Low velocity impact response of porous metal ceramic functionally graded rectangular plate
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摘要:
功能梯度材料(FGM)是由两种或两种以上不同性能的材料通过连续改变材料的组成和结构而合成的一种非均质复合材料。功能梯度材料因具有较高机械强度、抗冲击、耐高温性能等特点，被广泛应用于航空航天工程、汽车、海军工程等领域。与分层复合材料不同，其材料组分可沿特定方向以预定形式逐渐变化，并在材料变化方向上对陶瓷与金属进行精确分级，使其性能具有可裁剪性。但大多数有关功能梯度材料的研究工作是从理想模型出发，忽略了实际制造中孔隙的存在。由于孔隙可改变材料的力学性能，且低速冲击对结构局部位置产生侵彻，故开展具有孔隙的FGM板的低速冲击响应研究具有很高的研究意义。论文提出了一种基于赫兹弹性理论和一阶剪切变形板理论的数值分析模型，系统地研究了多孔金属陶瓷FGM矩形板在低速冲击下的动力学响应，论文有以下技术贡献：1.提出了用于预测冲击载荷下多孔FGM矩形板接触力和横向位移的响应解析解，揭示了低速冲击下多孔FGM矩形板的变形机理，拓展了FGM矩形板在工程领域中的潜在应用。2.讨论了低速冲击下孔隙率、功能梯度指数、冲击速度和宽厚比等参数对多孔金属陶瓷FGM矩形板冲击性能的影响，为能够制造各工作条件下适用的FGM板提供了新思路，如 图1 所示。低速冲击下多孔金属陶瓷功能梯度矩形板的理论模型 Abstract: In order to study the dynamic response of porous metal ceramic functionally graded rectangular plate under low velocity impact, a numerical analysis model based on Hertzian elastic theory and first-order shear deformation plate theory was presented, the analytical solution of response of porous cermet functionally graded rectangular plate under low velocity impact was obtained. According to Hamilton's principle, the equation of motion of functionally graded rectangular plate was derived, a spring-mass (S-M) model with two degrees of freedom was introduced to obtain the time-dependent contact forces during impact, using the Duhamel principle and Navier method to calculate the transverse displacement of porous functionally graded rectangular plate. The results obtained were compared with literature data to verify the validity. On this basis, the influence of related parameters on the impact resistance of functionally graded rectangular plate was compared and analyzed. The results show that with the decrease of porosity, functionally graded index and width to thickness ratio, the maximum transverse displacement of the functionally graded rectangular plate decreases, energy absorption and impact resistance are increased. -
图 1 多孔FGM矩形板低速冲击模型
Figure 1. Low velocity impact model of porous FGM rectangular plate
Vs is the impact velocity of the spherical impactor; a, b and h are the length, width and thickness of the functionally graded rectangular plate
图 2 冲击过程接触力与接触位移示意图
Figure 2. Schematic diagram of contact force and contact displacement during impact
$ {m_1} $ and $ {m_2} $ are the mass of the spherical impactor and the porous FGM rectangular plate; ${w_{\text{1}}}$ and ${w_{\text{2}}}$ are the displacement response of $ {m_1} $ and $ {m_2} $; $ \delta $ is the contact displacement; $F$ is the contact force
图 3 两自由度弹簧-质量模型
Figure 3. Two degree of freedom spring-mass model
$ {K_1} $ is the equivalent Hertz spring; $ {K_2} $ is the equivalent stiffness spring
图 5 多孔FGM矩形板不同孔隙率下的E分布
Figure 5. Young's modulus distribution of porous FGM rectangular plate with different porosity
图 6 孔隙率对FGM矩形板横向位移的影响
Figure 6. Effect of porosity on transverse displacement of FGM rectangular plate
图 7 多孔FGM矩形板不同功能梯度指数下的E分布
Figure 7. Young's modulus distribution of porous FGM rectangular plate under different functionally graded index
图 8 功能梯度指数对多孔FGM矩形板横向位移的影响
Figure 8. Effect of functionally graded index on transverse displacement of porous FGM rectangular plate
图 9 冲击速度对多孔FGM矩形板横向位移的影响
Figure 9. Effect of impact velocity on transverse displacement of porous FGM rectangular plate
图 10 宽厚比对多孔FGM矩形板横向位移的影响
Figure 10. Effect of width to thickness ratio on transverse displacement of porous FGM rectangular plate
表 1 多孔FGM矩形板各组分材料特性
Table 1. Material properties of components of porous FGM rectangular plate
Material Al6061 SiC Young's modulus/GPa 67 302 Poisson's ratio 0.33 0.17 Density/(kg·m−3) 2702 3100 
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表 2 含孔隙FGM板的无量纲频率验证
Table 2. Non-dimensional frequencies verification of porous FGM plate
$ \eta $ Method Porosity Non-dimensional frequency $ p = 0 $ $ p = 0.5 $ $ p = 1 $ 20 Present 0.2 0.03 0.0245 0.021 Razaei[27] 0.03 0.0245 0.021 10 Present 0.2 0.117 0.0959 0.0823 Razaei[27] 0.117 0.0961 0.0824 5 Present 0.2 0.4284 0.3542 0.3052 Razaei[27] 0.4313 0.3562 0.3068 Notes：$ \eta $ is the width to thickness ratio; $ p $ is the functionally graded index.
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表 3 不同冲击速度下理想FGM板接触力验证
Table 3. Verification of contact force for ideal FGM plate with different impact velocities
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