CN110018050B - Method for obtaining the modulus of elasticity of a plate-shaped component - Google Patents

Method for obtaining the modulus of elasticity of a plate-shaped component Download PDF

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CN110018050B
CN110018050B CN201910339395.0A CN201910339395A CN110018050B CN 110018050 B CN110018050 B CN 110018050B CN 201910339395 A CN201910339395 A CN 201910339395A CN 110018050 B CN110018050 B CN 110018050B
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elastic modulus
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林韦廷
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Hefei Lianbao Information Technology Co Ltd
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    • 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
    • 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
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    • 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
    • G01N2203/0058—Kind of property studied
    • 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
    • G01N2203/02—Details not specific for a particular testing method
    • G01N2203/026—Specifications of the specimen
    • G01N2203/0262—Shape of the specimen
    • G01N2203/0278—Thin specimens
    • G01N2203/0282—Two dimensional, e.g. tapes, webs, sheets, strips, disks or membranes

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Abstract

The invention discloses a method for acquiring the elastic modulus of a plate-shaped component, which comprises the following steps: s10: establishing three-point stress test conditions to obtain pressure P on the plate-shaped component and deflection Umax generated on the basis of the pressure P; s20: obtaining a relational expression about the elastic modulus E, the pressure P and the deflection Umax of the plate-shaped member based on a plate shell theory; s30: the elastic modulus E of the plate-like member is calculated based on the pressure P and the deflection Umax. According to the invention, the elastic modulus of the plate-shaped member is obtained by introducing a calculation formula of a plate shell theory, so that the size of a test sample is not limited by the width in test specifications such as ASTM (American society for testing and materials) and GB (GB), the measurable size range of a body to be measured is wider, the influence of the width of the plate-shaped member on the elastic modulus is considered in the calculation process of the elastic modulus, and the calculated elastic modulus is closer to the real elastic modulus.

Description

Method for obtaining the modulus of elasticity of a plate-shaped component
Technical Field
The invention relates to the technical field of engineering, in particular to a method for acquiring the elastic modulus of a plate-shaped member.
Background
It is easy to understand that, since the plate-shaped member applied to the electronic device is often subjected to an external force or an impact when the electronic device is used, it is necessary to know the mechanical properties of the plate-shaped member, and particularly, it is necessary to obtain the elastic modulus of the plate-shaped member.
In the prior art, the elastic modulus of the plate-shaped member is obtained by combining a three-point stress test and a theoretical formula, wherein the applied theoretical formula is mostly various transformed beam theoretical formulas, which results in that the elastic modulus of the obtained plate-shaped member is greatly different from the actual elastic modulus because the influence of the width of the plate-shaped member on the deformation is not considered (in test specifications such as ASTM and GB, the width of the plate-shaped member is limited to a small range, and further the elastic modulus of the plate-shaped member is considered to be not considered to consider a width parameter), namely, the accuracy of the elastic modulus of the obtained plate-shaped member is poor.
Disclosure of Invention
In view of the above technical problems in the prior art, embodiments of the present invention provide a method for obtaining an elastic modulus of a plate-shaped member.
In order to solve the technical problem, the embodiment of the invention adopts the following technical scheme:
a method for obtaining the modulus of elasticity of a plate-like member, comprising the steps of:
s10: establishing three-point stress test conditions to obtain pressure P on the plate-shaped component and deflection Umax generated on the basis of the pressure P;
s20: obtaining a relational expression about the elastic modulus E, the pressure P and the deflection Umax of the plate-shaped member based on a plate shell theory;
s30: the elastic modulus E of the plate-like member is calculated based on the pressure P and the deflection Umax.
Preferably, S10 includes the steps of:
s11: two supporting bars are arranged at intervals and in parallel;
s12: placing the plate-shaped member on the two support bars, wherein the extension direction of the support bars is consistent with the width direction of the plate-shaped member, and the arrangement direction of the two support bars is consistent with the length direction of the plate-shaped member;
s13: downward pressure P is applied to the upper plate surface of the plate-shaped member at the middle position between the two support bars, and the deflection Umax produced by the plate-shaped member under the pressure P is obtained by a displacement sensor positioned below the plate-shaped member.
Preferably, S20 includes the steps of:
s21: the strain field formula (1) in the shell theory is:
Figure BDA0002040236730000021
Figure BDA0002040236730000022
Figure BDA0002040236730000023
Figure BDA0002040236730000024
Figure BDA0002040236730000025
and stress field formula (2) in the plate-shell theory:
Figure BDA0002040236730000026
Figure BDA0002040236730000027
Figure BDA0002040236730000028
Figure BDA0002040236730000029
Figure BDA00020402367300000210
substituting into the strain energy formula (3) in the shell theory:
Vs＝1/2∫∫∫(σxεx+σyεy+σzεz+τxyεxy+τxzεxz+τyzεyz)dxdydz
s22: the displacement field formula (4) in the plate shell theory:
Figure BDA0002040236730000031
substituting into the working formula (5) in the plate shell theory:
Wload＝∫∫(-p(x,y)Uz)dxdy＝∫∫(-p(x,y)w)dxdy
s23: substituting formula (4) and formula (5) into the han m midton principle formula (6):
δ(1)＝∫(-Vs+Wload)dt＝0
deriving the Euler-Lagrange's equation by variational processing to obtain a deformation equation (7) for the rectangular sample:
Figure BDA0002040236730000032
wherein D ═ E h3)/(12(1-v2))
Wherein: p (x, y) is a pressure function; p (x, y) ═ P/b δ (x-L/2)
S24: the Levy formula (8):
w＝Σ{[Am e(-mπy)/L+y Bm e(-mπy)/L+Cm e(mπy)/L+y Dm e(mπy)/L+(2L3P Sin(mπ/2))/(b D m4π4)]Sin((mπx)/L)}
the boundary conditions of the three-point stress test are substituted to analyze Am, Bm, Cm and Dm, and then the Am to Dm and a formula (8) are substituted into a formula (7) to analyze:
Figure BDA0002040236730000041
wherein:
l is: a measured span length of the plate member.
b is as follows: the width of the plate-like member;
h is: the thickness of the plate-like member;
x is: coordinates corresponding to the length of the plate-like member;
y is: coordinates corresponding to the width of the plate-like member;
z is as follows: coordinates corresponding to the thickness of the plate-like member;
v is: the Poisson's ratio, taken as 0.35.
Compared with the prior art, the method for obtaining the elastic modulus of the plate-shaped member has the advantages that:
1. according to the invention, the elastic modulus of the plate-shaped member is obtained by introducing a calculation formula of a plate shell theory, so that the influence of the width of the plate-shaped member on the elastic modulus is considered in the calculation process of the elastic modulus, the size of a sample is not limited by the width in test specifications such as ASTM (American society for testing and materials) and GB (GB), the measurable size range of a body to be tested is wider, and the calculated elastic modulus is closer to the real elastic modulus.
2. The invention utilizes the Han Maitreton principle and the Euler-Lagrange daily variation equation formula to obtain the boundary condition parameter P and the relation formula between Umax and the elastic modulus, so that the elastic modulus can be obtained by the plate shell theory.
3. The invention analyzes the elastic modulus by using the Levy method, thereby simplifying the process of analyzing the elastic modulus.
Drawings
FIG. 1 is a test setup for establishing three-point test conditions.
Detailed Description
In order to make the technical solutions of the present invention better understood, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
An embodiment of the present invention discloses a method for obtaining an elastic modulus of a plate-like member 10, including the steps of:
s10: establishing three-point stress test conditions to obtain pressure P on the plate-shaped member 10 and deflection Umax generated on the basis of the pressure P;
s20: obtaining a relational expression about the elastic modulus E, the pressure P, and the deflection Umax of the plate-shaped member 10 based on a plate-shell theory;
s30: the elastic modulus E of the plate-like member 10 is calculated based on the pressure P and the deflection Umax.
Preferably, as in fig. 1, S10 includes the following steps:
s11: two supporting bars are arranged at intervals and in parallel;
s12: placing the plate-shaped member 10 on two support bars, wherein the extension direction of the support bars is consistent with the width direction of the plate-shaped member 10, and the arrangement direction of the two support bars is consistent with the length direction of the plate-shaped member 10;
s13: a downward pressure P is applied to the upper plate surface of the plate-shaped member 10 at a middle position between the two support bars, and a deflection Umax of the plate-shaped member 10 at the pressure P is obtained by a displacement sensor located below the plate-shaped member 10.
Preferably, S20 includes the steps of:
s21: the strain field formula (1) in the shell theory is:
Figure BDA0002040236730000051
Figure BDA0002040236730000052
Figure BDA0002040236730000053
Figure BDA0002040236730000054
Figure BDA0002040236730000055
and stress field formula (2) in the plate-shell theory:
Figure BDA0002040236730000056
Figure BDA0002040236730000061
Figure BDA0002040236730000062
Figure BDA0002040236730000063
Figure BDA0002040236730000064
substituting into the strain energy formula (3) in the shell theory:
Vs＝1/2∫∫∫(σxεx+σyεy+σzεz+τxyεxy+τxzεxz+τyzεyz)dxdydz
s22: the displacement field formula (4) in the plate shell theory:
Figure BDA0002040236730000065
substituting into the working formula (5) in the plate shell theory:
Wload＝∫∫(-p(x,y)Uz)dxdy＝∫∫(-p(x,y)w)dxdy
s23: substituting formula (4) and formula (5) into the han m midton principle formula (6):
δ(1)＝∫(-Vs+Wload)dt＝0
deriving the Euler-Lagrange's equation by variational processing to obtain a deformation equation (7) for the rectangular sample:
Figure BDA0002040236730000066
wherein D ═ E h3)/(12(1-v2))
Wherein: p (x, y) is a pressure function; p (x, y) ═ P/b δ (x-L/2)
S24: the Levy formula (8):
w＝Σ{[Am e(-mπy)/L+y Bm e(-mπy)/L+Cm e(mπy)/L+y Dm e(mπy)/L+(2L3P Sin(mπ/2))/(b D m4π4)]Sin((mπx)/L)}
the boundary conditions of the three-point stress test are substituted to analyze Am, Bm, Cm and Dm, and then the Am to Dm and a formula (8) are substituted into a formula (7) to analyze:
Figure BDA0002040236730000071
wherein:
l is: the measured span length of the plate member 10.
b is as follows: the width of the plate-like member 10;
h is: the thickness of the plate-like member 10;
x is: coordinates corresponding to the length of the plate-like member 10;
y is: coordinates corresponding to the width of the plate-like member 10;
z is as follows: coordinates corresponding to the thickness of the plate-like member 10;
v is: the Poisson's ratio, taken as 0.35.
The method for obtaining the modulus of elasticity of the plate-shaped member 10 provided by the present invention has the advantages that:
1. according to the invention, the elastic modulus of the plate-shaped member 10 is obtained by introducing a calculation formula of a plate-shell theory, so that the influence of the width of the plate-shaped member 10 on the elastic modulus is considered in the calculation process of the elastic modulus, the size of a sample is not limited by the width in test specifications such as ASTM (international test specification) and GB (GB), the size range of a measurable body to be tested is wider, and the calculated elastic modulus is closer to the real elastic modulus.
2. The invention utilizes the Han Maitreton principle and the Euler-Lagrange daily variation equation formula to obtain the boundary condition parameter P and the relation formula between Umax and the elastic modulus, so that the elastic modulus can be obtained by the plate shell theory.
3. The invention analyzes the elastic modulus by using the Levy method, thereby simplifying the process of analyzing the elastic modulus.
The above embodiments are only exemplary embodiments of the present invention, and are not intended to limit the present invention, and the scope of the present invention is defined by the claims. Various modifications and equivalents may be made by those skilled in the art within the spirit and scope of the present invention, and such modifications and equivalents should also be considered as falling within the scope of the present invention.

Claims (2)

1. A method for obtaining the modulus of elasticity of a plate-like member, characterized by comprising the steps of:
s10: establishing three-point stress test conditions to obtain pressure P on the plate-shaped component and deflection Umax generated on the basis of the pressure P;
s20: obtaining a relational expression about the elastic modulus E, the pressure P and the deflection Umax of the plate-shaped member based on a plate shell theory;
s30: calculating the elastic modulus E of the plate-shaped member based on the pressure P and the deflection Umax;
s20 includes the steps of:
s21: the strain field formula (1) in the shell theory is:
Figure FDA0003062964270000011
Figure FDA0003062964270000012
Figure FDA0003062964270000013
Figure FDA0003062964270000014
Figure FDA0003062964270000015
and stress field formula (2) in the plate-shell theory:
Figure FDA0003062964270000016
Figure FDA0003062964270000017
Figure FDA0003062964270000018
Figure FDA0003062964270000019
Figure FDA00030629642700000110
substituting into the strain energy formula (3) in the shell theory:
Vs＝1/2∫∫∫(σxεx+σyεy+σzεz+τxyεxy+τxzεxz+τyzεyz)dxdydz
s22: the displacement field formula (4) in the plate shell theory:
Figure FDA0003062964270000021
substituting into the working formula (5) in the plate shell theory:
Wload＝∫∫(-p(x,y)Uz)dxdy＝∫∫(-p(x,y)w)dxdy
s23: substituting formula (4) and formula (5) into the han m midton principle formula (6):
δ(1)＝∫(-Vs+Wload)dt＝0
deriving the Euler-Lagrange's equation by variational processing to obtain a deformation equation (7) for the rectangular sample:
Figure FDA0003062964270000022
wherein D ═ E h3)/(12(1-v2))
Wherein: p (x, y) is a pressure function; p (x, y) ═ P/b δ (x-L/2)
S24: the Levy formula (8):
w＝Σ{[Ame(-mπy)/L+yBme(-mπy)/L+Cme(mπy)/L+yDme(mπy)/L+(2L3P Sin(mπ/2))/(b D m4π4)]Sin((mπx)/L)}
the boundary conditions of the three-point stress test are substituted to analyze Am, Bm, Cm and Dm, and then the Am to Dm and a formula (8) are substituted into a formula (7) to analyze:
Figure FDA0003062964270000031
wherein:
l is: a measured span length of the plate member.
b is as follows: the width of the plate-like member;
h is: the thickness of the plate-like member;
x is: coordinates corresponding to the length of the plate-like member;
y is: coordinates corresponding to the width of the plate-like member;
z is as follows: coordinates corresponding to the thickness of the plate-like member;
v is: the Poisson's ratio, taken as 0.35.
2. The method for obtaining the elastic modulus of a plate-like member according to claim 1, wherein S10 comprises the steps of:
s11: two supporting bars are arranged at intervals and in parallel;
s12: placing the plate-shaped member on the two support bars, wherein the extension direction of the support bars is consistent with the width direction of the plate-shaped member, and the arrangement direction of the two support bars is consistent with the length direction of the plate-shaped member;
s13: downward pressure P is applied to the upper plate surface of the plate-shaped member at the middle position between the two support bars, and the deflection Umax produced by the plate-shaped member under the pressure P is obtained by a displacement sensor positioned below the plate-shaped member.
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Publication number Priority date Publication date Assignee Title
CN101144770A (en) * 2007-08-02 2008-03-19 上海交通大学 Method for measuring silicon base body and membrane base combination intensity
CN101788427A (en) * 2010-01-29 2010-07-28 湘潭大学 Device for detecting mechanical property of multifunctional film
CN103245437A (en) * 2012-02-13 2013-08-14 付康 System and method for determining nonlinear membrane stress
CN105547861A (en) * 2016-02-06 2016-05-04 南京林业大学 Method for enhancing capability of testing modulus of elasticity and precision of Poisson's ratio of wood by four-point bent beam
CN106596256A (en) * 2016-12-13 2017-04-26 西安科技大学 Apparatus suitable for measuring bending rigidity, elasticity modulus, shear modulus and bulk modulus
CN107729605A (en) * 2017-09-06 2018-02-23 北京航空航天大学 A kind of modification method of the layer stripping residual stress measurement value based on Plate Theory
CN109342225A (en) * 2018-12-05 2019-02-15 郑州大学 Elasticity modulus test method is drawn in Polypropylene Fiber Reinforced Cement Stabilized Macadam bending resistance

Patent Citations (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101144770A (en) * 2007-08-02 2008-03-19 上海交通大学 Method for measuring silicon base body and membrane base combination intensity
CN101788427A (en) * 2010-01-29 2010-07-28 湘潭大学 Device for detecting mechanical property of multifunctional film
CN103245437A (en) * 2012-02-13 2013-08-14 付康 System and method for determining nonlinear membrane stress
CN105547861A (en) * 2016-02-06 2016-05-04 南京林业大学 Method for enhancing capability of testing modulus of elasticity and precision of Poisson's ratio of wood by four-point bent beam
CN106596256A (en) * 2016-12-13 2017-04-26 西安科技大学 Apparatus suitable for measuring bending rigidity, elasticity modulus, shear modulus and bulk modulus
CN107729605A (en) * 2017-09-06 2018-02-23 北京航空航天大学 A kind of modification method of the layer stripping residual stress measurement value based on Plate Theory
CN109342225A (en) * 2018-12-05 2019-02-15 郑州大学 Elasticity modulus test method is drawn in Polypropylene Fiber Reinforced Cement Stabilized Macadam bending resistance

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