﻿ 一种固体推进剂药柱结构完整性的快速评估方法
 火炸药学报    2018, Vol. 41 Issue (2): 178-185   DOI: 10.14077/j.issn.1007-7812.2018.02.013 0

引用本文

ZHANG Lu, YU Rui, DENG Kang-qing, PANG Ai-min, YANG Ling. A Rapid Assessment Method of the Structural Integrity of Solid Propellant Grain[J]. Chinese Journal of Explosives & Propellants, 2018, 41(2): 178-185. DOI: 10.14077/j.issn.1007-7812.2018.02.013

文章历史

1 计算方法 1.1 固化降温过程中药柱等效模量的计算方法

 图 1 固化降温过程中温度随时间的变化曲线 Figure 1 The changing curve of temperature with time during cooling process after curing
 $\varepsilon (t) = \mathit{\Delta }{\varepsilon _1} + \mathit{\Delta }{\varepsilon _2} + \mathit{\Delta }{\varepsilon _3} + \cdots + \mathit{\Delta }{\varepsilon _n}$ (1)

 $E(t) = {E_{\rm{e}}} + \sum\limits_{i = 1}^n {{E_i}{{\rm{e}}^{\frac{{-t}}{{{t_i}}}}}}$ (2)

WLF方程为：

 $\log {\alpha _{\rm{T}}}(T) = \frac{{-{c_1}(T-{T_{\rm{s}}})}}{{{c_2} + T-{T_{\rm{s}}}}}$ (3)

 $\xi = \frac{1}{{{\alpha _{\rm{T}}}}}$ (4)

 ${E_{\rm{T}}}(t) = E(\xi ) = {E_{\rm{e}}} + \sum\limits_{i = 1}^n {{E_{\rm{i}}}\exp (-\frac{\xi }{{{t_i}}})}$ (5)

 $\xi = \int_0^t {\frac{{{\rm{d}}t}}{{{\alpha _{\rm{T}}}(T(t))}}}$ (6)

 $\sigma (t) = \sum\limits_{i = 1}^n {E({\xi _n}-{\xi _{i-1}}) \cdot \mathit{\Delta }} {\varepsilon _i}\;\;\;({\xi _0} = 0)$ (7)

 ${E_{{\rm{eq}}}} = \frac{{\sigma (t)}}{{\varepsilon (t)}} = \frac{{\sum\limits_{i = 1}^n {E({\xi _n}-{\xi _{i-1}}) \cdot \mathit{\Delta }{\varepsilon _i}} }}{{\mathit{\Delta }{\varepsilon _1} + \mathit{\Delta }{\varepsilon _2} + \mathit{\Delta }{\varepsilon _3} + \cdots + \mathit{\Delta }{\varepsilon _n}}}\;\;\;({\varepsilon _0} = 0)$ (8)

 $\varepsilon (t) = S{t_1} \cdot T + S{t_2}$ (9)

 $\mathit{\Delta }{T_1} = \mathit{\Delta }{T_2} = \mathit{\Delta }{T_3} = \cdots = \mathit{\Delta }{T_n}$ (10)

 $\mathit{\Delta }{\varepsilon _1} = \mathit{\Delta }{\varepsilon _2} = \mathit{\Delta }{\varepsilon _3} = \cdots = \mathit{\Delta }{\varepsilon _n}$ (11)

 ${E_{{\rm{eq}}}}(t) = \frac{1}{n}\sum\limits_{i = 1}^n {E({\xi _n}-{\xi _{i-1}})} \;\;\;\;({\xi _0} = 0)$ (12)

1.2 点火增压过程中药柱等效模量的计算方法

 图 2 点火增压过程中压强随时间变化曲线 Figure 2 The changing curve of pressure with time during ignition pressurization process

 $\varepsilon (t)/{p_{\rm{i}}} + S{p_1}/E(t) + S{p_2}$ (13)

i份的压力变化Δpit时刻所产生的应变为Δεi，可表示为：

 $\mathit{\Delta }{\varepsilon _i} = \mathit{\Delta }{\mathit{p}_i} \cdot (S{p_1}/E({t_n}-{t_{i-1}}) + S{p_2})\;\;({t_0} = 0)$ (14)

 $\mathit{\Delta }{\sigma _i} = E({t_n}-{t_{i-1}}) \cdot \mathit{\Delta }{\varepsilon _i}\;\;({t_0} = 0)$ (15)

 $\begin{array}{l} E(t) = \frac{{\sigma (t)}}{{\varepsilon (t)}} = \\ \frac{{E({t_n}) \cdot \mathit{\Delta }{\varepsilon _1} + E({t_n}-{t_1}) \cdot \mathit{\Delta }{\varepsilon _2} + \cdots + E({t_n}-{t_{n-1}}) \cdot \mathit{\Delta }{\varepsilon _n}}}{{\mathit{\Delta }{\varepsilon _1} + \mathit{\Delta }{\varepsilon _2} + \mathit{\Delta }{\varepsilon _3} + \cdots + \mathit{\Delta }{\varepsilon _n}}} \end{array}$ (16)

2 数值模拟 2.1 有限元模型的建立

 图 3 固体火箭发动机3D有限元模型 Figure 3 3D Finite element model of the solid rocket motor

2.2 推进剂力学性能参数

 $E(t) = {E_\infty } + \sum\limits_{i = 1}^n {{E_i}} \exp (-\frac{1}{{{t_i}}})$ (17)

 $\lg {\alpha _{\rm{T}}} = \frac{{-{C_1}(T-{T_{\rm{s}}})}}{{{C_2} + T-{T_{\rm{s}}}}}$ (18)

3 结果与讨论 3.1 温度载荷下药柱等效模量变化的评估 3.1.1 稳态温度载荷下药柱有效模量变化的评估

 图 4 药柱在0℃下的保温时间t及其等效时间ξ的关系曲线 Figure 4 Relation curve of time(t) vs. equivalent time(ξ) of grain at 0℃

 图 5 0℃下A点的松弛模量和等效模量分布曲线 Figure 5 Distribution curve of relaxation modulus and equivalent modulus at point A of grain at 0℃
3.1.2 固化降温过程中不同降温条件下药柱模量变化的评估

 $条件1: 直线降温{T_1}(t) = 60 - 1/3600 \cdot t$ (19)
 $条件2: 平方降温{T_2}(t) = - 40 + 1/1.29 \times {10^9} \cdot {(t - 3.6 \times {10^6})^2}$ (20)
 $条件3: 三次方降温{T_3}(t) = 60 - 1/4.6656 \times {10^{14}} \cdot {t^3}$ (21)

3种降温曲线如图 6所示。降温过程中3种降温曲线的等效时间ξ随降温时间t的分布情况如图 7所示，图 7中左侧纵坐标为直线降温和平方降温过程中等效时间大小变化的参考坐标，右侧纵坐标为三次方降温过程中等效时间大小变化的参考坐标。

 图 6 3种降温条件下温度随时间分布曲线 Figure 6 Distribution curves of temperature with time under three cooling conditions
 图 7 3种降温条件的降温时间及其等效时间分布曲线 Figure 7 Distribution curves of cooling time(t)and equivalent time(ξ) under three cooling conditions

 图 8 3种降温条件下两种评估方法的等效模量分布 Figure 8 Distribution of equivalent modulus obtained by two evaluation methods under three cooling conditions

3.1.3 固化降温过程中药柱结构完整性的快速评估

 ${\varepsilon _{{\rm{eq}}}}(t) =-0.001\;7800 \cdot T + 0.109\;22$ (22)

 图 9 3种降温条件下药柱A点的应力、应变曲线 Figure 9 Curves of von Mises strain vs. stress at point A of grain under three cooling conditions

3.2 压力载荷下药柱等效模量变化的评估 3.2.1 稳态压力载荷下药柱等效模量动态变化的评估

 图 10 2MPa下A点的松弛模量和等效模量分布曲线 Figure 10 Distribution curve of relaxation modulus and equivalent modulus at point A of grain at 2MPa

3.2.2 点火增压过程中不同增压条件下药柱模量变化的评估

 $条件1: 直线增压{p_1}(t) = 20 \cdot t$ (23)
 $条件2: 平方增压{p_2}(t) = 200 \cdot {t^2}$ (24)
 $条件3: 指数增压{p_3}(t) = 2 \cdot (1 - {{\rm{e}}^{ - 60t}})$ (25)

3种增压条件下压强随时间分布曲线如图 11所示。

 图 11 3种增压条件下压强随时间分布曲线 Figure 11 Distribution curves of pressure with time under three pressurized conditions

 ${\varepsilon _{{\rm{eq(}}t{\rm{)}}}}/{p_i} = 0.246\;41/E(t) + 0.00045502$ (26)

 图 12 3种增压条件下两种评估方法的等效模量分布曲线 Figure 12 Distribution curves of equivalent modulus obtained by two evaluation methods under three pressurized conditions

3.2.3 点火增压过程中药柱结构完整性的快速评估

 图 13 3种增压条件下药柱A点的应力、应变曲线 Figure 13 Curves of von Mises strain and stress at point A of grain under three pressurized conditions

4 结论

(1) 建立了固化降温和点火增压动态载荷作用下推进剂药柱等效模量的理论评估方法。具体根据温度和压力载荷加载历程进行分段，基于波尔兹曼叠加原理计算分段载荷对药柱应力、应变产生的叠加响应，推导出了固化降温和点火增压过程中推进剂药柱等效模量的计算公式。

(2) 提出了温度应变系数和压力应变系数的概念，建立了动态载荷工况下药柱等效应力和等效应变的计算模型。温度应变系数和压力应变系数可通过对有限元模拟计算的结果进行拟合得到，其可作为药柱本身的材料参数，不管在何种温度和压力载荷工况作用下，其温度和压力应变系数始终不变。

(3) 验证了动态模量评估方法和药柱等效应变、等效应力计算方法的准确性。通过与基于线黏弹性理论的模拟计算方法得到的药柱内部危险点位置的等效应力和等效应变进行对比，得到药柱等效应力和等效应变之间的差值都在5%以内，证明了该快速评估方法的可行性。

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A Rapid Assessment Method of the Structural Integrity of Solid Propellant Grain
ZHANG Lu, YU Rui, DENG Kang-qing, PANG Ai-min, YANG Ling
Hubei Institute of Aerospace Chemotechnology, Xiangyang Hubei 441003, China
Abstract: To realize the rapid assessment of the structural integrity of solid propellant grain, starting with the dynamic modulus of propellant grain, a fast real-time assessment method for the equivalent modulus of solid propellant grain under dynamic load condition is established. Then, the concepts of temperature strain coefficient and pressure strain coefficient are put forward, a calculation model of the equivalent strain and equivalent stress of grain under dynamic load condition is set up, and the change of equivalent strain and equivalent stress of the dangerous position of grain at all times can be obtained, and the results are compared with finite element simulation ones based on linear viscoelasticity theory. The results show that the maximum difference between the equivalent strain and the equivalent stress of the dangerous position of grain under three kinds of curing cooling curves obtained by two calculation methods are 4.60% and 4.74%, respectively and the maximum difference between the equivalent strain and the equivalent stress of the dangerous position of grain obtained by two calculation methods under ignition pressurization curves are 1.93% and 1.23%, respectively and the time required for the assessment method is greatly reduced, so the assessment method can be used for rapid assessment of structural integrity analysis of solid propellant grain.
Key words: propellant grain     curing and cooling     ignition pressurization     equivalent modulus     structural integrity