多孔金属陶瓷功能梯度矩形板的低速冲击响应

胡旭初; 付涛

doi:10.13801/j.cnki.fhclxb.20221223.003

多孔金属陶瓷功能梯度矩形板的低速冲击响应

doi: 10.13801/j.cnki.fhclxb.20221223.003

胡旭初,
付涛^,

昆明理工大学　机电工程学院，昆明 650500

基金项目: 国家自然科学基金(52205105)；云南省基础研究专项(202101AU070160；202201AT070145)

详细信息

通讯作者:
付涛，博士，讲师，硕士生导师，研究方向为智能材料与结构　E-mail: ftkmust@126.com

中图分类号: TB333
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出版历程
- 收稿日期: 2022-10-24
- 修回日期: 2022-11-29
- 录用日期: 2022-12-09
- 网络出版日期: 2022-12-26
- 刊出日期: 2023-10-15

Low velocity impact response of porous metal ceramic functionally graded rectangular plate

HU Xuchu,
FU Tao^,

Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650500, China

Funds: National Natural Science Foundation of China (52205105); Yunnan Fundamental Research Projects (202101AU070160; 202201AT070145)

摘要

摘要: 为研究多孔金属陶瓷功能梯度矩形板在低速冲击下的动力学响应，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，获得了低速冲击下多孔金属陶瓷功能梯度矩形板的响应解析解。根据Hamilton原理推导出功能梯度矩形板的运动方程，引入两自由度弹簧-质量(S-M)模型用于获得冲击过程中与时间相关的接触力，利用Duhamel原理和Navier法计算了多孔功能梯度矩形板的横向位移。所得结果与文献数据进行比对，验证了有效性。在此基础上，针对相关参数对功能梯度板抗冲击性能的影响进行了比对分析。结果表明：随着孔隙率、功能梯度指数和宽厚比的减小，最大横向位移减小，功能梯度矩形板的能量吸收和抗冲击性能增强。
- 赫兹弹性理论 /
- 一阶剪切变形板理论 /
- 功能梯度矩形板 /
- 弹簧-质量模型 /
- 低速冲击
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.
- Hertzian elastic theory /
- first-order shear deformation plate theory /
- functionally graded rectangular plate /
- spring-mass model /
- low velocity impact

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图 1 多孔功能梯度材料(FGM)矩形板低速冲击模型

Figure 1. Low velocity impact model of porous functionally graded material (FGM) rectangular plate

V_s—Impact velocity of the spherical impactor; a, b and h—Length, width and thickness of the FGM rectangular plate

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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

m₁ and m₂—Mass of the spherical impactor and the porous FGM rectangular plate; w₁ and w₂—Displacement response of m₁ and m₂; δ—Contact displacement; F—Contact force

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图 3 两自由度弹簧-质量模型

Figure 3. Two degree of freedom spring-mass model

$ {K_1} $—Equivalent Hertz spring; $ {K_2} $—Equivalent stiffness spring

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图 4 均质板的接触力验证

Figure 4. Verification of contact force of homogeneous plate

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图 5 多孔FGM矩形板不同孔隙率λ₀下的杨氏模量E分布

Figure 5. Young's modulus E distribution of porous FGM rectangular plate with different porosity λ₀

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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

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图 7 多孔FGM矩形板不同功能梯度指数p下的E分布

Figure 7. E distribution of porous FGM rectangular plate under different functionally graded index p

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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

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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

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图 10 宽厚比η对多孔FGM矩形板横向位移的影响

Figure 10. Effect of width to thickness ratio η on transverse displacement of porous FGM rectangular plate

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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	Young's modulus/GPa	Poisson's ratio	Density/ (kg·m⁻³)
Al6061	67	0.33	2702
SiC	302	0.17	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
$ \eta $	Method	Porosity	$ p = 0 $	$ p = 0.5 $	$ p = 1 $
20	Present	0.2	0.0300	0.0245	0.0210
20	Razaei et al^[27]	0.2	0.0300	0.0245	0.0210
10	Present	0.2	0.1170	0.0959	0.0823
10	Razaei et al^[27]	0.2	0.1170	0.0961	0.0824
5	Present	0.2	0.4284	0.3542	0.3052
5	Razaei et al^[27]	0.2	0.4313	0.3562	0.3068
Notes: $ \eta $—Width to thickness ratio; $ p $—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

V_s/(m·s⁻¹)	Maximum contact force/kN
V_s/(m·s⁻¹)	Khalili et al^[6]	Najafi et al^[29]	Present study	Maximum error/%
1	1.49	1.54	1.50	2.60
4	7.67	8.17	7.92	3.25
7	15.10	16.01	15.49	3.20
10	23.00	24.49	23.76	4.20

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