CN107766659A - It is a kind of suitable for the elastic constitutive model model of rubber type of material and its application - Google Patents
It is a kind of suitable for the elastic constitutive model model of rubber type of material and its application Download PDFInfo
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- G—PHYSICS
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- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
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- G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
- G01N33/00—Investigating or analysing materials by specific methods not covered by groups G01N1/00 - G01N31/00
- G01N33/44—Resins; rubber; leather
- G01N33/445—Rubber
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- G—PHYSICS
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- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
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Abstract
The invention discloses a kind of suitable for the elastic constitutive model model of rubber type of material and its application, belong to Solid Mechanics, Computational Mechanics and Experimental Mechanics technical field.Constitutive model proposed by the present invention can just obtain the model of accurate rubber-like imcompressible hyperelastic material only with single type experimental data, than currently model accuracy and reliability are all high, and it can only carry out simple uniaxial tensile test test and obtain more accurate, comprehensive elastomeric material characteristic model, need not use it is current domestic still be difficult to etc. biaxial stretch-formed and plane extension test, there is high engineering application value and Computational Mechanics prospect of the application.
Description
Technical field
The present invention relates to the mechanical property of each rubber-like materials, biological organization material is retouched in engineering and scientific research
State and model, and the association area of engineering and scientific application, particularly Computational Mechanics and Experimental Mechanics on this basis.
Background technology
The materials such as rubber, muscle, ligament can describe its mechanical characteristic with incompressible hyperelastic model, establish
One constitutive model that can completely describe such mechanics of materials is engineering and the significant problem of scientific research.Although currently carry
Out the invariant model based on strain, the model based on three principal elongation rates, and the model based on strand, but above-mentioned rubber
One glue, biological tissue's class material shortage complete and succinct model, are in particular in：Or the descriptive power of existing model
Not enough, it is impossible to mechanical characteristic of the material under various modifications state is fully described；Or model is excessively complicated, in the use of reality
In it is relatively difficult.Obtain one can most of accurate description such material mechanical characteristic, while model is again with sufficiently high
Reliability, using less data, the characteristic of material can be relatively accurately predicted if uniaxial tension data, is engineering and section
An active demand in grinding.
It is on May 10th, 2016 through retrieval, Chinese Patent Application No. 201610303626.9, the applying date, innovation and creation name
Referred to as：The super-elasticity constitutive model modeling method of elastomeric material；Chinese Patent Application No. 201610519914.8, the applying date be
On July 4th, 2016, invention and created name are：A kind of elastomeric material for considering correlation effect glues the modeling side of superlastic constitutive model
Method；Above-mentioned application case proposes the method (Parameters of constitutive model recognition methods) of structure material constitutive model, does not suggest that this
Structure model is in itself.Chinese Patent Application No. 201710045691.0, patent name：A kind of incompressible hyperelastic body of isotropism
Constitutive model and its application；It is that applicant discloses one kind with a patent of invention, this application case filed in 22 days January in 2017
The constitutive model of the incompressible hyperelastic body of isotropism, the constitutive model disclosed in this application case have good application power
And reliability, but model parameter is more, and upper relatively inconvenient in application, the claimed constitutive model of present patent application is then a kind of
New constitutive model, using less parameter, reach the precision and reliability similar with the model.
The content of the invention
1. invention technical problems to be solved
In order to solve the above-mentioned problems of the prior art, the invention provides a kind of elasticity suitable for rubber type of material
Constitutive model and its application, the present invention are a kind of new constitutive models, using minimum or less parameter, have prediction well
Ability and best reliability.
2. technical scheme
To reach above-mentioned purpose, technical scheme provided by the invention is：
A kind of incompressible hyperelastic body constitutive model of isotropism of the present invention, it is answering for arbitrary plane deformation state
The relational model of power-elongation function and planar stretch stress-elongation function, specially following form：
In formula, TiRepresent in any triaxial state of stress (λi,λj,λk) under λiDirection is (in the larger both direction of elongation
Any one direction) nominal stress；λi,λjFor that two principal elongation rate that numerical value in three principal elongation rates is larger；Tplanar
The nominal stress function that planar stretch experiment determines is expressed as, the function representation is principal elongation rate equidirectional with stress and letter
Number.
Further, under triaxial state of stress, the constitutive model is：
In formula, p is hydrostatic pressure, it is necessary to which the boundary condition of bond material determines.
Further, in uniaxial tensile test, uniaxial tension nominal stress function Tuniaxial(λ) should with planar stretch
Force function Tplanar(λ) is changed with following relation：
Further, in biaxial stretch-formed experiment is waited, biaxial stretch-formed nominal stress function T is waitedbiaxial(λ) and plane
Tensile stress function Tplanar(λ) is changed with following relation：
In formula, Tbiaxial(λ) the nominal stress function in biaxial stretch-formed experiment when elongation is λ such as is, k and g are material
Expect constant, particularly, for most of rubber type of material, two parameters can be taken as constant, the constitutive model now formed
Only three parameters.
Further, it is proposed that TuniaxialA kind of new form of (λ) function, greatly reduce the parameter of model：
Tplanar(λ)=a λ4+bλ1+cλ-2-a-b-c (5)
In formula, a, b, c represent model constants to be determined respectively.
A kind of application of elastic constitutive model model suitable for rubber type of material of the present invention, number is tested using any one
According to, be fitted material parameter, by above-mentioned model establish the incompressible hyperelastic body constitutive model of isotropism be used for engineering calculation,
Numerical simulation or material experiment.
3. beneficial effect
Using technical scheme provided by the invention, compared with existing known technology, there is following remarkable result：
(1) a kind of a kind of elastic constitutive model model suitable for rubber type of material of the invention, it is proposed that new planar stretch
Stress function form, there is less parameter, but there is similar precision and reliability.
(2) a kind of a kind of elastic constitutive model model suitable for rubber type of material of the invention, it is proposed that new single shaft model
It is more convenient in engineer applied to multiaxis model conversation form, the form.
(3) a kind of elastic constitutive model model suitable for rubber type of material of the invention, improve what is formed by above-mentioned two aspects
New model, it can be reached in existing model preferably or similar to precision of prediction but with more using less or even minimum (3)
Good reliability.
Brief description of the drawings
Fig. 1 is models fitting (Treloar, 1944) datagram using 5 parameters；
Fig. 2 is that (Kawabata et al., 1981) datagram is fitted using 5 parameter models；
Fig. 3 is using 3 parameter (k=0.25；G=0.4 model) is only with the result of planar stretch data fitting prediction
With (Treloar, 1944) data comparison figure；
Fig. 4 is using 3 parameter (k=0.25；G=0.4 model) is only with the result of uniaxial tension data fitting prediction
With (Treloar, 1944) data comparison figure.
Embodiment
To further appreciate that present disclosure, the present invention is described in detail in conjunction with the accompanying drawings and embodiments.
For existing super-elasticity constitutive model or it can not be fitted all types of empirical curves simultaneously well and (intend
It is low to close precision)；Or can not be reliably by a type experimental data (for example only using uniaxial tensile test data) Accurate Prediction
Other types of experimental data (such as twin shaft and plane stretching experiment data), the problem of model reliability is low, the present invention propose
A kind of elastic constitutive model model of new rubber type of material.
Embodiment 1
The present embodiment proposes the stress-elongation function and planar stretch stress-elongation of arbitrary plane deformation state
The relational model of function, by the relational model, it can directly establish rubber-like imcompressible hyperelastic material constitutive relation and know
Other model parameter, specially following form：
In formula, TiRepresent in any triaxial state of stress (λi,λj,λk) under λiDirection is (in the larger both direction of elongation
Any one direction) nominal stress；λi,λjFor that two principal elongation rate that numerical value in three principal elongation rates is larger；Tplanar
The nominal stress function that planar stretch experiment determines is expressed as, the function representation is principal elongation rate equidirectional with stress and letter
Number.
Formula (1) is the most typically form for the constitutive model that the present embodiment proposes, describes the rubber under plane stress state
Glue class imcompressible hyperelastic material constitutive relation.The model is with planar stretch stress function TplanarFor basic function, the function can
To take any suitable expression-form, including polynomial function, exponential function, power function etc..For more generally three axle
Stress state, the constitutive relation are：
In formula, p is hydrostatic pressure, it is necessary to which the boundary condition of bond material just can determine that.
And in uniaxial tensile test, uniaxial tension nominal stress function Tuniaxial(λ) and planar stretch stress function
Tplanar(λ) is changed with following relation：
In biaxial stretch-formed experiment is waited, the stress relation for waiting biaxial stretch-formed nominal stress and planar stretch is：
In formula, Tbiaxial(λ) the nominal stress function in biaxial stretch-formed experiment when elongation is λ such as is, k and g are material
Expect constant.
The present embodiment propose constitutive model can individually with any experimental data (uniaxial tension, planar stretch or
Etc. biaxial stretch-formed) fitting parameter, the stress response under random variation state then is released using the model, there is model accuracy
The characteristics of height, good reliability, because the present embodiment is proposed including uniaxial tensile test, planar stretch experiment and is waited biaxial stretch-formed
The model that stress function is mutually changed in any biaxial stretch-formed experimental state including these three experimental states is tested, as long as obtaining
A kind of experimental data, it can deduce that the material stress under other experimental state responds using the model, it is not necessary to as existing method
The middle material mechanics experiment for needing to carry out above-mentioned three types respectively, then simultaneously with these three experimental datas come model of fit.
Shown by the fitting to experimental data, only with a kind of experimental data such as uniaxial tensile test data, just can accurately obtained
Go out stress of the material under other deformation states.And the precision of prediction of the constitutive model of the present embodiment proposition is higher than known models,
Models fitting result is as Figure 1-Figure 4.The fitting effect of existing model can be found in (Steinmann P, Hossain M,
Possart G.Hyperelastic modelsfor rubber-like materials:Consistent tangent
operators and suitability for Treloar’s data.Arch.Appl.Mech.,2012,82(9):1183-
1217.)。
The present embodiment just can obtain the incompressible super-elasticity material of accurate rubber-like only with single type experimental data
Expect model, than currently model accuracy and reliability are all high, can only carry out simple uniaxial tensile test test and obtain
More accurate, comprehensive rubber-like imcompressible hyperelastic material characteristic model, it is not necessary to what the current country of use was still difficult to etc.
Biaxial stretch-formed and plane extension test, has high engineering application value and Computational Mechanics prospect of the application.
Detailed process with the constitutive model of the present embodiment is：
1) a type of material experiment, such as uniaxial tensile test either planar stretch experiment or biaxial stretch-formed are carried out
Experiment, nominal stress-extension curve in experimentation is obtained, as shown in fig. 1 uniaxial tension curve and plane stress strain curve；
2) model parameter is obtained using formula (3), formula (4) and the fitting of formula (5) one or both of which.
3) fitting parameter is substituted into remaining formula, obtains other kinds of empirical curve；
4) curve obtained with experiment and two curves being calculated are used to be fitted an existing constitutive relation
Model or for project analysis.
Embodiment 2
With embodiment 1, parameter identification is carried out with the constitutive model of the present embodiment substantially for the constitutive model of the present embodiment
Detailed process is：
1) material experiment for carrying out all three types obtains nominal stress-extension curve in experimentation, in Fig. 3
Shown uniaxial tension curve and plane stress strain curve；
2) substitute into formula (1) model and be directly fitted all parameters；
3) the complete constitutive model of material is obtained by formula (1) and formula (2), particularly can be as models such as viscous-super-elasticity
Hyperelastic model part.
Embodiment 3
The present embodiment by the model insertion of formula (1), formula (2), formula (3) and formula (4) into analysis softwares such as finite elements, or
In other materials mechanics property analysis software for calculation：
2) corresponding experimental data is based on, is fitted the parameter of the model；
3) material constitutive relation of similar formula (1) is gone out with the Construction of A Model of formula (4) using formula (1), formula (2), formula (3).
4) numerical simulation and stress analysis etc. are carried out using the constitutive relation.
Embodiment 4
Canonical parameter using Model Parameter k and g as commercial Application, the parameter is determined by experiment and combines any single shaft
Test data, for describing the mechanical property of such material.
Schematically the present invention and embodiments thereof are described above, this describes no restricted, institute in accompanying drawing
What is shown is also one of embodiments of the present invention, is actually not limited thereto.The model is a kind of incompressible super-elasticity mould
Type, isotropism imcompressible hyperelastic material is can be directly used for, viscous hyperelastic model can also be embedded into as basic model
Middle use, any of the above described application are included in the protection domain of this patent.
Claims (5)
- A kind of 1. elastic constitutive model model suitable for rubber type of material, it is characterised in that：The constitutive model deforms for arbitrary plane The relational model of the stress of state-elongation function and planar stretch stress-elongation function, specially following form：<mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>T</mi> <mn>3</mn> </msub> <mo>=</mo> <mi>&phi;</mi> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&lambda;</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>&lambda;</mi> <mi>i</mi> </msub> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mn>1</mn> <mo>/</mo> <msqrt> <msub> <mi>&lambda;</mi> <mi>j</mi> </msub> </msqrt> <mo>+</mo> <mi>k</mi> <mo>|</mo> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&lambda;</mi> <mi>j</mi> </msub> <mi>g</mi> </msup> <mo>|</mo> <mo>(</mo> <mrow> <msub> <mi>&lambda;</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>1</mn> <mo>/</mo> <msqrt> <msub> <mi>&lambda;</mi> <mi>j</mi> </msub> </msqrt> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>In formula, TiRepresent in any triaxial state of stress (λi,λj,λk) under λiThe nominal stress in direction；λi,λjFor three principal elongations That larger two principal elongation rate of numerical value in rate；TplanarIt is expressed as the nominal stress function that planar stretch experiment determines, the letter Number is expressed as principal elongation rate equidirectional with stress and function, and k and g are expressed as a material constant.
- A kind of 2. elastic constitutive model model suitable for rubber type of material according to claim 1, it is characterised in that：In single shaft In stretching experiment, uniaxial tension nominal stress function Tuniaxial(λ) and planar stretch stress function Tplanar(λ) is with following relation Conversion：<mrow> <msub> <mi>T</mi> <mrow> <mi>u</mi> <mi>n</mi> <mi>i</mi> <mi>a</mi> <mi>x</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&lambda;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&lambda;</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mroot> <mi>&lambda;</mi> <mn>4</mn> </mroot> <mo>+</mo> <mi>k</mi> <mo>|</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&lambda;</mi> <mrow> <mo>-</mo> <mn>0.5</mn> <mi>g</mi> </mrow> </msup> <mo>|</mo> <mo>(</mo> <mrow> <mi>&lambda;</mi> <mo>-</mo> <mroot> <mi>&lambda;</mi> <mn>4</mn> </mroot> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>In formula, Tuniaxial(λ) is the nominal stress function when elongation is λ in uniaxial tensile test, and k and g are a material Constant.
- A kind of 3. elastic constitutive model model suitable for rubber type of material according to claim 2, it is characterised in that：It is double waiting In axle stretching experiment, biaxial stretch-formed nominal stress function T is waitedbiaxial(λ) and plane tensile stress function Tplanar(λ) to Lower relation conversion：<mrow> <msub> <mi>T</mi> <mrow> <mi>b</mi> <mi>i</mi> <mi>a</mi> <mi>x</mi> <mi>i</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&lambda;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mi>l</mi> <mi>a</mi> <mi>n</mi> <mi>a</mi> <mi>r</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&lambda;</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mi>&lambda;</mi> </msqrt> <mo>+</mo> <mi>k</mi> <mo>|</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&lambda;</mi> <mi>g</mi> </msup> <mo>|</mo> <mo>(</mo> <mrow> <mi>&lambda;</mi> <mo>-</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mi>&lambda;</mi> </msqrt> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>In formula, Tbiaxial(λ) the nominal stress function in biaxial stretch-formed experiment when elongation is λ such as is, k and g are a material Constant.
- A kind of 4. elastic constitutive model model suitable for rubber type of material according to claim 3, it is characterised in that：Described Tplanar(λ) has following form：Tplanar(λ)=a λ4+bλ1+cλ-2-a-b-c (5)In formula, a, b, c are model constants to be determined.
- A kind of 5. application of the elastic constitutive model model as claimed in claim 4 suitable for rubber type of material, it is characterised in that：Make With any experimental data, material parameter is fitted, incompressible this structure of hyperelastic body of isotropism is established by above-mentioned model Model is used for engineering calculation, numerical simulation or material experiment.
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