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在静态热流平衡假设下,材料内部的能量守恒可以通过修正的傅里叶热传导公式表征[6]:
$ (\rho c)_{\mathrm{eq}} \frac{\partial T}{\partial t}=\nabla \cdot\left(\boldsymbol{\kappa}_{\mathrm{eq}} \cdot \nabla T\right)-\rho_{\mathrm{g}} c_{\mathrm{g}} \boldsymbol{u}_{\mathrm{g}} \cdot \nabla T+Q_{\mathrm{d}} \rho_{\mathrm{m}} \frac{\partial \varphi_{\mathrm{m}}}{\partial t} $
(1) 材料内部热解气体的质量守恒方程可描述为:
$ \frac{\partial\left(\rho_{\mathrm{g}} \varphi_{\mathrm{g}}\right)}{\mathrm{d} t}-\nabla \cdot\left(\boldsymbol{u}_{\mathrm{g}} \rho_{\mathrm{g}}\right)=-\frac{\partial \varphi_{\mathrm{m}}}{\partial t} \rho_{\mathrm{m}} \eta $
(2) 式中,${(\rho c)_{{\rm{eq}}}} = \sum\limits_{i = {\rm{f}}, {\rm{m}}, {\rm{c}}, {\rm{g}}} {{\varphi _i}} {\rho _i}{c_i}$,φ表示各组分的体积分数,ρ为材料等效密度,c为材料的等效比热容,下标i分别表示碳纤维(f)、基体(m)、残炭(c)和热解气体(g),各组分的热物参量见参考文献[6, 14];T表示材料温度,t表示激光辐照时间,$\nabla $表示梯度算符,κeq表示材料的等效热导率张量;ug为热解气体在材料中的流动速度;基体的热解热Qd=0.996×106J/kg[10];基体热解产生气体的质量分数η=0.7[6]。
激光辐照过程中基体的热解过程可采用阿伦纽斯方程描述[12-13]:
$ \frac{\partial \varphi_{\mathrm{m}}}{\partial t}=-A_{\mathrm{d}} \varphi_{\mathrm{m}, 0}\left(\frac{\varphi_{\mathrm{m}}-\varphi_{\mathrm{m}, \infty}}{\varphi_{\mathrm{m}, 0}}\right)^{n_{\mathrm{d}}} \exp \left(-\frac{E_{\mathrm{d}}}{R T}\right) $
(3) 式中,φm, 0和φm, ∞分别表示基体的初始体积分数和最终体积分数;指数前因子Ad=3.15×1011s-1,热解反应级数nd=1.344,热解活化能Ed=1.8173×105J/mol,气体常数R=8.314J/(mol·K)。
材料内部热解气体流动的动量守恒方程可由达西定律表征[6]:
$ \boldsymbol{u}_{\mathrm{g}}=\frac{\xi \nabla p_{\mathrm{g}}}{\mu_{\mathrm{g}} \varphi_{\mathrm{g}}} $
(4) 式中,ξ为材料内部空隙率,μg为热解气体的粘度,pg为热解气体在空隙内压强。
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5种结构CFRP的铺层方式如表 1所示。层叠型碳纤维增强环氧树脂复合材料一般由单向层板或编织层板叠加成型。层叠型CFRP的单层类型如图 1a所示,在单向层中,方向1表示平行于纤维方向,方向2表示垂直于纤维方向,方向3表示材料厚度方向;在编织层中, 方向4表示纤维方向,方向5表示材料厚度方向。
Table 1. Laminated structure of different types of CFRPs
type laminated structure surface dimensions/mm intrinsic property thickness/mm number of layers 1 [0°/90°/0°/90°] 20×20 — 0.56 4 2 [0°/90°/90°/0°] 20×20 — 0.56 4 3 [0/45°/90°/135°] 20×20 — 0.56 4 4 [1.5K/0°/90°/1.5K] 20×20 plain pattern θ/0.196 0.58 4 5 [3K/0°/3K] 20×20 plain pattern θ/0.145 0.58 3 CFRP由热物特性迥异的不同组分按不同的方式构成,其热导率取决于其内部的单层类型、铺层夹角以及铺层顺序。单向层板内的热导率可以通过混合定律描述[6],平行碳纤维方向的热导率可以由串联混合定律表征[6]:
$ \kappa_{1}=\sum\limits_{i=\mathrm{f}, \mathrm{m}, \mathrm{c}, \mathrm{g}} \varphi_{i} \kappa_{i} $
(5) 垂直碳纤维方向的热导率可以由并联混合定律表征[6]:
$ \kappa_{2}=\kappa_{3}=\left(\sum\limits_{i=\mathrm{f}, \mathrm{m}, \mathrm{c}, \mathrm{g}} \frac{\varphi_{i}}{\kappa_{i}}\right)^{-1} $
(6) 式中,κ表示材料的热导率,下标i分别表示碳纤维(f)、基体(m)、残炭(c)和热解气体(g), 各组分的热物参量见参考文献[6]和参考文献[14]。
编织层板内热导率可由DIMITRIENKO模型表征[14]:
$ \kappa_{4}=0.5 \kappa_{\mathrm{f}}\left(\frac{T}{T_{0}}\right)^{0.5}\left[B\left(1-\frac{\theta^{2}}{2}\right)+A C\left(1+\frac{\theta^{2}}{2}\right)\right] $
(7) $ \kappa_{5}=37.76 \kappa_{\mathrm{m}}\left(\frac{T}{T_{0}}\right)^{0.5}\left[\frac{A}{C}\left(1-\frac{\theta}{2}\right)+\frac{A^{2}}{2 B} \theta^{2}\right] $
(8) 式中,T0为初始温度,A=(0.59/64)b1,B=φf+A(1-φf),C=(1-φf+Aφf)-1,b1是中间变量,θ为编织层的固有特性。
考虑单向层的铺层夹角,如图 1b所示,在笛卡尔坐标系下,不同夹角的热导率张量见下:
在0°层,
$ {{\mathit{\boldsymbol{\kappa }}_{{\rm{eq}}}} = \left[ {\begin{array}{*{20}{c}} {{\kappa _1}}&0&0\\ 0&{{\kappa _2}}&0\\ 0&0&{{\kappa _3}} \end{array}} \right]} $
(9) 在90°层,
$ {{\mathit{\boldsymbol{\kappa }}_{{\rm{eq}}}} = \left[ {\begin{array}{*{20}{c}} {{\kappa _2}}&0&0\\ 0&{{\kappa _1}}&0\\ 0&0&{{\kappa _3}} \end{array}} \right]} $
(10) 在45°层,
$ \boldsymbol{\kappa}_{\mathrm{eq}}=\left[\begin{array}{ccc} \frac{\kappa_{1} \cos 45^{\circ}+\kappa_{2} \sin 45^{\circ}}{\cos 45^{\circ}+\sin 45^{\circ}} & \kappa_{1} & 0 \\ -\kappa_{2} & \frac{\kappa_{1} \sin 45^{\circ}+\kappa_{2} \cos 45^{\circ}}{\sin 45^{\circ}+\cos 45^{\circ}} & 0 \\ 0 & 0 & \kappa_{3} \end{array}\right] $
(11) 在135°层,
$ \boldsymbol{\kappa}_{\mathrm{eq}}=\left[\begin{array}{ccc} \frac{{\kappa}_{1} \sin 45^{\circ}+\kappa_{2} \cos 45^{\circ}}{\sin 45^{\circ}+\cos 45^{\circ}} & \kappa_{2} & 0 \\ -\kappa_{1} & \frac{\kappa_{1} \cos 45^{\circ}+\kappa_{2} \sin 45^{\circ}}{\cos 45^{\circ}+\sin 45^{\circ}} & 0 \\ 0 & 0 & \kappa_{3} \end{array}\right] $
(12) 在编织层,
$ {\mathit{\boldsymbol{\kappa }}_{{\rm{eq}}}} = \left[ {\begin{array}{*{20}{c}} {{\kappa _4}}&0&0\\ 0&{{\kappa _4}}&0\\ 0&0&{{\kappa _5}} \end{array}} \right] $
(13) 根据不同类型CFRP的铺层顺序即可进一步写出随材料厚度变化的热导率张量表达式。
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激光辐照下复合材料表面的烧蚀机制主要有氧化、气化以及剥蚀,其中氧化和剥蚀都与气流速度密切相关。本文中在数值模拟中施加切向气流的速率为10m/s。在材料表面施加切向气流可以有效减小热解气体对入射激光的屏蔽效应[8],因此, 可以忽略热解气体对激光的屏蔽效应进而简化理论模型。碳纤维复合材料在高速切向气流下才会发生显著的粒状剥蚀,因此也可以忽略纤维的剥蚀效应。通过对材料固有的氧化反应速率系数和氧气的扩散速率系数求协调平均数计算氧化烧蚀速率系数[20]:
$ {V_{{\rm{o}},i}} = {\left( {\frac{1}{{{V_{{\rm{r}},i}}}} + \frac{1}{{{V_{\rm{d}}}}}} \right)^{ - 1}} $
(14) 氧化反应速率系数为[21]:
$ \left\{ {\begin{array}{*{20}{l}} {{V_{{\rm{r}},{\rm{f}}}} = 4.3 \times {{10}^8}\sqrt T \exp \left( { - \frac{{29390}}{T}} \right)}\\ {{V_{{\rm{r}},{\rm{m}}}} = 1.35 \times {{10}^{10}}\sqrt T \exp \left( { - \frac{{31570}}{T}} \right)} \end{array}} \right. $
(15) 式中,Vd表示氧气在边界层的扩散速率[10],Vd=0.332Sc1/3Rex1/2D/dsp,其中Sc为施密特数;Rex为雷诺数;dsp为空气滑移长度。
氧化烧蚀速率可表示为:
$ {v_{\rm{o}}} = \frac{{\left[ {{V_{{\rm{o}},{\rm{f}}}}{\varphi _{\rm{f}}} + {V_{{\rm{o}},{\rm{m}}}}\left( {1 - {\varphi _{\rm{f}}}} \right)} \right] \cdot {M_{\rm{C}}}{C_{\rm{O}}}}}{{\sum\limits_{i = {\rm{f}},{\rm{m}},{\rm{c}},{\rm{g}}} {{\varphi _i}} {\rho _i}}} $
(16) 式中,MC为碳的摩尔质量,CO为氧气浓度。
气化速率通过赫兹-克努森方程表征[14]:
$ {v_{\rm{s}}} = \frac{{{A_{\rm{s}}}}}{{\sum\limits_{i = {\rm{f}},{\rm{m}},{\rm{c}},{\rm{g}}} {{\varphi _{\rm{i}}}} {\rho _i}}}{p_0}\exp \left[ {\frac{{{H_{\rm{C}}}{M_{\rm{C}}}}}{{{k_{\rm{B}}}T}}\left( {\frac{T}{{{T_{\rm{C}}}}} - 1} \right)} \right]{\left( {\frac{{{M_{\rm{C}}}}}{{2{\rm{ \mathsf{ π} }}{k_{\rm{B}}}T}}} \right)^{0.5}} $
(17) 式中,As为系数,p0为标准大气压,HC为碳的气化热,kB为玻尔兹曼常数,TC为碳的气化温度。
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初始温度T0均匀且等于环境温度300K。
激光辐照表面的边界条件为:
$ -\boldsymbol{\kappa}_{\mathrm{eq}} \frac{\partial T}{\partial z}=q_{\mathrm{laser}}-q_{\mathrm{conv}}-q_{\mathrm{rad}}-\rho_{\mathrm{eq}} v_{\mathrm{s}} H_{\mathrm{C}} $
(18) 式中,ρeq表示材料的等效密度。
激光垂直辐照在材料表面,激光功率密度在空间上呈高斯分布,即:
$ q_{\text {laser }}=\alpha \frac{2 P_{\text {laser }}}{\pi r_{\text {laser }}{ }^{2}} \exp \left[-2\left(\frac{x^{2}+y^{2}}{r_{\text {laser }}{ }^{2}}\right)\right] $
(19) 式中,材料对激光的吸收率α=0.8,激光平均功率Plaser=38W,激光光斑半径rlaser=2.1mm。
材料与气流之间的对流换热可由牛顿冷却定律描述:
$ q_{\text {conv }}=h_{\mathrm{f}}\left(T_{\mathrm{w}}-T_{0}\right) $
(20) 式中,hf为对流换热系数,Tw为材料表面温度。
材料与外部环境的辐射换热为:
$ q_{\mathrm{rad}}=\sigma \zeta\left(T_{0}{ }^{4}-T_{\mathrm{w}}{ }^{4}\right) $
(21) 式中,σ为斯蒂芬-玻尔曼兹常数,ζ为靶表面辐射系数。
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使用COMSOL软件通过有限元法对模型进行求解。在模型的不同部位采用了不同尺寸的网格,以适应计算数据的分布特点。在计算数据变化梯度较大的地方(如激光照射区域),需要一个相对密集的网格。网格划分采用预定义分布法,沿每条边按等差数列分布,采用正六面体单元。使用COMSOL的“变形几何”模拟了烧蚀引起的界面迁移,而使用COMSOL的“自适应网格细化”对求解器进行了优化。
激光辐照CFRP下温度对铺层结构的敏感性研究
Study on the sensitivity of temperature to laminated structure under laser irradiation of CFRPs
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摘要: 为了研究连续激光作用下层叠型碳纤维增强环氧树脂基复合材料(CFRP)的铺层结构对材料温度的影响, 采用有限元软件COMSOL模拟连续激光辐照5种典型层叠型CFRP的烧蚀过程。通过材料表面的温度偏差, 获得了5种典型层叠型CFRP的温度分布和演化规律; 通过对材料整体温度分布进行统计分析, 获得了材料温度均匀性随激光辐照时间的变化规律。结果表明, 由单向层叠加而成的CFRP在材料表面光斑边缘附近温度对铺层结构最为敏感, 呈非单调变化; 在5种典型的结构中45°夹角铺层结构的CFRP整体温度分布最为均匀。此研究结果有助于进一步研究激光辐照下复合材料的热力学破坏。Abstract: In order to study the influence of laminated structures of carbon fiber reinforced polymer (CFRP) on the temperature under continuous wave (CW) laser irradiation, the ablation processes of five typical laminated CFRPs irradiated by CW laser were simulated by using finite element software COMSOL. The temperature distribution and evolution of five typical laminated composites were obtained by the temperature deviation of the material surface. Meanwhile, based on analyzing the overall temperature distribution of the materials statistically, the variations of the temperature uniformity of the materials with the laser irradiation time were obtained. The results show that the surface temperature near the edge of the spot is most sensitive to the laminated structure in unidirectional ply CFRPs and the temperature curve is non-monotonic. Moreover, the temperature uniformity of CFRP with 45° angle laminated structure is the best in the five types. The results of this work can be referenced for the study of thermomechanics damage of CFRP induced by laser.
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Table 1. Laminated structure of different types of CFRPs
type laminated structure surface dimensions/mm intrinsic property thickness/mm number of layers 1 [0°/90°/0°/90°] 20×20 — 0.56 4 2 [0°/90°/90°/0°] 20×20 — 0.56 4 3 [0/45°/90°/135°] 20×20 — 0.56 4 4 [1.5K/0°/90°/1.5K] 20×20 plain pattern θ/0.196 0.58 4 5 [3K/0°/3K] 20×20 plain pattern θ/0.145 0.58 3 -
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