CN113092039A - Method for determining elastic energy of annular film under transversely uniformly distributed load - Google Patents

Method for determining elastic energy of annular film under transversely uniformly distributed load Download PDF

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CN113092039A
CN113092039A CN202110410485.1A CN202110410485A CN113092039A CN 113092039 A CN113092039 A CN 113092039A CN 202110410485 A CN202110410485 A CN 202110410485A CN 113092039 A CN113092039 A CN 113092039A
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annular film
uniformly distributed
annular
elastic energy
clamping device
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CN113092039B (en
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孙俊贻
李环宇
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Chongqing University
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Abstract

The invention discloses a method for determining the elastic energy of an annular film under transversely uniformly distributed load, which is characterized by comprising the following steps: and applying a transversely uniformly distributed load q to an initial flat annular film with a clamped inner edge and a fixedly clamped outer edge to enable the annular film to generate axisymmetric deformation, wherein the Young modulus of the annular film is E, the Poisson ratio is v, the thickness is h, the inner radius is b, the outer radius is a, the outer radius of the annular film inner edge clamping device is b, and the inner radius of the annular film outer edge fixing clamping device is a, so that after the dead weight of the annular film inner edge clamping device is neglected, based on static balance analysis of the axisymmetric deformation problem of the annular film, the elastic energy U after the axisymmetric deformation of the annular film can be determined by using the measured value of the transversely uniformly distributed load q.

Description

Method for determining elastic energy of annular film under transversely uniformly distributed load
Technical Field
The invention relates to a method for determining the elastic energy of an annular film with the inner edge clamped and the outer edge fixedly clamped under the action of transversely uniformly distributed loads.
Background
From the results of study, the analytical research results of the axial symmetric deformation problem of the annular film with the inner edge clamped and the outer edge fixedly clamped under the action of the transversely uniformly distributed load do not exist so far, and only the analytical research results of the axial symmetric deformation problem of the annular film with the rigid plate at the center under the action of the transversely uniformly distributed load exist. In the axial symmetry deformation problem of the annular film with the rigid plate at the center under the action of the transversely uniformly distributed load, the annular film and the rigid plate in the central area are simultaneously subjected to the action of the transversely uniformly distributed load. In the axial symmetry deformation problem of the annular film with the inner edge clamped and the outer edge fixedly clamped under the action of the transversely uniformly distributed loads, the transversely uniformly distributed loads only act on the annular film, and the central area inside the inner edge of the annular film does not have the action of the transversely uniformly distributed loads. Obviously, these two axisymmetric deformation problems are not the same. Based on the analytical research result of the axial symmetric deformation problem of the annular film with the rigid plate at the center under the action of the transversely uniformly distributed load, the invention patent of 'a method for determining the elastic energy of the annular film with the rigid plate at the center under the uniformly distributed load' (patent number: 201610266626.6) is applied. However, the analytical solution of the axial symmetry deformation problem of the annular film with the clamped inner edge and the clamped outer edge under the action of the transversely uniform load is not only significant for the design and analysis of engineering structures, but also can provide a larger research and development space for many technical application fields, such as the research and development of adhesion energy measurement of a film/substrate system, the research and development of various instruments and meters, various sensors and the like. Therefore, if this analytical solution can be obtained, this is certainly a very valuable task.
Disclosure of Invention
The invention is dedicated to the analytical research of the axial symmetric deformation problem of the annular film with the clamped inner edge and the fixedly clamped outer edge under the action of the transversely uniformly distributed load, obtains the analytical solution of the axial symmetric deformation problem based on the static balance analysis of the axial symmetric deformation problem of the annular film with the clamped inner edge and the fixedly clamped outer edge under the action of the transversely uniformly distributed load, and provides the method for determining the elastic energy of the annular film under the transversely uniformly distributed load on the basis.
The method for determining the elastic energy of the annular film under the transversely uniformly distributed load comprises the following steps: applying a transversely uniform load q to an initially flat annular film with clamped inner edge and fixedly clamped outer edge to generate axisymmetric deformation, wherein the Young's elastic modulus of the annular film is E, the Poisson ratio is v, the thickness is h, the inner radius is b, the outer radius is a, the outer radius of an annular film inner edge clamping device is b, and the inner radius of an annular film outer edge fixed clamping device is a, so that after the dead weight of the annular film inner edge clamping device is ignored, based on the static balance analysis of the axisymmetric deformation problem of the annular film, the analytical relationship between the applied transversely uniform load q and the elastic energy U of the annular film after axisymmetric deformation can be obtained
Figure BDA0003023967190000021
Wherein the content of the first and second substances,
Figure BDA0003023967190000022
Figure BDA0003023967190000023
Figure BDA0003023967190000024
Figure BDA0003023967190000025
Figure BDA0003023967190000026
Figure BDA0003023967190000027
Figure BDA0003023967190000031
Figure BDA0003023967190000032
β＝(1+α)/2，
and b0、b1Is given by the equation
Figure BDA0003023967190000033
And
Figure BDA0003023967190000034
determining, wherein,
Figure BDA0003023967190000035
Figure BDA0003023967190000036
Figure BDA0003023967190000037
Figure BDA0003023967190000038
Figure BDA0003023967190000041
thus, the elastic energy U of the annular film after axial symmetric deformation can be determined as long as the value of the load q uniformly distributed in the transverse direction is accurately measured, wherein the units of a, b and h are millimeters (mm), and the units of E, q are newtons per square millimeter (N/mm)2) U is in Newton-mm (N-mm), and v, b0、b1、b2、b3、b4、b5、b6、c0、c1、c2、c3、c4、c5、c6Q, alpha and beta are dimensionless quantities.
Drawings
FIG. 1 is a schematic diagram showing the axial symmetric deformation problem of a ring-shaped film with an inner edge clamped and an outer edge fixedly clamped under a transversely uniform load, wherein 1 isAn axially symmetrically deformed annular membrane, 2 an annular membrane inner edge clamp, 3 an annular membrane outer edge holding clamp, 4 a geometric middle plane of the initially flat annular membrane, 5 a support for holding the annular membrane outer edge holding clamp, while a denotes an outer radius of the annular membrane and an inner radius of the annular membrane outer edge holding clamp, b denotes an inner radius of the annular membrane and an outer radius of the annular membrane inner edge clamp, o denotes an origin of a coordinate system, r denotes a radial coordinate, w denotes a transverse coordinate (also denotes a deflection of the axially symmetrically deformed annular membrane), q denotes a transversely uniform load acting on the annular membrane, w denotes a transverse uniform load acting on the annular membrane, andmshowing the maximum deflection after axisymmetric deformation of the annular membrane.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
as shown in figure 1, an initially flat annular membrane clamped at its inner edge and at its outer edge is subjected to an axially symmetrical deformation by applying a transversely uniform load q, wherein the annular membrane has a Young's modulus E of 7.84N/mm2The poisson ratio v is 0.47, the thickness h is 0.2mm, the inner radius b is 5mm, the outer radius a is 20mm, the outer radius b of the annular film inner edge clamping device is 5mm, the inner radius a of the annular film outer edge fixing clamping device is 20mm, and the load q is 0.0003N/mm2Then, after neglecting the dead weight of the clamping device at the inner edge of the annular film, the method provided by the invention is adopted, and the formula is expressed by
Figure BDA0003023967190000051
Figure BDA0003023967190000052
Figure BDA0003023967190000053
Figure BDA0003023967190000054
Figure BDA0003023967190000055
Figure BDA0003023967190000056
Figure BDA0003023967190000057
Figure BDA0003023967190000058
Figure BDA0003023967190000059
β＝(1+α)/2
To obtain b0＝0.00915326、b1-0.00423391 and b2＝0.00245419、b3＝-0.01485462、b4＝0.026999038、b5＝-0.05415672、b60.09728856, then
Figure BDA0003023967190000061
Figure BDA0003023967190000062
Figure BDA0003023967190000063
Figure BDA0003023967190000064
Figure BDA0003023967190000065
Figure BDA0003023967190000066
Figure BDA0003023967190000067
To obtain c00.06391313 and c1＝-0.10973839、c2＝-0.14661498、c3＝-0.01756738、c4＝-0.05236632、c5＝-0.01959143、c6-0.01303530, finally by the equation
Figure BDA0003023967190000071
Determining that the annular film is uniformly distributed with a load q equal to 0.0003N/mm in the transverse direction2The elastic energy under the action is U-0.342644383N mm.

Claims (1)

1. The method for determining the elastic energy of the annular film under the transversely uniformly distributed load is characterized by comprising the following steps of: applying a transversely uniform load q to an initially flat annular film with clamped inner edge and fixedly clamped outer edge to generate axisymmetric deformation, wherein the Young's modulus of elasticity of the annular film is E, the Poisson ratio is v, the thickness is h, the inner radius is b, the outer radius is a, the outer radius of a clamping device at the inner edge of the annular film is b, and the inner radius of a fixing clamping device at the outer edge of the annular film is a, so that after the dead weight of the clamping device at the inner edge of the annular film is ignored, based on the static balance analysis of the axisymmetric deformation problem of the annular film, the measured value of the transversely uniform load q is utilized, and the equation is used for calculating the axial symmetric deformation problem of the annular
Figure FDA0003023967180000011
Figure FDA0003023967180000012
Figure FDA0003023967180000013
Figure FDA0003023967180000014
Figure FDA0003023967180000015
Figure FDA0003023967180000016
Figure FDA0003023967180000021
Figure FDA0003023967180000022
Figure FDA0003023967180000023
β＝(1+α)/2
Determination of b0、b1And b2、b3、b4、b5、b6Then by the equation
Figure FDA0003023967180000024
Figure FDA0003023967180000025
Figure FDA0003023967180000026
Figure FDA0003023967180000027
Figure FDA0003023967180000028
Figure FDA0003023967180000031
Figure FDA0003023967180000032
Determination of c0And c1、c2、c3、c4、c5、c6Is finally given by the equation
Figure FDA0003023967180000033
Determining the elastic energy U of the annular film under the action of transversely uniformly distributed loads q, wherein the units of a, b and h are millimeters (mm), and the units of E, q are newtons per square millimeter (m) ((N/mm2) U is in Newton-mm (N-mm), and v, b0、b1、b2、b3、b4、b5、b6、c0、c1、c2、c3、c4、c5、c6Q, alpha and beta are dimensionless quantities.
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CN113551978A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining maximum stress of annular film with rigid inner edge
CN113551977A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining the deflection of a ring-shaped film with a rigid inner edge
CN113551976A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining elastic energy of annular film with rigid inner edge

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Publication number Priority date Publication date Assignee Title
CN113551978A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining maximum stress of annular film with rigid inner edge
CN113551977A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining the deflection of a ring-shaped film with a rigid inner edge
CN113551976A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining elastic energy of annular film with rigid inner edge

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