CN106442129A - Determining method for prestress thin-film elastic energy under transversely and uniformly distributed loads - Google Patents

Abstract

The invention discloses a determining method for prestess thin-film elastic energy under transversely and uniformly distributed loads. A uniformly-distributed load q is transversely applied to a prestress circular thin film with the fixedly-clamped periphery, wherein the thickness of the circular thin film is h, the Young's elastic modulus is E, the Poisson ratio is nu, the prestress is sigma 0, the radius is a, static balance analysis based on the axial symmetry deformation problem is achieved, and by means of a measured value of the uniformly-distributed load q, the elastic energy U obtained after the thin film deforms can be determined.

Description

The laterally determination method of uniform load lower prestress circular membrane elasticity energy

Technical field

The present invention relates to the determination of the prestressing force circular membrane elasticity energy that laterally Uniform Loads following peripheral fixes to clamp Method.

Background technology

The laterally analytic solutions of the circular membrane On Axisymmetric Deformation of A that Uniform Loads following peripheral fixes to clamp, to sensing The development of device and instrument and meter is significant.However, in terms of the result of document Investigation, have at present and be applied to thin film and turn Angle θ is larger but the circular membrane axial symmetry that do not have the horizontal uniform load following peripheral of prestressed situation to fix to clamp in thin film The analytic solutions of problem on deformation, for example, patent of invention (" the big corner circular membrane bullet under a kind of uniform load declared before applicant The determination method of property strain energy ", number of patent application：201510194410.9) employed in analytic solutions it is simply that suppose circular Do not have in thin film prestressed under the conditions of obtain, thus it is only applicable in thin film do not have prestressed situation；Also have applicable Carry the circle that prestressing force but the horizontal uniform load following peripheral of the less situation of thin film rotational angle theta fix to clamp thin in thin film The analytic solutions of film On Axisymmetric Deformation of A, for example, (" one kind determines uniform load to another patent of invention declared before applicant The method of lower prestress circular membrane elasticity energy ", number of patent application：201410238597.3) employed in analytic solutions it is simply that It is assumed that in circular membrane carry prestressing force but thin film rotational angle theta less under the conditions of obtain, i.e. the solution procedure of this analytic solutions In consider and carry prestressed situation in thin film but employ thin film rotational angle theta and approximately meet sin θ=tan θ's it is assumed that many institutes It is known,It is therefore apparent that only under the less situation of thin film rotational angle theta, just can haveApproximate establishment, thus this analytic solutions be only applicable in thin film carry prestressing force but thin film rotational angle theta is less Situation；But so far, have not yet seen be applied to thin film rotational angle theta larger, carry prestressed situation thin film simultaneously again The analytic solutions of circular membrane On Axisymmetric Deformation of A that fix to clamp of horizontal uniform load following peripheral.

Content of the invention

This invention address that the analysis research of thin film problem, the horizontal uniform load following peripheral of Analytical Solution fixes to clamp The On Axisymmetric Deformation of A of prestressing force circular membrane, and obtain the analytic solutions of this problem.In solution procedure, the present invention considers With prestressed situation and employ in thin film(abandoning the hypothesis of sin θ=tan θ), Thus the analytic solutions being obtained be applied to thin film rotational angle theta larger, carry prestressed situation thin film simultaneously again.Based on this solution Analysis solution, The present invention gives the determination method of the prestressing force circular membrane elasticity energy U that laterally uniform load following peripheral fixes to clamp.

The laterally determination method of uniform load lower prestress circular membrane elasticity energy：The prestressing force that periphery is fixed to clamp is circular Thin film is laterally applied to uniform load q, wherein the thickness of circular membrane be h, radius be a, Young's modulus of elasticity be E, Poisson Than for ν, prestressing force be σ₀, the standing balance analysis based on this On Axisymmetric Deformation of A is it is possible to obtain this prestressing force circle Elasticity energy U after deformation of thin membrane and the parsing relation of uniform load q being applied

Wherein, π is pi,

And the value of centre parameter c is by equation

Determine, wherein,

So, as long as accurately measuring the value of applied uniform load q it is possible to determine this prestressing force circular membrane Elasticity energy U after deformation.Wherein, middle parameter c does not have unit, and other all parameters all adopt the International System of Units.

Brief description

The loading organigram of the prestressing force circular membrane that Fig. 1 fixes to clamp for uniform load following peripheral, wherein, 1- justifies Shape thin film, 2- clamping device, symbol therein is that a represents the inside radius of clamping device and the radius of circular membrane, and r represents footpath To coordinate, w (r) represents the lateral coordinates at point r, and q represents horizontal uniform load, w_mRepresent the maximum defluxion of circular membrane, θ represents Corner after deformation of thin membrane.

Specific embodiment

Below in conjunction with the accompanying drawings technical scheme is described in further detail：

The prestressing force round rubber thin film that periphery is fixed to clamp is laterally applied to uniform load q, wherein, round rubber Thickness h=the 0.5mm of thin film, radius a=20mm, Young's modulus of lasticity E=7.84MPa, Poisson's ratio ν=0.47, prestressing force σ₀ =0.02MPa, and record q=0.01MPa.Using the method given by the present invention, by equation

C=0.2853500604 then can be obtained, wherein,

Finally, by equation

Elasticity energy U=29.43214781 × 10 after the deformation of this circular membrane then can be obtained^-3J, wherein,

If additionally, using patent of invention (" a kind of determination uniform load lower prestress circular membrane declared before applicant The method of elasticity energy ", number of patent application：201410238597.3) method in, then after can obtaining the deformation of this circular membrane Elasticity can U=28.70780828 × 10^-3J it can be seen that, be differentiated between two methods.This difference is by applicant Patent of invention (" a kind of method determining uniform load lower prestress circular membrane elasticity energy ", the number of patent application before declared： 201410238597.3) assumed condition (assuming sin θ=tan θ) in causes.

Claims (1)

1. horizontal uniform load lower prestress circular membrane elasticity can determination method it is characterised in that：Periphery is fixed to clamp Prestressing force circular membrane is laterally applied to uniform load q, wherein the thickness of circular membrane be h, radius be a, Young springform Measure for E, Poisson's ratio be ν, prestressing force be σ₀, record the value of applied uniform load q, this circular membrane determined by below equation Elasticity energy U after deformation：

$U=\frac{{\mathrm{\πa}}^{10/3}{c}^{1/3}{q}^{4/3}}{2^{1/3}{E}^{1/3}{h}^{1/3}}\[g\left(c\right)-h\left(c\right)\],$

Wherein, π is pi,

$\begin{array}{c}g\left(c\right)=1+\left(\frac{1}{4}+\frac{1}{2}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c+\left(\frac{5}{36}+\frac{11}{18}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{1}{2}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\right)c^2\\ +\left(\frac{55}{576}+\frac{49}{72}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{61}{48}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{5}{8}\frac{2a^2{q}^2}{h^2{E}^{2}c}\right)c^3+(\frac{7}{96}+\frac{2647}{3600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{4051}{1800}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{253}{100}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7}{8}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4+(\frac{205}{3456}+\frac{101639}{129600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{443119}{129600}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{8521}{1350}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7141}{1440}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{21}{16}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5\\ +(\frac{17051}{338688}+\frac{2639243}{3175200}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{866077}{181440}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{13308283}{1058400}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1904549}{117600}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{113623}{11760}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33}{16}\frac{4a^4{q}^4}{h^4{E}^4{c}^2})c^6+(\frac{2864485}{65028096}\\ +\frac{5947093}{6773760}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{91705651}{14515200}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{1115512751}{50803200}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1366411591}{33868800}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{740195}{18816}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33661}{1792}\frac{4a^4{q}^4}{h^4{E}^4{c}^2}\\ +\frac{429}{128}\frac{2^{7/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

$\begin{array}{c}h\left(c\right)=\frac{1}{2}+\frac{1}{3}\left(\frac{1}{4}+\frac{1}{2}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c+\frac{1}{4}\left(\frac{5}{36}+\frac{11}{18}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{1}{2}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\right)c^2\\ +\frac{1}{5}\left(\frac{55}{576}+\frac{49}{72}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{61}{48}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{5}{8}\frac{2a^2{q}^2}{h^2{E}^{2}c}\right)c^3+\frac{1}{6}(\frac{7}{96}+\frac{2647}{3600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{4051}{1800}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{253}{100}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7}{8}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4+\frac{1}{7}(\frac{205}{3456}+\frac{101639}{129600}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +\frac{443119}{129600}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{8521}{1350}\frac{2a^2{q}^2}{h^2{E}^{2}c}+\frac{7141}{1440}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{21}{16}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5\\ +\frac{1}{8}(\frac{17051}{338688}+\frac{2639243}{3175200}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{866077}{181440}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{13308283}{1058400}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1904549}{117600}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{113623}{11760}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33}{16}\frac{4a^4{q}^4}{h^4{E}^4{c}^2})c^6+\frac{1}{9}(\frac{2864485}{65028096}\\ +\frac{5947093}{6773760}\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+\frac{91705651}{14515200}\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+\frac{1115512751}{50803200}\frac{2a^2{q}^2}{h^2{E}^{2}c}\\ +\frac{1366411591}{33868800}\frac{2^{4/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+\frac{740195}{18816}\frac{2^{5/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+\frac{33661}{1792}\frac{4a^4{q}^4}{h^4{E}^4{c}^2}\\ +\frac{429}{128}\frac{2^{7/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

And the value of centre parameter c is by equation

$\[2{\mathrm{cf}}^{\′}\left(c\right)+(1-\mathrm{\ν})f\left(c\right)\]-(1-\mathrm{\ν})\frac{2^{5/3}{h}^{2/3}{c}^{1/3}}{a^{2/3}{q}^{2/3}{E}^{1/3}}{\mathrm{\σ}}_0=0$

Determine, wherein,

$\begin{array}{c}f\left(c\right)=1-\frac{1}{2}c-\frac{1}{6}\left(1+2\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c^2-\frac{1}{144}(13+56\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +48\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}})c^3-\frac{1}{1440}(85+584\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+1104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +1152\frac{a^2{q}^2}{h^2{E}^{2}c})c^4-\frac{1}{21600}(925+8904\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+27104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +62592\frac{a^2{q}^2}{h^2{E}^{2}c}+23040\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^5-\frac{1}{907200}(30125+377688\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +1619968\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+6038784\frac{a^2{q}^2}{h^2{E}^{2}c}+4907520\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +1382400\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^6-\frac{1}{203212800}(5481025+85400640\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +478992416\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+2520947712\frac{a^2{q}^2}{h^2{E}^{2}c}+3305599488\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +2047057920\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+928972800\frac{a^4{q}^4}{h^4{E}^4{c}^2})c^7\end{array},$

$\begin{array}{c}{f}^{\′}\left(c\right)=-\frac{1}{2}-\frac{1}{3}\left(1+2\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\right)c-\frac{1}{48}(13+56\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +48\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}})c^2-\frac{1}{360}(85+584\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+1104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +1152\frac{a^2{q}^2}{h^2{E}^{2}c})c^3-\frac{1}{4320}(925+8904\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+27104\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +62592\frac{a^2{q}^2}{h^2{E}^{2}c}+23040\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}})c^4-\frac{1}{151200}(30125+377688\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +1619968\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+6038784\frac{a^2{q}^2}{h^2{E}^{2}c}+4907520\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +1382400\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}})c^5-\frac{1}{29030400}(5481025+85400640\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}\\ +478992416\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}+2520947712\frac{a^2{q}^2}{h^2{E}^{2}c}+3305599488\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}\\ +2047057920\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}+928972800\frac{a^4{q}^4}{h^4{E}^4{c}^2})c^6-\frac{1}{914457600}(165851725\\ +3108156432\×\frac{2^{1/3}{a}^{2/3}{q}^{2/3}}{h^{2/3}{E}^{2/3}{c}^{1/3}}+21730770208\×\frac{2^{2/3}{a}^{4/3}{q}^{4/3}}{h^{4/3}{E}^{4/3}{c}^{2/3}}\\ +149806363136\frac{a^2{q}^2}{h^2{E}^{2}c}+277407668736\×\frac{2^{1/3}{a}^{8/3}{q}^{8/3}}{h^{8/3}{E}^{8/3}{c}^{4/3}}+276724703232\×\frac{2^{2/3}{a}^{10/3}{q}^{10/3}}{h^{10/3}{E}^{10/3}{c}^{5/3}}\\ +274790154240\frac{a^4{q}^4}{h^4{E}^4{c}^2}+52022476800\×\frac{2^{1/3}{a}^{14/3}{q}^{14/3}}{h^{14/3}{E}^{14/3}{c}^{7/3}})c^7\end{array},$

Middle parameter c does not have unit, and other all parameters all adopt the International System of Units.