CN112903216B - Method for determining elastic energy of circular prestressed film with limited maximum deflection under air pressure - Google Patents

Method for determining elastic energy of circular prestressed film with limited maximum deflection under air pressure Download PDF

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CN112903216B
CN112903216B CN202110063861.4A CN202110063861A CN112903216B CN 112903216 B CN112903216 B CN 112903216B CN 202110063861 A CN202110063861 A CN 202110063861A CN 112903216 B CN112903216 B CN 112903216B
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何晓婷
常浩
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Chongqing University
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    • G—PHYSICS
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    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M5/00—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings
    • G01M5/0041—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by determining deflection or stress
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    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
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    • 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
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    • G01N2203/0069—Fatigue, creep, strain-stress relations or elastic constants
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Abstract

The invention discloses a method for determining the elastic energy of a circular prestressed film with limited maximum deflection under air pressure, which comprises the following steps: e for Young's modulus of elasticity and Poisson's ratio ofV, radius a, thickness h and prestress sigma 0 The initially flat and peripherally clamped circular pre-stressed membrane of (a) is subjected to a gas pressure q, the circular pre-stressed membrane is subjected to axisymmetric deformation under the gas pressure, and a circular smooth contact area with a radius b is formed with a rigid flat plate parallel to the initially flat circular pre-stressed membrane, wherein the rigid flat plate is at a distance H from the initially flat circular pre-stressed membrane, and then based on a static equilibrium analysis of the axisymmetric deformation of the circular pre-stressed membrane, the elastic energy U of the circularly pre-stressed membrane after axisymmetric deformation can be determined by using a measured value of the gas pressure q.

Description

Method for determining elastic energy of circular prestressed film with limited maximum deflection under air pressure
Technical Field
The invention relates to a method for determining the elastic performance of a circular prestressed film with a clamped periphery under the action of gas pressure under the condition of limited maximum deflection.
Background
Analytical solutions to the problem of axisymmetric deformations of peripherally clamped circular membranes under conditions of limited maximum deflection have applications in many engineering and technical fields, for example for studying adhesion energy measurements of membrane/substrate systems, and for developing various instruments and sensors. From the results of study and study, the present analytical research results only show that the circular membrane without prestress is clamped at the periphery under the action of gas pressure, and the axial symmetry deformation problem is under the condition of maximum deflection limitation. For example, the analytical solution used in the invention of "method for determining elastic properties of circular thin film with limited maximum deflection under gas pressure" (application No. 202010199390.5) is obtained without considering the circular thin film with prestress (i.e., with initial in-plane stress), and thus this analytical solution is not suitable for the circular thin film with prestress. However, in practice, when clamping an initially flat film along a circular periphery, it is easy to cause the circular, periphery-clamped film to be stretched or compressed (often referred to as in-plane stretching or compression) in the plane of the initially flat film, so that the circular, periphery-clamped film is already subjected to an initial in-plane stretching (or compression) stress, often referred to as pre-stress, before the gas pressure is applied. Further, even when the circular film is subjected to the peripheral clamping, the circular film after the peripheral clamping is not caused to have the initial in-plane tensile (or compressive) stress, but since the film material generally has the property of expanding with heat and contracting with cold, if the temperature at the time of applying the gas pressure is greatly different from the temperature at the time of peripheral clamping the circular film, in this case, the circular film without the in-plane tensile (or compressive) stress at the time of peripheral clamping becomes a circular film with the in-plane tensile (or compressive) stress at the time of peripheral clamping, and when the gas pressure is further applied thereto, it is a so-called circular prestressed film. However, no analytical research has been carried out to date on the problem of axisymmetric deformation of a circumferentially clamped circular prestressed thin film under maximum deflection-limited conditions under gas pressure. Therefore, the analytic research of the axial symmetry deformation problem of the circular prestressed thin film clamped at the periphery under the action of gas pressure under the condition of limited maximum deflection is further carried out, and a larger research and development space can be provided for the technical field of engineering.
Disclosure of Invention
The invention is dedicated to the analytical research of the axial symmetry deformation problem of the circular prestressed thin film clamped at the periphery under the action of gas pressure under the condition of limited maximum deflection, obtains the analytical solution of the axial symmetry deformation problem based on the static balance analysis of the axial symmetry deformation of the circular prestressed thin film clamped at the periphery under the action of gas pressure under the condition of limited maximum deflection, and provides the method for determining the elastic performance of the circular prestressed thin film under the action of gas pressure under the condition of limited maximum deflection.
The method for determining the elastic energy of the circular prestressed film with the limited maximum deflection under the air pressure comprises the following steps: for Young's modulus of elasticity E, poisson's ratio v, radius a, thickness h and prestress sigma 0 The circular pre-stressed film which is initially flat and is clamped at the periphery applies gas pressure q, so that the circular pre-stressed film generates axisymmetric deformation under the action of the gas pressure, and forms a circular smooth contact area with the radius b with a rigid flat plate which is parallel to the initially flat circular pre-stressed film, wherein the rigid flat plate is away from the initially flat circular pre-stressed film by the distance H, and then based on the static balance analysis of the axisymmetric deformation of the circular pre-stressed film, the analytic relationship between the applied gas pressure q and the elastic energy U of the circular pre-stressed film after the axisymmetric deformation is obtained as follows
Figure BDA0002903419170000021
Wherein the content of the first and second substances,
Figure BDA0002903419170000022
Figure BDA0002903419170000023
Figure BDA0002903419170000024
Figure BDA0002903419170000025
Figure BDA0002903419170000026
Figure BDA0002903419170000031
Figure BDA0002903419170000032
Figure BDA0002903419170000033
Figure BDA0002903419170000034
and b, c therein 0 、c 1 、d 0 Is given by the equation
Figure BDA0002903419170000035
Figure BDA0002903419170000036
Figure BDA0002903419170000037
And
Figure BDA0002903419170000041
determining the number of the first and second groups, wherein,
Figure BDA0002903419170000042
Figure BDA0002903419170000043
Figure BDA0002903419170000044
Figure BDA0002903419170000045
Figure BDA0002903419170000051
thus, the elastic energy U of the circular prestressed thin film after axisymmetric deformation can be determined by measuring the value of the gas pressure q, wherein the units of a, b, H and H are all millimeters (mm), E, q and sigma 0 All units of (2) are Newton per square millimeter (N/mm) 2 ) U has the unit Newton-millimeter (N.mm), and v, 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, β are dimensionless quantities.
Drawings
Fig. 1 is a schematic view of axisymmetrical deformation of a circular pre-stressed film clamped at the periphery under the maximum deflection limit condition by gas pressure, wherein 1 is the circularly pre-stressed film after axisymmetrical deformation, 2 is a rigid flat plate, 3 is a clamping device, 4 is a geometric middle plane of the initially flat circular pre-stressed film, 5 is a pedestal, a represents the radius of the circularly pre-stressed film and the inner radius of the clamping device, b represents the radius of a circular smooth contact area formed between the circularly pre-stressed film after axisymmetrical deformation and the rigid flat plate, r represents the distance from any point on the circularly pre-stressed film after axisymmetrical deformation to a straight line perpendicular to the geometric middle plane of the initially flat circular pre-stressed film and passing through the centroid of the geometric middle plane, w (r) represents the film deflection, q represents the gas pressure acting on the surface of the circularly pre-stressed film, and H represents the distance between the initially flat circular pre-stressed film and the rigid flat plate.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
as shown in FIG. 1, E =7.84N/mm for Young's modulus of elasticity 2 Poisson's ratio ν =0.47, radius a =10mm, thickness h =1mm, prestress σ 0 ＝0.2N/mm 2 The initially flat and peripherally clamped circular pre-stressed membrane of (a) is subjected to a gas pressure q, which causes the circular pre-stressed membrane to deform axisymmetrically under the action of the gas pressure and form a circular smooth contact area of radius b with a rigid plate parallel to the initially flat circular pre-stressed membrane, wherein the rigid plate is at a distance of H =2.5mm from the initially flat circular pre-stressed membrane, and the gas pressure q =0.1N/mm is measured 2 Then the method given by the invention is adopted, and the equation is expressed
Figure BDA0002903419170000061
Figure BDA0002903419170000062
Figure BDA0002903419170000063
Figure BDA0002903419170000064
Figure BDA0002903419170000065
Figure BDA0002903419170000066
Figure BDA0002903419170000071
Figure BDA0002903419170000072
Figure BDA0002903419170000081
Figure BDA0002903419170000082
Figure BDA0002903419170000083
Figure BDA0002903419170000084
Figure BDA0002903419170000091
Figure BDA0002903419170000092
Figure BDA0002903419170000101
Figure BDA0002903419170000102
Figure BDA0002903419170000103
Figure BDA0002903419170000104
B =1.90870mm, c are obtained 0 ＝0.926624、c 1 ＝-0.361658、d 0 =0.187758 and c 2 ＝-0.500042、c 3 ＝-0.174924、c 4 ＝-0.251664、c 5 ＝-0.177005、c 6 ＝-0.228971、d 1 ＝-0.288278、d 2 ＝-0.353777、d 3 ＝-0.112867、d 4 ＝-0.181197、d 5 ＝-0.082964、d 6 = -0.217546, final equation
Figure BDA0002903419170000105
The elastic energy of the circular prestressed film after axisymmetrical deformation is determined to be U =45.179795N · mm.

Claims (1)

1. Circular prestressed film with limited maximum deflection under air pressureA method for determining elastic properties, characterized by: for Young's modulus of elasticity E, poisson's ratio v, radius a, thickness h and prestress sigma 0 The initially flat and peripherally clamped circular pre-stressed membrane of (a) applying a gas pressure q to cause the circular pre-stressed membrane to deform axisymmetrically under the gas pressure and form a circular smooth contact area of radius b with a rigid plate parallel to the initially flat circular pre-stressed membrane, wherein the rigid plate is at a distance H from the initially flat circular pre-stressed membrane, then based on the circular pre-stressed membrane axisymmetricallyDeformation ofUsing the measured value of the gas pressure q, from the equation
Figure FDA0002903419160000011
Figure FDA0002903419160000012
Figure FDA0002903419160000017
Figure FDA0002903419160000013
Figure FDA0002903419160000014
Figure FDA0002903419160000015
Figure FDA0002903419160000016
Figure FDA0002903419160000021
Figure FDA0002903419160000031
Figure FDA0002903419160000032
Figure FDA0002903419160000033
Figure FDA0002903419160000041
Figure FDA0002903419160000042
Figure FDA0002903419160000043
Figure FDA0002903419160000051
Figure FDA0002903419160000052
Figure FDA0002903419160000053
Figure FDA0002903419160000054
Determination of 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 Is finally given by the equation
Figure FDA0002903419160000055
Determining the elastic energy U of the circular prestressed film after axisymmetric deformation, wherein the units of a, b, H and H are all millimeters (mm), E, q and sigma 0 All units of (2) are Newton per square millimeter (N/mm) 2 ) U has the unit Newton-millimeter (N-mm), and v, 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, β are dimensionless quantities.
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