功能梯度材料结构由于容重轻,机械强度高,常常应用于飞机、滑翔机等的航空板中。本文研究了Pasternak地基对FGM圆板在横向载荷下非线性弯曲行为的影响。假设功能梯度材料性质只沿梁厚度方向并且按成分含量的幂指数形式变化;为消除材料非均匀性导致的拉弯耦合效应,引入了物理中面,给出了地基弹性模量kw和地基剪切模量kp对FGM圆板变形的关系,分析了材料梯度指数对不同地基板的影响。结果表明:地基弹性模量是影响FGM板变形的主要因素,kw越大,板的变形越小;材料中金属成分越多,板的变形越大;kw越大,梯度指数对FGM圆板变形的非线性影响越小。 Functionally gradient materials (FGM) are often used in aircraft, gliders and other aircraft plates due to their light bulk density and high mechanical strength. The influence of Pasternak foundation on the nonlinear bending behavior of FGM circular plate under transverse load was studied in this paper. It is assumed that the properties of FGM only change along the thickness of plate and change according to the power law distribution with the volume fraction of the constituents. In order to eliminate the tension-bending coupling effect caused by material inhomogeneity, the Physical middle plane was introduced. The relationships between elastic modulus of foundation and shear modulus of foundation on deformation of FGM circular plate were given, and the influences of material gradient index on different ground plates were analyzed. The results show that the elastic modulus of foundation is the main factor affecting the deformation of FGM plate, and the larger the elastic modulus kw is, the smaller the deformation of FGM plate is; The more metal components in the materials, the greater the deformation of the plate will be; The larger kw is, the smaller the nonlinear effect of gradient index on the deformation of FGM circular plate is.
功能梯度材料结构由于容重轻,机械强度高,常常应用于飞机、滑翔机等的航空板中。本文研究了Pasternak地基对FGM圆板在横向载荷下非线性弯曲行为的影响。假设功能梯度材料性质只沿梁厚度方向并且按成分含量的幂指数形式变化;为消除材料非均匀性导致的拉弯耦合效应,引入了物理中面,给出了地基弹性模量kw和地基剪切模量kp对FGM圆板变形的关系,分析了材料梯度指数对不同地基板的影响。结果表明:地基弹性模量是影响FGM板变形的主要因素,kw越大,板的变形越小;材料中金属成分越多,板的变形越大;kw越大,梯度指数对FGM圆板变形的非线性影响越小。
功能梯度材料,圆板,Pasternak地基,弯曲,非线性
Yu Chen1, Fanzhuan Yang2, Xin Yan1, Qinglu Li1*
1Department of Engineering Mechanics, Lanzhou University of Technology, Lanzhou Gansu
2Road and Bridge Engineering College, Xinjiang Vocational & Technical College of Communications, Urumqi Xinjiang
Received: Nov. 17th, 2020; accepted: Dec. 24th, 2020; published: Dec. 31st, 2020
Functionally gradient materials (FGM) are often used in aircraft, gliders and other aircraft plates due to their light bulk density and high mechanical strength. The influence of Pasternak foundation on the nonlinear bending behavior of FGM circular plate under transverse load was studied in this paper. It is assumed that the properties of FGM only change along the thickness of plate and change according to the power law distribution with the volume fraction of the constituents. In order to eliminate the tension-bending coupling effect caused by material inhomogeneity, the Physical middle plane was introduced. The relationships between elastic modulus of foundation and shear modulus of foundation on deformation of FGM circular plate were given, and the influences of material gradient index on different ground plates were analyzed. The results show that the elastic modulus of foundation is the main factor affecting the deformation of FGM plate, and the larger the elastic modulus kw is, the smaller the deformation of FGM plate is; The more metal components in the materials, the greater the deformation of the plate will be; The larger kw is, the smaller the nonlinear effect of gradient index on the deformation of FGM circular plate is.
Keywords:Functionally Graded Material, Circular Plate, Pasternak Foundation, Bending, Nonlinear
Copyright © 2020 by author(s) and Hans Publishers Inc.
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航空胶合板由材质均匀、强度高的木材旋制的薄单板。鉴于功能梯度材料(FGM)具有耐高温,抗腐蚀,强度高,韧性好等优点 [
国内外学者采用不同的弹性地基模型,研究了地基系数对板屈曲或弯曲的力学行为做了大量的工作。文献 [
工程实际中,由于水平力作用必然会导致地基与结构接触面上存在切向方向的相互作用,因此没有考虑切向力的Winkler模型与实际问题不符。为了克服Winkler模型不足,马立博等 [
在上述研究的基础之上,考虑水平摩阻的Pasternak地基模型被许多学者所采用。边学成 [
大量的文献调研表明:各种弹性地基上梁结构的静动态力学响应分析较多,相比之下,弹性地基上板的弯曲和屈曲分析较少,尤其是FGM圆板在Pasternak弹性地基上的非线性弯曲行为的研究鲜见。鉴于此,本文给出了Pasternak弹性地基上FGM圆板在横向均匀载荷下的非线性弯曲控制方程,引入物理中面的概念消除了非均匀材料引起的拉弯耦合效应,数值计算了双参数弹性地基系数对FGM圆板弯曲响应的影响。
考虑置于Pasternak双参数弹性地基上的周边夹紧FGM薄圆板了,其厚度为h,半径为a,圆板上受到横向均布载荷q的作用。采用笛卡尔坐标系 ( r , θ , z ) ,这里设r为径向且并由板中心向外为正, θ 为环向,z轴正方向垂直于平面向上,如图1所示。圆板由陶瓷和金属复合而成的功能梯度材料制成。
图1. Pasternak地基上的FGM圆板
由于FGM材料的非均匀性,导致几何中面和物理中面不重合,为消除FGM材料中拉弯耦合效应,引入如下物理中面 [
z 0 = ∫ − 0.5 h 0.5 h z E ( z ) d z ∫ − 0.5 h 0.5 h E ( z ) d z (3)
根据经典板理论,圆板的位移函数表示为:
u r ( r , z ) = u ( r ) − ( z − z 0 ) w , r , u z ( r , z ) = w ( r ) (4)
非线性几何关系为:
{ ε r = ε r 0 + z κ r = u , r + w , r / 2 − ( z − z 0 ) w , r r ε θ = ε θ 0 + ( z − z 0 ) κ r = u / r − ( z − z 0 ) w , r / r (5)
考虑对称性,FGM圆板的本构方程为:
( σ r σ θ ) = E ( z ) 1 − ν 2 [ 1 ν ν 1 ] ( ε r ε θ ) (6)
应力沿着板厚度方向积分,可得薄膜力和弯矩为
( N i , M i ) = ∫ − h / 2 h / 2 σ i ( 1 , ( z − z 0 ) ) d z , ( i = r , θ ) (7)
式中 ε r , ε θ 和 σ r , σ θ 分别为径向和环向的应变和应力; N i ( i = r , θ ) 为薄膜力, M i ( i = r , θ ) 为弯矩。将式(5)代入式(6),考虑到式(4),得到
( N r , M r ) = ( A , B ) [ u , r + 1 2 w , r 2 + ν u r ] − ( B , D ) ( w , r r + ν r w , r ) (8)
( N θ , M θ ) = ( A , B ) [ u , r + ν 2 w , r 2 + u r ] − ( B , D ) ( ν w , r r + 1 r w , r ) (9)
其中A、B、D分别为拉伸、拉弯耦合和弯曲,刚度定义为
( A , B , D ) = ∫ − h / 2 h / 2 ( 1 , ( z − z 0 ) , ( z − z 0 ) 2 ) E ( z ) 1 − ν 2 d z (10)
Pastrnak模型的压力–位移关系假定为
p = k w w − k p w ″ (11)
这里 k w 是地基弹性模量, k p 是地基剪切模量。
弹性地基上圆板静力平衡方程如下 [
N r − N θ + r N r , r = 0 (12)
M r , r r 2 + 2 M r , r / r − M θ , r / r + ( r N r w , r ) , r / r + k w w + k p ( w , r r + w , r / r ) = q (13)
将式(5)~(8)代入方程(12)和(13),整理后可获得位移形式的基本控制方程,然后再引入如下无量纲变换
( X , W ) = ( r , w ) / a , U = u a h 2 , η = a h , F 1 = B D h , F 2 = A D h 2 , F 3 = B A h ,
( K w , K p ) = ( k w b 4 , k p b 2 ) / D , Q = a 4 D h q (14)
得到无量纲控制方程表示如下:
U , X X + U , X / X − U , X / X 2 + W , X W , X X + ( 1 − ν ) W , X 2 / 2 X = F 3 W , X X X + W , X X / X − W , X / X 2 (15)
W , X X X X + 2 W , X X X / X − W , X X / X 2 + W , X X X / X 3 = F 2 ( ν U , X W , X / X + ν W , X 3 / 2 / X + U W , X / X 2 + ( U , X W , X X + W , X 2 W , X X / 2 + ν U W , X X / X ) / ( 1 − F 1 F ) − F 1 ( W , X X 2 + W , X 2 / X 2 + 2 ν W , X W , X X / X ) / ( 1 − F 1 F 3 ) + { K w W − K p ( W , X / X + W , X X ) + Q } / ( 1 − F 1 F 3 ) (16)
无量纲边界条件为:
X = 0 处, W = f i n i t e , U = 0 , W , X = 0 , lim X → 0 ( W , X X X X + W , X X / X ) = 0 (17a)
X = 1 处, W = 0 , U = 0 , W , X = 0 (17b)
分析中考虑由铝(Al)和氧化锆(ZrO2)复合而成的FGM圆板,其弹性模量分别为 E m = 70 G P a , E c = 151 G P a ,泊松比均为 ν i ( i = c , m ) = 0.3 。当 ( K w , K p ) = ( 50 , 10 ) 时,为Pasternak地基支撑圆板,当 ( K w , K p ) = ( 50 , 0 ) 时,退化为Winkler地基支撑,当 ( K w , K p ) = ( 0 , 0 ) 时,退化为无地基支撑。首先,给定梯度指数为 p = 2 ,图2考察了不同地基下圆板中心挠度随载荷的变化情况,显然,给定相同载荷下,Pasternak地基上FGM圆板的变形最小,无地基支撑圆板的最大,Winkler地基上FGM圆板的变形介于Pasternak地基和无地基之间。三种情况下FGM圆板的变形随横向载荷的关系均为非线性递增。
图2. 各地基下Q-W(0)关系曲线(p = 2)
下面考察Pasternak地基对FGM圆板弯曲行为的影响。给定 p = 2 , K p = 5 , K w 取不同值时中心挠度即最大挠度如图3所示。不难看出,随着
图4为给定
图5中给定了Pasternak地基
图6给出了不同地基下随梯度指数对FGM圆板最大挠度的影响。不出意外,随着p的增加,中心无量纲挠度也增加;随着弹性地基系数
图3. Kw对Q-W(0)关系的影响
图4. Kp对Q-W(0)关系的影响
图5. 不同梯度下Pasternak地基圆板Q-W(0)关系
图6. 不同地基下W(0)-p变化关系
引入物理中面的概念,消除了材料非均匀性导致的拉弯耦合效应,分析了Pasternak双参数弹性地基上FGM航空圆板非线性弯曲弯曲力学行为。数值计算结果表明:
(1) 材料梯度指数一定时,Pasternak地基下FGM板的变形小于相同工况下的Winkler地基板,板的挠度随载荷的增加非线性递增。
(2) 相比地基剪切模量
(3) 地基系数一定时,随着梯度指数的增加,板的变形也随之增加;
陈宇,杨凡转,闫鑫,李清禄. Pasternak地基对FGM圆板非线性弯曲的影响The Influence of Pasternak Foundation on the Nonlinear Bending of FGM Circular Plate[J]. 力学研究, 2020, 09(04): 201-208. https://doi.org/10.12677/IJM.2020.94022