{matrix} {\overset{̅}{a}}_{0} = \frac{a_{0}}{h}, \overset{̅}{a} = \frac{a}{h}, \overset{̅}{b} = \frac{b}{h} \\ \overset{̅}{k} = \frac{k_{2}}{k_{1}} \end{matrix} \end{matrix}\}$ （8）

其中, $Fo$ 为傅里叶数, $Bi$ 为毕渥数,分别表征无量纲时间和导热/对流热阻比. 采用拉普拉斯变换法,并利用留数定理求解式(1) $~$ 式(7),得到涂层--基体系统的瞬态温度场如下

$\begin{matrix} \begin{matrix} {\overset{̅}{θ}}_{1} = 1 + \sum_{n}^{=} (Bi \cdot π^{2} 4 λ_{n} G_{n} H_{n} + I_{n} J_{n}) e^{- λ_{n}^{2} Fo} / Δ' (λ_{n}) \\ G_{n} = \frac{\overset{̅}{k}}{\sqrt[]{\overset{__}{D}}} [Y_{1} (\frac{λ_{n} \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) J_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}})] - \\ J_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) Y_{1} (\frac{λ_{n} \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) \end{matrix} \\ \begin{matrix} Y_{0} (λ_{n} \overset{̅}{a}) J_{0} (λ_{n} \overset{̅}{r}) - J_{0} (λ_{n} \overset{̅}{a}) Y_{0} (λ_{n} \overset{̅}{r}) \\ I_{n} = Y_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) J_{0} (\frac{λ_{n} \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) - J_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) Y_{0} (\frac{λ_{n} \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) \\ \begin{array}{l} J_{n} = Y_{1} (λ_{n} \overset{̅}{a}) J_{0} (λ_{n} \overset{̅}{r}) - J_{1} (λ_{n} \overset{̅}{a}) Y_{0} (λ_{n} \overset{̅}{r}) \end{array} \end{matrix} \end{matrix}\}$ (9)

$\begin{matrix} {\overset{̅}{θ}}_{2} = 1 + \sum_{n}^{=} (- Bi \cdot \frac{π^{2}}{4 λ_{n}}) L_{n} K_{n} e^{- λ_{n}^{2} Fo} / Δ' (λ_{n}) \\ L_{n} = Y_{0} (\frac{λ_{n} \overset{̅}{r}}{\sqrt[]{\overset{__}{D}}}) J_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) - J_{0} (\frac{λ_{n} \overset{̅}{r}}{\sqrt[]{\overset{__}{D}}}) Y_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) \\ K_{n} = J_{1} (λ_{n} \overset{̅}{a}) Y_{0} (λ_{n} \overset{̅}{a}) - J_{0} (λ_{n} \overset{̅}{a}) Y_{1} (λ_{n} \overset{̅}{a}) \end{matrix}\}$ （10）

式(9), 式(10)中, $J_{0} (\cdot)$ , $J_{1} (\cdot)$ 分别是零阶、一阶第一类贝塞尔函数, $Y_{0} (\cdot)$ , $Y_{1} (\cdot)$ 分别是零阶、一阶第二类贝塞尔函数. $λ_{n}$ 是以下超越方程之根

$Δ (λ) = \frac{\overset{̅}{k} π^{2} λ}{4 \sqrt[]{\overset{__}{D}}} {λ [J_{1} (λ {\overset{̅}{a}}_{0}) Y_{0} (λ \overset{̅}{a}) - J_{0} (λ \overset{̅}{a}) Y_{1} (λ {\overset{̅}{a}}_{0})] +$

$Bi \cdot [J_{0} (λ {\overset{̅}{a}}_{0}) Y_{0} (λ \overset{̅}{a}) - J_{0} (λ \overset{̅}{a}) Y_{0} (λ {\overset{̅}{a}}_{0})]} \cdot$

$[J_{1} (\frac{λ \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) - Y_{1} (\frac{λ_{n} \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) - J_{1} (\frac{λ \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) Y_{1} (\frac{λ \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}})] +$

$π^{2} 4 λ {λ \cdot [Y_{1} (λ {\overset{̅}{a}}_{0}) J_{1} (λ \overset{̅}{a}) - Y_{1} (λ \overset{̅}{a}) J_{1} (λ {\overset{̅}{a}}_{0})] +$

$Bi \cdot [Y_{0} (λ {\overset{̅}{a}}_{0}) J_{1} (λ \overset{̅}{a}) - Y_{1} (λ \overset{̅}{a}) J_{0} (λ {\overset{̅}{a}}_{0})]} \cdot$

$[J_{0} (\frac{λ \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}}) Y_{1} (\frac{λ \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) - J_{1} (\frac{λ \overset{̅}{b}}{\sqrt[]{\overset{__}{D}}}) Y_{0} (\frac{λ \overset{̅}{a}}{\sqrt[]{\overset{__}{D}}})] = 0$ （11）

根据热--弹性理论,在平面应变情形下,筒壁的环向热应力 $(σ_{θ})$ 为^[25]

$\frac{σ_{θ 1}}{σ_{0}} = \frac{- 1}{{\overset{̅}{r}}^{2}} \int_{{\overset{̅}{a}}_{0}}^{\overset{̅}{r}} {\overset{̅}{θ}}_{1} (\overset{̅}{ρ}, Fo) \overset{̅}{ρ} d \overset{̅}{ρ} + {\overset{̅}{θ}}_{1} (\overset{̅}{r}, Fo) - \frac{1 - v_{1}}{α_{1} (1 + v_{1})} (\frac{C_{11}}{1 - 2 v_{1}} + \frac{C_{21}}{{\overset{̅}{r}}^{2}})$ （12）

$\frac{σ_{θ 2}}{σ_{0}} = - \frac{α_{2} E_{2} (1 - v_{1})}{α_{1} E_{1} (1 - v_{2})} \frac{1}{{\overset{̅}{r}}^{2}} \int_{\overset{̅}{a}}^{\overset{̅}{r}} {\overset{̅}{θ}}_{2} (\overset{̅}{ρ}, Fo) \overset{̅}{ρ} d \overset{̅}{ρ} +$

$\frac{α_{2} E_{2} (1 - v_{1})}{α_{1} E_{1} (1 - v_{2})} {\overset{̅}{θ}}_{2} (\overset{̅}{r}, Fo) -$

$\frac{E_{2} (1 - v_{1}) E_{1}}{α_{1} (1 + v_{2})} (\frac{C_{12}}{1 - 2 v_{1}} + \frac{C_{22}}{{\overset{̅}{r}}^{2}})$ (13)

其中

$σ_{0} = - E_{1} α_{1} θ_{0} / (1 - v_{1})$ (14)

$C_{11} = 1 - 2 v_{1} {\overset{̅}{a}}_{0}^{2} Δ_{1} Δ_{0}$ (15)

$C_{12} = \frac{α_{2} (1 + v_{2}) (1 - 2 v_{2}) B}{(1 - v_{2}) \overset{__}{b^{2}}} + \frac{1 - 2 v_{2}}{\overset{__}{b^{2}}} \frac{Δ_{2}}{Δ_{0}}$ (16)

$C_{21} = \frac{Δ_{1}}{Δ_{0}}, C_{22} = \frac{Δ_{2}}{Δ_{0}}$ (17)

$Δ_{0} = \frac{- E_{2} ({\overset{̅}{a}}^{2} - {\overset{̅}{b}}^{2}) [(1 - 2 v_{1}) {\overset{̅}{a}}^{2} + {\overset{̅}{a}}_{0}^{2}]}{(1 + v_{2}) {\overset{̅}{a}}_{0}^{2} {\overset{̅}{a}}^{4} \overset{__}{b^{2}}} + \frac{E_{1} ({\overset{̅}{a}}^{2} - {\overset{̅}{a}}_{0}^{2}) [(1 - 2 v_{2}) {\overset{̅}{a}}^{2} + {\overset{̅}{b}}^{2}]}{(1 + v_{1}) {\overset{̅}{a}}_{0}^{2} {\overset{̅}{a}}^{4} \overset{__}{b^{2}}}$ (18)

$Δ_{1} = \frac{- E_{2} ({\overset{̅}{a}}^{2} - {\overset{̅}{b}}^{2})}{(1 - v_{1}) (1 - v_{2}^{2}) {\overset{̅}{a}}^{4} \overset{__}{b^{4}}} [- (1 + v_{1}) (1 - v_{2}) α_{1} {\overset{̅}{b}}^{2} A +$

$(1 - v_{1}) (1 + v_{2}) (1 - 2 v_{2}) α_{2} {\overset{̅}{a}}^{2} B] + [α_{2} E_{2} {\overset{̅}{a}}^{2} (1 - v_{1}) B +$ $α_{1} E_{1} {\overset{̅}{b}}^{2} (1 - v_{2}) A] \frac{[(1 - 2 v_{2}) {\overset{̅}{a}}^{2} + {\overset{̅}{b}}^{2}]}{(1 - v_{1}) (1 - v_{2}) {\overset{̅}{a}}^{4} \overset{__}{b^{4}}}$ (19)

$Δ_{2} = \frac{E_{1} ({\overset{̅}{a}}^{2} - {\overset{̅}{a}}_{0}^{2})}{(1 - v_{1}^{2}) (1 - v_{2}) {\overset{̅}{a}}_{0}^{2} {\overset{̅}{a}}^{4} \overset{__}{b^{2}}} \cdot$

$[(1 + v_{1}) (1 - v_{2}) α_{1} {\overset{̅}{b}}^{2} A - (1 - v_{1}) (1 + v_{2}) (1 - 2 v_{2}) α_{2} {\overset{̅}{a}}^{2} B] +$ $[α_{2} E_{2} {\overset{̅}{a}}^{2} (1 - v_{1}) B + α_{1} E_{1} {\overset{̅}{b}}^{2} (1 - v_{2}) A] \cdot$

$[(1 - 2 v_{1}) {\overset{̅}{a}}^{2} + {\overset{̅}{a}}_{0}^{2}] (1 - v_{1}) (1 - v_{2}) a_{0}^{2} {\overset{̅}{a}}^{4} \overset{__}{b^{2}}$ (20)

$A = \int_{{\overset{̅}{a}}_{0}}^{\overset{̅}{a}} {\overset{̅}{θ}}_{1} (\overset{̅}{r}, Fo) \overset{̅}{r} d \overset{̅}{r}, B = \int_{\overset{̅}{a}}^{\bar{b}} {\overset{̅}{θ}}_{2} (\overset{̅}{r}, Fo) \overset{̅}{r} d \overset{̅}{r}$ (21)

以上各式中, $E$ , $α$ , $ν$ 分别为杨氏模量,热膨胀系数和泊松比.

2 热应力强度因子

根据权函数理论^[26,27],应力强度因子( $K_{I})$ 可通过假想裂纹处的应力与权函数乘积的积分得到,即

$K_{I} = \int_{0}^{a_{1}} W (x, a_{1}) σ_{θ} (x) dx$ (22)

其中, $x$ 轴的原点位于裂纹口部(见图1),权函数( $W)$ 可表示成如下级数形式

$W (x, a_{1}) = \sqrt[]{\frac{2}{π a_{1}}} [\frac{1}{\sqrt[]{1 - x / a_{1}}} + \sum_{j}^{=} M_{j} (1 - x / a_{1})^{j - 1 / 2}]$ (23)

上述表达式通常只需取前四项就具有足够的精度^[21],因此,待定系数 $M_{j} (j$ =1,2,3)可由两个参考应力强度因子和一个附加条件确定. 附加条件通常取权函数在裂纹口部( $r = a_{0})$ 的曲率为零^[21,22],即

$\frac{\partial^{2} W}{\partial x^{2}} = 0 (x = 0)$ (24)

选取以下参考载荷：其一在裂纹口部处作用一对等值、反向的集中力 $P$ ;其二在裂纹面上作用均布载荷 $σ$ .在参考载荷作用下,涂层边缘裂纹的应力强度因子分别为

$K_{IP} = \frac{2 P}{\sqrt[]{π h}} F_{P}$ (25)

$K_{I σ} = σ \sqrt[]{π h} F_{σ}$ (26)

$F_{P}$ , $F_{σ}$ 称为形状因子(shape factor).

将式(23)、式(25)、式(26)代入式(22)和式(24), 可得待定系数^[21]

$\begin{matrix} M_{1} = - \frac{15 \sqrt[]{2}}{8} \sqrt[]{\frac{a_{1}}{h}} F_{P} + \frac{35 \sqrt[]{2} π}{16} \sqrt[]{\frac{h}{a_{1}}} F_{σ} - 7 \\ M_{2} = \frac{15 \sqrt[]{2}}{4} \sqrt[]{\frac{a_{1}}{h}} F_{P} - \frac{35 \sqrt[]{2} π}{12} \sqrt[]{\frac{h}{a_{1}}} F_{σ} + \frac{25}{3} \\ M_{3} = - \frac{7 \sqrt[]{2}}{8} \sqrt[]{\frac{a_{1}}{h}} F_{P} + \frac{35 \sqrt[]{2} π}{48} \sqrt[]{\frac{h}{a_{1}}} F_{σ} - \frac{7}{3} \end{matrix}\}$ (27)

本文研究的涂层/基体材料为碳化硅(SiC)/镍(Ni),对应的无量纲材料参数列于表1. 选取修正八节点二次平面应变单元, 用有限元方法计算参考载荷作用下的应力强度因子 $K_{IP}$ 和 $K_{I σ}$ ,全部有限元分析由商业软件ANSYS 11.0^[29]标准代码执行. 根据对称性,只需考虑二分之一个模型即可. 为保证计算精度,除细化裂纹尖端附近的网格外,还采用奇异单元模拟裂纹尖端附近应力场的奇异性. 有限元整体网格及裂纹尖端附近的局部网格分别如图2(a)和图2(b)所示.

表1 材料参数^{[

28
]}

Table 1 Properties of the used material pair^{[

28
]}

$D_{2} / D_{1}$	$k_{2} / k_{1}$	$E_{2} / E_{1}$	$ν_{1}$	$ν_{2}$	$α_{2} / α_{1}$
.611	3.606	0.644	0.195	0.312	1.797

新窗口打开

选择不同的几何参数 $a / h$ , $b / h$ 和材料参数 $E_{2} / E_{1}$ , $ν_{1}$ , $ν_{2}$ ,将有限元计算的 $K_{IP}$ 和 $K_{I σ}$ 代入式(25) 和式(26) 得到形状因子 $F_{P}$ 和 $F_{σ}$ .然后由式(27)得到 $M_{1}$ , $M_{2}$ , $M_{3}$ ,再代入式(23)得出权函数. 最后,将式(12)和式(23)代入式(22)得到应力强度因子. 就本文研究对象而言,形状因子的拟合表达式为(可信度 $R^{2} = 0.995$ )

$\frac{a_{1}}{h} = η$ （28）

（29）

（30）

显示原图|下载原图ZIP|生成PPT

图2 有限元网格划分

Fig. 2 Schematic of finite element meshing

3 结果与讨论

3.1 温度场和热应力场

无量纲温度 $θ_{i} / θ_{0}$ 沿圆筒径向的分布规律如图3所示,图中毕渥数 $Bi = 100$ , 1,基体/涂层厚度比( $b - a) / h = 25$ , 10. 可以看出,总体上各曲线具有相似的变化趋势：在任意时刻(即给定傅里叶数 $Fo$ ),无量纲温度沿径向单调降低;随着时间的延长,曲线逐渐抬升,直至热稳定状态. 在涂层与基体界面( $r = a)$ 处,由于界面两侧材料具有不同的热阻,曲线呈现不同的斜率.

显示原图|下载原图ZIP|生成PPT

图3 温度分布曲线

Fig. 3 Temperature distribution curves

其他条件相同时,基体/涂层厚度比为10时的温度分布曲线比25抬升得更快,这是由于前者基体相对较薄,达到稳定状态所需时间更短. 还可看出： $Bi = 100$ 时的无量纲温度比 $Bi = 1$ 时高,例如：当毕渥数 $Bi = 100$ 时,涂层表面处无量纲温度近似为1, 而当 $Bi = 1$ 时无量纲温度明显小于1,这是由于前者近似于热震情形(thermal shock),加载热量立即反映在涂层表面温度上,而后者完成热扩散需要足够时间.

图4为无量纲环向热应力 $σ_{θ} / σ_{0}$ 沿圆筒径向的分布规律,图中毕渥数 $Bi = 100$ , 1,基体/涂层厚度比( $b - a) / h = 25$ , 10. 由图可知,在热对流开始后的短时间内,涂层及紧邻界面之基体的热应力为拉应力,由于热应力的自平衡特性,靠近外壁之基体为压应力. 随着时间(傅里叶数 $Fo$ )的延长,涂层拉应力逐渐降低,最后转换为压应力,这是因为在接近热稳定状态时,镍基体与碳化硅涂层的温差很小,但前者的热膨胀系数远大于后者(约1.8倍). 另外,由于涂层和基体材料的物性差异,环向热应力( $σ_{θ})$ 在涂层/基体界面处( $r = a)$ 不连续.

显示原图|下载原图ZIP|生成PPT

图4 环向应力分布曲线

Fig. 4 Hoop stress distribution curves

还可看到,热应力沿涂层厚度分布不均,其分布梯度随时间延长而递减. 对 $Bi = 100$ 情形,涂层表面拉应力在初始时刻( $Fo = 0$ )达到峰值 $σ_{peak} = σ_{0}$ (式(14)),与涂层受热震时的应力相同^[19]. 可以预见,当此峰值超过涂层材料的强度极限时,涂层表面将萌生边缘裂纹. 对 $Bi = 1$ 情形,涂层表面处的峰值应力 $σ_{peak}$ 不仅低于 $σ_{0}$ ,且延后产生. 其他条件相同时,热应力的峰值随着基体/涂层厚度比的增大而增大,因而导致涂层更易开裂.

3.2 应力强度因子

将本文得到的参考应力强度因子与已有文献的对应值进行对比,最大相对误差仅4.64%,证实本文所确定的权函数具有足够精度^[30]. 无量纲应力强度因子 $K_{I} / K_{0}$ 随傅里叶数 $Fo$ 的变化规律如图5所示,其中 $Bi = 100$ , 1, 基体/涂层厚度比( $b - a) / h = 25$ , 10, $K_{0} = σ_{0} (π h)^{1 / 2}$ .对比各图可知：在不同毕渥数 $Bi$ 或基体/涂层厚度比的情形下,各曲线的总体变化趋势相似. 给定裂纹长度,边缘裂纹的应力强度因子首先快速增大,达到峰值 $K_{peak}$ 后缓慢下降或趋于平稳. 随着裂纹长度 $a_{1} / h$ 的增大,应力强度因子 $K_{I} / K_{0}$ 单调递减,这是由于裂纹长度增大时,裂纹尖端附近“潜在”热应力(即无裂纹应力)降低(见图4). 值得注意的是：在任意时刻,应力强度因子随着基体/涂层厚度比的增大而增大. 因此,增大涂层厚度或减小基体厚度可增强涂层抵抗瞬态热载荷的能力.

显示原图|下载原图ZIP|生成PPT

图5 应力强度因子时间历程

Fig. 5 The time history of SIF

对比应力分布曲线(图4)可知,应力峰值 $σ_{peak}$ 与应力强度因子峰值 $K_{peak}$ 并非同时发生. 例如, 当毕渥数 $Bi = 100$ 时,涂层表面的应力峰值发生在初始时刻 $Fo = 0$ ,而应力强度因子峰值则延迟一段时间出现. 因此,对筒壁涂层边缘裂纹来说,最危险时刻既不发生在对流加载的初始时刻,也不发生在热稳态时刻,而出现在时间历程的中间时刻. 由此可见,用热稳态分析来代替热瞬态分析的做法将严重低估热载荷的破坏程度. 还可看到,毕渥数显著影响热应力强度因子, 其他条件相同时,提高毕渥数不仅使热应力强度因子的峰值增大,而且令 $K_{peak}$ 的发生时刻提前.

4 结论

(1) 边缘裂纹的热应力强度因子峰值既非发生在热载荷初始时刻,也非发生在热稳态时刻,而出现在时间历程的中间时刻. 用热稳态分析来代替热瞬态分析将严重低估瞬态热载荷的破坏程度.

(2) 裂纹长度、基体/涂层厚度比及热流强度显著影响热应力强度因子. 其他条件相同时,随着裂纹长度的增大,筒壁边缘裂纹的热应力强度因子降低;增大涂层厚度或减小基体厚度可增强涂层抵抗瞬态热载荷的能力;增大热流强度不仅使热应力强度因子的峰值增大,而且令峰值的发生时刻提前.

The authors have declared that no competing interests exist.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]	Hutchinson JW , Suo Z . Mixed mode cracking in layered materials . Advances in Applied Mechanics, 1991 , 29 : 63 - 191
[2]	Miller RA . Thermal barrier coatings for aircraft engines: History and directions . Journal of Thermal Spray Technology, 1997 , 6 ( 1 ): 35 - 42
[3]	Underwood JH , Park AP , Vigllante GN , et al . Thermal damage, cracking and rapid erosion of cannon bore coatings . Journal of Pressure Vessel Technology, 2003 , 125 ( 3 ): 299 - 304
[4]	周益春 , 刘奇星 , 杨丽等 . 热障涂层的破坏机理与寿命预测 . 固体力学学报, 2010 , 31 ( 5 ): 504 - 531 ( Zhou Yichun , Liu Qixing , Yang Li , et al . Failure mechanisms and life prediction of thermal barrier coatings . Acta Mechanica Solida Sinica, 2010 , 31 ( 5 ): 504 - 531 (in Chinese))
[5]	徐颖强 , 孙戬 , 李万钟等 . 基于圆筒模型的热障涂层安定分析 . 力学学报, 2015 , 47 ( 5 ): 779 - 788 ( Xu Yinqiang , Sun Jian , Li Wangzhong , et al . Shakedown analysis of thermal barrier coatings based on cylinder model . Chinese Journal of Theoretical and Applied Mechanics, 2015 , 47 ( 5 ): 779 - 788 (in Chinese))
[6]	Li HX , Chen GN , Zhang K , et al . Degradation failure features of chromium-plated gun barrels with a laser-discrete-quenched substrate . Surface and Coatings Technology, 2007 , 201 ( 24 ): 9558 - 9564
[7]	Li DJ , Tan M , Liu GQ , et al . Preparation and characterization of ZrB/AlN multilayers by N+ beam assisted deposition . Surface and Coatings Technology, 2011 , 205 ( 13-14 ): 3791 - 3797
[8]	Chen XG , Yang Y , Yan DR , et al . Microstructure and properties of in situ nanostructured ceramic matrix composite coating prepared by plasma spraying . Journal of Materials Science, 2011 , 46 ( 23 ): 7369 - 7376
[9]	龚伟 , 周黎明 , 王恩泽等 . Q235钢基体表面微晶玻璃功能梯度涂层接触应力的数值模拟 . 中国表面工程, 2014 , 27 ( 2 ): 46 - 51 ( Gong Wei , Zhou Liming , Wang Enze , et al . Numerical simulation of the contact stress of functionally gradient glass-ceramic coatings on Q235 Ssteel . China Surface Engineering, 2014 , 27 ( 2 ): 46 - 51 (in Chinese))
[10]	陈光南 . 材料力学性能的激光调控 . 中国激光, 2011 , 38 ( 6 ): 43 - 53 ( Chen Guangnan . Regulation of the mechanical properties of materials by laser . Chinese Journal of Lasers, 2011 , 38 ( 6 ): 43 - 53 (in Chinese))
[11]	Chen XJ , Yan Q , Ma Q . Influence of the laser pre-quenched substrate on an electroplated chromium coating/steel substrate . Applied Surface Science, 2017 , 405 : 273 - 279
[12]	Rizk AA . Stress intensity factor for an edge crack in two bonded dissimilar materials under convective cooling . Theoretical and Applied Fracture Mechanics, 2008 , 49 ( 3 ): 251 - 267
[13]	Zhou B , Kokini K . Effect of surface pre-crack morphology on the fracture of thermal barrier coatings under thermal shock . Acta Materialia, 2004 , 52 ( 14 ): 4189 - 4197
[14]	李俊 , 冯伟哲 , 高效伟 . 一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法 . 力学学报, 2016 , 48 ( 2 ): 387 - 398 ( Li Jun , Feng Weizhe , Gao Xiaowei . A dual boundary integral equation method based on direct evaluation of higher order singular integral for crack problems . Chinese Journal of Theoretical and Applied Mechanics, 2016 , 48 ( 2 ): 387 - 398 (in Chinese))
[15]	Shen G , Glinka G . Determination of weight functions from reference stress intensity factors . Theoretical and Applied Fracture Mechanics, 1991 , 15 ( 3 ): 237 - 245
[16]	Fett T . Direct determination of weight functions from reference loading cases and geometrical conditions . Engineering Fracture Mechanics, 1992 , 42 ( 3 ): 435 - 444
[17]	Wu XR , Xu W . Strip yield crack analysis for multiple site damage in infinite and finite panels - A weight function approach . Engineering Fracture Mechanics, 2011 , 78 ( 14 ): 2585 - 2596
[18]	Chen XJ , You Y . Weight functions for multiple edge cracks in a coating . Engineering Fracture Mechanics, 2014 , 116 : 31 - 40
[19]	童第华 , 吴学仁 , 胡本润等 . 半无限板边缘裂纹的权函数解法与评价 . 力学学报, 2017 , 49 ( 4 ): 848 - 857 ( Tong Dihua , Wu Xueren , Hu Benrun , et al . Weight function methods and assessment for an edge crack in a semi-infinite plate . Chinese Journal of Theoretical and Applied Mechanics, 2017 , 49 ( 4 ): 848 - 857 (in Chinese))
[20]	Jin XC , Zeng YJ , Ding SD , et al . Weight function of stress intensity factor for single radial crack emanating from hollow cylinder . Engineering Fracture Mechanics, 2017 , 170 : 77 - 86
[21]	Fett T , Diegele E , Munz D , et al . Weight functions for edge cracks in thin surface layers . International Journal of Fracture, 1996 , 81 ( ??): 205 - 215
[22]	Fett T , Munz D , Yang YY . Direct adjustment procedure for weight functions of graded materials . Fatigue & Fracture of Engineering Materials & Structures, 2000 , 23 ( 3 ): 191 - 198
[23]	Carslaw HS , Jaeger JC. Conduction of Heat in Solids. 2nd Ed. London : Oxford University Press , 1986
[24]	谈庆明.量纲分析 . 第1版. 合肥 : 中国科学技术大学出版社 , 2005 ( Tan Qingming . Dimensional Analysis.1st Ed. Hefei : Press of University of Science and Technology of China , 2005 (in Chinese))
[25]	Boley BA , Weiner JH . Theory of Thermal Stresses. 1st Ed. New York:Dover Pulilication Inc., 2011
[26]	Bueckner HF . A novel principle for the computation of stress intensity factors . Zeitschrift für Angewandte Mathematik und Mechanik, 1970 , 50 ( 1 ): 529 - 546
[27]	Rice JR . Some remarks on elastic crack-tip stress fields. International Journal of Solids and Structures, 1972 , 8 ( 6 ): 751 - 758
[28]	Wang BL , Mai YW , Zhang XH . Thermal shock resistance of functionally graded materials . Acta Materialia, 2004 , 52 ( 17 ): 4961 - 4972
[29]	ANSYS Release 11.0, ANSYS Inc. , Canonsburg, Release 11.0, ANSYS Inc., Canonsburg, PA, 2009
[30]	Chen XJ , You Y . Weight functions for multiple axial cracks in a coated hollow cylinder . Archive of Applied Mechanics, 2015 , 85 ( 5 ): 617 - 628

Mixed mode cracking in layered materials

1991

Thermal barrier coatings for aircraft engines: History and directions

1997

Thermal damage, cracking and rapid erosion of cannon bore coatings

2003

热障涂层的破坏机理与寿命预测

2010

热障涂层的破坏机理与寿命预测

2010

基于圆筒模型的热障涂层安定分析

2015

基于圆筒模型的热障涂层安定分析

2015

Degradation failure features of chromium-plated gun barrels with a laser-discrete-quenched substrate

2007

Preparation and characterization of ZrB/AlN multilayers by N+ beam assisted deposition

2011

Microstructure and properties of in situ nanostructured ceramic matrix composite coating prepared by plasma spraying

2011

Q235钢基体表面微晶玻璃功能梯度涂层接触应力的数值模拟

2014

Q235钢基体表面微晶玻璃功能梯度涂层接触应力的数值模拟

2014

材料力学性能的激光调控

2011

材料力学性能的激光调控

2011

Influence of the laser pre-quenched substrate on an electroplated chromium coating/steel substrate

2017

Stress intensity factor for an edge crack in two bonded dissimilar materials under convective cooling

2008

Effect of surface pre-crack morphology on the fracture of thermal barrier coatings under thermal shock

2004

一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法

2016

一种基于直接计算高阶奇异积分的断裂力学双边界积分方程分析法

2016

Determination of weight functions from reference stress intensity factors

1991

Direct determination of weight functions from reference loading cases and geometrical conditions

1992

Strip yield crack analysis for multiple site damage in infinite and finite panels - A weight function approach

2011

Weight functions for multiple edge cracks in a coating

2014

半无限板边缘裂纹的权函数解法与评价

2017

半无限板边缘裂纹的权函数解法与评价

2017

Weight function of stress intensity factor for single radial crack emanating from hollow cylinder

2017

Weight functions for edge cracks in thin surface layers

1996

Direct adjustment procedure for weight functions of graded materials

2000

1986

2005

Theory of Thermal Stresses. 1st Ed.

2011

A novel principle for the computation of stress intensity factors

1970

Some remarks on elastic crack-tip stress fields. International

1972

Thermal shock resistance of functionally graded materials

2004

Canonsburg, Release 11.0, ANSYS Inc., Canonsburg,

2009

Weight functions for multiple axial cracks in a coated hollow cylinder

2015

〈

〉

热对流作用下筒壁涂层的边裂行为

EDGE CRACKING BEHAVIOR OF A COATED HOLLOW CYLINDER DUE TO THERMAL CONVECTION

引言

1 温度场及热应力场

2 热应力强度因子

3 结果与讨论

3.1 温度场和热应力场

3.2 应力强度因子

4 结 论

参考文献 文献选项 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

4 结论

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子