Thermal scale effect analysis of enhanced Reddy’s laminated composite based on new modified couple stress theory
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摘要: 基于新修正偶应力理论，提出复合材料增强型Reddy层合板热尺度效应模型。该模型只含有一个材料长度参数
$\ell $ ，同时将首次引入厚度方向的旋转变量。通过虚功原理推导出平衡方程，并且利用纳维方法，分析热载作用下细观复合材料层合/夹层方板的位移和应力。数值计算表明：该模型能够很好地捕捉板的热尺度效应，随着材料长度参数增大，板的热尺度效应就会增强，另外随着板的跨厚比增加，板的热尺度效应会减弱，但减弱程度会下降。-
关键词:
- 增强型Reddy层合板 /
- 夹层板 /
- 热尺度效应 /
- 偶应力 /
- 材料长度参数
Abstract: Based on new modified couple stress theory, a model of thermal scale effect was proposed for the laminated composite plates of enhanced Reddy theory. The model contained only one materials length parameter$\ell $ , and the rotation variable was introduced through thickness direction for the first time. The equilibrium equations for the model were presented from the principle of virtual work, and displacements and stresses of micro-laminated composite/sandwich plates were analyzed by using Navier’s method under thermal loading. Numerical results show that the model can capture the thermal scale effect of plates well. As the material length parameter increases, the scale effects of plates are enhanced. Meanwhile, the scale effects are weakened with increasing of span-thickness ratio of plates, but there is a decline in the weakening extend. -
图 1 细观层合板[0°/90°/0°]沿厚度方向分布的位移和应力(a/h=5)
Figure 1. Distribution of displacements and stresses of micro-composite laminated plates [0°/90°/0°] through thickness (a/h=5)
l—Materials length parameter; h—Thickness of plates
图 2 细观夹层板[0°/core/0°]沿厚度方向分布的位移和应力(a/h=4)
Figure 2. Distribution of displacements and stresses of micro-composite sandwich plates [0°/core/0°] through thickness (a/h=4)
表 1 热载作用下细观层合板[0°/90°/0°]位移与应力的对比(括号内数值为
$\eta $ )Table 1. Comparison of displacements and stresses of micro-composite laminated plates [0°/90°/0°] under thermal loading (There is
$\eta $ in the bracket)${a/h}$ ${\ell /h}$ $\bar u\left( {0,\dfrac{b}{2}, - \dfrac{h}{2}} \right)$ $\bar v\left( {\dfrac{a}{2},0, - \dfrac{h}{2}} \right)$ $\bar w\left( {\dfrac{a}{2},\dfrac{b}{2},0} \right)$ ${\bar \sigma _x}\left( {\dfrac{a}{2},\dfrac{b}{2},\dfrac{h}{2}} \right)$ ${\bar \sigma _y}\left( {\dfrac{a}{2},\dfrac{b}{2}, - \dfrac{h}{2}} \right)$ ${\bar \tau _{x{\textit{z}}}}\left( {0,\dfrac{b}{2},0} \right)$ ${\bar \tau _{y{\textit{z}}}}\left( {\dfrac{a}{2},0,0} \right)$ ${\bar \tau _{xy}}\left( {0,0, - \dfrac{h}{2}} \right)$ 5 0 0.1511/0.1327 0.3139/0.2866 0.6315/0.5410 0.7447/0.7837 1.0272/0.8050 0.0686/0.0696 −0.0802/−0.0644 0.1461/0.1317 1/4 0.1430(5.36) 0.2971(5.35) 0.5754(8.88) 0.6599(11.39) 1.0444(1.67) 0.0583(15.01) −0.0864(7.73) 0.1383(5.34) 1/2 0.1240(17.94) 0.2575(17.97) 0.4433(29.80) 0.4615(38.03) 1.0849(5.62) 0.0344(49.84) −0.1009(25.81) 0.1199(17.93) 1 0.0864(42.82) 0.1769(43.64) 0.1735(72.53) 0.0678(90.90) 1.1670(13.61) −0.0123(117.9) −0.1302(62.34) 0.0827(43.39) 10 0 0.3149/0.2625 0.4161/0.3542 2.2205/1.8368 0.7658/0.7511 1.1139/0.8722 0.0469/0.0454 −0.0576/−0.0444 0.1148/0.0969 1/4 0.3019(4.13) 0.3989(4.13) 2.1114(4.91) 0.6996(8.64) 1.1236(0.87) 0.0422(10.02) −0.0594(3.13) 0.1101(4.09) 1/2 0.2689(14.61) 0.3554(14.59) 1.8354(17.34) 0.5319(30.54) 1.1482(3.08) 0.0302(35.61) −0.0639(10.94) 0.0981(14.55) 1 0.1896(39.79) 0.2507(39.75) 1.1709(47.27) 0.1295(83.09) 1.2072(8.38) 0.0013(97.23) −0.0747(29.69) 0.0692(39.72) 20 0 0.6403/0.5232 0.6941/0.5714 8.3894/6.8426 0.7779/0.7392 1.1420/0.8954 0.0255/0.0244 −0.0318/−0.0243 0.1048/0.0860 1/4 0.6162(3.76) 0.6680(3.76) 8.0583(3.95) 0.7172(7.8) 1.1497(0.67) 0.0233(8.63) −0.0325(2.2) 0.1009(3.72) 1/2 0.5538(13.51) 0.6005(13.49) 7.2014(14.16) 0.5603(27.97) 1.1699(2.44) 0.0174(31.76) −0.0343(7.86) 0.0907(13.45) 1 0.3954(38.25) 0.4291(38.18) 5.0251(40.1) 0.1619(79.19) 1.2209(6.91) 0.0025(90.2) −0.0390(22.64) 0.0648(38.17) Notes: ${a/h}$—Span-thickness ratio of micro-composite laminated plates; ${\ell /h}$—Length-thickness ratio of micro-composite laminated plates. 
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表 2 热载作用下细观夹层板[0°/core/0°]位移与应力的对比(括号内数值为
$\eta $ )Table 2. Comparison of displacements and stresses of micro-composite sandwich plates [0°/core/0°] under thermal loading (There is
$\eta $ in the bracket )${a/h}$ ${\ell /h}$ $\bar u\left( {0,\dfrac{b}{2}, - \dfrac{h}{2}} \right)$ $\bar v\left( {\dfrac{a}{2},0, - \dfrac{h}{2}} \right)$ $\bar w\left( {\dfrac{a}{2},\dfrac{b}{2},0} \right)$ ${\bar \sigma _x}\left( {\dfrac{a}{2},\dfrac{b}{2},\dfrac{h}{2}} \right)$ ${\bar \sigma _y}\left( {\dfrac{a}{2},\dfrac{b}{2}, - \dfrac{h}{2}} \right)$ ${\bar \tau _{x{\textit{z}}}}\left( {0,\dfrac{b}{2},0} \right)$ ${\bar \tau _{y{\textit{z}}}}\left( {\dfrac{a}{2},0,0} \right)$ ${\bar \tau _{xy}}\left( {0,0, - \dfrac{h}{2}} \right)$ 4 0 0.5384/0.3310 10.5594/9.3526 12.0011/8.7933 29.3719/27.0127 18.5871/17.5300 1.8548/1.2821 −1.5962/−1.4914 5.4476/4.7534 1/4 0.3579(33.53) 9.8409(6.8) 9.6279(19.77) 25.5814(12.91) 19.2884(3.77) 1.6175(12.79) −1.7410(9.07) 5.0063(8.1) 1/2 −0.0140(102.6) 8.4783(19.71) 4.7786(60.18) 17.8069(39.37) 20.6277(10.98) 1.1402(38.53) −2.0107(25.97) 4.1549(23.73) 1 −0.6743(225.2) 7.1226(32.54) −3.4920(129.1) 4.3304(85.26) 22.0484(18.62) 0.4040(78.22) −2.2279(39.58) 3.1653(41.9) 8 0 1.2860/0.6431 9.4853/8.2103 24.2632/18.8468 29.6706/26.9638 23.7887/20.4417 1.4998/1.1408 −1.3898/−1.2466 2.6437/2.1730 1/4 1.1531(10.33) 9.1360(3.68) 22.4860(7.32) 28.3031(4.61) 23.9660(0.75) 1.4253(4.97) −1.4111(1.53) 2.5253(4.48) 1/2 0.8237(35.95) 8.2853(12.65) 18.0807(25.48) 24.9140(16.03) 24.3989(2.57) 1.2411(17.25) −1.4629(5.26) 2.2357(15.43) 1 0.0586(95.44) 6.4712(31.78) 7.8705(67.56) 17.0682(42.47) 25.3314(6.49) 0.8193(45.37) −1.5731(13.19) 1.6026(39.38) 12 0 2.1399/0.9690 8.3364/6.7533 36.4750/25.6378 30.4641/25.4640 25.5342/23.1161 1.1282/0.8838 −1.0635/−0.9493 1.7142/1.2636 1/4 2.0200(5.6) 8.1096(2.72) 34.8332(4.5) 29.6502(2.67) 25.6144(0.31) 1.0949(2.95) −1.0704(0.65) 1.6575(3.31) 1/2 1.7053(20.31) 7.5184(9.81) 30.5234(16.32) 27.5143(9.68) 25.8236(1.13) 1.0074(10.71) −1.0882(2.32) 1.5092(11.96) 1 0.8724(59.23) 6.0002(28.02) 19.1150(47.59) 21.8667(28.22) 26.3635(3.25) 0.7769(31.14) −1.1341(6.64) 1.1245(34.4) Notes: ${a/h}$—Span-thickness ratio of micro-composite sandwich plates; ${\ell /h}$—Length-thickness ratio of micro-composite sandwich plates
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