﻿ 基于圆筒模型的热障涂层安定分析
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 力学学报  2015, Vol. 47 Issue (5): 779-788  DOI: 10.6052/0459-1879-15-073 0

### 引用本文 [复制中英文]

[复制中文]
Xu Yingqiang, Sun Jian, Li Wanzhong, Zhang Yichi, Chen Yaya. SHAKEDOWN ANALYSIS OF THERMAL BARRIER COATINGS BASED ON CYLINDER MODEL[J]. Chinese Journal of Ship Research, 2015, 47(5): 779-788. DOI: 10.6052/0459-1879-15-073.
[复制英文]

### 文章历史

2015-03-05收稿
2015-07-23录用
2015–07–27网络版发表

1 圆筒模型安定性分析 1.1 热障涂层界面区圆筒模型

 图 1 热障涂层界面形貌及几何模型示意图 Fig. 1 Interface geometry of TBCs and schematic of cylinder
1.2 多层圆筒安定性

 ${\sigma _{ij,j}} + {Q_i} = {\bf{0}}\;\;(V内)$ (1a)
 ${\varepsilon _{ij}} = \left( {{u_{i,j}} + {u_{j,i}}} \right)/2\;\;(V内)$ (1b)
 ${\sigma _{ij}}{n_j} = {\hat p_i}\;\;({S_P}\;{\rm{上}}{\mkern 1mu} )$ (1c)
 ${u_i} = {\widehat u_i}\;\;({S_u}\;{\rm{上}}{\mkern 1mu} )$ (1d)

 ${\left( {{\sigma _{ij,j}}} \right)_k} = 0\;\;({V_k}\;{\rm{内}})$ (2a)
 ${\left( {{\varepsilon _{ij}}} \right)_k} = \left( {{{\left( {{u_{i,j}}} \right)}_k} + {{\left( {{u_{j,i}}} \right)}_k}} \right)/2\;\;({V_k}\;{\rm{内}})$ (2b)
 ${\left( {{\varepsilon _{ij}}} \right)_k} = {\left( {{{\hat \varepsilon }_{ij}}} \right)_k}\;\;({\left( {{S_u}} \right)_k}{\kern 1pt} {\kern 1pt} {\rm{上}})$ (2c)
 ${\left( {{u_i}} \right)_k} = {\left( {{{\widehat u}_i}} \right)_k}\;\;({\left( {{S_u}} \right)_k}{\rm{上}})$ (2d)

 $\varepsilon _{ij}^{\rm{E}} = {A_{ijkl}}{\sigma _{kl}}{\mkern 1mu} ,\;\varepsilon _{ij}^{\rm{T}} = {a_{ij}}T$ (3)

 ${\varepsilon _{ij}} = \varepsilon _{ij}^{\rm{E}} + \varepsilon _{ij}^{\rm{P}} + \varepsilon _{ij}^{\rm{T}}$ (4)

 $F = f\left( {{\sigma _{ij}}} \right) - K(T) = 0$ (5)

 $\dot \varepsilon _{ij}^{\rm{P}} = \dot \lambda \partial F/\partial {\sigma _{ij}}$ (6)

 $\dot \lambda \ge 0{\mkern 1mu} ,\;\;\;\dot \lambda F = 0$ (7)

 $D = {\sigma _{ij}}\dot \varepsilon _{ij}^{\rm{P}} = D\left( {\dot \varepsilon _{ij}^{\rm{P}},T} \right)$ (8)

 $K(T) = {K_0}h(T){\mkern 1mu} ,\;\;{K_0} = K\left( 0 \right){\mkern 1mu} ,\;\;h\left( 0 \right) = 1$ (9)

 $D = {D_0}\left( {\dot \varepsilon _{ij}^{\rm{P}}} \right)h\left( T \right)$ (10)

 ${\sigma _{ij}} = \sigma _{ij}^{\rm{E}} + {\rho _{ij}}$ (11)

 ${\rho _{ij}} = {\rm{L}}\left( {\varepsilon _{ij}^{\rm{P}}} \right)$ (12)

 $\sigma _{ij}^{\rm{E}}\left( {r,t} \right) = \sum\limits_{s = 1}^{{n_s}} {{a_s}(t)} \sigma _{ij}^{{{\rm{E}}_s}}\left( r \right)$ (13)

 ${\varepsilon _{ij}} = {A_{ijkl}}\sigma _{kl}^{\rm{E}} + \varepsilon _{ij}^{\rm{T}} + \varepsilon _{ij}^{\rm{P}} + {A_{ijkl}}{\rho _{kl}}$ (14)

 ${A_{ijkl}}\sigma _{kl}^{\rm{E}} + \varepsilon _{ij}^{\rm{T}} = \left( {u_{i,j}^{\rm{E}} + u_{j,i}^{\rm{E}}} \right)/2$ (15)
 $\varepsilon _{ij}^{\rm{P}} + {A_{ijkl}}{\rho _{kl}} = \left( {u_{i,j}^{\rm{R}} + u_{ij,i}^{\rm{R}}} \right)/2$ (16)
 ${u_i} = u_i^{\rm{E}} + u_i^{\rm{R}}$ (17)

$u_i^{\rm{E}}$可表示为

 $u_i^{\rm{E}}\left( {r,t} \right) = \sum\limits_{s = 1}^n {{a_s}\left( t \right)} u_i^{{{\rm{E}}_s}}\left( r \right)$ (18)

 ${p_i}\left( {r,t} \right) = \sum\limits_{s = 1}^n {{a_s}\left( t \right)} p_i^s\left( r \right){\mkern 1mu} ,\;\;r \in {S_p}$ (19)
 ${Q_i}\left( {r,t} \right) = \sum\limits_{s = 1}^n {{a_s}\left( t \right)} Q_i^s\left( r \right){\mkern 1mu} ,\;\;r \in V$ (20)
 $T\left( {r,t} \right) = \sum\limits_{s = 1}^n {{a_s}\left( t \right)} {T^s}\left( r \right){\mkern 1mu} ,\;\;r \in V$ (21)

 $\Delta \dot \varepsilon _{ij}^{ * {\rm{P}}} = \int_{{t_0}}^{{t_0} + \tau } {\dot \varepsilon _{ij}^{ * {\rm{P}}}} = \frac{1}{2}\left( {u_{i,j}^ * + u_{j,i}^ * } \right)$ (22)

 $\begin{array}{*{20}{l}} {\int_{{t_0}}^{{t_0} + {\tau _s}} {\left[ {\sum\limits_{s = 1}^n {{a_s}\left( t \right)} \left( {\int_V {Q_i^s\dot u_i^*dV} + \int_{{S_P}} {\hat p_i^s\dot u_i^*dS} } \right) + } \right.\left. {\int_V {{a_{ij}}T\dot \rho _{ij}^*dV} } \right]} dt > }\\ {\int_{{t_0}}^{{t_0} + \tau } {\int_V {D\left( {\dot \varepsilon _{ij}^{*{\rm{P}}},T} \right)} dVdt} } \end{array}$ (23)

 $\begin{array}{l} \sigma _{ij}^E = \sigma _{ij}^{EE} + \rho _{ij}^{\rm{T}} = \\ \sum\limits_{s = 1}^{{n_s}} {{a_s}(t)} \left( {\sigma _{ij}^{{\rm{EEs}}}(x) + \rho _{ij}^{{\rm{Ts}}}(x)} \right) \end{array}$ (24)

 $\int_V {{a_{ij}}T\dot \rho _{ij}^*dV} = \int_V {\varepsilon _{ij}^{\rm{T}}\dot \rho _{ij}^*dV} = - \int_V {\rho _{ij}^{\rm{T}}{A_{ijkl}}\dot \rho _{kl}^*dV} = \int_V {\rho _{ij}^{\rm{T}}\dot \varepsilon _{ij}^{*{\rm{P}}}dV}$ (25)

 $\begin{array}{*{20}{l}} {\int_{{t_0}}^{{t_0} + \tau } {\int_V {\sigma _{ij}^{\rm{E}}\left( {r,t} \right)\dot \varepsilon _{ij}^{*{\rm{P}}}\left( {r,t} \right)dVdt > } } }\\ {\int_{{t_0}}^{{t_0} + \tau } {\int_V {D\left( {\dot \varepsilon _{ij}^{*{\rm{P}}},t} \right)dVdt} } } \end{array}$ (26)

 $h(T) = A - BT$ (27)

 $\begin{array}{*{20}{l}} {D\left( {\dot \varepsilon _{ij}^{*{\rm{P}}},T} \right) = {D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right)h\left( T \right) = }\\ {A{D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right) - BT{D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right)} \end{array}$ (28)

 $\begin{array}{*{20}{l}} {\eta \int_{{t_0}}^{{t_0} + \tau } {\int_V {\left[ {\sigma _{ij}^{\rm{E}}\left( {r,t} \right)\dot \varepsilon _{ij}^{*{\rm{P}}}\left( {r,t} \right) + BT{D_0}\left( {\dot \varepsilon _{ij}^{*P}} \right)} \right]dVdt \le } } }\\ {\int_{{t_0}}^{{t_0} + \tau } {\int_V {A{D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right)dVdt} } } \end{array}$ (29)

 $\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right) = \varepsilon _{ij}^{ * {\rm{P}}}\left( {r,{t_0} + \tau } \right) - \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right)$ (30)

 $\varepsilon _{ij}^{ * {\rm{P}}}\left( {r,t} \right) = \lambda \left( {r,t} \right)\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right)$ (31)

 $\dot \lambda \left( {r,t} \right) \ge 0{\mkern 1mu} ,\;\lambda \left( {r,{t_0}} \right) = 0{\mkern 1mu} ,\;\lambda \left( {r,{t_0} + \tau } \right) = 1$ (32)

 $\begin{array}{*{20}{l}} {\int_{{t_0}}^{{t_0} + \tau } {{D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right)dt} = }\\ {\int_{{t_0}}^{{t_0} + \tau } {{D_0}\left( {\dot \lambda \Delta \varepsilon _{ij}^{*{\rm{P}}}} \right)dt = {D_0}\left( {\Delta \varepsilon _{ij}^{*{\rm{P}}}} \right)} } \end{array}$ (33)

 $\begin{array}{*{20}{l}} {\int_{{t_0}}^{{t_0} + \tau } {\int_V {\sum\limits_{s = 1}^n {{a_s}\left( t \right)} } } \left[ {\sigma _{ij}^{{{\rm{E}}_s}}\left( r \right)\dot \varepsilon _{ij}^{*P}\left( {r,t} \right) + } \right.}\\ {\left. {B{T^s}\left( r \right){D_0}\left( {\dot \varepsilon _{ij}^{*{\rm{P}}}} \right)} \right]dV{\rm{d}}t \le }\\ {\int_V {A{D_0}\left( {\Delta \varepsilon _{ij}^{*{\rm{P}}}} \right)dV} } \end{array}$ (34)

 $\begin{array}{l} \int_{{t_0}}^{{t_0} + \tau } {\int_V {\sum\limits_{s = 1}^n {{a_s}\left( t \right)} } } \left[{\sigma _{ij}^{{{\rm{E}}_s}}\left( r \right)\dot \lambda \left( {r,t} \right)\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right) + } \right.\\ \left. {B{T^s}\left( r \right)\dot \lambda \left( {r,t} \right){D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right)} \right)} \right]dVdt \end{array}$ (35)

 $\begin{array}{l} \sum\limits_{s = 1}^n {{a_s}(t)} \left[{\sigma _{ij}^{Es}\left( r \right)\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right) + } \right.\\ \left. {B{T^s}\left( r \right)\dot \lambda \left( {r,t} \right){D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right)} \right)} \right] \end{array}$ (36)

 $\begin{array}{l} f\left( r \right) = \mathop {\max }\limits_{{a_s} \in G} \{ \sum\limits_{s = 1}^n {{a_s}(t)} \left[{\sigma _{ij}^{{{\rm{E}}_s}}\left( r \right)\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right) + } \right.\\ \left. {B{T^s}\left( r \right){D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right)} \right)} \right]\} \end{array}$ (37)

 $\int_V {f\left( r \right)dV} = \int_V {A{D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}} \right)dV}$ (38)

 $a_s^ - \le {a_s} \le a_s^ +$ (39)

 $\int_V {\sum\limits_{s = 1}^n {{\beta _s}\left( r \right){y_s}\left( r \right)dV} } = \int_V {A{D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}} \right)dV}$ (40)

 $\left. \begin{array}{l} {\beta _s}(r) = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} a_s^ + ,{y_s}(r) > 0\\ a_s^ - ,{y_s}(r) < 0 \end{array} \end{array}} \right.\\ {y_s}(r) = \sigma _{ij}^{{{\rm{E}}_s}}(r)\Delta \varepsilon _{ij}^{ * {\rm{P}}}(r) + B{T^s}(r){D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}(r)} \right)n \end{array} \right\}$ (41)

2 热障涂层安定性分析 2.1 材料参数及载荷

 图 2 随温度变化的氧化层屈服强度[27] Fig. 2 Yield strength of TGO change with temperature[27]

 $\max \left\{ {|{\sigma _r}|,|{\sigma _\phi }|,|{\sigma _\phi } - {\sigma _r}|} \right\} = 2K(T)$ (42)

 $K(T) = {K_0}\left( {A - BT} \right){\mkern 1mu} ,\quad 2{K_0} = {\sigma _s}$ (43)

2.2 屈服强度与温度的双线性模型

 $\left. {\begin{array}{*{20}{l}} \begin{array}{l} \Delta {\varepsilon _r} = \int_{{t_0}}^{{t_0} + \tau } {{{\dot \varepsilon }_r}dt} = \int_{{t_0}}^{{t_0} + \tau } {\frac{{\partial \mathop u\limits^. }}{{\partial r}}dt} = - \frac{{\Delta c}}{{{r^2}}}\\ \Delta {\varepsilon _\varphi } = \int_{{t_0}}^{{t_0} + \tau } {{{\dot \varepsilon }_\varphi }dt} = \int_{{t_0}}^{{t_0} + \tau } {\frac{{\mathop u\limits^. }}{r}dt} = \frac{{\Delta c}}{{{r^2}}} \end{array} \end{array}} \right\}$ (44)

 $\begin{array}{l} {D_0} = {\sigma _r}\left( { - \frac{{\Delta c}}{{{r^2}}}} \right) + {\sigma _\varphi }\left( {\frac{{\Delta c}}{{{r^2}}}} \right) = \\ \frac{{\Delta c}}{{{r^2}}}\left( {{\sigma _\varphi } - {\sigma _r}} \right) = 2{K_0}\frac{{\Delta c}}{{{r^2}}} \end{array}$ (45)

 $\begin{array}{l} \sigma _{ij}^{{{\rm{E}}_s}}\left( r \right)\Delta \varepsilon _{ij}^{ * {\rm{P}}}\left( r \right) = \\ \sigma _r^{\rm{E}}\left( r \right)\Delta {\varepsilon _r}\left( r \right) + \sigma _\phi ^{\rm{E}}\left( r \right)\Delta {\varepsilon _\phi }\left( r \right) \end{array}$ (46)

 $\begin{array}{l} {A_{\rm{D}}} = \int_V {} {D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}} \right)dV - \int_V {} \sum\limits_{s = 1}^n {} _{}^{}{\beta _s}\left( r \right){y_s}\left( r \right)dV = \\ \int_{{a_i}}^{{a_{i + 1}}} {{D_0}\left( {\Delta \varepsilon _{ij}^{ * {\rm{P}}}} \right)rdr} - \int_{{a_i}}^{{a_{i + 1}}} {\sum\limits_{s = 1}^n {{\beta _s}\left( r \right){y_s}\left( r \right)rdr} } \end{array}$ (47)

 图 3 屈服强度与温度双线性简化关系示意图 Fig. 3 Simplified schematic between yield strength and temperature

(1) 如图3所示，图中$B,C$和$D$点位于同一直线，$O,A$和$D$点位于同一直线且直线$AB$平行于横坐标轴$OC$. 设材料屈服特 性随温度的变化关系服从$D$-$B$-$C$直线关系，分别沿$D$-$B$线段和$D$-$B$-$C$线段计算式(47)，此时式(44)中$K_0$, $A$和$B$唯一，计算得到该情况下的式(47)结果，分别记为$A_{\rm D}$1和$A_{\rm D}$2；

(2) 设材料屈服特性随温度的变化关系服从$A$-$B$直线关系，沿$A$-$B$线段计算式(47)，此时式(43)中$K_0$和$B$唯一，计算得到该情况下的式(47)结果记为$A_{\rm D}$3；

(3) 由步骤1和步骤2得到的$A_{\rm D}$1,$A_{\rm D}$2和$A_{\rm D}$3计算式(43)沿$A$-$B$-$D$线段积分结果，为

 ${A_{\rm{D}}} = {A_{\rm{D}}}2 - ({A_{\rm{D}}}1 - {A_{\rm{D}}}3)$

(4) 令$A_{\rm D} = 0$，求得$T_{\rm w}$，即为安定极限.

2.3 安定性分析结果

(1) 圆筒模型基体曲率对涂层安定极限的影响

 图 4 不同基体曲率下热障涂层极限载荷 Fig. 4 Limit load for TBCs with different curvature of SUB

(2) 材料厚度对热障涂层安定极限的影响

 图 5 热障涂层极限载荷与涂层厚度之间的关系 Fig. 5 Relationship between limit load and thickness of TBCs

(3)热障涂层蠕变应力仿真

 图 6 内凹模型蠕变对应力场的影响 Fig. 6 The influence of creep on stress in concave model
 图 7 外凸模型蠕变对应力场的影响 Fig. 7 The influence of creep on stress in concave model
3 结 论

(1)借助本文给出的安定性分析方法，利用增量破坏准则处理对时间的积分，将材料屈服强度随温度的变化关系简化为双线性模型，能够得到多层结构的安定极限，且方法简单便于实际工程应用；

(2)热障涂层安定极限值明显高于弹性设计值，且界面外凸区域安定极限高于内凹区域极限值，结构优先在内凹处失效，与试验所得结果一致. 同时，圆筒涂层基体曲率半径和涂层厚度越大，安定极限越高；

(3)仿真分析结果表明蠕变引起的应力释放效果明显，本文建立的热障涂层安定分析方法，对进一步研究考虑蠕变因素影响热障涂层安定性具有重要意义.