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CN111426568A - Method for determining deflection of circular film with limited maximum deflection under gas pressure - Google Patents

Method for determining deflection of circular film with limited maximum deflection under gas pressure Download PDF

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CN111426568A
CN111426568A CN202010199461.1A CN202010199461A CN111426568A CN 111426568 A CN111426568 A CN 111426568A CN 202010199461 A CN202010199461 A CN 202010199461A CN 111426568 A CN111426568 A CN 111426568A
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circular film
gas pressure
circular
film
deflection
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何晓婷
梅东
李雪
赵智航
孙俊贻
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Chongqing University
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    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/08—Investigating strength properties of solid materials by application of mechanical stress by applying steady tensile or compressive forces
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    • G01N3/12—Pressure testing
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    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
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    • G01N2203/0019—Compressive
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2203/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N2203/003—Generation of the force
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    • G01N2203/0044—Pneumatic means

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Abstract

本发明公开了一种气体压力下最大挠度受限的圆形薄膜的挠度的确定方法，其特征在于：对杨氏弹性模量为E、泊松比为ν、半径为a、厚度为h的周边固定夹紧的最初平坦的圆形薄膜施加气体压力q，让圆形薄膜产生轴对称变形、并与一块平行于最初平坦的圆形薄膜的刚性平板形成一个半径为b的圆形光滑接触区域，其中刚性平板与最初平坦的圆形薄膜相距H，那么基于该圆形薄膜轴对称变形问题的静力平衡分析，利用气体压力q的测量值，就可以得到轴对称变形后的圆形薄膜在区域b＜r＜a(r表示最初平坦的圆形薄膜上的任意一点到其圆心的距离)内各点的挠度w(r)。

Figure 202010199461

The invention discloses a method for determining the deflection of a circular thin film with limited maximum deflection under gas pressure. The gas pressure q is applied to the initially flat circular film fixed and clamped around the periphery, so that the circular film is deformed axisymmetrically and forms a circular smooth contact area with a radius b with a rigid flat plate parallel to the initially flat circular film , where the distance between the rigid plate and the initially flat circular film is H, then based on the static equilibrium analysis of the axisymmetric deformation of the circular film, using the measured value of the gas pressure q, the circular film after the axisymmetric deformation can be obtained in The deflection w(r) of each point in the region b<r<a (r represents the distance from any point on the initially flat circular film to the center of the circle).

Figure 202010199461

Description

气体压力下最大挠度受限的圆形薄膜的挠度的确定方法Method for determining deflection of circular membranes with limited maximum deflection under gas pressure

技术领域technical field

本发明涉及一种气体压力作用下周边固定夹紧的圆形薄膜在最大挠度受限制条件下的挠度的确定方法。The invention relates to a method for determining the deflection of a circular film fixed and clamped around the periphery under the action of gas pressure under the condition that the maximum deflection is limited.

背景技术Background technique

周边固定夹紧的圆形薄膜在最大挠度受限制条件下的轴对称变形，在许多工程技术领域都有应用，例如，用来研究薄膜/基层系统的粘附能测量、研制各类传感器等。然而从文献查新结果看，目前仅有横向均布载荷作用下周边固定夹紧的圆形薄膜在最大挠度受限制条件下轴对称变形问题的解析研究成果，例如，发明专利“最大挠度受限制状态下圆形薄膜挠度的确定方法”(专利号：2017100325346)便只能解决在始终垂直于最初平坦的圆形薄膜的几何中面的均布载荷(即横向均布载荷)作用下最大挠度受限制的周边固定夹紧的圆形薄膜的轴对称变形问题。然而，在工程技术领域中常会遇到气体压力作用下周边固定夹紧的圆形薄膜的轴对称变形问题，例如，在研究薄膜/基层系统粘附能的约束鼓泡试验中，通常采用气体加压的方式来使得薄膜和基层脱离。众所周知，在气体压力作用下周边固定夹紧的圆形薄膜的轴对称变形过程中，气体压力的作用方向始终垂直于轴对称变形的圆形薄膜的表面，而非始终垂直于最初平坦的圆形薄膜的几何中面，所以不能用横向均布载荷来代替气体载荷，因此，需要对气体压力作用下周边固定夹紧的圆形薄膜在最大挠度受限制条件下的轴对称变形问题进行解析研究，并获得该力学问题的解析解，这项工作可以为工程技术领域提供更大的研发空间，具有重要的实际意义，这也正是本发明所要解决的技术问题。The axisymmetric deformation of a circular film that is fixed and clamped around the periphery under the condition that the maximum deflection is limited has applications in many engineering and technical fields, for example, it is used to study the adhesion energy measurement of film/substrate systems, and develop various sensors. However, judging from the results of literature search, there are only analytical research results on the axisymmetric deformation of the circular film fixed and clamped around the periphery under the condition of limited maximum deflection under the lateral uniform load. For example, the invention patent "The maximum deflection is limited The method for determining the deflection of a circular film in a state of Axisymmetric deformation of circular membranes clamped by perimeter fixed clamps. However, in the field of engineering technology, the problem of axisymmetric deformation of a fixed and clamped circular film around the periphery is often encountered under the action of gas pressure. Press the way to release the film and the base layer. It is well known that in the process of axisymmetric deformation of a circular film that is fixed and clamped at the periphery under the action of gas pressure, the direction of action of the gas pressure is always perpendicular to the surface of the axisymmetrically deformed circular film, not always perpendicular to the initially flat circular film. Because of the geometric midplane of the film, the gas load cannot be replaced by a laterally uniform load. Therefore, it is necessary to analyze the axisymmetric deformation problem of a circular film that is fixed and clamped around the periphery under the action of gas pressure under the condition that the maximum deflection is limited. And obtain the analytical solution of the mechanical problem, this work can provide a larger research and development space for the engineering technology field, and has important practical significance, which is also the technical problem to be solved by the present invention.

发明内容SUMMARY OF THE INVENTION

本发明致力于气体压力作用下周边固定夹紧的圆形薄膜在最大挠度受限制条件下的轴对称变形问题的解析研究，基于静力平衡分析，得到了该圆形薄膜轴对称变形问题的解析解，并在此基础上给出了气体压力下最大挠度受限的圆形薄膜的挠度的确定方法。The present invention is devoted to the analytical research on the axisymmetric deformation problem of a circular membrane fixed and clamped around the periphery under the condition of limited maximum deflection under the action of gas pressure. On this basis, a method for determining the deflection of a circular membrane with limited maximum deflection under gas pressure is given.

气体压力下最大挠度受限的圆形薄膜的挠度的确定方法，其特征在于：对杨氏弹性模量为E、泊松比为ν、半径为a、厚度为h的周边固定夹紧的最初平坦的圆形薄膜施加气体压力q，让圆形薄膜产生轴对称变形、并与一块平行于最初平坦的圆形薄膜的刚性平板形成一个半径为b的圆形光滑接触区域，其中刚性平板与最初平坦的圆形薄膜相距H，那么基于该圆形薄膜轴对称变形问题的静力平衡分析，就可以得到所施加的气体压力q与轴对称变形后的圆形薄膜在区域b＜r＜a(r表示最初平坦的圆形薄膜上的任意一点到其圆心的距离)内各点的挠度w(r)之间的解析关系The method for determining the deflection of a circular film with limited maximum deflection under gas pressure is characterized in that: the initial clamping of the periphery with Young's modulus of elasticity E, Poisson's ratio ν, radius a, and thickness h is fixed and clamped. The gas pressure q is applied to the flat circular film, which causes the circular film to deform axisymmetrically and form a circular smooth contact area with a radius b with a rigid flat plate parallel to the initially flat circular film, where the rigid flat plate is in contact with the original flat circular film. The flat circular film is separated by H, then based on the static equilibrium analysis of the axisymmetric deformation of the circular film, it can be obtained that the applied gas pressure q and the circular film after the axisymmetric deformation are in the region b<r<a( r represents the distance from any point on the initially flat circular film to the center of the circle) The analytical relationship between the deflections w(r) of each point

Figure BDA0002418852700000021
Figure BDA0002418852700000021

其中，in,

Figure BDA0002418852700000022
Figure BDA0002418852700000022

Figure BDA0002418852700000023
Figure BDA0002418852700000023

Figure BDA0002418852700000024
Figure BDA0002418852700000024

Figure BDA0002418852700000025
Figure BDA0002418852700000025

Figure BDA0002418852700000026
Figure BDA0002418852700000026

Figure BDA0002418852700000031
Figure BDA0002418852700000031

Figure BDA0002418852700000032
Figure BDA0002418852700000032

Figure BDA0002418852700000033
Figure BDA0002418852700000033

Figure BDA0002418852700000034
Figure BDA0002418852700000034

而b、c0、c1、d0的值由方程And the values of b, c 0 , c 1 , and d 0 are determined by the equation

Figure BDA0002418852700000035
Figure BDA0002418852700000035

Figure BDA0002418852700000036
Figure BDA0002418852700000036

Figure BDA0002418852700000037
Figure BDA0002418852700000037

和and

Figure BDA0002418852700000041
Figure BDA0002418852700000041

确定，其中，OK, where,

Figure BDA0002418852700000045
Figure BDA0002418852700000045

Figure BDA0002418852700000042
Figure BDA0002418852700000042

Figure BDA0002418852700000043
Figure BDA0002418852700000043

Figure BDA0002418852700000044
Figure BDA0002418852700000044

Figure BDA0002418852700000051
Figure BDA0002418852700000051

这样，只要准确测得气体压力q的值，就可以把轴对称变形后的圆形薄膜在区域b＜r＜a内各点的挠度w(r)确定下来，其中，r、a、b、h、H、w(r)的单位均为毫米(mm)，E、q的单位均为牛顿每平方毫米(N/mm2)，而ν、c0、c1、c2、c3、c4、c5、c6、d0、d1、d2、d3、d4、d5、d6、Q、e和β均为无量纲的量。In this way, as long as the value of the gas pressure q is accurately measured, the deflection w(r) of each point in the region b<r<a of the axisymmetrically deformed circular film can be determined, where r, a, b, The units of h, H, w(r) are all millimeters (mm), the units of E and q are Newtons per square millimeter (N/mm 2 ), and the units of ν, c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , d 0 , d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , Q, e, and β are all dimensionless quantities.

附图说明Description of drawings

图1为气体压力作用下周边固定夹紧的圆形薄膜在最大挠度受限制条件下的轴对称变形的示意图，其中，1是轴对称变形后的圆形薄膜，2是刚性平板，3是夹紧装置，4是最初平坦的圆形薄膜的几何中面，5是支座，而a表示圆形薄膜的半径和夹紧装置的内半径，b表示轴对称变形后的圆形薄膜与刚性平板之间形成的圆形光滑接触区域的半径，o表示坐标系的原点，r表示径向坐标，w(r)表示横向坐标(也表示轴对称变形后的圆形薄膜上任一点的挠度)，q表示气体压力，H表示最初平坦的圆形薄膜与刚性平板之间的距离。Figure 1 is a schematic diagram of the axisymmetric deformation of a circular film fixed and clamped around the periphery under the action of gas pressure under the condition that the maximum deflection is limited, wherein, 1 is the circular film after axisymmetric deformation, 2 is a rigid plate, and 3 is a clamp The clamping device, 4 is the geometric mid-plane of the initially flat circular film, 5 is the support, and a represents the radius of the circular film and the inner radius of the clamping device, and b represents the axisymmetrically deformed circular film and the rigid plate The radius of the circular smooth contact area formed between them, o represents the origin of the coordinate system, r represents the radial coordinate, w(r) represents the transverse coordinate (also represents the deflection of any point on the circular film after axisymmetric deformation), q is the gas pressure and H is the distance between the initially flat circular membrane and the rigid flat plate.

具体实施方式Detailed ways

下面结合具体案例对本发明的技术方案作进一步的说明：Below in conjunction with specific case, the technical scheme of the present invention is further described:

如图1所示，对杨氏弹性模量E＝7.84N/mm2、泊松比ν＝0.47、半径a＝10mm、厚度h＝1mm的周边固定夹紧的最初平坦的圆形薄膜施加气体压力q，让圆形薄膜产生轴对称变形、并与一块平行于最初平坦的圆形薄膜的刚性平板形成一个半径为b的圆形光滑接触区域，其中刚性平板与最初平坦的圆形薄膜之间的距离H＝2.5mm，测得气体压力q＝0.1N/mm2，采用本发明所给出的方法，由以下方程As shown in Fig. 1, a gas was applied to an initially flat circular film with Young's modulus of elasticity E = 7.84 N/mm 2 , Poisson's ratio ν = 0.47, radius a = 10 mm, and thickness h = 1 mm fixed and clamped around the periphery The pressure q causes the circular membrane to deform axisymmetrically and form a circular smooth contact area of radius b with a rigid plate parallel to the initially flat circular membrane, where the rigid plate and the initially flat circular membrane are in contact with each other. The distance H=2.5mm, the measured gas pressure q=0.1N/mm 2 , using the method provided by the present invention, the following equation

Figure BDA0002418852700000061
Figure BDA0002418852700000061

Figure BDA0002418852700000062
Figure BDA0002418852700000062

Figure BDA0002418852700000063
Figure BDA0002418852700000063

Figure BDA0002418852700000064
Figure BDA0002418852700000064

Figure BDA0002418852700000065
Figure BDA0002418852700000065

Figure BDA0002418852700000066
Figure BDA0002418852700000066

Figure BDA0002418852700000071
Figure BDA0002418852700000071

Figure BDA0002418852700000072
Figure BDA0002418852700000072

Figure BDA0002418852700000081
Figure BDA0002418852700000081

Figure BDA0002418852700000082
Figure BDA0002418852700000082

Figure BDA0002418852700000083
Figure BDA0002418852700000083

Figure BDA0002418852700000091
Figure BDA0002418852700000091

Figure BDA0002418852700000092
Figure BDA0002418852700000092

Figure BDA0002418852700000093
Figure BDA0002418852700000093

Figure BDA0002418852700000101
Figure BDA0002418852700000101

Figure BDA0002418852700000102
Figure BDA0002418852700000102

Figure BDA0002418852700000103
Figure BDA0002418852700000103

Figure BDA0002418852700000104
Figure BDA0002418852700000104

得到b＝2.74904mm、c0＝0.81217、c1＝-0.38986、d0＝0.18878以及c2＝-0.63549、c3＝-0.28727、c4＝-0.42793、c5＝-0.45402、c6＝-0.58449、d1＝-0.31945、d2＝-0.44174、d3＝-0.16481、d4＝-0.33085、d5＝-0.21219、d6＝-0.50953，最后，由方程Obtained b=2.74904mm, c0 = 0.81217, c1 = -0.38986 , d0 = 0.18878 and c2= -0.63549 , c3 = -0.28727 , c4=-0.42793, c5= -0.45402 , c6=- 0.58449, d 1 =-0.31945, d 2 =-0.44174, d 3 =-0.16481, d 4 =-0.33085, d 5 =-0.21219, d 6 =-0.50953, and finally, by the equation

Figure BDA0002418852700000105
Figure BDA0002418852700000105

得到轴对称变形后的圆形薄膜在区域2.74904mm＜r＜10mm内各点的挠度(mm)：Obtain the deflection (mm) of each point in the region 2.74904mm<r<10mm of the circular film after the axisymmetric deformation:

Figure BDA0002418852700000106
Figure BDA0002418852700000106

Claims (1)

1.气体压力下最大挠度受限的圆形薄膜的挠度的确定方法，其特征在于：对杨氏弹性模量为E、泊松比为ν、半径为a、厚度为h的周边固定夹紧的最初平坦的圆形薄膜施加气体压力q，让圆形薄膜产生轴对称变形、并与一块平行于最初平坦的圆形薄膜的刚性平板形成一个半径为b的圆形光滑接触区域，其中刚性平板与最初平坦的圆形薄膜相距H，那么基于该圆形薄膜轴对称变形问题的静力平衡分析，利用气体压力q的测量值，由以下方程1. The method for determining the deflection of a circular film with a limited maximum deflection under gas pressure, characterized in that: the periphery of which Young's modulus of elasticity is E, Poisson's ratio is ν, radius is a, and thickness is h is fixed and clamped The gas pressure q is applied to the initially flat circular film, which causes the circular film to deform axisymmetrically and form a circular smooth contact area of radius b with a rigid plate parallel to the initially flat circular film, where the rigid plate distance H from the initially flat circular membrane, then based on a static equilibrium analysis of the axisymmetric deformation problem for this circular membrane, using the measured value of the gas pressure q, given by the following equation
Figure FDA0002418852690000011
Figure FDA0002418852690000011
Figure FDA0002418852690000012
Figure FDA0002418852690000012
Figure FDA0002418852690000013
Figure FDA0002418852690000013
Figure FDA0002418852690000014
Figure FDA0002418852690000014
Figure FDA0002418852690000015
Figure FDA0002418852690000015
Figure FDA0002418852690000016
Figure FDA0002418852690000016
Figure FDA0002418852690000017
Figure FDA0002418852690000017
Figure FDA0002418852690000021
Figure FDA0002418852690000021
Figure FDA0002418852690000031
Figure FDA0002418852690000031
Figure FDA0002418852690000032
Figure FDA0002418852690000032
Figure FDA0002418852690000033
Figure FDA0002418852690000033
Figure FDA0002418852690000041
Figure FDA0002418852690000041
Figure FDA0002418852690000042
Figure FDA0002418852690000042
Figure FDA0002418852690000043
Figure FDA0002418852690000043
Figure FDA0002418852690000051
Figure FDA0002418852690000051
Figure FDA0002418852690000052
Figure FDA0002418852690000052
Figure FDA0002418852690000053
Figure FDA0002418852690000053
Figure FDA0002418852690000054
Figure FDA0002418852690000054
 确定参量b、c0、c1、d0以及c2、c3、c4、c5、c6、d1、d2、d3、d4、d5、d6的值，最后，由方程Determine the values of the parameters b, c 0 , c 1 , d 0 and c 2 , c 3 , c 4 , c 5 , c 6 , d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , and finally, by the equation
Figure FDA0002418852690000055
Figure FDA0002418852690000055
 确定轴对称变形后的圆形薄膜在区域b＜r＜a(r表示最初平坦的圆形薄膜上的任意一点到其圆心的距离)内各点的挠度w(r)，其中，r、a、b、h、H、w(r)的单位均为毫米(mm)，E、q的单位均为牛顿每平方毫米(N/mm2)，而ν、c0、c1、c2、c3、c4、c5、c6、d0、d1、d2、d3、d4、d5、d6、Q、e和β均为无量纲的量。Determine the deflection w(r) of each point of the circular film after axisymmetric deformation in the region b<r<a (r represents the distance from any point on the initially flat circular film to the center of the circle), where r, a , b, h, H, w(r) are in millimeters (mm), E, q are in Newton per square millimeter (N/mm 2 ), and ν, c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , d 0 , d 1 , d 2 , d 3 , d 4 , d 5 , d 6 , Q, e, and β are all dimensionless quantities.
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Cited By (4)

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CN112880950A (en) * 2021-01-18 2021-06-01 重庆大学 Method for determining deflection of circular prestressed film with limited maximum deflection under air pressure
CN113075048A (en) * 2021-03-30 2021-07-06 重庆大学 Method for determining maximum deflection of circular film under gas pressure
CN113720688A (en) * 2021-08-17 2021-11-30 重庆大学 Method for determining elastic energy of circular film in contact with rigid plate under gas pressure
CN115290424A (en) * 2022-07-26 2022-11-04 重庆大学 A method for determining the deflection of circular membranes under gas pressure

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Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112880950A (en) * 2021-01-18 2021-06-01 重庆大学 Method for determining deflection of circular prestressed film with limited maximum deflection under air pressure
CN113075048A (en) * 2021-03-30 2021-07-06 重庆大学 Method for determining maximum deflection of circular film under gas pressure
CN113720688A (en) * 2021-08-17 2021-11-30 重庆大学 Method for determining elastic energy of circular film in contact with rigid plate under gas pressure
CN115290424A (en) * 2022-07-26 2022-11-04 重庆大学 A method for determining the deflection of circular membranes under gas pressure

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