CN117497069B - Construction method and device of super-elastic constitutive model of high polymer material - Google Patents
Construction method and device of super-elastic constitutive model of high polymer material Download PDFInfo
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Abstract
The invention discloses a method and a device for constructing a super-elastic constitutive model of a high polymer material, belonging to the technical field of material mechanics, wherein the method comprises the following steps: the constitutive parameters of the high polymer material with practical physical significance are utilized: calculating deformation free energy caused by elongation of the cross-linked chain and free energy caused by elongation of the free chain according to the modulus of the cross-linked molecular chain, the modulus of the free molecular chain, the number of the cross-linked molecular chain Coulomb monomers and the motion hardening coefficient of the free molecular chain; and further obtaining the mapping relation between the Cauchy stress of the high polymer material and Zuo Kexi-Green deformation tensor so as to represent the super-elastic constitutive model of the high polymer material. The invention introduces the free chain motion hardening coefficient, thereby comprehensively considering the influence of the cross-linked chain and the free chain elongation on the mechanical property of the high polymer material. The free chain motion hardening coefficient is introduced to enable the calculated free energy to be closer to the actual situation. The mechanical property of the high polymer material can be predicted by the constitutive parameters which can be measured through experiments.
Description
Technical Field
The invention belongs to the technical field of material mechanics, and particularly relates to a method and a device for constructing a super-elastic constitutive model of a high polymer material.
Background
The high molecular polymer material refers to a high molecular weight compound formed by connecting repeated structural units through covalent bonds, and is short for high polymer. An amorphous polymer is one that has an insufficiently regular structure, and lacks or has no ability to crystallize. Common amorphous polymers include natural or artificial rubber materials, hydrogel materials, solid propellant matrix materials and the like, and have been widely used in the fields of vehicle engineering, biological medicine, aerospace and the like. Amorphous high polymer materials exhibit superelasticity under static or quasi-static loading, i.e., can undergo significant nonlinear elastic deformation under external forces. This nonlinear, recoverable large deformation behavior is the basis for the widespread use of polymers.
Currently, the characterization of the superelastic behavior of a high polymer material mainly comprises two types of models: one is a superelastic model based on the theory of only image, and the other is a superelastic model based on statistical physics. Although the superelastic model based on the image-only theory can make a better approximation to the tensile mechanical test curve of the high polymer material, the model parameters of the superelastic model have no practical physical significance and cannot be obtained through a physical and chemical experiment. The superelastic model based on statistical physics relates the macroscopic mechanical behavior of the high polymer material with microscopic molecular chain configuration evolution, and has clearer physical meaning compared with the superelastic model based on the image-only theory.
The eight-chain model proposed by the most commonly used super-elastic model based on statistical physics at present considers the non-affine elongation of cross-linked molecular chains in a high polymer material, and the two parameters contained in the eight-chain model have relatively clear physical significance. However, the parameters of the eight-chain model are still quite different from those obtained by actual physical and chemical experiments, and meanwhile, the eight-chain model is difficult to describe the super-elastic mechanical response of the high polymer material in various deformation states.
Disclosure of Invention
In order to meet the above-mentioned defects or improvement demands of the prior art, the invention provides a construction method and a construction device of a super-elastic constitutive model of a high polymer material, which aims at calculating the total free energy W of the high polymer material by using constitutive parameters with clear physical meanings and a left-hand Cauchy-Green deformation tensor B, and obtaining the mapping relation between Cauchy stress sigma and Zuo Kexi-Green deformation tensor B by using the total free energy W to bias a first invariant I 1 of Zuo Kexi-Green deformation tensor B so as to characterize the super-elastic constitutive model of the high polymer material. Therefore, the technical problem that the superelastic mechanical response of the high polymer material in various deformation states cannot be described in the existing superelastic constitutive model and constitutive parameters cannot be measured through experimental means is solved.
In order to achieve the above object, according to one aspect of the present invention, there is provided a method for constructing a super elastic constitutive model of a high polymer material, comprising:
step 1: obtaining constitutive parameters of a high polymer material through a physical and chemical experiment, wherein the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
Step 2: calculating the deformation free energy W cc caused by the elongation of the cross-linked chains in the high polymer material and the free energy W fc caused by the elongation of the free chains in the high polymer material based on the constitutive parameters of the high polymer material and the Leuchy-green deformation tensor B; summing the two to obtain an overall free energy W;
step 3: and performing bias derivation on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the total free energy W to obtain a mapping relation between the Cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to characterize a super-elastic constitutive model of the high polymer material.
In one embodiment, the step 2 includes:
S21: establishing an evolution model of end-to-end distance R cc of the cross-linked chains in the high polymer material along with deformation based on the free molecular chain motion hardening coefficient alpha and the first invariant I 1 of the Zuo Kexi-Green deformation tensor B And the evolution model/>, of the free chain end-to-end distance R fc with deformation
S22: evolution model with deformation by utilizing end-to-end distance R cc of cross-linked chainCalculating deformation free energy W cc caused by the elongation of the crosslinking chain; evolution model/>, using the end-to-end distance R fc of the free chain as a function of deformationCalculating the free energy W fc caused by the elongation of the free chain;
S23: the deformation free energy W cc caused by the elongation of the cross-linked chain and the free energy W fc caused by the elongation of the free chain are summed to obtain the total free energy W.
In one embodiment, the step S21 includes:
S211: using the formula Establishing an evolution model of the end-to-end distance R cc of the cross-linked chain along with deformationI 1 is the first invariant of the left Cauchy-Green deformation tensor B;
S212: using the formula Establishing an evolution model/>, of the free chain, wherein the evolution model of the free chain is formed by the end-to-end distance R fc along with deformation
In one embodiment, the step S22 includes:
s221: evolution model with deformation by utilizing end-to-end distance R cc of cross-linked chain Calculating deformation free energy W cc caused by elongation of the crosslinked chain according to the modulus G c of the crosslinked molecular chain and the number N of the crosslinked molecular chain Coulomb monomers;
S222: evolution model with deformation by utilizing end-to-end distance R fc of free chain And the free molecular chain modulus G e acquires the free energy W fc caused by free chain elongation in the polymer material.
In one embodiment, the step S221 includes:
Using the formula The deformation free energy W cc due to the elongation of the crosslinked chain is calculated, wherein the first intermediate variable/>Y, then the second intermediate variableTan -1 is the arctangent function.
In one embodiment, the step S222 includes:
Using the formula The free energy W fc resulting from the free chain elongation is calculated.
In one embodiment, the step 1 includes:
Testing by nuclear magnetic resonance spectroscopy to obtain the crosslinked molecular chain modulus G c;
the free molecular chain modulus G e is obtained through a swelling method test;
And obtaining the number N of the crosslinked molecular chain Coulomb monomers and the free molecular chain motion hardening coefficient alpha through molecular dynamics simulation or curve fitting of a material tensile mechanical test.
According to another aspect of the present invention, there is provided an apparatus for constructing a superelastic constitutive model of a high polymer material, comprising:
the experimental module is used for obtaining constitutive parameters of the high polymer material through a physical and chemical experiment, and the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
A calculation module, configured to calculate a deformation free energy W cc caused by the elongation of the cross-linked chain in the polymer material and a free energy W fc caused by the elongation of the free chain in the polymer material based on the constitutive parameters of the polymer material and the cauchy-green deformation tensor B; summing the two to obtain an overall free energy W;
The modeling module is used for performing bias derivation on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the overall free energy W to obtain a mapping relation between the Cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to characterize a super-elastic constitutive model of the high polymer material.
According to another aspect of the invention there is provided an electronic device comprising a memory storing a computer program and a processor implementing the steps of the above method when the processor executes the computer program.
According to another aspect of the present invention there is provided a computer readable storage medium having stored thereon a computer program which when executed by a processor performs the steps of the above method.
In general, the above technical solutions conceived by the present invention, compared with the prior art, enable the following beneficial effects to be obtained:
(1) The invention utilizes the constitutive parameters of the high polymer material with specific practical physical significance: the deformation free energy W cc caused by the elongation of the cross-linked chain and the free energy W fc caused by the elongation of the free chain are calculated by the modulus G c of the cross-linked molecular chain, the modulus G e of the free molecular chain, the number N of the cross-linked molecular chain Coulomb monomers and the motion hardening coefficient alpha of the free molecular chain; and further obtaining the mapping relation between the Cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to characterize the super-elastic constitutive model of the high polymer material. The invention considers the influence of the elongation of the cross-linked chain and the free chain on the mechanical behavior of the high polymer material, introduces the motion hardening coefficient alpha of the free chain, further calculates and obtains the deformation free energy corresponding to the cross-linked chain and the free chain, and changes the free energy to obtain the final super-elastic constitutive relation. The introduction of the free chain motion hardening coefficient alpha enables the method provided by the invention to fully consider different motion evolution behaviors of the cross-linked chain and the free chain, so that the calculated free energy is closer to the actual situation.
(2) The scheme utilizes the evolution model of the built end-to-end distance R cc of the cross-linked chain along with deformationCalculating deformation free energy W cc caused by the elongation of the crosslinking chain; evolution model/>, using the established end-to-end distance R fc of the free chain as a function of deformationCalculating the free energy W fc caused by the elongation of the free chain; summing the two to obtain the total free energy W; the overall free energy W is only related to the Leuchy-Green deformation tensor I 1, and the high polymer material can be regarded as incompressible material, and the deformation free energy W is deflected on the I 1 to obtain the Cauchy stress sigma of the high polymer, so that the calculation complexity is low.
(3) The present embodiment uses the formulaEstablishing an evolution model/>, of the free chain, wherein the evolution model of the free chain is formed by the end-to-end distance R fc along with deformationEvolution model/>The method is only related to the Leuchy-Grignard deformation tensor I 1 and the free molecular chain motion hardening coefficient alpha, can simply, conveniently and effectively embody the kinematic characteristics of the free molecular chain and the relation between the kinematic characteristics and the physical and chemical properties of the high polymer material, and realizes the comprehensive description of the evolution process of the molecular chain in the high polymer material along with macroscopic deformation.
(4) The embodiment utilizes an evolution model of the end-to-end distance R cc of the crosslinking chain along with deformationCalculating deformation free energy W cc caused by elongation of the crosslinked chain according to the modulus G c of the crosslinked molecular chain and the number N of the crosslinked molecular chain Coulomb monomers; evolution model/>, using the end-to-end distance R fc of the free chain as a function of deformationAnd the free molecular chain modulus G e acquires the free energy W fc caused by the elongation of the free chain in the high polymer material, so that the contribution of various molecular chain interactions in the high polymer material to the free energy and the influence of the interactions on the mechanical property can be described, and the high-efficiency and accurate calculation of the free energy in the high polymer material is realized.
(5) The scheme is tested by nuclear magnetic resonance spectroscopy to obtain the crosslinked molecular chain modulus G c; the free molecular chain modulus G e is obtained through a swelling method test; the molecular dynamics simulation or the curve fitting of the tensile mechanical test of the material is adopted to obtain the number N of the crosslinked molecular chain Coulomb monomers and the free molecular chain motion hardening coefficient alpha, so that the experimental cost can be reduced, the damage of an experimental sample is avoided, the parameters of the constitutive model are rapidly and accurately obtained, and the advanced prediction of the mechanical property of the high polymer material is realized.
Drawings
FIG. 1 is a flow chart of a method for modeling a superelastic constitutive model of a polymeric material according to an embodiment of the present invention.
FIG. 2 is a schematic diagram showing the results of uniaxial stretching experiments of natural rubber in an embodiment of the invention.
FIG. 3 is a schematic diagram showing the results of the biaxial stretching test of natural rubber according to an embodiment of the present invention.
FIG. 4 is a schematic diagram showing the results of the natural rubber shear stretching test according to an embodiment of the present invention.
FIG. 5 is a schematic illustration of the results of uniaxial tensile testing of NEPE propellant matrices in accordance with one embodiment of the present invention.
FIG. 6 is a graphical representation of the results of a series of uniaxial stretching experiments on NEPE propellant matrices with varying curing parameters in accordance with one embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
Example 1
As shown in FIG. 1, the invention provides a modeling method for constructing a super-elastic constitutive model of a high polymer material, which comprises the following steps:
step 1: obtaining constitutive parameters of the high polymer material through a physical experiment, wherein the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
step 2: based on the constitutive parameters of the high polymer material and the Leuchy-Green deformation tensor B, the deformation free energy W cc caused by the elongation of the cross-linked chains in the high polymer material and the free energy W fc caused by the elongation of the free chains in the high polymer material are calculated; summing the two to obtain an overall free energy W;
Step 3: and performing bias guide on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the total free energy W to obtain a mapping relation between the cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to represent the super-elastic constitutive model of the high polymer material.
Example 2
In this embodiment, step 2 includes: s21: establishing an evolution model of end-to-end distance R cc of a cross-linked chain in a high polymer material along with deformation based on a free molecular chain motion hardening coefficient alpha and a first invariant I 1 of a Leuchy-Grin deformation tensor BAnd the evolution model/>, of the free chain end-to-end distance R fc with deformationS22: evolution model/>, using end-to-end distance R cc of the crosslink as a function of deformationCalculating deformation free energy W cc caused by crosslinking chain elongation; evolution model/>, using end-to-end distance R fc of free chain as a function of deformationCalculating free energy W fc caused by free chain elongation; s23: the deformation free energy W cc caused by the elongation of the cross-linked chain and the free energy W fc caused by the elongation of the free chain are summed to obtain the total free energy W.
Example 3
In this embodiment, S21 includes: s211: using the formulaEstablishing an evolution model/>, of a cross-linked chain, wherein the evolution model of the end-to-end distance R cc is deformed along with deformationI 1 is the first invariant of the left Cauchy-Green deformation tensor B; s212: using the formulaEstablishing an evolution model/>, of the free chain, wherein the evolution model of the free chain is formed by the end-to-end distance R fc along with deformation
Considering that both ends of a crosslinking chain in the high polymer material have crosslinking points, and the crosslinking points are mutually connected to form a three-dimensional crosslinking network, the three molecular chains are assumed to be uniformly distributed in three main directions, and are replaced by three molecular chains, one ends of the three molecular chains are connected to the same crosslinking point, and the other ends of the three molecular chains move along with the elastic background of the material, so that an evolution model of the end-to-end distance R cc of the crosslinking chain in the high polymer material along with deformation can be established: In the/> For the end-to-end distance of the crosslink in the initial state, I 1 is the first invariant of the left Cauchy-Green deformation tensor B.
The free chain has no cross-linking point connection at two ends, but when the free chain is long, the molecular chain always tends to curl, and strong topological constraint exists locally, so that the movement mode of the free chain is different from that of the cross-linking chain. The length of the free chains and the topological constraint strength between the free chains affect the elongation speed of the free chains, so that the elongation speed of the free chains can be larger or smaller than that of the cross-linked chains, thus introducing a free chain motion hardening coefficient alpha, and establishing an evolution model of the end-to-end distance R fc of the free chains in the polymer material along with deformation:
In the/> Is the end-to-end distance of the crosslink in the initial state.
Example 4
In this embodiment, S22 includes: s221: evolution model with deformation of end-to-end distance R cc by using cross-linked chainThe deformation free energy W cc caused by the elongation of the crosslinking chain is calculated by the crosslinking molecular chain modulus G c and the crosslinking molecular chain Coulomb monomer number N; s222: evolution model/>, using end-to-end distance R fc of free chain as a function of deformationAnd the free molecular chain modulus G e to obtain the free energy W fc caused by the free chain elongation in the polymer material.
Example 5
In the present embodiment, S221 includes: using the formulaThe deformation free energy W cc due to the elongation of the crosslinked chain was calculated, wherein the first intermediate variable/>Y, then the second intermediate variableTan -1 is the arctangent function.
Specifically, the inter-chain interactions between the crosslinks are weak, so only intra-chain interactions of the crosslinks are considered. The cross-linking points at both ends of the cross-link chain are such that their elongation is limited, and thus the probability density function of the cross-link chain end-to-end distance within the polymeric material can be expressed as:
Wherein c is a normalization constant, N is the number of Coulomb monomers of the crosslinking chain, b is the length of the Coulomb monomers of the crosslinking chain, and the initial end-to-end distance of the crosslinking chain is Beta satisfies:
where L is langevin's equation and β=l -1(Rcc/Nb) is the inverse langevin's equation. The hyperbolic functions referred to above are respectively:
wherein the numerical approximation of the inverse langevin function can be written as: According to the boltzmann formula Δe= -ktln P (Δe is the free energy, k is the boltzmann constant, T is the absolute temperature, P is the probability density function), considering the density of cross-linked molecular chains within the polymeric material as n cc, the deformation-related free energy W cc contributed by the cross-linked chains can be calculated and numerically approximated as: /(I) In the middle ofTan -1 is the arctangent function.
Example 6
In this embodiment, S222 includes: using the formulaThe free energy W fc due to free chain elongation was calculated.
Specifically, both intra-and inter-chain interactions of the free chain need to be considered, and thus the probability density function description in modified gaussian form is used:
Wherein F is a normalization constant, Is the free link end-to-end distance in the initial state.
Wherein, according to boltzmann formula Δe= -ktln P (Δe is free energy, k is boltzmann constant, T is absolute temperature, and P is probability density function), considering the density of free molecular chains in the polymer material is n fc, the free energy W fc related to deformation, which the free chains contribute to, can be calculated:
Further, since the deformation free energy W is related to I 1 only, and the high polymer material can be regarded as an incompressible material, biasing the deformation free energy W to I 1 gives the cauchy stress σ of the high polymer: wherein B is the levocetirizine deformation tensor, which is related to the three main elongations lambda 1,λ2,λ3 of the polymeric material: /(I) 1 Is the unit tensor, p is the unknown Lagrangian operator introduced by incompressibility, and can be eliminated by subtracting the cauchy stresses in different principal directions.
Example 7
In this embodiment, step 1 includes: testing by nuclear magnetic resonance spectroscopy to obtain a crosslinked molecular chain modulus G c; the free molecular chain modulus G e is obtained through a swelling method test; and obtaining the number N of the crosslinked molecular chain Coulomb monomers and the free molecular chain motion hardening coefficient alpha through molecular dynamics simulation or curve fitting of a material tensile mechanical test.
The crosslinking molecular chain modulus G c＝ncckT,ncc is crosslinking molecular chain density, can be obtained by testing through a nuclear magnetic resonance spectrometry, and is characterized in that k is Boltzmann constant, the unit is Joule per Kelvin, T is absolute temperature, and the unit is Kelvin. The free molecular chain modulus G e＝nfckT,nfc is the free molecular chain density and can be obtained by means of a swelling method and the like. The number N of the cross-linked molecular chain Coulomb monomers and the free molecular chain motion hardening coefficient alpha can be obtained through molecular dynamics simulation or curve fitting of a material tensile mechanical test.
On the one hand, the parameters of the super-elastic constitutive model provided by the invention can be measured through a physical and chemical experiment, and have real physical significance, so that the physical and chemical properties of the high polymer material are reflected, and on the other hand, compared with the existing constitutive model, the super-elastic constitutive model provided by the invention can provide more accurate description for the high polymer mechanical behavior under a complex stress state. The superiority of the super-elastic constitutive model provided by the invention is illustrated by adopting the mechanical test data of the natural rubber material under uniaxial stretching, biaxial stretching and pure shearing, the uniaxial stretching data and the physical and chemical experimental test data of the solid propellant matrix material.
The modeling method of the super-elastic constitutive model of the high polymer material comprehensively considers the influence of the extension of the cross-linked chain and the free chain on the mechanical behavior of the high polymer material, introduces the motion hardening coefficient alpha of the free chain, establishes a nonlinear motion evolution model of the cross-linked chain and the free chain, respectively substitutes the nonlinear motion evolution model into corresponding probability density functions, calculates to obtain deformation free energy corresponding to the cross-linked chain and the free chain, and changes the free energy to obtain the final super-elastic constitutive relation. The introduction of the free chain motion hardening coefficient alpha enables the method provided by the invention to fully consider different motion evolution behaviors of the cross-linked chain and the free chain, and the introduction of the two probability density functions enables the calculated free energy to be closer to the actual situation, so that the parameters in the obtained super-elastic constitutive relation can be directly obtained through a physical and chemical experiment, and then the method is called as a constitutive model based on the physical and chemical experiment. The application range of the constitutive model based on the physical and chemical experiments comprises various polymer materials such as natural or artificial synthetic rubber materials, hydrogel materials, solid propellant matrix materials and the like.
The experimental procedure of the above embodiment of the invention is described by way of example below:
Firstly, adopting uniaxial stretching data of a natural rubber material to fit a Arruda-Boyce model and a model based on a physical and chemical experiment respectively, so as to obtain a Arruda-Boyce model parameter of G A-B＝0.27,NA-B =26.5, and a model parameter based on the physical and chemical experiment of G c＝0.156,Ge =0.258, N=23.3 and alpha=0.5. And respectively predicting the biaxial stretching and pure shearing curves of the natural rubber material by adopting two models by utilizing the parameters. Original experimental data, arruda-Boyce model prediction curves, and model prediction curves based on physicochemical experiments are drawn in FIG. 2, FIG. 3 and FIG. 4. The average deviation of the model prediction curve based on the physical and chemical experiments relative to the experimental curve under various deformation states is 5.8%,21.3% and 19.2%, and the average deviation of the Arruda-Boyce model prediction curve relative to the experimental curve under various deformation states is 10.1%,30.0% and 28.6%, so that the model based on the physical and chemical experiments shows better performance than the classical Arruda-Boyce model in predicting the mechanical response condition of the material under the complex stress by the uniaxial tensile experimental data.
Secondly, adopting NEPE propellant matrix material uniaxial tension data to fit Arruda-Boyce model and model based on physicochemical experiment respectively, obtaining parameters of Arruda-Boyce model as G A-B＝0.204,NA-B =37.7, model parameters based on physicochemical experiment as G c＝0.037,Ge =0.087, N=27.1 and alpha=0.95. Original experimental data, arruda-Boyce model predictive curves, model predictive curves based on physicochemical experiments are plotted in FIG. 5. The average deviation of the model prediction curve relative to the experimental curve based on the materialization experiment is 6.5%, and the average deviation of the Arruda-Boyce model prediction curve relative to the experimental curve is 10.8%, so that the model prediction capability based on the materialization experiment is obviously stronger than that of the Arruda-Boyce model. In addition, the crosslinking chain density of the same batch of samples of the experimental sample is n cc＝1.4×10-5mol·cm-3 by nuclear magnetic resonance spectrometry, and the experimental value is calculated according to G c＝ncc kTWhereas the model based on the physicochemical experiment gave a predicted value of G c =0.037, the deviation from the experimental value was 8.8%, indicating that G c in the model based on the physicochemical experiment can give a prediction close to the experimental value.
Finally, fitting a model based on a physical and chemical experiment by adopting a series of NEPE propellant matrix material uniaxial tension data with different curing parameters R to obtain a series of model parameters as shown in table1, and calculating an experimental value according to the equivalent free chain density n fc of the samples in the same batch measured by a swelling methodThe experimental values are listed in the following table, and model predicted value G e and experimental value/>, based on physicochemical experimentThe average error of (2) is not more than 1%. The uniaxial stretching experiment curve and the model prediction curve based on the physicochemical experiment are shown in fig. 6. The model predictive curve almost coincides with the uniaxial stretching experimental curve, while the parameter G e coincides with the experimental value/>The error of (2) is very small, and the model based on the physical and chemical experiment can describe the mechanical behavior of the material well, and simultaneously gives a parameter predicted value very close to the experimental result.
TABLE 1
Example 8
According to another aspect of the present invention, there is provided an apparatus for constructing a superelastic constitutive model of a high polymer material, comprising:
The experimental module is used for obtaining constitutive parameters of the high polymer material through physical experiments, wherein the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
The calculation module is used for calculating deformation free energy W cc caused by the elongation of the cross-linked chain in the high polymer material and free energy W fc caused by the elongation of the free chain in the high polymer material based on the constitutive parameters of the high polymer material and the Leuchy-green deformation tensor B; summing the two to obtain an overall free energy W;
The modeling module is used for carrying out bias derivation on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the total free energy W to obtain a mapping relation between the cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to represent the super-elastic constitutive model of the high polymer material.
Example 9
According to another aspect of the invention there is provided an electronic device comprising a memory storing a computer program and a processor implementing the steps of the above method when the processor executes the computer program.
Example 10
According to another aspect of the present invention there is provided a computer readable storage medium having stored thereon a computer program which when executed by a processor performs the steps of the above method.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (5)
1. The method for constructing the super-elastic constitutive model of the high polymer material is characterized by comprising the following steps of:
step 1: obtaining constitutive parameters of a high polymer material through a physical and chemical experiment, wherein the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
Step 2: calculating the deformation free energy W cc caused by the elongation of the cross-linked chains in the high polymer material and the free energy W fc caused by the elongation of the free chains in the high polymer material based on the constitutive parameters of the high polymer material and the Leuchy-green deformation tensor B; summing the two to obtain an overall free energy W;
Step 3: performing bias derivation on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the total free energy W to obtain a mapping relation between the Cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to represent a super-elastic constitutive model of the high polymer material;
The step 2 comprises the following steps:
S21: establishing an evolution model of end-to-end distance R cc of the cross-linked chains in the high polymer material along with deformation based on the free molecular chain motion hardening coefficient alpha and the first invariant I 1 of the Zuo Kexi-Green deformation tensor B And the evolution model/>, of the free chain end-to-end distance R fc with deformation
S22: evolution model with deformation by utilizing end-to-end distance R cc of cross-linked chainCalculating deformation free energy W cc caused by the elongation of the crosslinking chain; evolution model/>, using the end-to-end distance R fc of the free chain as a function of deformationCalculating the free energy W fc caused by the elongation of the free chain;
S23: summing the deformation free energy W cc caused by the elongation of the cross-linked chain and the free energy W fc caused by the elongation of the free chain to obtain the total free energy W;
The S21 includes:
S211: using the formula Establishing an evolution model of the end-to-end distance R cc of the cross-linked chain along with deformationI 1 is the first invariant of the left Cauchy-Green deformation tensor B; /(I)Is the end-to-end distance of the crosslink in the initial state;
S212: using the formula Establishing an evolution model/>, of the free chain, wherein the evolution model of the free chain is formed by the end-to-end distance R fc along with deformation Is the end-to-end distance of the cross-linked chain in the initial state
The S22 includes:
s221: evolution model with deformation by utilizing end-to-end distance R cc of cross-linked chain Calculating deformation free energy W cc caused by elongation of the crosslinked chain according to the modulus G c of the crosslinked molecular chain and the number N of the crosslinked molecular chain Coulomb monomers;
S222: evolution model with deformation by utilizing end-to-end distance R fc of free chain The free molecular chain modulus G e and the free molecular chain motion hardening coefficient alpha acquire free energy W fc caused by free chain elongation in the high polymer material;
The S221 includes: the deformation free energy W cc due to the elongation of the crosslinked chain is calculated using the formula W cc＝Gc. U (y), wherein the first intermediate variable is noted Y, then the second intermediate variableTan -1 is the arctangent function;
the S222 includes: using the formula The free energy W fc resulting from the free chain elongation is calculated.
2. The method for constructing a superelastic constitutive model of a polymeric material according to claim 1, wherein said step 1 comprises:
Testing by nuclear magnetic resonance spectroscopy to obtain the crosslinked molecular chain modulus G c;
the free molecular chain modulus G e is obtained through a swelling method test;
And obtaining the number N of the crosslinked molecular chain Coulomb monomers and the free molecular chain motion hardening coefficient alpha through molecular dynamics simulation or curve fitting of a material tensile mechanical test.
3. A construction apparatus for a super-elastic constitutive model of a high polymer material, characterized by performing the construction method of the super-elastic constitutive model of a high polymer material according to claim 1, comprising:
the experimental module is used for obtaining constitutive parameters of the high polymer material through a physical and chemical experiment, and the constitutive parameters comprise: the modulus of the cross-linked molecular chain G c, the modulus of the free molecular chain G e, the number of the cross-linked molecular chain Coulomb monomers N and the motion hardening coefficient alpha of the free molecular chain;
A calculation module, configured to calculate a deformation free energy W cc caused by the elongation of the cross-linked chain in the polymer material and a free energy W fc caused by the elongation of the free chain in the polymer material based on the constitutive parameters of the polymer material and the cauchy-green deformation tensor B; summing the two to obtain an overall free energy W;
The modeling module is used for performing bias derivation on the first invariant I 1 of the Zuo Kexi-Green deformation tensor B by utilizing the overall free energy W to obtain a mapping relation between the Cauchy stress sigma of the high polymer material and the Zuo Kexi-Green deformation tensor B so as to characterize a super-elastic constitutive model of the high polymer material.
4. An electronic device comprising a memory and a processor, the memory storing a computer program, characterized in that the processor implements the steps of the method of claim 1 or 2 when executing the computer program.
5. A computer readable storage medium, on which a computer program is stored, characterized in that the computer program, when being executed by a processor, implements the steps of the method of claim 1 or 2.
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