CN113987763B - Construction method and application of polyethylene hyperbola constitutive model - Google Patents
Construction method and application of polyethylene hyperbola constitutive model Download PDFInfo
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- 239000004698 Polyethylene Substances 0.000 title claims abstract description 104
- 229920000573 polyethylene Polymers 0.000 title claims abstract description 104
- -1 polyethylene Polymers 0.000 title claims abstract description 103
- 238000010276 construction Methods 0.000 title claims abstract description 18
- 239000000463 material Substances 0.000 claims abstract description 114
- 238000012360 testing method Methods 0.000 claims abstract description 52
- 238000009864 tensile test Methods 0.000 claims abstract description 36
- 238000000034 method Methods 0.000 claims description 79
- 230000008569 process Effects 0.000 claims description 30
- 230000001419 dependent effect Effects 0.000 claims description 13
- 238000006243 chemical reaction Methods 0.000 claims description 8
- 230000008859 change Effects 0.000 claims description 6
- 238000012886 linear function Methods 0.000 claims description 5
- 238000004364 calculation method Methods 0.000 claims description 3
- 238000012545 processing Methods 0.000 claims description 2
- 229910000831 Steel Inorganic materials 0.000 claims 3
- 239000010959 steel Substances 0.000 claims 3
- 230000003213 activating effect Effects 0.000 claims 1
- 229920001903 high density polyethylene Polymers 0.000 abstract description 9
- 239000004700 high-density polyethylene Substances 0.000 abstract description 9
- 230000006399 behavior Effects 0.000 abstract 1
- 229920003023 plastic Polymers 0.000 description 11
- 239000004033 plastic Substances 0.000 description 11
- 230000004913 activation Effects 0.000 description 4
- 238000001994 activation Methods 0.000 description 4
- 238000004458 analytical method Methods 0.000 description 4
- 230000004807 localization Effects 0.000 description 3
- 239000002861 polymer material Substances 0.000 description 3
- 230000004044 response Effects 0.000 description 3
- 230000007774 longterm Effects 0.000 description 2
- 238000012986 modification Methods 0.000 description 2
- 230000004048 modification Effects 0.000 description 2
- 229940070527 tourmaline Drugs 0.000 description 2
- 229910052613 tourmaline Inorganic materials 0.000 description 2
- 239000011032 tourmaline Substances 0.000 description 2
- VGGSQFUCUMXWEO-UHFFFAOYSA-N Ethene Chemical compound C=C VGGSQFUCUMXWEO-UHFFFAOYSA-N 0.000 description 1
- 239000005977 Ethylene Substances 0.000 description 1
- 241000863480 Vinca Species 0.000 description 1
- 230000002776 aggregation Effects 0.000 description 1
- 238000004220 aggregation Methods 0.000 description 1
- 238000013459 approach Methods 0.000 description 1
- 238000007596 consolidation process Methods 0.000 description 1
- 238000009826 distribution Methods 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 238000011156 evaluation Methods 0.000 description 1
- 238000000605 extraction Methods 0.000 description 1
- 238000009533 lab test Methods 0.000 description 1
- 238000004519 manufacturing process Methods 0.000 description 1
- 238000003801 milling Methods 0.000 description 1
- 239000002994 raw material Substances 0.000 description 1
- 230000009467 reduction Effects 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 230000000717 retained effect Effects 0.000 description 1
- 238000012031 short term test Methods 0.000 description 1
- 238000007725 thermal activation Methods 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
- G06F17/10—Complex mathematical operations
- G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
- G06F17/13—Differential equations
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2113/00—Details relating to the application field
- G06F2113/14—Pipes
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
Abstract
The high-density polyethylene pipeline is subjected to medium and low pressure, the deformation rate is very small, and because the test period is long and the cost is high, few laboratories develop strain control tests at very low rate, and no very reliable prediction model exists at present. The invention discloses a construction method of a polyethylene hyperbola constitutive model, which is characterized in that tensile tests under different strain rates are carried out on a polyethylene pipe, mechanical behaviors related to the strain rates are analyzed, and the dependence rules of material yield stress, initial elastic modulus and yield strain and the strain rate are determined, so that a yield strain model expressed by a relation of the yield stress and the strain rate, a relation of the initial elastic modulus and the strain rate, failure material parameters and the yield stress and the initial elastic modulus of the polyethylene pipe to be tested is obtained, and the polyethylene hyperbola constitutive model reflecting the relation of the yield stress, the initial elastic modulus and the yield strain and the strain rate is constructed. The invention has ingenious conception, saves test time and greatly reduces test and test cost.
Description
Technical Field
The invention relates to the technical field of hyperbolic constitutive model construction, in particular to a construction method and application of a polyethylene hyperbolic constitutive model.
Background
High Density Polyethylene (HDPE) is a viscoelastic-plastic material and is widely used in low pressure gas distribution pipelines in cities. HDPE pipes are usually subjected to medium and low pressures, the deformation rate is very small, long-term service behavior and life evaluation of the pipes are of great concern, but no very reliable unified prediction model exists at present. In engineering application, people are used to test the strength of materials according to test standards and check the strength of a pipeline according to the test standards, however, as high polymer materials such as HDPE (high-density polyethylene) have viscoelastic-plastic properties, stress-strain response is related to time or loading rate, a great deal of researches show that the bearing strength of the high polymer materials such as HDPE is reduced along with the reduction of the strain rate, and for the situation of extremely low strain rate of practical engineering, the long-term performance of the materials is not conservative by directly adopting test results under the loading rate condition of a test room, so that very slow strain control test is required to be carried out, or the stress-strain behavior under the extremely low strain rate is predicted according to short-term test data of the test room. Because of the length and expense of testing, few laboratories develop strain control tests at very low rates, and it is therefore necessary to predict stress-strain response at very low strain rates based on short-term laboratory tests and efficient constitutive models.
Disclosure of Invention
Based on the method, the invention provides a construction method of a hyperbolic constitutive model of polyethylene, which is characterized in that a quasi-static tensile test in different strain rate ranges is carried out on a polyethylene gas pipe, the mechanical behavior related to the strain rate is analyzed, the stress-strain relationship related to the rate is described by determining the yield stress, the initial elastic modulus and the yield stress and strain rate dependency rule of a material, and the yield strength, the initial elastic modulus, the yield stress and the like and the tensile mechanical property under the extremely low strain rate condition are predicted by adopting a hyperbolic constitutive model through a short-time normal strain rate tensile test.
The invention aims at realizing the following technical invention:
the construction method of the polyethylene hyperbola constitutive model comprises the following steps:
s1: setting failure stress according to the rate-related stress-strain hyperbola constitutive modelWherein R is f Is a failure material parameter, and R f <1, and the failure stress is equal to the yield stress, i.e.>Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula deduction:
in sigma true Is true stress, epsilon true Is true strain, E 0 For initial modulus of elasticity, σ y In order to be a yield stress,r is the strain rate f Is a failure material parameter;
the true stress at yield based on polyethylene is equal to the yield stress, i.eWill->Substituting the rate-dependent hyperbolic constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain to obtain a yield strain model expressed by the yield stress and the initial elastic modulus:
in the method, in the process of the invention,is the relation between yield stress and strain rate, +.>For the power law relation of initial modulus of elasticity and strain rate, +.>R is the strain rate f Is a failure material parameter;
carrying out tensile tests on the polyethylene to be tested under different strain rates, obtaining true stress and true strain under different strain rates through data conversion, and then constructing a relation between the yield stress and the strain rate of the polyethylene to be tested according to the true stress and the true strain under different strain rates;
s2: constructing a power law relation between the initial elastic modulus and the strain rate of the polyethylene to be tested according to the yield stress, the initial elastic modulus and the rate-related hyperbolic constitutive model expressed by true stress-true strain obtained in the step S1 and the relation between the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: substituting the relation between the yield stress and the strain rate obtained in the step S1, the power law relation between the initial elastic modulus and the strain rate obtained in the step S2 and the value of the failure material parameter into the yield strain model expressed by the yield stress and the initial elastic modulus obtained in the step S1 to obtain the polyethylene hyperbolic constitutive model expressed by the yield strain and the strain rate.
Compared with the prior art, the method has the advantages that the dependence rules among the yield stress, the initial elastic modulus and the strain rate of the polyethylene are determined, the stress-strain relation related to the rate is described based on the hyperbolic constitutive model, and the hyperbolic constitutive model of the polyethylene reflecting the relation between the yield stress and the strain rate of the polyethylene is constructed through a short-time constant strain rate tensile test, so that the tensile mechanical properties of the polyethylene, such as the yield strength, the initial elastic modulus, the yield strain and the like, at different strain rates, particularly under the extremely low strain rate condition are predicted.
Further, in step S1, the process of constructing the rate-related hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain is as follows:
the following rate-related stress-strain hyperbola constitutive model is first:
performing simplification processing to obtain a simplified rate-related stress-strain hyperbola constitutive model:
wherein epsilon is the axial strain,for strain rate, a and b are strain rate related material parameters, respectively;
the epsilon tends to infinity, and the ultimate stress of the polyethylene is obtained
Setting the failure stress of the polyethylene to be tested asR f Is a failure material parameter, and R f <1；
And performing differential operation on the simplified rate-related stress-strain hyperbola constitutive model to obtain tangential modulus of the simplified rate-related stress-strain hyperbola constitutive model:
epsilon was brought to 0 to give the initial modulus of elasticity of the polyethylene
The simplified rate-dependent stress-strain hyperbola constitutive model can be written as:
in sigma true Is true stress, epsilon true Is true strain, E 0 For initial modulus of elasticity, σ f In order to fail the stress of the material,r is the strain rate f Is a failure material parameter;
stress to failureDeriving a rate-dependent hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain from the simplified rate-dependent stress-strain hyperbolic constitutive model:
in sigma true Is true stress, epsilon true Is true strain, E 0 For initial modulus of elasticity, σ y In order to be a yield stress,r is the strain rate f Is a failure material parameter.
Further, in step S1, the process of constructing the relation between the yield stress and the strain rate is as follows:
measuring poisson ratio of the polyethylene to be measured, and then converting engineering stress and engineering strain obtained by carrying out tensile test on the polyethylene to be measured under different strain rates into true stress and true strain by the following equation to obtain true stress-true strain curves under different strain rates:
ε true ＝ln(1+ε)
σ true ＝σ(1+ε) 2μ
wherein ε true Is true strain, ε is engineering strain, σ true True stress, sigma engineering stress, mu poisson ratio;
obtaining yield stress of the polyethylene to be tested under different strain rates according to the true stress-true strain curve under the different strain rates;
according to the Erying model (the tourmaline model), a linear function relation of yield stress and logarithmic strain rate is deduced:
in the method, in the process of the invention,to reference the viscoplastic strain rate, k B Is Boltzmann constant, T is absolute temperature, Q is activation energy, and V is activation volume; a is a first material parameter, B is a second material parameter;
and then, fitting the yield stress of the polyethylene to be measured under different strain rates, and determining the value of the first material parameter and the value of the second material parameter in the linear function relation of the yield stress and the logarithmic strain rate, thereby obtaining the relation of the yield stress and the strain rate.
Further, in step S2, the yield stress is related to the strain rate as follows:
in sigma y In order to be a yield stress,is the strain rate.
Further, in step S2, the construction process of the relation between the initial elastic modulus and the strain rate and the determination process of the failure material parameter are:
defining the ratio of true strain to true stress as instantaneous compliance, and further defining the rate-dependent hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain obtained in step S1:
conversion to a relationship of instantaneous compliance to true strain:
wherein ε true Is true strain, sigma true Is true stress, sigma y For yield stress, E 0 For the initial modulus of elasticity, the elastic modulus,r is the strain rate f Is a failure material parameter;
substituting the relation of the yield stress change along with the strain rate obtained in the step S1 and the relation of the real stress and the real strain under different strain rates obtained in the step S1 into the relation of the instantaneous compliance and the real strain to obtain an instantaneous compliance-real strain curve under different strain rates, and then performing linear fitting to determine the initial elastic modulus and the failure material parameters of the polyethylene to be tested under different strain rates;
and then constructing and obtaining a power law relation between the initial elastic modulus and the strain rate according to the law that the initial elastic modulus changes along with different strain rates:
wherein E is 0 For the initial modulus of elasticity, the elastic modulus,the strain rate is C, the third material parameter is C, and D is the fourth material parameter; wherein the values of C and D are determined according to a rule fit of the initial elastic modulus with the change of strain rate.
Further, in step S2, the power law relation between the initial elastic modulus and the strain rate is:
wherein E is 0 For the initial modulus of elasticity, the elastic modulus,is the strain rate.
Further, in step S3, the polyethylene hyperbolic constitutive model reflecting the relation between the yield strain and the strain rate is:
in the method, in the process of the invention,representing the yield stress sigma y I.e. +.>The method is a relation between yield stress and strain rate of polyethylene, wherein A is a first material parameter, and B is a second material parameter; />Representing the primary modulus of elasticity E 0 I.e. +.>The method is a power law relation between the initial elastic modulus and the strain rate of the polyethylene, wherein C is a third material parameter, and D is a fourth material parameter; r is R f For failure ofMaterial parameters, and R f <1。
Further, the polyethylene hyperbola constitutive model reflecting the relation between yield strain and strain rate is as follows:
wherein ε y In order to be a yield strain,is the strain rate.
Further, the method for constructing the polyethylene hyperbola constitutive model further comprises the step S4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a yield strain calculated value, comparing the yield strain calculated value with a yield strain test value obtained through a tensile test, judging whether the error between the calculated value and the test value is within an acceptable range, and if the error is within the acceptable range, predicting the reliability of the result by the polyethylene hyperbolic constitutive model;
alternatively, the relation between the yield stress and the strain rate obtained in the step S1, the power law relation between the initial elastic modulus and the strain rate obtained in the step S2 and the failure material parameter R are calculated f Substituting the values of the stress and the initial elastic modulus into the rate-related hyperbola constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain obtained in the step S1, obtaining the rate-related hyperbola constitutive model expressed by the strain rate and the true stress-true strain, substituting any given strain rate into the rate-related hyperbola constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain to calculate a true stress calculated value, comparing the true stress calculated value with a true stress test value converted from engineering stress measured by a tensile test, and judging the reliability of the prediction result of the polyethylene hyperbola constitutive model.
The invention also provides an application of the polyethylene hyperbola constitutive model constructed according to the construction method:
and predicting the yield stress, the initial elastic modulus and the yield strain of the polyethylene by substituting any strain rate into the hyperbolic constitutive model of the polyethylene, so as to evaluate the tensile mechanical property of the polyethylene to be tested under the strain rate condition.
For a better understanding and implementation, the present invention is described in detail below with reference to the drawings.
Drawings
FIG. 1 (a) is an engineering stress-engineering strain curve obtained from a test obtained by subjecting a test piece to a constant strain rate tensile test in accordance with an embodiment of the present invention; FIG. 1 (b) is an engineering stress-engineering strain curve of a specimen of an embodiment of the present invention before stress softening and strain localization occur during a tensile test;
FIG. 2 shows a sample of an embodiment of the invention at a strain rate of 10 -3 s -1 An axial strain-time curve and a transverse strain-time curve measured under the condition;
FIG. 3 is a graph of transverse strain versus axial strain for a test specimen according to an embodiment of the present invention;
FIG. 4 is a graph showing the true stress-true strain curves of samples according to embodiments of the present invention at different strain rates;
FIG. 5 is a graph showing the linear relationship of yield stress at different strain rates during tensile testing for samples according to embodiments of the present invention;
FIG. 6 is a graph of instantaneous compliance versus true strain curve and model fit for samples at different strain rates according to an embodiment of the present invention;
FIG. 7 shows the initial elastic modulus E of a sample according to an embodiment of the present invention 0 A curve that varies with strain rate;
FIG. 8 is a graph of yield strain as a function of strain rate for a test specimen according to an embodiment of the present invention;
FIG. 9 is a schematic flow chart of a method for constructing a polyethylene hyperbolic constitutive model of the invention.
Detailed Description
In the prior art, polyethylene tubing performance is typically evaluated by tensile testing measured stress-strain behavior at different strain rates.However, when the tensile test is to be performed under a low strain rate condition, a long test is required, when the strain rate isWhen the tensile test is performed under the condition, it takes more than ten hours, and the strain rate is lower than 10 -5 s -1 The tensile test is performed under the condition, and a great deal of time and cost are required. Therefore, the applicant establishes a relation between the strain rate and the stress-strain relation of the polyethylene to be tested by carrying out tensile tests under different strain rate conditions and extrapolating the stress-strain constitutive model related to the rate, so as to construct the polyethylene hyperbola constitutive model. Therefore, by substituting any given strain rate into the constructed hyperbolic constitutive model of the polyethylene, the yield strain of the polyethylene to be measured can be predicted, so that the performance of the polyethylene pipe under different strain rates can be predicted.
The construction process of the polyethylene hyperbolic constitutive model is further described below through the explanation of the experimental process.
The PE100 buried polyethylene gas pipeline is obtained and then is processed into a sample by numerical control milling, the model of the pipe is SDR11/Dn315 multiplied by 28.6, the raw material brand is P6006, and the sample is produced by Hebei plastic pipeline manufacturing Limited liability company. The sample size meets the requirements of ISO 527-2:2012.
The specimen was then subjected to a constant strain rate tensile test by a CSS44020 electronic tensile tester (manufactured by vinca tester institute). The test was carried out at room temperature (23 ℃) and relative humidity 50% RH, the test specimens were deformed under tension at a constant strain rate, the axial strain was measured by an extensometer, and the engineering strain rates applied to the test specimens were 5X 10, respectively -2 s -1 、10 -2 s -1 、10 -3 s -1 、10 -4 s -1 And 10 -5 s -1 . To measure the poisson's ratio of a sample, both the axial and transverse strains during tensile deformation were measured using Digital Image Correlation (DIC) techniques to obtain the poisson's ratio of the sample.
Referring to fig. 1 (a), fig. 1 (a) is an engineering stress-engineering strain curve obtained by a constant strain rate tensile test of a test specimen. It can be seen from 1 (a) that when the strain increases to a certain value, stress softening and strain localization occur due to necking, after which the engineering stress-engineering strain curve does not truly reflect the constitutive properties of the material. Thus, the applicant only retained the engineering stress-engineering strain curve before necking for subsequent analysis, as shown in fig. 1 (b), which is the engineering stress-engineering strain curve before stress softening and strain localization of the test specimen during tensile testing.
Referring to FIG. 2, FIG. 2 shows that the strain rate of the sample is 10 -3 s -1 An axial strain-time curve and a transverse strain-time curve measured under the conditions. As can be seen from fig. 2, the axial strain measured by the extensometer is consistent with the axial strain measured by DIC technology, so in other constant strain rate tensile tests, applicant only used the extensometer to measure the axial strain for stress-strain analysis.
The poisson's ratio of polyethylene can be measured by the ratio of transverse strain to axial strain, and the transverse strain as a function of axial strain in fig. 2 is plotted to give an axial strain-time curve and a transverse strain-time curve as shown in fig. 3. As can be seen from fig. 3, during stretching the transverse strain of the specimen varied in proportion to the axial strain, the data were linearly fitted, and the poisson's ratio of the material was 0.456 as a result of the slope of the fit.
As can be seen from FIGS. 1 (a) and 1 (b), the strain rate is 10 -5 s -1 ～5×10 -2 s -1 Under the condition that the strain experienced by the sample before necking and yielding reaches 0.07-0.17, the deformation is large, so that the applicant carries out conversion through the equation (1) and the equation (2) on test data corresponding to the engineering stress-engineering strain curve before yielding in the constant strain rate uniaxial tensile test shown in the figure 1 (b) to obtain the true stress-true strain curve of the sample under different strain rates shown in the figure 4.
ε true ＝ln(1+ε) (1)
σ true ＝σ(1+ε) 2μ (2)
Wherein ε true Is true strain, ε is engineering strain, σ true Is true stress, sigma is engineering stress, and mu is Poisson's ratio.
As can be seen from fig. 1 (a), the elongation process of the test specimen exhibits a necked yield behavior, and the peak stress before necking (i.e., the inflection point) is considered to be the yield stress, and the corresponding strain is the yield strain, all of which are related to the load strain rate. Referring to FIG. 5, FIG. 5 shows the yield stress versus strain rate for a sample during a tensile test, and shows the true yield stress σ y Linearly increasing with increasing logarithmic strain rate. The strain rate dependence is embodied in a plurality of high polymer materials and is characterized in the strain rate<0.1s -1 Under the quasi-static loading condition, an Eyrining model (an tourmaline model) is satisfied.
The Eyring model was first used to describe chemical reaction rate, but was applied to viscoelastic-plastic mechanical analysis of high molecular materials soon after extraction, and it was considered that viscoelastic is a thermal activation rate process, and yield stress and logarithmic strain rate at plastic yield satisfy a linear function, namely
In the method, in the process of the invention,to reference the viscoplastic strain rate, k B Is Boltzmann constant, T is absolute temperature, Q is activation energy, and V is activation volume; a is a first material parameter, and B is a second material parameter.
Fitting the test data of fig. 5 according to equation (3) yields the yield stress as a function of strain rate as shown in fig. 5:
in sigma y In order to be a yield stress,is the strain rate.
As can be seen from equations (3) and (4), the value of the first material parameter a is 34.62 and the value of the second material parameter B is 3.18.
For the triaxial mechanical response of consolidation non-drainage of a geotechnical medium, kondner and Duncan et al propose a rate-dependent stress-strain hyperbolic constitutive model characterized by a hyperbolic function, which is:
in sigma 1 -σ 3 Epsilon is the axial strain and a and b are the strain rate related material parameters, respectively, for the primary stress difference.
Further, merry and Scott apply equation (5) to strain rate related mechanical behavior analysis of polymeric materials. Suleiman et al further developed a focal hyperbolic constitutive model taking into account the linear dependence of parameters a and b.
For uniaxial stretching or compressive mechanical behavior, formula (5) is simplified to:
wherein epsilon is the axial strain,is the strain rate.
From equation (6), it can be seen that when the axial strain ε approaches infinity, the ultimate stress of the material is obtainedSaid ultimate stress is not present for a real material, so that the applicant sets the material's failure stress to +.>Wherein R is f <1 is the failure material parameter.
Further, the differential operation is performed on equation (6), and the tangential modulus of the model is obtained as follows:
wherein epsilon is the axial strain,for strain rate, a and b are strain rate related material parameters, respectively.
As can be seen from equation (7), when the axial strain ε tends to 0, the initial elastic modulus of the test piece
As previously described, applicants analyzed the uniaxial tensile stress-strain behavior of the test specimens using true stress and true strain. Taking into account parametersAnd->According to the above-deduced ultimate strain of the materialInitial modulus of elasticity of the material +.>The hyperbolic constitutive model given in equation (6) can be written as:
in sigma true Is true stress, epsilon true For yielding strain, sigma f For failure stress, E 0 For the initial modulus of elasticity, the elastic modulus,r is the strain rate f Is a failure material parameter.
Since in engineering polyethylene yielding means failure, the applicant has setCorresponding strain to failure epsilon f Is epsilon f ＝ε y The rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus, and true stress-true strain can be deduced according to equation (8):
in sigma ture Is true stress, epsilon true Is true strain, sigma y For yield stress, E 0 For the initial modulus of elasticity, the elastic modulus,is the strain rate;
and, a yield strain model expressed by yield stress and initial elastic modulus:
wherein ε y For yielding strain, sigma y For yield stress, E 0 For the initial modulus of elasticity, the elastic modulus,r is the strain rate f Is a failure material parameter;
further, equation (9) can be rewritten as:
wherein ε true Is true strain, sigma ture Is true stress, sigma y For yield stress, E 0 For the initial modulus of elasticity, the elastic modulus,r is the strain rate f Is a failure material parameter.
The applicant has determined the ratio of true strain to true stress (. Epsilon.) in equation (11) true /σ true ) Defined as "instantaneous compliance", the true stress-true strain curve is converted to instantaneous compliance (ε) by data conversion true /σ true ) True strain epsilon true The curves give instantaneous compliance-true strain curves for the test specimens at different strain rates as shown in figure 6. As can be seen in FIG. 6, at 10 -5 s -1 Up to 5X 10 -2 s -1 The instantaneous compliance and true strain in the strain rate range substantially satisfy a linear relationship and can therefore be described using the model given in equation (11).
Substituting equation (4) into equation (11), and linearly fitting the data in FIG. 6 with equation (11) to determine the initial modulus of elasticity of the test specimen at different strain ratesAnd failure material parameter R f As shown in table 1 below. As can be seen from Table 1, the failure material parameter R f As a constant, the fitting result is shown as a solid line in fig. 8.
TABLE 1 model parameters E 0 And R is f
Referring to FIG. 7, FIG. 7 shows the initial elastic modulus E of the test specimen 0 A curve of change with strain rate. As can be seen from fig. 7, in the double logarithmic coordinate system, the initial elastic modulus increases linearly with the increase of the strain rate, and thus can be described by a power law model, the relation between the initial elastic modulus and the strain rate:
wherein E is 0 For the initial modulus of elasticity, the elastic modulus,is the strain rate.
Further, equation (12) may be expressed as:
wherein E is 0 For the initial modulus of elasticity, the elastic modulus,for strain rate, C is the third material parameter and D is the fourth material parameter.
Setting the failure material parameter R f =0.9, equation (4) and equation (12) are substituted into equation (10), a hyperbolic constitutive model of the polyethylene sample is obtained:
wherein ε y In order to be a yield strain,is the strain rate.
Referring to fig. 8, fig. 8 is a graph showing yield strain as a function of strain rate for a test specimen according to an embodiment of the present invention. The solid line in the figure is the calculated yield strain value calculated by equation (13), and the scatter in the figure is at 5×10 -2 s -1 、10 -2 s -1 、10 -3 s -1 、10 -4 s -1 And 10 -5 s -1 Yield strain test value measured by tensile test at 8 strain rate. As can be seen from FIG. 8, for any given strain rate, the aggregation by equation (13)And comparing the yield strain calculated by the ethylene hyperbolic constitutive model with a test value obtained by a tensile test, wherein the model calculated value is better matched with the test value.
Parameter R f =0.9, equation (4) and equation (12) are substituted into equation (9), then the rate-dependent hyperbolic constitutive model expressed by strain rate and true stress-true strain is:
referring to fig. 6, the solid line in the graph is the calculated value of true stress obtained by equation (14), the scattered points in the graph are the test values of true stress converted from the engineering stress measured by the tensile test, and as can be seen from fig. 6, the calculated values of the model and the test values are also well matched.
In conclusion, the prediction result of the polyethylene hyperbola constitutive model is proved to be reliable.
In practical application, any strain rate can be substituted into the hyperbolic constitutive model in the equation (13) to calculate a yield strain calculated value, and the yield strain calculated value is compared with a test value obtained through a tensile test to judge the reliability of the prediction result of the polyethylene hyperbolic constitutive model. The engineering strain measured by the tensile test of the sample under any strain rate can be converted into the true strain, the true strain is substituted into the equation (14), the true stress calculated value is calculated, and then the true stress calculated value is compared with the true stress test value converted by the engineering stress measured by the tensile test, so that the reliability of the polyethylene hyperbolic constitutive model prediction result is judged. It should be noted that, the user may set an acceptable range of the error between the calculated value and the test value according to the need, and if the error is within the acceptable range, the prediction result of the polyethylene hyperbolic constitutive model is reliable.
Referring to fig. 9, fig. 9 is a flow chart summarizing a method for constructing a hyperbolic constitutive model of polyethylene according to the present invention based on the whole test process, wherein the method comprises the following steps:
s1: setting failure stress according to the rate-related stress-strain hyperbola constitutive modelWherein R is f Is a failure material parameter, and R f <1, and the failure stress is equal to the yield stress, i.e.>Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula deduction:
in sigma true Is true stress, epsilon true Is true strain, E 0 For initial modulus of elasticity, σ y In order to be a yield stress,r is the strain rate f Is a failure material parameter;
the true stress at yield based on polyethylene is equal to the yield stress, i.eWill->Substituting the rate-dependent hyperbolic constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain to obtain a yield strain model expressed by the yield stress and the initial elastic modulus:
in the method, in the process of the invention,is the relation between yield stress and strain rate, +.>For the power law relation of initial modulus of elasticity and strain rate, +.>R is the strain rate f Is a failure material parameter;
carrying out tensile tests on the polyethylene to be tested under different strain rates, obtaining true stress and true strain under different strain rates through data conversion, and then constructing a relation between the yield stress and the strain rate of the polyethylene to be tested according to the true stress and the true strain under different strain rates;
s2: constructing a power law relation between the initial elastic modulus and the strain rate of the polyethylene to be tested according to the yield stress, the initial elastic modulus and the rate-related hyperbolic constitutive model expressed by true stress-true strain obtained in the step S1 and the relation between the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: substituting the relation between the yield stress and the strain rate obtained in the step S1, the power law relation between the initial elastic modulus and the strain rate obtained in the step S2 and the value of the failure material parameter into the yield strain model expressed by the yield stress and the initial elastic modulus obtained in the step S1 to obtain a polyethylene hyperbolic constitutive model expressed by the yield strain and the strain rate:
in the method, in the process of the invention,representing the yield stress sigma y I.e. +.>The method is a relation between yield stress and strain rate of polyethylene, wherein A is a first material parameter, and B is a second material parameter; />Representing the primary modulus of elasticity E 0 I.e. +.>The method is a power law relation between the initial elastic modulus and the strain rate of the polyethylene, wherein C is a third material parameter, and D is a fourth material parameter; r is R f Is a failure material parameter, and R f <1。
Further, the construction method further comprises the step of verifying the reliability of the polyethylene hyperbola constitutive model:
s4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a yield strain calculated value, comparing the yield strain calculated value with a yield strain test value obtained through a tensile test, judging whether the error between the calculated value and the test value is within an acceptable range, and if the error is within the acceptable range, predicting the reliability of the result by the polyethylene hyperbolic constitutive model.
Alternatively, the relation between the yield stress and the strain rate obtained in the step S1, the power law relation between the initial elastic modulus and the strain rate obtained in the step S2 and the failure material parameter R are calculated f Substituting the rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain obtained in the step S1 to obtain the rate-related hyperbolic constitutive model expressed by strain rate and true stress-true strain:
in sigma true Is true stress, epsilon true In order to be a yield strain,a is a first material parameter, B is a second material parameter, R f The material parameter is a failure material parameter, C is a third material parameter, and D is a fourth material parameter;
and substituting any given strain rate into the rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain to calculate a true stress calculated value, and comparing the true stress calculated value with a true stress test value converted from engineering stress measured by a tensile test to judge the reliability of the prediction result of the polyethylene hyperbolic constitutive model.
Based on the hyperbolic constitutive model of polyethylene, which reflects the relation between the yield strain and the strain rate, obtained by the construction method, any given strain rate is substituted into the hyperbolic constitutive model of equation (13), so that the yield stress, the initial elastic modulus and the yield strain of the high-density polyethylene pipe sample under the strain rate can be obtained, and the tensile mechanical properties such as the yield strength, the initial elastic modulus, the yield strain and the like of the high-density polyethylene pipe material under different strain rate conditions are predicted.
It should be noted that, the method for constructing the polyethylene hyperbola constitutive model is applicable to other viscoelastic-plastic materials, and the steps of the method are the same as those of the specific embodiment of the invention, except that part of parameters need to be determined by re-fitting according to test results, and the method comprises the following steps:
(1) When the method is used for constructing hyperbolic constitutive models of other viscoelastic plastic materials, the values of the first material parameter A and the second material parameter B in the equation (3) are different from those of the specific embodiment, and the first material parameter A and the second material parameter B of the viscoelastic plastic material to be measured are required to be obtained by meeting the linear function relation of the yield stress and the logarithmic strain rate of the material.
(2) When used in the construction of hyperbolic constitutive models of other viscoelastic-plastic materials, the initial elastic modulus E in equation (8) 0 And failure material parameter R f Different from the specific embodiment, the initial elastic modulus E is determined by fitting an instantaneous compliance-true strain curve and a model of the material at different strain rates 0 And failure material parameter R f 。
(3) Equation (12) initial elastic modulus E when used in the construction of hyperbolic constitutive models of other viscoelastic-plastic materials 0 And strain rateUnlike the present embodiment, the relation of (a) is based on the initial modulus of elasticity E for the material 0 Deriving initial elastic modulus E from results obtained by fitting a change curve with strain rate 0 And strain rate->Is a function of the empirical relationship of (a).
From the above (1) (2) (3), it can be seen that when used for the construction of hyperbolic constitutive models of other viscoelastic-plastic materials, the hyperbolic constitutive model of equation (13) is:
compared with the prior art, the method has the advantages that the dependence rules among the yield stress, the initial elastic modulus and the strain rate of the polyethylene are determined, the stress-strain relation related to the rate is described based on the hyperbolic constitutive model, and the hyperbolic constitutive model of the polyethylene reflecting the relation between the yield strain and the strain rate of the material is constructed through a short-time constant-strain-rate tensile test, so that the yield strength, the initial elastic modulus, the yield strain and the tensile mechanical property of the polyethylene under different strain rates, particularly under the extremely low strain rate condition are predicted.
The present invention is not limited to the above-described embodiments, but, if various modifications or variations of the present invention are not departing from the spirit and scope of the present invention, the present invention is intended to include such modifications and variations as fall within the scope of the claims and the equivalents thereof.
Claims (10)
1. The construction method of the polyethylene hyperbola constitutive model is characterized by comprising the following steps of:
s1: setting failure stress according to the rate-related stress-strain hyperbola constitutive modelWherein, the method comprises the steps of, wherein,R f is a failure material parameter, and->And let the failure stress equal to the yield stress, i.e +.>Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula deduction:
in the method, in the process of the invention,is true stress->Is the true strain of the steel sheet,E 0 for the initial modulus of elasticity, the elastic modulus,σ y for yield stress>In order for the strain rate to be a function of,R f is a failure material parameter;
the true stress at yield based on polyethylene is equal to the yield stress, i.e=/>Will->=/>Substituting the rate-dependent hyperbolic constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain to obtain a yield strain model expressed by the yield stress and the initial elastic modulus:
in the method, in the process of the invention,is the relation between yield stress and strain rate, +.>For the power law relation of initial modulus of elasticity and strain rate, +.>In order for the strain rate to be a function of,R f is a failure material parameter;
carrying out tensile tests on the polyethylene to be tested under different strain rates, obtaining true stress and true strain under different strain rates through data conversion, and then constructing a relation between the yield stress and the strain rate of the polyethylene to be tested according to the true stress and the true strain under different strain rates;
s2: constructing a power law relation between the initial elastic modulus and the strain rate of the polyethylene to be tested according to the yield stress, the initial elastic modulus and the rate-related hyperbolic constitutive model expressed by true stress-true strain obtained in the step S1 and the relation between the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: substituting the relation between the yield stress and the strain rate obtained in the step S1, the power law relation between the initial elastic modulus and the strain rate obtained in the step S2 and the value of the failure material parameter into the yield strain model expressed by the yield stress and the initial elastic modulus obtained in the step S1 to obtain the polyethylene hyperbolic constitutive model expressed by the yield strain and the strain rate.
2. The method for constructing a polyethylene hyperbolic constitutive model according to claim 1, wherein the method comprises the following steps:
in step S1, the process of constructing the rate-related hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain is as follows:
the following rate-related stress-strain hyperbola constitutive model is first:
performing simplification processing to obtain a simplified rate-related stress-strain hyperbola constitutive model:
in the method, in the process of the invention,for axial strain>For strain rate->And->Material parameters related to strain rate respectively;
will beApproaching infinity, obtaining the ultimate stress of the polyethylene +.>；
Setting the failure stress of the polyethylene to be tested as，R f Is a failure material parameter, and->；
And performing differential operation on the simplified rate-related stress-strain hyperbola constitutive model to obtain tangential modulus of the simplified rate-related stress-strain hyperbola constitutive model:
will beTending to 0, giving the polyethylene an initial modulus of elasticity +.>；
The simplified rate-dependent stress-strain hyperbola constitutive model can be written as:
in the method, in the process of the invention,is true stress->Is the true strain of the steel sheet,E 0 for initial modulus of elasticity, +.>For failure stress->In order for the strain rate to be a function of,R f is a failure material parameter;
stress to failureDeriving a rate-dependent hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain from the simplified rate-dependent stress-strain hyperbolic constitutive model:
in the method, in the process of the invention,is true stress->Is the true strain of the steel sheet,E 0 for the initial modulus of elasticity, the elastic modulus,σ y for yield stress>In order for the strain rate to be a function of,R f
is a failure material parameter.
3. The method for constructing a polyethylene hyperbolic constitutive model according to claim 1, wherein the method comprises the following steps:
in step S1, the process of constructing the relation between the yield stress and the strain rate is as follows:
measuring poisson ratio of the polyethylene to be measured, and then converting engineering stress and engineering strain obtained by carrying out tensile test on the polyethylene to be measured under different strain rates into true stress and true strain by the following equation to obtain true stress-true strain curves under different strain rates:
in the method, in the process of the invention,for true strain, ++>For engineering strain->Is true stress->For engineering stress->Is poisson's ratio;
obtaining yield stress of the polyethylene to be tested under different strain rates according to the true stress-true strain curve under the different strain rates;
and deducing a linear functional relation between the yield stress and the logarithmic strain rate according to the Erying model:
in the method, in the process of the invention,for reference of the viscoplastic strain rate, +.>For the Boltzmann constant,Tabsolute temperature>For activating energy, < >>Is an activated volume; a is a first material parameter, B is a second material parameter;
and then, fitting the yield stress of the polyethylene to be measured under different strain rates, and determining the value of the first material parameter and the value of the second material parameter in the linear function relation of the yield stress and the logarithmic strain rate, thereby obtaining the relation of the yield stress and the strain rate.
4. A method for constructing a polyethylene hyperbolic constitutive model according to claim 3, wherein:
in step S2, the yield stress is related to the strain rate as follows:
in sigma y In order to be a yield stress,is the strain rate.
5. The method for constructing a polyethylene hyperbolic constitutive model according to claim 1, wherein the method comprises the following steps:
in step S2, the construction process of the relation between the initial elastic modulus and the strain rate and the determination process of the failure material parameter are:
defining the ratio of true strain to true stress as instantaneous compliance, and further defining the rate-dependent hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain obtained in step S1:
conversion to a relationship of instantaneous compliance to true strain:
in the method, in the process of the invention,for true strain, ++>Is a true stress, and is a true stress,σ y in order to be a yield stress,E 0 for initial modulus of elasticity, +.>In order for the strain rate to be a function of,R f is a failure material parameter;
substituting the relation of the yield stress change along with the strain rate obtained in the step S1 and the relation of the real stress and the real strain under different strain rates obtained in the step S1 into the relation of the instantaneous compliance and the real strain to obtain an instantaneous compliance-real strain curve under different strain rates, and then performing linear fitting to determine the initial elastic modulus and the failure material parameters of the polyethylene to be tested under different strain rates;
and then constructing and obtaining a power law relation between the initial elastic modulus and the strain rate according to the law that the initial elastic modulus changes along with different strain rates:
in the method, in the process of the invention,E 0 for the initial modulus of elasticity, the elastic modulus,in order for the strain rate to be a function of,Cas a parameter of the third material, a second material,Dis a fourth material parameter; wherein,CandDaccording to the value of the initial bulletAnd the rule fitting of the change of the modulus of performance along with the strain rate is determined.
6. The method for constructing a polyethylene hyperbolic constitutive model according to claim 5, wherein the method comprises the following steps:
in step S2, the power law relation between the initial elastic modulus and the strain rate is:
in the method, in the process of the invention,E 0 for the initial modulus of elasticity, the elastic modulus,is the strain rate.
7. The method for constructing a polyethylene hyperbolic constitutive model according to claim 1, wherein the method comprises the following steps:
in step S3, the polyethylene hyperbolic constitutive model reflecting the relation between the yield strain and the strain rate is:
in the method, in the process of the invention,representing yield stress +.>I.e. +.>Which is a relationship between yield stress and strain rate of polyethylene, wherein,Aas a first material parameter,Bis a second material parameter; />Representing the primary modulus of elasticity>I.e. +.>Which is a power law relation of initial elastic modulus and strain rate of the polyethylene,Cas a parameter of the third material, a second material,Dis a fourth material parameter;R f is a failure material parameter, and->。
8. The method for constructing a polyethylene hyperbolic constitutive model according to claim 7, wherein the method comprises the following steps:
the polyethylene hyperbola constitutive model reflecting the relation between yield strain and strain rate is as follows:
wherein ε y In order to be a yield strain,is the strain rate.
9. The method for constructing a polyethylene hyperbolic constitutive model according to claim 1, wherein the method comprises the following steps:
further comprising step S4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a yield strain calculated value, comparing the yield strain calculated value with a yield strain test value obtained through a tensile test, judging whether the error between the calculated value and the test value is within an acceptable range, and if the error is within the acceptable range, predicting the reliability of the result by the polyethylene hyperbolic constitutive model;
alternatively, the relation between the yield stress and the strain rate obtained in step S1, the initial elastic modulus obtained in step S2 and the stress are calculatedPower law relation of variable rate and said failure material parameterR f Substituting the values of the stress and the initial elastic modulus into the rate-related hyperbola constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain obtained in the step S1, obtaining the rate-related hyperbola constitutive model expressed by the strain rate and the true stress-true strain, substituting any given strain rate into the rate-related hyperbola constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain to calculate a true stress calculated value, comparing the true stress calculated value with a true stress test value converted from engineering stress measured by a tensile test, and judging the reliability of the prediction result of the polyethylene hyperbola constitutive model.
10. An application of a polyethylene hyperbolic constitutive model constructed according to the construction method of any one of claims 1-9, which is characterized in that:
and predicting the yield stress, the initial elastic modulus and the yield strain of the polyethylene by substituting any strain rate into the hyperbolic constitutive model of the polyethylene.
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