CN109342192B - Method for determining maximum deflection of circular film under limitation of elasticity of maximum deflection - Google Patents

Method for determining maximum deflection of circular film under limitation of elasticity of maximum deflection Download PDF

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CN109342192B
CN109342192B CN201811540012.8A CN201811540012A CN109342192B CN 109342192 B CN109342192 B CN 109342192B CN 201811540012 A CN201811540012 A CN 201811540012A CN 109342192 B CN109342192 B CN 109342192B
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circular
circular plate
film
circular film
rigid
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CN109342192A (en
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孙俊贻
李雪
郭莹
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Shandong Industry Research Institute Zhongke High End Chemical Industry Technology Research Institute Co ltd
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    • G—PHYSICS
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    • 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
    • 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
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    • G01N2203/0069—Fatigue, creep, strain-stress relations or elastic constants
    • G01N2203/0075—Strain-stress relations or elastic constants
    • 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
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    • G01N2203/067—Parameter measured for estimating the property
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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
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/06—Indicating or recording means; Sensing means
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Abstract

The invention discloses a method for determining the maximum deflection of a circular film under the limitation of elasticity, which comprises the following steps: e for Young's modulus of elasticity, v for Poisson's ratio, h for thickness, a for radius, and q for self-weight per unit area0Initially flat and peripherally fixedly clamped circular membraneTransversely applying a uniform load q to make the circular film produce axial symmetric deformation and further drive a rigid circular plate with radius a, centre of circle of which is on same axis with centre of circle of the circular film and always parallel to the initially flat circular film, smooth surface and parallel movement wmAnd the axisymmetrically deformed circular membrane forms an interaction contact area with a radius b with the rigid circular plate, and the parallel moving rigid circular plate compresses a spring with a stiffness coefficient k from an original length L by wmThen, after the dead weight of the rigid circular plate is ignored, based on the static balance analysis of the axisymmetric deformation of the circular film, the maximum deflection w after the axisymmetric deformation of the circular film can be determined by using the measured value of the load qm。

Description

Method for determining maximum deflection of circular film under limitation of elasticity of maximum deflection
Technical Field
The invention relates to a method for determining the maximum deflection of a circular film under the action of transversely uniformly distributed loads, wherein the periphery of the circular film is fixedly clamped under the condition that the maximum deflection is limited by elasticity.
Background
The axisymmetric deformation of a circular film with a periphery fixedly clamped under the condition that the maximum deflection is limited by elasticity under the action of transversely uniformly distributed loads is utilized and applied in a plurality of engineering technical fields, for example, the axisymmetric deformation is used for researching the adhesion energy measurement of a film/base layer system, and researching various instruments and meters, various sensors and the like. Because the film is usually very thin, the film often generates very large deflection under the action of transversely uniformly distributed load, so that the film has strong geometric nonlinearity, which brings difficulty to analytical research, and therefore, from the new checking result of the existing literature, the axial symmetry deformation problem of the circular film with the periphery fixedly clamped under the condition that the maximum deflection is limited by elasticity under the action of transversely uniformly distributed load is not solved analytically at present. Therefore, it is necessary to perform related analytical research work, which provides convenience and a larger research and development space for the research and development of engineering technology.
Disclosure of Invention
The invention is dedicated to the analytical research of the axial symmetry deformation problem of the circular film fixedly clamped at the periphery under the condition that the maximum deflection is limited by elasticity under the action of the transversely uniformly distributed load, obtains the analytical solution of the axial symmetry deformation problem based on static balance analysis, and provides the method for determining the maximum deflection of the circular film fixedly clamped at the periphery under the condition that the maximum deflection is limited by elasticity under the action of the transversely uniformly distributed load.
The method for determining the maximum deflection of the circular film under the limitation of elasticity comprises the following steps: chinese PopulusE is the modulus of elasticity, v is the Poisson's ratio, h is the thickness, a is the radius, q is the dead weight per unit area0The initially flat circular film with the fixed and clamped periphery transversely applies an evenly distributed load q, so that the circular film generates axisymmetric deformation to push a rigid circular plate with the radius of a, the circle center of the rigid circular plate is on the same axis with the circle center of the circular film, the rigid circular plate is always parallel to the initially flat circular film, the surface of the rigid circular plate is smooth, and the rigid circular plate moves in parallel wmAnd the axisymmetrically deformed circular membrane forms an interaction contact area with a radius b with the rigid circular plate, and the parallel moving rigid circular plate compresses a spring with a stiffness coefficient k from an original length L by wmThen, after the dead weight of the rigid circular plate is neglected, based on the static balance analysis of the axisymmetric deformation of the circular film, the applied load q and the maximum deflection w of the axisymmetric deformation of the circular film can be obtainedmAnalytic relationship between
Figure BDA0001907808930000011
Wherein,
Figure BDA0001907808930000021
Figure BDA0001907808930000022
Figure BDA0001907808930000023
Figure BDA0001907808930000024
Figure BDA0001907808930000025
Figure BDA0001907808930000031
Figure BDA0001907808930000041
and b, c0、c1Is given by the equation
Figure BDA0001907808930000042
Figure BDA0001907808930000043
And
Figure BDA0001907808930000044
determining, wherein,
Figure BDA0001907808930000051
Figure BDA0001907808930000052
Figure BDA0001907808930000053
Figure BDA0001907808930000054
Figure BDA0001907808930000061
thus, the maximum deflection w after the axial symmetry deformation of the circular film can be realized by only accurately measuring the value of the load qmDetermined, wherein the parameters a, b, h, L, wmAll units of (A) are millimeters (mm), reference E, q0Q is in newtons per square millimeter (N/mm)2) The unit of the parameter k is Newton per millimeter (N/mm), while the parameters v, c0、c1、c2、c3、c4、c5、c6、d0、d1、d2、d3、d4、d5、d6All are dimensionless quantities.
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FIG. 1 is a schematic diagram showing the problem of axisymmetric deformation of a circular film fixedly clamped at its periphery under elastic limit of maximum deflection under a laterally uniform load, wherein 1 is the circular film after axisymmetric deformation, 2 is a rigid circular plate, 3 is a spring, 4 is a clamping device, 5 is a support, 6 is a geometric middle plane of the initially flat circular film, a represents the radii of the circular film and the rigid circular plate and the inner radius of the clamping device, b represents the contact radius of the circular film with the rigid circular plate after axisymmetric deformation, L represents the original length of the spring, q represents the laterally uniform load, o represents the origin of coordinates, r represents a radial coordinate, w represents a lateral coordinate, and w represents a lateral coordinatemShowing the maximum deflection after axisymmetric deformation of the circular film and the distance that the rigid circular plate is moved in parallel and the length that the spring is compressed.
Detailed Description
The technical solution of the present invention is further described with reference to fig. 1 below:
as shown in FIG. 1, the Young's modulus E is 7.84N/mm using a clamp with an inner radius a of 20mm2Poisson's ratio v is 0.47, thickness h is 1mm, unit area dead weight q0＝7.64×10-7N/mm2Then the clamping device is fixed on the support to form a peripheral fixedly clamped circular film structure with a radius a of 20mm, and a transverse clamping force is applied to the circular film structureUniformly distributing load q, making the circular film produce axial symmetric deformation and further pushing a rigid circular plate with smooth surface, whose radius a is 20mm, and whose centre is on same axis with that of the circular film, and is always parallel to the initially flat circular filmmAnd an interaction contact area with radius b is formed between the circular film after the axisymmetric deformation and the rigid circular plate, and the rigid circular plate moving in parallel compresses a spring with stiffness coefficient k of 1.5N/mm from the original length L of 50mm by wmThe measured load q is 0.01N/mm2The method provided by the invention is adopted and is represented by the following equation
Figure BDA0001907808930000071
Figure BDA0001907808930000072
Figure BDA0001907808930000073
Figure BDA0001907808930000074
Figure BDA0001907808930000075
Figure BDA0001907808930000076
Figure BDA0001907808930000077
Figure BDA0001907808930000081
Figure BDA0001907808930000082
Figure BDA0001907808930000091
Figure BDA0001907808930000092
Figure BDA0001907808930000093
Figure BDA0001907808930000101
Figure BDA0001907808930000102
Figure BDA0001907808930000111
B is 4.018560mm, c0＝0.401570、c1-0.073751 and d0＝0.396980、d1＝-0.663931、d2＝-0.753233、d3＝-0.081738、d4＝-0.231329、d5＝0.059966、d6-0.241261. Finally, the equation
Figure BDA0001907808930000112
Obtaining the maximum deflection w after the axisymmetric deformation of the circular filmm＝3.178515mm。

Claims (1)

1. The method for determining the maximum deflection of the circular film under the limitation of elasticity is characterized by comprising the following steps of: e for Young's modulus of elasticity, v for Poisson's ratio, h for thickness, a for radius, and q for self-weight per unit area0The initially flat circular film with the fixed and clamped periphery transversely applies an evenly distributed load q, so that the circular film generates axisymmetric deformation to push a rigid circular plate with the radius of a, the circle center of the rigid circular plate is on the same axis with the circle center of the circular film, the rigid circular plate is always parallel to the initially flat circular film, the surface of the rigid circular plate is smooth, and the rigid circular plate moves in parallel wmAnd the axisymmetrically deformed circular membrane forms an interaction contact area with a radius b with the rigid circular plate, and the parallel moving rigid circular plate compresses a spring with a stiffness coefficient k from an original length L by wmThen, after neglecting the dead weight of the rigid circular plate, based on the static equilibrium analysis of the axisymmetric deformation of the circular film, using the measured value of the load q, from the following equation
Figure FDA0001907808920000011
Figure FDA0001907808920000012
Figure FDA0001907808920000013
Figure FDA0001907808920000014
Figure FDA0001907808920000015
Figure FDA0001907808920000016
Figure FDA0001907808920000017
Figure FDA0001907808920000021
Figure FDA0001907808920000022
Figure FDA0001907808920000031
Figure FDA0001907808920000032
Figure FDA0001907808920000033
Figure FDA0001907808920000041
Figure FDA0001907808920000042
Figure FDA0001907808920000051
Determining the variables b, c0、c1And d0、d1、d2、d3、d4、d5、d6And finally, from the equation
Figure FDA0001907808920000052
Determining the maximum deflection w after the axisymmetric deformation of the circular filmmWherein the parameters a, b, h, L, wmAll units of (A) are millimeters (mm), reference E, q0Q is in newtons per square millimeter (N/mm)2) The unit of the parameter k is Newton per millimeter (N/mm), while the parameters v, c0、c1、c2、c3、c4、c5、c6、d0、d1、d2、d3、d4、d5、d6All are dimensionless quantities.
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CN110320101B (en) * 2019-06-05 2021-06-29 重庆大学 Method for determining maximum deflection of prestressed circular film under action of liquid
CN110231215B (en) * 2019-06-05 2021-06-08 重庆大学 Method for determining maximum deflection of prestressed circular film with deflection limited by elasticity
CN113434986A (en) * 2021-07-14 2021-09-24 重庆大学 Method for determining deflection of annular thin film with rigid connection between inner edge and circular thin plate

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