CN106815442B - Method for constructing isotropic incompressible superelastic body constitutive model and application thereof - Google Patents

Method for constructing isotropic incompressible superelastic body constitutive model and application thereof Download PDF

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CN106815442B
CN106815442B CN201710045691.0A CN201710045691A CN106815442B CN 106815442 B CN106815442 B CN 106815442B CN 201710045691 A CN201710045691 A CN 201710045691A CN 106815442 B CN106815442 B CN 106815442B
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魏志刚
王孝义
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Anhui University of Technology AHUT
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Abstract

The invention discloses a method for constructing an isotropic incompressible super elastomer constitutive model and application thereof, belonging to the technical field of solid mechanics, computational mechanics and experimental mechanics. The constitutive model provided by the invention can obtain a more accurate model of the rubber incompressible superelasticity material only by adopting a single type of experimental data, has higher precision and reliability than the existing models at present, can obtain a more accurate and comprehensive rubber material characteristic model only by carrying out simple uniaxial tensile experimental tests, does not need to adopt equal biaxial tensile and plane tensile tests which are difficult to carry out at present in China, and has extremely high engineering application value and computational mechanics application prospect.

Description

Method for constructing isotropic incompressible superelastic body constitutive model and application thereof
Technical Field
The invention relates to the description and modeling of the mechanical properties of various rubber materials and biological tissue materials in engineering and scientific research, and engineering and scientific application on the basis of the description and modeling, in particular to the related fields of computational mechanics and experimental mechanics.
Background
The mechanical properties of materials such as rubber, muscle, ligament and the like can be described by using an incompressible superelasticity model, and the establishment of a constitutive model capable of completely describing the mechanics of the materials is a significant problem in engineering and scientific research. Although a strain-based invariant model, a model based on three main elongations, and a molecular chain-based model have been proposed at present, the above rubber and biological tissue materials lack a complete and concise model, which is specifically shown in the following: or the description capability of the existing model is not enough, and the mechanical properties of the material in various deformation states cannot be completely described; or the model is too complex to be used practically.
In addition, generally, at least three types of experiments of uniaxial stretching, plane stretching and equibiaxial stretching are needed for establishing a model of the rubber incompressible superelasticity material, but only the first experiment, namely the uniaxial stretching experiment, is mature in practice and theory and is close to an ideal state at present, certain defects exist in the latter two experiments theoretically, the experiments are very difficult due to the limitation of experimental equipment and conditions, the two experiments can be better performed without laboratories in China, the experiments can be performed by depending on overseas AXEL professional laboratories in North America basically, and the experiment cost and the expense are very high.
With the development of industry, the requirements on mechanical property models of rubber incompressible superelasticity materials are higher and higher, and through search, the invention has the following inventive names in Chinese patent application No. 201610303626.9, application date of 2016, 5 and 10 days: a modeling method of a superelasticity constitutive model of a rubber material; the Chinese patent application No. 201610519914.8, the application date is 2016, 7, 4, the name of the invention is: a modeling method of a rubber material viscous superelasticity constitutive model considering correlation effects; the above applications all propose a method for constructing a material constitutive model (constitutive model parameter identification method), do not propose the constitutive model itself, and currently, there is no published constitutive relation application in China.
Disclosure of Invention
1. Technical problem to be solved by the invention
In order to solve the problems in the prior art, the invention provides a method for constructing an isotropic incompressible super-elastomer constitutive model and application thereof.
2. Technical scheme
In order to achieve the purpose, the technical scheme provided by the invention is as follows:
the invention relates to a method for constructing an isotropic incompressible super-elastomer constitutive model, which is a relation model of a stress-elongation function and a plane tensile stress-elongation function in any plane deformation state, and the method specifically comprises the following steps:
Figure GDA0002257716080000021
in the formula, TiIndicating stress conditions (λ) in any three axesi,λj,λk) Lower lambdaiA nominal stress in the direction; t isi(λi,λj) Indicates a state of stress (λ) in the planei,λj,1/λiλj) Lower lambdaiA nominal stress in the direction; lambda [ alpha ]i,λj,λkThe main elongations in three stress directions are respectively, and subscripts i, j and k are the combined arrangement of numbers 1, 2 and 3; k is a parameter of the material to be identified, k being at λjWhen the value is more than 1 or less than 1, two different values can be taken; t isplanarExpressed as a function of the nominal stress at an elongation of lambda measured in a mechanical property test of a plane tensile material.
Further, in a triaxial stress state, the constitutive model is:
Figure GDA0002257716080000022
where p is the hydrostatic pressure, the boundary conditions of the bonding material need to be determined.
Further, in the uniaxial tensile experiment, the nominal stress function T of uniaxial tensionuniaxial(lambda) and plane tensile stress function Tplanar(λ) is transformed with the following relation:
Figure GDA0002257716080000023
further, in the equibiaxial stretching experiment, the nominal stress function T of equibiaxial stretchingbiaxial(lambda) and the plane tensile stress function Tplanar(λ) is transformed with the following relation:
Figure GDA0002257716080000024
in the formula, Tbiaxial(λ) is the name of the elongation at λ in the equibiaxial stretching testThe stress function is defined, and g is a material constant.
The application of the method for constructing the isotropic incompressible super elastomer constitutive model disclosed by the invention is to use any experimental data, fit material parameters and establish the isotropic incompressible super elastomer constitutive model by means of the model for engineering calculation, numerical simulation or material experiments.
3. Advantageous effects
Compared with the prior art, the technical scheme provided by the invention has the following remarkable effects:
(1) the invention takes two main elongations as model variables, establishes the relationship between various experimental type data, and constructs a brand new constitutive model on the basis, wherein the model is an incompressible superelasticity model, can be directly used for isotropic incompressible superelasticity materials, can also be used as a basic model to be embedded into a viscous superelasticity model for use, can describe the anisotropy of the materials, and has higher reliability than the existing model;
(2) according to the method for constructing the isotropic incompressible superelastic body constitutive model, the mechanical response of the rubber material in all deformation states can be reliably predicted by using experimental type experimental data, equal biaxial stretching and plane stretching tests which are difficult to perform at home at present are not needed, the expensive material multi-axis testing cost is saved, the method has extremely high engineering application value and computational mechanics use prospect, and is a new discovery of the constitutive characteristics of the rubber incompressible superelastic material.
Drawings
FIG. 1 is a schematic diagram of the fitting results of the isotropic incompressible superelastic constitutive model according to the present invention;
fig. 2 is a plot of nominal stress-elongation (Treloar) obtained in example 1.
Detailed Description
For a further understanding of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
All types of experimental curves are not well fitted to the existing superelastic constitutive model (i.e., the fitting accuracy is low); or other types of experimental data (such as equibiaxial and plane stretching experimental data) cannot be reliably and accurately predicted from one type of experimental data (such as uniaxial stretching experimental data only), the invention provides a novel isotropic incompressible super elastomer constitutive model and a parameter identification method thereof.
Example 1
The embodiment provides a relation model of a stress-elongation function and a plane tensile stress-elongation function of any plane deformation state, and through the relation model, the constitutive relation of the rubber incompressible superelasticity material can be directly established and model parameters can be identified, and the relation model is specifically in the following form:
Figure GDA0002257716080000031
in the formula, TiIndicating stress conditions (λ) in any three axesi,λj,λk) Lower lambdaiA nominal stress in the direction; t isi(λi,λj) Indicates a state of stress (λ) in the planei,λj,1/λiλj) Lower lambdaiNominal stress in direction (force/initial area); lambda [ alpha ]i,λj,λkMain elongation (lambda) for three stress directionsi,λjFirst and second principal elongations in a plane stress state), subscripts i, j, k being an arrangement of numbers 1, 2, 3; k is a material parameter to be identified, which can be made equal to 0 in the general case, at an elongation λjThe same value or different values can be adopted when the value is more than 1 and less than 1; t isplanarExpressed as a function of the nominal stress at an elongation of lambda measured in a mechanical property test of a plane tensile material.
Equation (1) is the most general form of the constitutive model proposed in this example, and describes the constitutive relation of the rubber-like incompressible superelastic material under the plane stress state. The model uses a plane tensile stress function TplanarFor basis functions, the function may take any suitable form of expression, including polynomial functions, fingersA number function, a power function, and the like. For a more general triaxial stress state, the constitutive relation is:
Figure GDA0002257716080000041
where p is the hydrostatic pressure and requires the boundary conditions of the bonding material to be determined.
And in the uniaxial tensile experiment, the nominal stress function T of uniaxial tensionuniaxial(lambda) and plane tensile stress function Tplanar(λ) is transformed with the following relation:
Figure GDA0002257716080000042
in the equibiaxial stretching experiment, the stress relationship between the nominal stress and the plane stretching of equibiaxial stretching is as follows:
Figure GDA0002257716080000043
in the formula, Tbiaxial(λ) is the nominal stress function at an elongation of λ in an equibiaxial tensile test, and g is a material constant.
The constitutive model provided by the embodiment can independently use any experimental data (uniaxial tension, plane tension or equal biaxial tension) to fit parameters, and then the model is used for deducing stress response in any deformation state, so that the method has the characteristics of high model precision and good reliability. The fitting of the experimental data shows that the stress of the material in other deformation states can be accurately obtained only by adopting one type of experimental data such as uniaxial tensile experimental data. The prediction accuracy of the constitutive model provided in this embodiment is higher than that of the known model, the model fitting result is shown in fig. 1, fig. 1 is the fitting result of the model using only one kind of tensile experimental data, such as uniaxial tensile data, and the fitting results of other models are shown in the literature (hui-xiao, liu xiu, lising, roman wave, selection strategy of carbon black filled rubber superelasticity constitutive model, engineering mechanics, 2014,31 (5): 34-42.).
The embodiment can obtain the more accurate rubber incompressible superelasticity material model only by adopting single type of experimental data, has higher precision and reliability than the existing model at present, can obtain the more accurate and comprehensive rubber incompressible superelasticity material characteristic model only by carrying out simple uniaxial tensile experimental tests, does not need to adopt equal biaxial tensile and plane tensile tests which are difficult to carry out at home at present, and has extremely high engineering application value and computational mechanics application prospect.
The specific process of applying the constitutive model of this embodiment is as follows:
1) performing one type of material experiment, such as uniaxial tensile experiment or planar tensile experiment or equibiaxial tensile experiment, to obtain a nominal stress-elongation curve during the experiment, such as the uniaxial tensile curve and the planar tensile curve shown in fig. 2;
2) fitting the experimentally obtained nominal stress curve using any feasible functional form to obtain the nominal stress function expression T with elongation as a variable for uniaxial, planar or equibiaxial stretchinguniaxial(lambda) or Tplanar(lambda) or Tbiaxial(λ)。
3) If a non-planar tensile test is performed, a planar tensile stress function is obtained according to equation (3) or equation (4) (where k, g is 0), assuming that the obtained planar tensile stress function is:
Figure GDA0002257716080000051
Figure GDA0002257716080000052
Figure GDA0002257716080000053
wherein a isiThe material parameters are obtained by fitting a nominal stress-strain curve obtained through experiments.
4) Then using formula (1) to pair TplanarSubstituting the (lambda) function to obtain Ti(λi,λj)-(λ3/λi)T3. Finally, substituting into equation (1) (let k be 0 in equation (1)) can obtain the relationship of material stress-strain under triaxial stress, i.e., the constitutive function.
Figure GDA0002257716080000054
Figure GDA0002257716080000055
Or directly fitting other material constitutive models such as an Ogden model and the like by using the three function curves (namely the formula (5), the formula (6) and the formula (7)) obtained in the step 3).
Example 2
The incompressible super elastomer constitutive model of this embodiment is substantially the same as embodiment 1, and the specific process of using the constitutive model of this embodiment to perform parameter identification is as follows:
1) all three types of material experiments were performed to obtain nominal stress-elongation curves during the experiment, such as uniaxial tensile curve and planar tensile curve shown in fig. 2;
2) the substitution formula (1) model directly fits all parameters;
3) the complete constitutive model of the material is obtained by the formulas (1) and (2), and the complete constitutive model can be particularly used as a super-elastic model part of a visco-super-elastic model and the like.
Example 3
In this embodiment, the models of formula (1), formula (2), formula (3) and formula (4) are embedded into analysis software such as finite element or other material mechanical property analysis and calculation software:
1) construction of T using polynomials or other functional formsuniaxial(lambda) or Tplanar(lambda) or Tbiaxial(λ);
2) fitting parameters of the model based on corresponding experimental data;
3) the material constitutive relation similar to the formula (1) is constructed by using the models of the formula (1), the formula (2), the formula (3) and the formula (4).
4) Numerical simulation, stress analysis and the like are performed by using the constitutive relation.
The present invention and its embodiments have been described above schematically, without limitation, and what is shown in the drawings is only one of the embodiments of the present invention and is not actually limited thereto. The model is an incompressible superelastic model, and can be used directly in an isotropic incompressible superelastic material, or can be used as a basic model to be embedded in a visco-superelastic model, and any of the above applications is within the scope of this patent.

Claims (5)

1. A method for constructing an isotropic incompressible superelastic constitutive model is characterized by comprising the following steps: the constitutive model is a relation model of a stress-elongation function and a plane tensile stress-elongation function in any plane deformation state, and is specifically in the following form:
Figure FDA0002436030580000011
in the formula, TiIndicating stress state λ in any three axesi，λj，λkLower lambdaiA nominal stress in the direction; t isi(λi,λj) Indicating a state of stress in plane λi，λj，1/λiλjLower lambdaiA nominal stress in the direction; lambda [ alpha ]i,λj,λkThe main elongations in three stress directions are respectively, and subscripts i, j and k are the combined arrangement of numbers 1, 2 and 3; m is a parameter of the material to be identified, m being at λjWhen the value is more than 1 or less than 1, two different values can be taken; t isplanarExpressed as a planeNominal stress function at elongation λ measured in tensile material mechanical property tests.
2. The method of constructing an isotropic incompressible superelastic constitutive model according to claim 1, wherein: under the triaxial stress state, the constitutive model is as follows:
Figure FDA0002436030580000012
where p is the hydrostatic pressure, the boundary conditions of the bonding material need to be determined.
3. The method of constructing an isotropic incompressible superelastic constitutive model according to claim 2, wherein: in uniaxial tensile experiments, the nominal stress function T of uniaxial tensionuniaxial(lambda) and plane tensile stress function Tplanar(λ) is transformed with the following relation:
Figure FDA0002436030580000013
4. a method of constructing an isotropic incompressible superelastic constitutive model according to claim 3, wherein: nominal stress function T of equibiaxial stretching in equibiaxial stretching experimentsbiaxial(lambda) and the plane tensile stress function Tplanar(λ) is transformed with the following relation:
Figure FDA0002436030580000014
in the formula, Tbiaxial(λ) is the nominal stress function at an elongation of λ in an equibiaxial tensile test, and g is a material constant.
5. Use of the isotropic incompressible superelastic constitutive model according to claim 4, wherein: and (3) fitting material parameters by using any experimental data, and establishing an isotropic incompressible super elastomer constitutive model by means of the model for engineering calculation, numerical simulation or material experiment.
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