CN111442982A - Method for determining maximum stress of circular film under uniformly distributed load - Google Patents

Method for determining maximum stress of circular film under uniformly distributed load Download PDF

Info

Publication number
CN111442982A
CN111442982A CN202010190003.1A CN202010190003A CN111442982A CN 111442982 A CN111442982 A CN 111442982A CN 202010190003 A CN202010190003 A CN 202010190003A CN 111442982 A CN111442982 A CN 111442982A
Authority
CN
China
Prior art keywords
circular film
maximum stress
circular
film
load
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
CN202010190003.1A
Other languages
Chinese (zh)
Inventor
何晓婷
李雪
赵智航
孙俊贻
郭莹
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Chongqing University
Original Assignee
Chongqing University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Chongqing University filed Critical Chongqing University
Priority to CN202010190003.1A priority Critical patent/CN111442982A/en
Publication of CN111442982A publication Critical patent/CN111442982A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • G—PHYSICS
    • G01—MEASURING; TESTING
    • 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
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/18—Complex mathematical operations for evaluating statistical data, e.g. average values, frequency distributions, probability functions, regression analysis
    • 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
    • G01N2203/0014—Type of force applied
    • G01N2203/0016—Tensile or compressive
    • G01N2203/0019—Compressive

Landscapes

  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Data Mining & Analysis (AREA)
  • Mathematical Analysis (AREA)
  • Pure & Applied Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Mathematical Physics (AREA)
  • Mathematical Optimization (AREA)
  • Computational Mathematics (AREA)
  • Life Sciences & Earth Sciences (AREA)
  • Bioinformatics & Computational Biology (AREA)
  • General Engineering & Computer Science (AREA)
  • Bioinformatics & Cheminformatics (AREA)
  • Investigating Strength Of Materials By Application Of Mechanical Stress (AREA)
  • Pathology (AREA)
  • Operations Research (AREA)
  • Immunology (AREA)
  • Algebra (AREA)
  • Biochemistry (AREA)
  • Probability & Statistics with Applications (AREA)
  • Health & Medical Sciences (AREA)
  • Chemical & Material Sciences (AREA)
  • Databases & Information Systems (AREA)
  • Software Systems (AREA)
  • Evolutionary Biology (AREA)
  • General Health & Medical Sciences (AREA)
  • Analytical Chemistry (AREA)

Abstract

The invention discloses a method for determining the maximum stress of a circular film under uniformly distributed loads, which comprises the following steps: transversely applying an evenly distributed load q to a circular film which is fixedly clamped at the periphery and has the radius of a, the thickness of h, the Young modulus of elasticity of E and the Poisson ratio of v to ensure that the circular film generates axisymmetric deformation, and determining the maximum stress sigma after the axisymmetric deformation of the circular film by utilizing the measurement value of the load q based on the static balance analysis of the axisymmetric deformation problem of the circular filmm。

Description

Method for determining maximum stress of circular film under uniformly distributed load
Technical Field
The invention relates to a method for determining the maximum stress of a circular film which is fixedly clamped at the periphery under the action of transversely uniformly distributed loads.
Background
The axisymmetric deformation of a circular membrane, which is peripherally and fixedly clamped under the action of a transversely uniformly distributed load, has applications in many engineering technology fields, for example, to study the adhesion energy measurement of membrane/substrate systems, and to develop various instruments and meters, various sensors, and the like. From the results of the study, in the process of solving the problem of axisymmetric deformation of the circular film, the commonly-called assumption of small rotation angle of the film (i.e. the assumption that the rotation angle theta of the film satisfies sin theta ≈ tan theta) is abandoned to improve the calculation accuracy, for example, the invention patent "the determination of the maximum stress of the circular film with large rotation angle under uniform loadThe analytical solution used in the method of determination (patent No. Z L201510194408.1) is obtained by giving up the assumption of small film rotation angle, but the influence of film deflection is not considered when establishing the in-plane equilibrium equation of the mechanical problem, so that an approximate in-plane equilibrium equation d (r σ) is establishedr)/dr-σtWhere r denotes the radial coordinate of the circular membrane, σrAnd σtThe radial and hoop stresses of a circular film are indicated, respectively. When the external acting load is larger and the film deflection is larger, the film deflection has larger influence on the in-plane equilibrium equation, obviously, the in-plane equilibrium equation neglecting the influence of the film deflection is still adopted when the mechanical problem is solved, so that the obtained analytic solution has larger error, therefore, the influence of the film deflection is considered when the in-plane equilibrium equation of the mechanical problem is established, and a more accurate in-plane equilibrium equation d (r sigma) is establishedr)/dr-σt[1+(dw/dr)2]0. The analytic solution obtained based on the accurate in-plane equilibrium equation can be suitable for the conditions of large external acting load and large film deflection, which is undoubtedly a very valuable work, and the application range of axisymmetric deformation of the circular film with the periphery fixedly clamped under the action of transversely uniformly distributed load can be enlarged, which is also the technical problem to be solved by the invention.
Disclosure of Invention
The invention is dedicated to the analytical research of the axisymmetric deformation problem of the circular film with the periphery fixedly clamped under the action of the transversely uniformly distributed load, obtains a more accurate analytical solution of the axisymmetric deformation problem based on more precise static balance analysis, and provides a method for determining the maximum stress of the circular film under the uniformly distributed load on the basis.
The method for determining the maximum stress of the circular film under uniformly distributed load comprises the following steps: transversely applying an evenly distributed load q to a circular film which is fixedly clamped at the periphery and has the radius of a, the thickness of h, the Young's modulus of elasticity of E and the Poisson ratio of v to ensure that the circular film generates axisymmetric deformation, and obtaining the applied load q and the most axially symmetrically deformed circular film based on the static balance analysis of the axisymmetric deformation problem of the circular filmLarge stress sigmamAnalytic relationship between
Figure BDA0002415378190000011
Wherein the content of the first and second substances,
Figure BDA0002415378190000021
Figure BDA0002415378190000022
Figure BDA0002415378190000023
Figure BDA0002415378190000024
Figure BDA0002415378190000025
Figure BDA0002415378190000026
d0＝b0，
Figure BDA0002415378190000027
Figure BDA0002415378190000028
Figure BDA0002415378190000031
Figure BDA0002415378190000032
Figure BDA0002415378190000033
Figure BDA0002415378190000034
Figure BDA0002415378190000035
and b0Is given by the equation
Figure BDA0002415378190000036
And (4) determining.
Thus, the maximum stress sigma after the axial symmetric deformation of the circular film can be obtained by accurately measuring the value of the load qmWherein the units of a and h are millimeter (mm), E, q, and sigmamAll units of (2) are Newton per square millimeter (N/mm)2) And v, b0、b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
Drawings
FIG. 1 is a schematic view of axisymmetrical deformation of a circular film whose periphery is fixedly clamped under the action of a transversely uniformly distributed load, wherein 1 is the circular film after axisymmetrical deformation, 2 is a clamping device, 3 is the circular film before deformation, a represents the radius of the circular film and the inner radius of the clamping device, q represents the transversely uniformly distributed load, wmShowing the maximum deflection after axisymmetric deformation of the 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/mm2Poisson ratio v is 0.47, transversely applying an evenly distributed load q to the circular film fixedly clamped at the periphery to enable the circular film to generate axisymmetric deformation, and measuring the load q to be 0.1N/mm2By using the method given in the invention, the equation
Figure BDA0002415378190000041
Figure BDA0002415378190000042
Figure BDA0002415378190000043
Figure BDA0002415378190000044
Figure BDA0002415378190000045
Figure BDA0002415378190000046
Figure BDA0002415378190000051
d0＝b0，
Figure BDA0002415378190000052
Figure BDA0002415378190000053
Figure BDA0002415378190000054
Figure BDA0002415378190000055
Figure BDA0002415378190000056
Figure BDA0002415378190000057
Figure BDA0002415378190000061
To obtain b01.351319 and b2＝1.295919、b4＝0.702265、b6＝-0.582036、b8＝-1.046604、b10＝0.785018、b12＝2.543903、d0＝1.351319、d2＝0.543440、d4＝0.304119、d6＝-0.061103、d8＝-0.297796、d10＝-0.016376、d120.549795, then
Figure BDA0002415378190000062
Determining that the load q uniformly distributed in the transverse direction of the round film is 0.1N/mm2Maximum stress under influence σm＝10.5943N/mm2。
In order to reflect the error caused by the approximate in-plane equilibrium equation, and to embody the beneficial effects of the present invention, the applicant also calculated the peripheral fixedly clamped circular film with the transverse uniform load q equal to 0.1N/mm by the previous method (a method for determining the maximum stress of the circular film with large rotation angle under uniform load, patent number Z L201510194408.1)2Maximum stress sigma after axisymmetric deformation under actionm＝15.6220N/mm2The error of the maximum stress of the film calculated by the two methods is about 47.45%, which is far beyond the calculation error range allowed by the engineering structure design (i.e. less than 15%). The invention patent adopts an in-plane equilibrium equation neglecting the influence of the deflection of the film when solving the mechanics problem, so the method for calculating the maximum stress of the film is providedMust be used when the applied uniform load q is small. The method adopts an in-plane equilibrium equation considering the influence of the deflection of the film when solving the mechanical problem, so that the method provided by the invention can be more suitable for the situation that the applied uniform load q is larger and the deflection w is larger, thereby eliminating the limitation that the applied transverse load q cannot be overlarge, and the technical effect is obvious.

Claims (1)

1. The method for determining the maximum stress of the circular film under uniformly distributed load is characterized by comprising the following steps: transversely applying an evenly distributed load q to a circular thin film which is fixedly clamped at the periphery and has the radius of a, the thickness of h, the Young modulus of elasticity of E and the Poisson ratio of v to ensure that the circular thin film generates axisymmetric deformation, and then carrying out static balance analysis based on the axisymmetric deformation problem of the circular thin film by using the measured value of the load q and using the following equation
Figure FDA0002415378180000011
Figure FDA0002415378180000012
Figure FDA0002415378180000013
Figure FDA0002415378180000014
Figure FDA0002415378180000015
Figure FDA0002415378180000016
Figure FDA0002415378180000017
d0＝b0，
Figure FDA0002415378180000021
Figure FDA0002415378180000022
Figure FDA0002415378180000023
Figure FDA0002415378180000024
Figure FDA0002415378180000025
Figure FDA0002415378180000026
Figure FDA0002415378180000027
Determination of b0And b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12And finally, from the equation
Figure FDA0002415378180000028
Determining the maximum stress sigma after axisymmetric deformation of a circular filmmWherein the units of a and h are millimeter (mm), E, q and sigmamAll units of (2) are Newton per square millimeter (N/mm)2) And v, b0、b2、b4、b6、b8、b10、b12、d0、d2、d4、d6、d8、d10、d12And Q are dimensionless quantities.
CN202010190003.1A 2020-03-18 2020-03-18 Method for determining maximum stress of circular film under uniformly distributed load Pending CN111442982A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202010190003.1A CN111442982A (en) 2020-03-18 2020-03-18 Method for determining maximum stress of circular film under uniformly distributed load

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202010190003.1A CN111442982A (en) 2020-03-18 2020-03-18 Method for determining maximum stress of circular film under uniformly distributed load

Publications (1)

Publication Number Publication Date
CN111442982A true CN111442982A (en) 2020-07-24

Family

ID=71648728

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202010190003.1A Pending CN111442982A (en) 2020-03-18 2020-03-18 Method for determining maximum stress of circular film under uniformly distributed load

Country Status (1)

Country Link
CN (1) CN111442982A (en)

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112858001A (en) * 2021-01-18 2021-05-28 重庆大学 Method for determining maximum stress of circular prestressed thin film under uniformly distributed load
CN112903218A (en) * 2021-01-18 2021-06-04 重庆大学 Method for determining maximum stress of prestressed circular film with limited maximum deflection under air pressure
CN113551978A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining maximum stress of annular film with rigid inner edge
CN113720689A (en) * 2021-08-17 2021-11-30 重庆大学 Method for determining the maximum stress of a circular membrane in contact with a rigid plate under gas pressure

Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2006029999A (en) * 2004-07-16 2006-02-02 Axiom Co Ltd Viscoelastic characteristic measuring instrument for film-like viscoelastic object
CN101629859A (en) * 2009-05-04 2010-01-20 付康 System and method for determining thin-film stress based on deformation measurement and numerical reverse
US20120100719A1 (en) * 2009-06-18 2012-04-26 Commisariat A L'Energie Atomique ET Aux Ene Alt Method for making a planar membrane
CN103983515A (en) * 2014-05-30 2014-08-13 重庆大学 Method for determining resilience of prestressed circular film under uniformly-distributed load
CN103983380A (en) * 2014-05-30 2014-08-13 重庆大学 Method for determining maximum stress value of prestress circular thin film under uniformly-distributed loads
CN110286040A (en) * 2019-06-05 2019-09-27 重庆大学 A kind of determination method of the maximum stress of liquid effects lower prestress circular membrane

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2006029999A (en) * 2004-07-16 2006-02-02 Axiom Co Ltd Viscoelastic characteristic measuring instrument for film-like viscoelastic object
CN101629859A (en) * 2009-05-04 2010-01-20 付康 System and method for determining thin-film stress based on deformation measurement and numerical reverse
US20120100719A1 (en) * 2009-06-18 2012-04-26 Commisariat A L'Energie Atomique ET Aux Ene Alt Method for making a planar membrane
CN103983515A (en) * 2014-05-30 2014-08-13 重庆大学 Method for determining resilience of prestressed circular film under uniformly-distributed load
CN103983380A (en) * 2014-05-30 2014-08-13 重庆大学 Method for determining maximum stress value of prestress circular thin film under uniformly-distributed loads
CN110286040A (en) * 2019-06-05 2019-09-27 重庆大学 A kind of determination method of the maximum stress of liquid effects lower prestress circular membrane

Non-Patent Citations (9)

* Cited by examiner, † Cited by third party
Title
XUE LI 等: "A New Solution to Well-Known Hencky Problem:Improvement of In-Plane Equilibrium Equation", 《MATHEMATICS》 *
YONG-SHENG LIAN 等: "Application of perturbation idea to well-known Hencky problem: Aperturbation solution without small-rotation-angle assumption", 《MECHANICS RESEARCH COMMUNICATIONS》 *
何晓婷 等: "均布荷载下受有预加张力圆薄膜的轴对称变形", 《重庆大学学报》 *
吴建梁: "沿周边径向张拉圆薄膜的应力分布", 《重庆工商大学学报(自然科学版)》 *
张迅炜 等: "非线性变厚度圆薄板轴对称弯曲问题的变分解答", 《机械设计与制造》 *
段浩 等: "大挠度橡胶圆板的数值解及对其Hencky解的修正", 《应用力学学报》 *
郭建军 等: "圆薄膜在均布荷载下大挠度变形的精确解", 《重庆科技学院学报(自然科学版)》 *
钱伟长 等: "圆薄膜中心部份受均布载荷产生的对称变形", 《应用数学和力学》 *
陆晔: "横向均布载荷下圆形预应力薄膜大挠度问题研究", 《中国优秀硕士学位论文全文数据库 基础科学辑》 *

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN112858001A (en) * 2021-01-18 2021-05-28 重庆大学 Method for determining maximum stress of circular prestressed thin film under uniformly distributed load
CN112903218A (en) * 2021-01-18 2021-06-04 重庆大学 Method for determining maximum stress of prestressed circular film with limited maximum deflection under air pressure
CN113551978A (en) * 2021-07-30 2021-10-26 重庆大学 Method for determining maximum stress of annular film with rigid inner edge
CN113720689A (en) * 2021-08-17 2021-11-30 重庆大学 Method for determining the maximum stress of a circular membrane in contact with a rigid plate under gas pressure

Similar Documents

Publication Publication Date Title
CN111442982A (en) Method for determining maximum stress of circular film under uniformly distributed load
CN111442977B (en) Method for determining maximum stress of circular film under action of transversely uniformly distributed load
CN111442976B (en) Method for determining maximum deflection of circular film under action of transversely uniformly distributed load
CN111442984A (en) Method for determining maximum stress of circular film under transversely uniformly distributed load
CN111442985A (en) Method for determining maximum deflection of circular film under transversely uniformly distributed load
CN109323924B (en) Method for determining maximum stress of circular film under limitation of elasticity on maximum deflection
CN111442981A (en) Method for determining elastic strain energy of circular film under uniformly distributed load
CN111426567A (en) Method for determining maximum stress of circular film with limited maximum deflection under air pressure
CN113092041B (en) Method for determining maximum deflection of annular film under transversely uniformly distributed load
CN109991083B (en) Method for determining maximum stress under condition of large rotation angle of circular film limited by elasticity
CN112730071A (en) Method for determining elastic energy of circular prestressed film under gas pressure
CN110031300B (en) Elastic performance determination method under condition of large rotation angle of circular film limited by elasticity
CN111474040A (en) Method for determining elastic energy of prestressed round film under action of uniformly distributed load
CN111442978B (en) Method for determining elastic strain energy of circular film under action of transversely uniformly distributed load
CN111442980A (en) Method for determining maximum deflection of circular film under uniformly distributed load
CN111426568A (en) Method for determining deflection of circular film with limited maximum deflection under gas pressure
CN111426566A (en) Method for determining elastic energy of circular film with limited maximum deflection under gas pressure
CN111474038A (en) Method for determining maximum deflection of prestressed circular film under uniformly distributed load
CN111442983A (en) Method for determining elastic strain energy of circular film under transversely uniformly distributed load
CN113092040B (en) Method for determining maximum stress of annular film under transversely uniformly distributed load
CN106644248A (en) Method for determining maximum stress of circular membrane under limited maximum deflection
CN103837400A (en) Method for determining thin film deflection value of periphery-clamped circular thin film under concentrated force
CN111474039A (en) Method for determining maximum deflection of prestressed circular film under action of uniformly distributed load
CN111474042A (en) Method for determining maximum stress of prestressed circular thin film under uniformly distributed load
CN112858001A (en) Method for determining maximum stress of circular prestressed thin film under uniformly distributed load

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
RJ01 Rejection of invention patent application after publication

Application publication date: 20200724