CN111089793A - Hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01Determination method - Google Patents
Hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01Determination method Download PDFInfo
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- 229920000459 Nitrile rubber Polymers 0.000 title claims abstract description 49
- 238000000034 method Methods 0.000 title claims abstract description 24
- 239000000463 material Substances 0.000 claims abstract description 51
- 239000005060 rubber Substances 0.000 claims abstract description 48
- 229920001971 elastomer Polymers 0.000 claims abstract description 41
- 238000012360 testing method Methods 0.000 claims abstract description 38
- 230000032683 aging Effects 0.000 claims abstract description 18
- 150000002825 nitriles Chemical class 0.000 claims abstract description 13
- 238000009864 tensile test Methods 0.000 claims abstract description 12
- 239000005062 Polybutadiene Substances 0.000 claims abstract description 5
- 238000011160 research Methods 0.000 claims description 10
- 238000004458 analytical method Methods 0.000 claims description 8
- 230000035882 stress Effects 0.000 claims description 8
- 230000009021 linear effect Effects 0.000 claims description 4
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 claims description 3
- 238000002474 experimental method Methods 0.000 claims description 3
- 238000012417 linear regression Methods 0.000 claims description 2
- 238000004364 calculation method Methods 0.000 description 3
- 238000011161 development Methods 0.000 description 3
- 230000018109 developmental process Effects 0.000 description 3
- 238000004088 simulation Methods 0.000 description 3
- 238000013461 design Methods 0.000 description 2
- 241000272522 Anas Species 0.000 description 1
- 230000005489 elastic deformation Effects 0.000 description 1
- 239000013013 elastic material Substances 0.000 description 1
- 239000000806 elastomer Substances 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 230000009022 nonlinear effect Effects 0.000 description 1
- 238000007789 sealing Methods 0.000 description 1
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Abstract
The invention provides a two-parameter constitutive model C of hydrogenated nitrile butadiene rubber10、C01The determination method comprises the following steps: step 1, obtaining a superelasticity stress-strain relation of a hydrogenated nitrile rubber material under different working environments through a quasi-static elastic characteristic test of a rubber tensile test piece, and fitting the superelasticity constitutive relation of the hydrogenated nitrile rubber material; step 2, according to C01And C10The law between the ratio and the temperature change is used for obtaining the ratio C of the two-parameter constitutive model of the hydrogenated nitrile rubber01/C10A formula that varies with temperature; step 3, fitting all the elastic modulus data to obtain a formula of the change of the elastic modulus E of the hydrogenated nitrile rubber along with the temperature and the aging time; step 4, solving and determining two-parameter constitutive model parameter C of hydrogenated nitrile butadiene rubber10、C01. The hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01Determining methodThe method avoids the complicated process of determining the parameters by testing or curve fitting.
Description
Technical Field
The invention relates to the technical field of material analysis, in particular to a hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01And determining a method.
Background
The rubber material is a super-elastic material, a model reflecting the stress-strain relationship of the rubber material is called a constitutive model, the research on the constitutive relationship of the rubber material is continuously developed since the 19 th century, and a plurality of constitutive models based on different theories are established, wherein the Mooney-Rivlin model is widely applied due to the fact that the model is simple and the calculation parameters are few.
In recent years, the research of nonlinear constitutive models describing rubber-like superelastic materials has been greatly advanced, and the development of computer technology and finite element analysis software has made it possible to analyze rubber materials using these more accurate constitutive models. Researchers at home and abroad carry out analysis and calculation on the contact pressure and the distribution rule of the packer rubber cylinder and the inner wall of the casing based on ANAS, ABAQUS, MARC and the like, detect and verify whether various indexes of the packer rubber cylinder can meet design requirements or not, and have important significance for optimizing the design of the packer and improving the success rate of field use.
With the development of large nonlinear finite element software and the improvement of the cost performance of a computer, a wide development prospect is provided for the engineering simulation of rubber products. The research on tires, air springs, sealing elements and the like is more and more carried out by utilizing large nonlinear finite element software. However, due to the nonlinearity of the rubber constitutive relation, and the large deformation and contact nonlinear boundary conditions of the rubber product during application, the engineering simulation of the rubber product becomes very difficult. The accuracy of the simulation result is closely related to the simplification degree of the researched problem, the accuracy of the adopted rubber constitutive relation model and the accuracy of the material constant test in the model. Therefore, the method takes a Mooney-Rivlin model commonly used in rubber as an example, and researches are carried out on the determination of constitutive model parameters (Rivlin coefficients) of the hydrogenated nitrile rubber under different environments.
It is difficult to determine the non-linear properties of elastomeric materials, but several constitutive theories based on strain energy density for large elastic deformations have been developed and applied to superelastic materials. These constitutive equations are mainly of two types: the first category considers strain energy density to be a polynomial function of the principal strain invariant. When the material is incompressible, this material model is commonly referred to as a Rivlin material. If only one term is used, the model is called Mooney-Rivlin material. The second category considers the strain energy density as an independent function of the three main elongations. Such as the Ogden, Peng, and Peng-Landel material models.
For rubber-based physical nonlinear materials, the Mooney-Rivlin model is commonly used and described as follows,
in the formula: w is the strain energy density, CijIs the Rivlin coefficient, I1、I2A first Green strain invariant and a second Green strain invariant.
I1＝λ1 2+λ2 2+λ3 2(2)
I2＝(λ1λ2)2+(λ2λ3)2+(λ3λ1)2(3)
If a two-parameter Mooney-Rivlin model is used, equation (1) becomes:
W＝C10(I1-3)+C01(I2-3) (4)
in the formula, C10And C01Are Rivlin coefficients, all of which are positive constants. For most rubbers, reasonable approximations are obtained at strains within 150%.
Mooney-Rivlin model for two parameters, i.e. equation W ═ C10(I1-3)+C01(I2-3), assuming the material is incompressible, I is obtained3＝(λ1λ2λ3)21. In the special case of uniaxial stretching, the stress-strain equation for the Mooney-Rivlin material model can be expressed as:
the equation can be expressed as σ/[2(λ - λ)-2)]Plotting the 1/lambda, expressing the test points in the respective coordinate systems and regressing these test points to a straight line, C01The slope of this line, C10The intercept of this line.
The constitutive model adopted is a two-parameter Mooney-Rivlin model, and for a rubber material, the elastic modulus E and the shear modulus G have the following relationship
The rubber incompressibility gives a Poisson ratio μ of 0.5, so that E is 3G
In the two parameter Mooney-Rivlin model,
G＝2(C10+C01) (7)
E＝6(C10+C01) (8)
in elastomers other thanIn linear finite element analysis, a Mooney-Rivlin strain energy function is a widely applied constitutive relation, the influence of high-temperature aging on parameters of a rubber constitutive model is not considered in the methods for obtaining the parameters in the current domestic and foreign researches, and if the constitutive model parameter values under different temperatures and aging durations are obtained, a series of high-temperature aging tests are required to be carried out for determination. For this purpose, we have invented a new two-parameter constitutive model C of hydrogenated nitrile rubber10、C01The determination method solves the technical problems.
Disclosure of Invention
The invention aims to provide a Mooney-Rivlin two-constitutive model material parameter C of a rubber material under different temperatures and different high-temperature aging durations of hydrogenated nitrile rubber10And C01The method of (1).
The object of the invention can be achieved by the following technical measures: hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01Determination of the two-parameter constitutive model C of the hydrogenated nitrile rubber10、C01The determination method comprises the following steps: step 1, obtaining a superelasticity stress-strain relation of a hydrogenated nitrile rubber material under different working environments through a quasi-static elastic characteristic test of a rubber tensile test piece, and fitting the superelasticity constitutive relation of the hydrogenated nitrile rubber material; step 2, according to C01And C10The law between the ratio and the temperature change is used for obtaining the ratio C of the two-parameter constitutive model of the hydrogenated nitrile rubber01/C10A formula that varies with temperature; step 3, fitting all the elastic modulus data to obtain a formula of the change of the elastic modulus E of the hydrogenated nitrile rubber along with the temperature and the aging time; step 4, solving and determining two-parameter constitutive model parameter C of hydrogenated nitrile butadiene rubber10、C01。
The object of the invention can also be achieved by the following technical measures:
in step 1, obtaining the superelasticity stress-strain relationship of the hydrogenated nitrile rubber material under different working environments through a quasi-static elastic characteristic test of a rubber tensile test piece: preparing a dumbbell-shaped test piece by adopting the rubber material with the same batch in the structure of the target rubber material; carrying out quasi-static loading on a hydraulic servo experiment table, recording the loaded load and the deformation of a test piece, and measuring stress values sigma under different elongation ratios lambda; the tensile stress-elongation ratio relationship of the rubber material of 100-200% was obtained by the tensile test of the dumbbell-shaped test piece.
In step 1, the stress-elongation ratio test data of 100-:
wherein sigma is stress, lambda is elongation ratio, lambda is 1+ epsilon, and epsilon is strain of the material; expressing the test points in the respective coordinate systems and regressing the test points to a straight line, C01The slope of this line, C10For the intercept of the straight line, the rubber material C under different working environments is obtained10And C01The specific numerical values of (a); characteristic parameter C10And C01Is the Mooney-Rivlin superelasticity material model parameter required to be input in quasi-static finite element analysis.
In step 2, the test data of step 1 are analyzed through research, and the Mooney-Rivlin constitutive model parameter C of the hydrogenated nitrile rubber is found10And C01The rule between the ratio and the temperature change is that C under different working environments10And C01Obtaining a two-parameter constitutive model ratio C of the hydrogenated nitrile rubber by adopting multivariate linear fitting to the ratio data01/C10Formula as a function of temperature:
Y＝0.66-7.3×10-3T+3.8×10-5T2(9)
in the formula: Y-C01/C10(ii) a T-temperature (. degree.C.).
In step 3, obtaining the elastic modulus of the hydrogenated nitrile butadiene rubber under different working environments through the quasi-static elastic property test of the rubber tensile test piece in step 1, and fitting all the obtained elastic modulus data by MATLAB data processing software by adopting nlifit nonlinear parameter function to obtain a formula of the change of the elastic modulus of the hydrogenated nitrile butadiene rubber along with temperature and aging time:
E(T,L)＝11.68×(T-0.125)×(L0.197) (10)
in the formula: e-modulus of elasticity (MPa); t-temperature (. degree. C.); l-aging duration (h).
In step 4, calculating to obtain C under different temperature environments by using a fitting formula (9)10And C01The ratio of (A) to (B); calculating the elastic modulus of the hydrogenated nitrile rubber under the environments with different temperatures and different aging durations by using a formula (10); finally combining the formula E to 6 (C)10+C01) Solving for determining C10And C01。
Hydrogenated nitrile rubber two-parameter constitutive model C in the invention10、C01The determination method summarizes the Mooney-Rivlin constitutive model parameter C of the hydrogenated nitrile rubber through research and analysis on the basis of a large number of tests on different temperatures and different aging durations10And C01The law with the temperature change is that C of the hydrogenated nitrile rubber at different temperatures is calculated according to a fitting formula10And C01Determining the elastic modulus of the hydrogenated nitrile rubber material under different temperatures and aging durations, and determining a Mooney-Rivlin two-parameter constitutive model C of the hydrogenated nitrile rubber at a certain temperature and aging duration through a fitting formula10And C01The complicated process of determining the parameters by testing or curve fitting is avoided.
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FIG. 1 shows a two-parameter constitutive model C of hydrogenated nitrile rubber according to the present invention10、C01A flow chart of one embodiment of a method of determining.
Detailed Description
In order to make the aforementioned and other objects, features and advantages of the present invention comprehensible, preferred embodiments accompanied with figures are described in detail below.
As shown in FIG. 1, FIG. 1 shows a two-parameter constitutive model C of hydrogenated nitrile rubber of the present invention10、C01A flow chart of the method is determined.
First, fitting the superelasticity constitutive relation of hydrogenated nitrile rubber material
(1) Firstly, acquiring the superelasticity stress-strain relation of a hydrogenated nitrile rubber material under different working environments through a quasi-static elastic characteristic test of a rubber tensile test piece: the dumbbell-shaped test piece is prepared by adopting the rubber material with the same batch as that in the target rubber material structure. And (3) carrying out quasi-static loading on a hydraulic servo experiment table, recording the loaded load and the deformation of the test piece, and measuring stress values sigma under different elongation ratios lambda. Obtaining the relation of 100-200% of tensile stress-elongation ratio of the rubber material through a tensile test of a dumbbell-shaped test piece;
(2) from the measured stress σ at different elongation ratios λ, the 100-200% stress-elongation ratio test data were fit to the following form using multiple linear regression by ORIGIN (function mapping software)
Where σ is the stress, λ is the elongation ratio, λ ═ 1+ ε, and ε is the strain of the material. Expressing the test points in the respective coordinate systems and regressing the test points to a straight line, C01The slope of this line, C10For the intercept of the straight line, the rubber material can be obtained under different working environments C10And C01The specific numerical value of (1). Characteristic parameter C10And C01Is the Mooney-Rivlin superelasticity material model parameter required to be input in quasi-static finite element analysis.
Secondly, obtaining the ratio C of the two-parameter constitutive model of the hydrogenated nitrile butadiene rubber01/C10Formula of variation with temperature
Through research and analysis of the test data of the first step, the parameter C of the Mooney-Rivlin constitutive model of the hydrogenated nitrile rubber is found10And C01The rule between the ratio and the temperature change is that C under different working environments10And C01Obtaining the ratio C of the two-parameter constitutive model of the hydrogenated nitrile rubber by ORIGIN through multivariate linear fitting01/C10Formula as a function of temperature:
Y＝0.66-7.3×10-3T+3.8×10-5T2(9)
in the formula: Y-C01/C10(ii) a T-temperature (. degree.C.).
Thirdly, obtaining a formula of the change of the elastic modulus of the hydrogenated nitrile rubber along with the temperature and the aging time
Through the quasi-static elastic property test of the rubber tensile test piece in the first step, the elastic modulus of the hydrogenated nitrile rubber under different working environments can be obtained, all the obtained elastic modulus data are subjected to nlinlit nonlinear parameter function fitting through MATLAB data processing software, and a formula of the elastic modulus of the hydrogenated nitrile rubber changing along with the temperature and the aging time is obtained:
E(T,L)＝11.68×(T-0.125)×(L0.197) (10)
in the formula: e-modulus of elasticity (MPa); t-temperature (. degree. C.); l-aging duration (h).
Fourthly, a hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01Method determination
Calculating to obtain C under different temperature environments by using a fitting formula (9)10And C01The ratio of (A) to (B); calculating the elastic modulus of the hydrogenated nitrile rubber under the environments with different temperatures and different aging durations by using a formula (10); finally, the formula (8) is combined to solve and determine C10And C01。
The invention aims at solving the Mooney-Rivlin two-constitutive model material parameter C of a rubber material in the range of aging time of 80-240h at the temperature of 25-150 ℃ by using hydrogenated nitrile rubber10And C01If the method exceeds the range, larger errors may exist, and if higher-precision calculation is needed, corresponding tests and researches still need to be carried out to serve as a foundation.
Claims (6)
1. Hydrogenated nitrile butadiene rubber two-parameter constitutive model C10、C01The determination method is characterized in that the hydrogenated nitrile rubber two-parameter constitutive model C10、C01The determination method comprises the following steps:
step 1, obtaining a superelasticity stress-strain relation of a hydrogenated nitrile rubber material under different working environments through a quasi-static elastic characteristic test of a rubber tensile test piece, and fitting the superelasticity constitutive relation of the hydrogenated nitrile rubber material;
step 2, according to C01And C10The law between the ratio and the temperature change is used for obtaining the ratio C of the two-parameter constitutive model of the hydrogenated nitrile rubber01/C10A formula that varies with temperature;
step 3, fitting all the elastic modulus data to obtain a formula of the change of the elastic modulus E of the hydrogenated nitrile rubber along with the temperature and the aging time;
step 4, solving and determining two-parameter constitutive model parameter C of hydrogenated nitrile butadiene rubber10、C01。
2. Hydrogenated nitrile rubber two-parameter constitutive model C according to claim 110、C01The determination method is characterized in that in the step 1, the superelasticity stress-strain relation of the hydrogenated nitrile rubber material under different working environments is obtained through a quasi-static elastic characteristic test of a rubber tensile test piece: preparing a dumbbell-shaped test piece by adopting the rubber material with the same batch in the structure of the target rubber material; carrying out quasi-static loading on a hydraulic servo experiment table, recording the loaded load and the deformation of a test piece, and measuring stress values sigma under different elongation ratios lambda; the tensile stress-elongation ratio relationship of the rubber material of 100-200% was obtained by the tensile test of the dumbbell-shaped test piece.
3. Hydrogenated nitrile rubber two-parameter constitutive model C according to claim 210、C01The determination method is characterized in that in step 1, according to the measured stress sigma under different elongation ratios lambda, the multiple linear regression is adopted to fit the test data of the stress-elongation ratio of 100-200% into the following form:
wherein sigma is stress, lambda is elongation ratio, lambda is 1+ epsilon, and epsilon is strain of the material; expressing the test points in the corresponding coordinatesIn the series, and returning these test points to a straight line, C01The slope of this line, C10For the intercept of the straight line, the rubber material C under different working environments is obtained10And C01The specific numerical values of (a); characteristic parameter C10And C01Is the Mooney-Rivlin superelasticity material model parameter required to be input in quasi-static finite element analysis.
4. Hydrogenated nitrile rubber two-parameter constitutive model C according to claim 110、C01The determination method is characterized in that in step 2, the test data of step 1 are analyzed through research to find the Mooney-Rivlin constitutive model parameter C of the hydrogenated nitrile rubber10And C01The rule between the ratio and the temperature change is that C under different working environments10And C01Obtaining a two-parameter constitutive model ratio C of the hydrogenated nitrile rubber by adopting multivariate linear fitting to the ratio data01/C10Formula as a function of temperature:
Y＝0.66-7.3×10-3T+3.8×10-5T2(9)
in the formula: Y-C01/C10(ii) a T-temperature (. degree. C.).
5. Hydrogenated nitrile rubber two-parameter constitutive model C according to claim 110、C01The determination method is characterized in that in step 3, the elastic modulus of the hydrogenated nitrile butadiene rubber under different working environments is obtained through the quasi-static elastic property test of the rubber tensile test piece in step 1, all the obtained elastic modulus data are fitted by MATLAB data processing software by adopting nlifit nonlinear parameter function, and a formula of the change of the elastic modulus of the hydrogenated nitrile butadiene rubber along with temperature and aging time is obtained:
E(T，L)＝11.68×(T-0.125)×(L0.197) (10)
in the formula: e-modulus of elasticity (MPa); t-temperature (. degree. C.); l-aging duration (h).
6. Hydrogenated nitrile rubber according to claim 1, having two parametersConstitutive model C10、C01The determination method is characterized in that in step 4, C under different temperature environments is calculated by using a fitting formula (9)10And C01The ratio of (A) to (B); calculating the elastic modulus of the hydrogenated nitrile rubber under the environments with different temperatures and different aging durations by using a formula (10); finally combining the formula E to 6 (C)10+C01) Solving for determining C10And C01。
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Application publication date: 20200501 |