CN111474041A - Method for determining elastic energy of prestressed circular film under uniformly distributed load - Google Patents

Method for determining elastic energy of prestressed circular film under uniformly distributed load Download PDF

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CN111474041A
CN111474041A CN202010325634.XA CN202010325634A CN111474041A CN 111474041 A CN111474041 A CN 111474041A CN 202010325634 A CN202010325634 A CN 202010325634A CN 111474041 A CN111474041 A CN 111474041A
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prestressed
prestressed circular
elastic energy
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许劲
朱杰东
李卷利
龚河洛
刘孟秋
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Abstract

The invention discloses a method for determining the elastic energy of a prestressed circular film under uniformly distributed loads, which comprises the following steps: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma 0The prestressed circular film fixedly clamped at the periphery transversely applies an evenly distributed load q to generate axisymmetric deformation, and based on static balance analysis of the axisymmetric deformation problem of the prestressed circular film, the elastic energy U of the prestressed circular film after axisymmetric deformation can be determined by using the measured value of the load q.

Description

Method for determining elastic energy of prestressed circular film under uniformly distributed load
Technical Field
The invention relates to a method for determining the elastic energy of a prestressed circular film with the periphery fixedly clamped under the action of transversely uniformly distributed loads.
Background
the analytical solutions for the problem of axisymmetric deformation of a circular membrane structure clamped fixedly at the periphery under transversely uniform load, such as for developing various instruments and sensors, and for studying the measurement of the adhesion energy of a membrane/substrate system, have been applied in many engineering fields, and as a result of the study of new results, there are two analytical solutions, one being the analytical solution used in the invention patent "a method for determining the elastic energy of a circular membrane under uniform load" (patent No. Z L201410238597.3), and the other being the analytical solution used in the invention patent "a method for determining the elastic energy of a circular membrane under transversely uniform load" (patent No. Z L201610802144.8), both of which take into account the fact that the membrane has an initial stress (i.e. the so-called pre-stress), wherein the analytical solution used in the first patent is obtained under the so-called small corner assumption of the membrane (i.e. the assumption that the corner θ of the membrane is so small as to satisfy the sin tan), and the second being the analytical solution obtained under the so-called small corner stress equation (i.e. the small corner equation) which has not been applied under the assumption that the initial stress is obtained under the small corner of the invention.
From the results of study, the problem of axisymmetrical deformation of a circular film which is fixedly clamped at the periphery under the action of transversely uniform load is not solved so far, and the analytical solution which gives up the assumption of small corner of the film and considers the influence of the film deflection on the establishment of an in-plane equilibrium equation and is suitable for the situation that the film has initial stress (namely prestress) is not provided.
Disclosure of Invention
The invention is dedicated to the analytical research of the axial symmetric deformation problem of the prestressed circular film with the periphery fixedly clamped under the action of the transversely uniformly distributed load, obtains the analytical solution of the axial symmetric deformation problem under the condition of giving up the assumption of a small corner of the film and simultaneously considering the influence of the film deflection on the establishment of an in-plane equilibrium equation, and provides the method for determining the elastic performance of the prestressed circular film under the uniformly distributed load on the basis.
The method for determining the elastic energy of the prestressed round film under uniformly distributed load comprises the following steps: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma 0The prestressed circular film with the fixed and clamped periphery transversely applies an evenly distributed load q to generate axisymmetric deformation, and the analytical relationship between the applied load q and the elastic energy U of the prestressed circular film after axisymmetric deformation can be obtained based on the static balance analysis of the axisymmetric deformation problem of the prestressed circular film
Figure BDA0002461796000000021
Wherein the content of the first and second substances,
Figure BDA0002461796000000022
Figure BDA0002461796000000023
Figure BDA0002461796000000024
Figure BDA0002461796000000025
Figure BDA0002461796000000026
Figure BDA0002461796000000027
Figure BDA0002461796000000028
And b 0、c0Is given by the equation
Figure BDA0002461796000000029
And
Figure BDA00024617960000000210
Determining, wherein,
Figure BDA0002461796000000031
Figure BDA0002461796000000032
Figure BDA0002461796000000033
Figure BDA0002461796000000034
Figure BDA0002461796000000035
Figure BDA0002461796000000036
d0＝b0，
Figure BDA0002461796000000037
Figure BDA0002461796000000038
Figure BDA0002461796000000041
Figure BDA0002461796000000042
Figure BDA0002461796000000043
Figure BDA0002461796000000044
Thus, the elastic energy U of the prestressed circular film after axisymmetric deformation can be determined by measuring the value of the load q, wherein the units of a and h are millimeters (mm), E, q and sigma 0All units of (2) are Newton per square millimeter (N/mm) 2) U has the unit Newton-millimeter (N-mm), and v, b 0、b2、b4、b6、b8、b10、b12、c0、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
Drawings
FIG. 1 is a schematic view showing the axisymmetrical deformation of a prestressed circular thin film whose periphery is fixedly clamped under the action of a transversely uniformly distributed load, wherein 1 is the prestressed circular thin film after the axisymmetrical deformation, 2 is a clamping device, 3 is the prestressed circular thin film before the deformation, a represents the radius of the prestressed circular thin film and the inner radius of the clamping device, q represents the transversely uniformly distributed load, w represents the transverse uniformly distributed load, and mShowing the maximum deflection after axisymmetric deformation of the prestressed circular film.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
As shown in fig. 1, the radius a is 20mm, the thickness h is 0.06mm, and the young's modulus E is 7.84N/mm 2Poisson ratio v is 0.47, prestress sigma 0＝2N/mm2The prestressed circular film with the fixed and clamped periphery transversely applies an evenly distributed load q, measuring the load q is 0.1N/mm 2By using the method given in the invention, the equation
Figure BDA0002461796000000051
Figure BDA0002461796000000052
Figure BDA0002461796000000053
Figure BDA0002461796000000054
Figure BDA0002461796000000055
Figure BDA0002461796000000056
Figure BDA0002461796000000057
Figure BDA0002461796000000058
Figure BDA0002461796000000061
Figure BDA0002461796000000062
Figure BDA0002461796000000063
Figure BDA0002461796000000064
Figure BDA0002461796000000065
Figure BDA0002461796000000066
d0＝b0，
Figure BDA0002461796000000067
Figure BDA0002461796000000068
Figure BDA0002461796000000069
Figure BDA00024617960000000610
Figure BDA0002461796000000071
Figure BDA0002461796000000072
Figure BDA0002461796000000073
To obtain b 01.371399 and c 01.126905 and b 2＝1.279175、b4＝0.673654、b6＝-0.543665、b8＝-0.948286、b10＝0.695263、b12＝2.178050、c2＝-0.775066、c4＝-0.104131、c6＝0.211316、c8＝0.147241、c10＝-0.263354、c12＝-0.342911、d0＝1.371399、d2＝0.542178、d4＝0.294528、d6＝-0.060732、d8＝-0.275129、d10＝-0.009216、d120.483802, then
Figure BDA0002461796000000074
Determining that the horizontal uniform load q of the prestressed round film is 0.1N/mm 2The elastic energy U under the action is 1744.36N mm.
in addition, under the same conditions, if the method of the invention for determining the elastic strain energy of a circular film under uniform load (application No. 202010190001.2) is adopted, the assumption that the small corner of the film is given, and the influence of the deflection of the film on the establishment of an in-plane equilibrium equation is taken into consideration, but the method of the invention for determining the elastic strain energy of the circular film under uniform load (application No. 202010190001.2) is not adopted, the elastic energy U is 1907.08N mm, and compared with the invention, the method for determining the elastic energy of the circular film under uniform load (patent No. Z L201410238597.3) is adopted, the method for determining the elastic energy of the circular film under uniform load (patent No. 1170.40N mm is adopted, the method for determining the elastic strain energy of the circular film under uniform load (patent No. Z L201610802144.8) is adopted, the assumption that the initial stress (prestress) is adopted, the method for determining the elastic strain energy of the circular film under uniform load (patent No. 1170.40N mm is adopted, the method for determining the elastic strain energy of the circular film under uniform load (application No. Z L201610802144.8) is adopted, the assumption that the invention for determining the elastic strain energy of the film under uniform load and the invention for the elastic strain energy of the invention, the method for determining the elastic strain energy of the invention for determining the small corner under uniform load (application No. Z L201610802144.8) is adopted, the assumption that the method for determining the elastic strain energy of the invention for the circular film under uniform load and the method for determining the method for the elastic strain energy of the invention for the invention) is adopted, the method for the invention for determining the method for determining the small corner under uniform load (application No. 1720.50) and the invention for determining.

Claims (1)

1. The method for determining the elastic energy of the prestressed circular film under uniformly distributed load is characterized by comprising the following steps: the pair radius is a, the thickness is h, the Young's modulus is E, the Poisson ratio is v, and the prestress is sigma 0Transversely applying a uniformly distributed load q to the prestressed circular film fixedly clamped at the periphery to generate axisymmetric deformation, and then carrying out static balance analysis on the prestressed circular film axisymmetric deformation problem by using the measured value of the load q and using the following equation
Figure FDA0002461795990000011
Figure FDA0002461795990000012
Figure FDA0002461795990000013
Figure FDA0002461795990000014
Figure FDA0002461795990000015
Figure FDA0002461795990000016
Figure FDA0002461795990000017
Figure FDA0002461795990000021
Figure FDA0002461795990000022
Figure FDA0002461795990000023
Figure FDA0002461795990000024
Figure FDA0002461795990000025
Figure FDA0002461795990000026
Figure FDA0002461795990000027
d0＝b0，
Figure FDA0002461795990000028
Figure FDA0002461795990000029
Figure FDA0002461795990000031
Figure FDA0002461795990000032
Figure FDA0002461795990000033
Figure FDA0002461795990000034
Figure FDA0002461795990000035
Determination of b 0And c 0And b 2、b4、b6、b8、b10、b12、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And finally, from the equation
Figure FDA0002461795990000036
Determining elastic energy U of the prestressed circular film after axisymmetric deformation, wherein the units of a and h are millimeters (mm), E, q and sigma 0All units of (2) are Newton per square millimeter (N/mm) 2) U has the unit Newton-millimeter (N-mm), and v, b 0、b2、b4、b6、b8、b10、b12、c0、c2、c4、c6、c8、c10、c12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
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CN112903216A (en) * 2021-01-18 2021-06-04 重庆大学 Method for determining elastic energy of circular prestressed film with limited maximum deflection under air pressure

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Publication number Priority date Publication date Assignee Title
CN112903216A (en) * 2021-01-18 2021-06-04 重庆大学 Method for determining elastic energy of circular prestressed film with limited maximum deflection under air pressure

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