3D Analytical Solutions of Mechatronic Coupling for Functionally Graded Piezoelectric Material Plates With a Circular Hole
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摘要: 在推广后的England-Spencer功能梯度板理论基础上，该研究进一步将功能梯度弹性材料推广到了功能梯度压电材料，研究了含有圆孔无限大横观各向同性功能梯度压电材料板在机械荷载作用下的三维响应. 该板理论将三维问题转化为二维问题，利用复变函数解法，通过求解四个解析函数获得具体边值问题的三维解析解. 针对在无穷远处受机械荷载作用的含圆孔功能梯度压电材料板，利用边界条件确定了四个解析函数的具体表达形式. 通过数值算例讨论了材料参数沿板厚方向呈指数函数梯度变化时，边界条件和相关参数对圆孔边上三维应力的影响. 该解析方法可为分析功能梯度压电板的三维孔口问题提供有效解析求解手段.Abstract: Based on the extended England-Spencer functionally graded plate theory, the functionally graded elastic material (FGM) was extended to functionally graded piezoelectric material (FGPM) and the responses of the transversely isotropic FGPM plate containing a circular hole were investigated. By virtue of this theory, a 3D boundary value problem can be transformed into a 2D one. The 3D analytical solution was obtained from 4 analytical functions based on the complex variable function method. For a FGPM plate with a circular hole and subjected to mechanical loads, the specific expressions of the 4 analytical functions were determined according to the boundary conditions. Numerical examples were presented to discuss the effects of boundary conditions and relevant parameters on the 3D stresses along the circular hole edge with the material parameters varying exponentially along the plate thickness. The analytical method provides an effective tool for accurate analysis of 3D hole-related issues in functionally graded piezoelectric plates.
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Key words:
- functionally graded piezoelectric material /
- circular hole /
- mechatronic coupling /
- complex function /
- analytic solution
edited-byedited-by1) (我刊编委陈伟球推荐) -
图 1 含圆孔无限大FGPM板示意图
Figure 1. Schematic diagram of an infinite FGPM material plate with a circular hole
图 3 无穷远处受单向拉伸作用时孔边无量纲环向应力分布
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 3. Distributions of dimensionless circumferential stresses around the hole in a plate under uniaxial tension at infinity
图 4 无穷远处受单边弯曲作用时孔边无量纲环向应力分布
Figure 4. Distributions of dimensionless circumferential stresses around the hole in a plate under unilateral bending at infinity
图 5 无穷远处受扭转作用时孔边无量纲环向应力分布
Figure 5. Distributions of dimensionless circumferential stresses around the hole in a plate under torsion at infinity
图 6 无穷远处受双向拉伸作用时孔边无量纲环向应力分布
Figure 6. Distributions of dimensionless circumferential stresses around the hole in a plate under biaxial tension at infinity
图 7 无穷远处受双边弯曲作用时孔边无量纲环向应力分布
Figure 7. Distributions of dimensionless circumferential stresses around the hole in a plate under bilateral bending at infinity
图 8 弹性常数和无量纲环向应力沿板厚方向变化曲线
Figure 8. Variation of elastic constants and dimensionless hoop stress along the thickness
表 1 横观各向同性HPM板孔边的力矩Mθ/M0对比
Table 1. Torque ratios Mθ/M0 at the edge of a hole in a transversely isotropic HPM plate
θ 0 0.523 1.046 1.569 2.092 2.615 3.141 the present -0.083 0.462 1.561 2.130 1.561 0.462 -0.083 ref. [12] -0.092 0.445 1.542 2.105 1.542 0.445 -0.092 
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表 2 横观各向同性FGM板孔边的力矩Mθ/M对比
Table 2. Torque ratios Mθ/M at the edge of a hole in a transversely isotropic FGM plate
θ 0 0.523 1.046 1.569 2.092 2.615 3.141 the present 0.224 0.667 1.552 1.996 1.552 0.667 0.224 ref. [17] 0.224 0.667 1.552 1.996 1.552 0.667 0.224
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表 3 材料参数的变化形式对孔边环向应力集中的影响
Table 3. The impacts of material parameter variations on circumferential stress concentrations around the hole
material parameter variation P1 P2 P3 P4 maxσα FGM plate 3.801 3.618 3.408 3.348 FGPM plate 3.567 3.565 3.363 3.285 magnitude of stress concentration relief of the FGPM plate compared to the FGM plate 6.16% 1.46% 1.32% 1.88%
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