CN113987763A - Construction method and application of polyethylene hyperbolic constitutive model - Google Patents

Construction method and application of polyethylene hyperbolic constitutive model Download PDF

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CN113987763A
CN113987763A CN202111186262.8A CN202111186262A CN113987763A CN 113987763 A CN113987763 A CN 113987763A CN 202111186262 A CN202111186262 A CN 202111186262A CN 113987763 A CN113987763 A CN 113987763A
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李茂东
杨波
李彦
罗文波
王志刚
郑佩根
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Guangzhou Special Pressure Equipment Inspection and Research Institute
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Abstract

The high-density polyethylene pipeline is under the action of medium and low pressure, the deformation rate is very low, and due to the long test period and high cost, few laboratories carry out strain control tests at extremely low rate, and at present, no very reliable prediction model exists. The invention discloses a method for constructing a polyethylene hyperbolic constitutive model, which comprises the steps of carrying out tensile tests on polyethylene pipes under different strain rates, analyzing mechanical behaviors related to the strain rates, determining the yield stress, the initial elastic modulus and the correlation rule of the yield strain and the strain rates of materials, and obtaining a relational expression of the yield stress and the strain rates of polyethylene to be tested, a relational expression of the initial elastic modulus and the strain rates, failure material parameters and a yield strain model expressed by the yield stress and the initial elastic modulus, so that the polyethylene hyperbolic constitutive model reflecting the relationship of the yield stress, the initial elastic modulus and the yield strain and the strain rates is constructed. The invention has ingenious conception, saves the test time and greatly reduces the test cost.

Description

Construction method and application of polyethylene hyperbolic constitutive model
Technical Field
The invention relates to the technical field of construction of a hyperbola constitutive model, in particular to a construction method and application of a polyethylene hyperbola constitutive model.
Background
High density polyethylene (HDPE for short) is a viscoelastic plastic material and is widely used in urban medium and low pressure gas transmission and distribution pipelines at present. The HDPE pipeline is usually acted by medium and low pressure, the deformation rate of the HDPE pipeline is very small, the long-term service behavior and the service life evaluation of the pipeline are concerned, but a very reliable unified prediction model does not exist at present. In engineering application, people are used to test the strength of materials according to test standards and check the strength of pipelines according to the test standards, however, as high polymer materials such as HDPE have visco-elastic plasticity and the stress-strain response is related to time or loading rate, a great deal of research shows that the bearing strength of the high polymer materials such as HDPE is reduced along with the reduction of strain rate, for the situation of extremely low strain rate of practical engineering, the long-term performance of the materials is not conservative by directly adopting the test result under the condition of loading rate of a test chamber, so that a very slow strain control test needs 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 chamber. For reasons of test duration and expense, few laboratories have conducted strain control tests at very low rates, and it is therefore necessary to predict the stress-strain response at very low strain rates based on short-term laboratory tests and effective constitutive models.
Disclosure of Invention
Based on the above, the invention provides a method for constructing a polyethylene hyperbolic constitutive model, which comprises the steps of carrying out quasi-static tensile tests on a polyethylene gas pipe in different strain rate ranges, analyzing mechanical behaviors related to the strain rate, determining the yield stress, the initial elastic modulus and the correlation rule of the yield strain and the strain rate of the material, and describing the stress-strain relation related to the rate by using the hyperbolic constitutive model, so as to predict the yield strength, the initial elastic modulus, the yield strain and other tensile mechanical properties under the condition of extremely low strain rate through the short-time constant strain rate tensile test.
The invention is realized by the following technical inventions:
a method for constructing a polyethylene hyperbolic constitutive model comprises the following steps:
s1: setting failure stress according to rate-dependent stress-strain hyperbolic constitutive model
Figure BDA0003299349350000011
Wherein R isfIs a parameter of the failed material, and Rf<1 and setting the failure stress equal to the yield stress, i.e.
Figure BDA0003299349350000012
Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula derivation:
Figure BDA0003299349350000021
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σyIn order to be able to obtain a yield stress,
Figure BDA0003299349350000022
is the strain rate, RfIs a failure material parameter;
based on polyethylene having a true stress at yield equal to the yield stress, i.e.
Figure BDA0003299349350000023
Will be provided with
Figure BDA0003299349350000024
Substituting said pass yield stress, onsetThe modulus of elasticity and the rate expressed as true stress-true strain are related to a hyperbolic constitutive model, yielding a yield strain model expressed by yield stress and initial modulus of elasticity:
Figure BDA0003299349350000025
in the formula (I), the compound is shown in the specification,
Figure BDA0003299349350000026
is a relation between yield stress and strain rate,
Figure BDA0003299349350000027
is a power law relation of initial elastic modulus and strain rate,
Figure BDA0003299349350000028
is the strain rate, RfIs a failure material parameter;
performing a tensile test on 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 relational expression of yield stress and 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 of 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 represented by the true stress-true strain obtained in the step S1 and the relation of the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: and substituting the yield stress-strain rate relation obtained in the step S1, the initial elastic modulus-strain rate power law relation obtained in the step S2 and the value of the failure material parameter into the yield strain model represented by the yield stress and the initial elastic modulus obtained in the step S1 to obtain a polyethylene hyperbolic constitutive model represented by the yield strain and the strain rate.
Compared with the prior art, the method has the advantages that the yield stress, the initial elastic modulus and the strain rate of the polyethylene are determined according to the correlation rule among the yield stress, the initial elastic modulus and the strain rate, the stress-strain relation related to the rate is described on the basis of the hyperbolic constitutive model, and the polyethylene hyperbolic constitutive model reflecting the relation between the yield strain 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 under different strain rates, particularly under the condition of extremely low strain rate, are predicted.
Further, in step S1, the construction process of the rate-dependent hyperbolic constitutive model represented by the yield stress, the initial elastic modulus, and the true stress-true strain is as follows:
firstly, the following rate-dependent stress-strain hyperbolic constitutive model is prepared:
Figure BDA0003299349350000031
simplified processing is carried out to obtain a simplified rate-dependent stress-strain hyperbolic constitutive model:
Figure BDA0003299349350000032
wherein epsilon is axial strain,
Figure BDA0003299349350000033
for strain rate, a and b are respectively strain rate related material parameters;
the epsilon tends to be infinite to obtain the ultimate stress of the polyethylene
Figure BDA0003299349350000034
Setting the failure stress of the polyethylene to be tested as
Figure BDA0003299349350000035
RfIs a parameter of the failed material, and Rf<1；
And carrying out differential operation on the simplified rate-related stress-strain hyperbolic constitutive model to obtain the tangent modulus of the simplified rate-related stress-strain hyperbolic constitutive model:
Figure BDA0003299349350000036
driving epsilon to 0 to obtain the initial elastic modulus of the polyethylene
Figure BDA0003299349350000037
The simplified rate-dependent stress-strain hyperbolic constitutive model can be written as:
Figure BDA0003299349350000038
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σfIn order to be a failure stress,
Figure BDA0003299349350000039
is the strain rate, RfIs a failure material parameter;
stress at failure
Figure BDA00032993493500000310
Deriving from said simplified rate-dependent stress-strain hyperbolic constitutive model a rate-dependent hyperbolic constitutive model represented by yield stress, initial modulus of elasticity and true stress-true strain:
Figure BDA00032993493500000311
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σyIn order to be able to obtain a yield stress,
Figure BDA00032993493500000312
is the strain rate, RfIs a failure material parameter.
Further, in step S1, the relation between the yield stress and the strain rate is constructed by:
measuring Poisson's ratio of polyethylene to be measured, and converting engineering stress and engineering strain obtained by performing a tensile test on the polyethylene to be measured under different strain rates into true stress and true strain through the following equations to obtain true stress-true strain curves under different strain rates:
εtrue＝ln(1+ε)
σtrue＝σ(1+ε)2μ
in the formula, epsilontrueIs true strain, ε is engineering strain, σtrueIs true stress, σ is engineering stress, μ is poisson's ratio;
then obtaining the yield stress of the polyethylene to be tested under different strain rates according to the true stress-true strain curve under different strain rates;
according to an Erying model (Ailin model), a linear function relation of yield stress and logarithmic strain rate is deduced:
Figure BDA0003299349350000041
in the formula (I), the compound is shown in the specification,
Figure BDA0003299349350000042
for reference viscoplastic strain rate, kBIs 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 fitting the yield stress of the polyethylene to be tested under different strain rates to determine the value of the first material parameter and the value of the second material parameter in a linear function relation of the yield stress and the logarithmic strain rate, so as to obtain a relation of the yield stress and the strain rate.
Further, in step S2, the relation of the yield stress with the change of strain rate is:
Figure BDA0003299349350000043
in the formula, σyIn order to be able to obtain a yield stress,
Figure BDA0003299349350000044
is the strain rate.
Further, in step S2, the building process of the relation between the initial elastic modulus and the strain rate and the determining process of the failure material parameter are:
defining the ratio of true strain to true stress as the instantaneous compliance, and then correlating the yield stress, initial modulus of elasticity, and true stress-true strain rate obtained in step S1 with a hyperbolic constitutive model:
Figure BDA0003299349350000045
conversion to instantaneous compliance versus true strain:
Figure BDA0003299349350000046
in the formula, epsilontrueIs true strain, σtrueIs true stress, σyTo yield stress, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000047
is the strain rate, RfIs a failure material parameter;
substituting the yield stress relation obtained in the step S1 with the strain rate change and the real stress and the real strain obtained in the step S1 under different strain rates into the relation between 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 according to the law that the initial elastic modulus changes along with different strain rates, constructing and obtaining a power law relational expression of the initial elastic modulus and the strain rate:
Figure BDA0003299349350000051
in the formula, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000052
is the strain rate, C is the third material parameter, D is the fourth material parameter; and C and D are determined according to the rule that the initial elastic modulus changes along with the strain rate.
Further, in step S2, the power law relation between the initial elastic modulus and the strain rate is:
Figure BDA0003299349350000053
in the formula, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000054
is the strain rate.
Further, in step S3, the hyperbolic constitutive model of the polyethylene reflecting the relation between yield strain and strain rate is:
Figure BDA0003299349350000055
in the formula (I), the compound is shown in the specification,
Figure BDA0003299349350000056
represents the yield stress σyI.e. by
Figure BDA0003299349350000057
It is the yield stress and stress of polyethyleneA relationship to the variability, wherein A is a first material parameter and B is a second material parameter;
Figure BDA0003299349350000058
denotes the primary modulus of elasticity E0I.e. by
Figure BDA0003299349350000059
The power law relation between the initial elastic modulus and the strain rate of the polyethylene is shown, C is a third material parameter, and D is a fourth material parameter; rfIs a parameter of the failed material, and Rf<1。
Further, the hyperbolic constitutive model of the polyethylene reflecting the relation between yield strain and strain rate is as follows:
Figure BDA00032993493500000510
wherein epsilonyIn order to be able to yield to the strain,
Figure BDA00032993493500000511
is the strain rate.
Further, the method for constructing the polyethylene hyperbolic constitutive model further comprises the step S4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a calculated value of yield strain, comparing the 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, judging that the prediction result of the polyethylene hyperbolic constitutive model is reliable;
or, 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 RfSubstituting the value of (d) into the rate-dependent hyperbolic constitutive model expressed by the yield stress, the initial elastic modulus and the true stress-true strain obtained in step S1, obtaining a rate-dependent hyperbolic constitutive model expressed by the strain rate and the true stress-true strain, and then subjecting the strain toSubstituting a given strain rate into the hyperbolic constitutive model related to the rate represented by the yield stress, the initial elastic modulus and the true stress-true strain to calculate a true stress calculation value, and comparing the true stress calculation value with a true stress test value converted from the engineering stress measured by a tensile test to judge the reliability of the prediction result of the polyethylene hyperbolic constitutive model.
The invention also provides an application of the polyethylene hyperbolic constitutive model constructed according to the construction method, which comprises the following steps:
and (3) predicting the yield stress, the initial elastic modulus and the yield strain of the polyethylene by substituting any strain rate into the polyethylene hyperbolic constitutive model, thereby evaluating the tensile mechanical property of the polyethylene to be tested under the condition of the strain rate.
For a better understanding and practice, the invention is described in detail below with reference to the accompanying drawings.
Drawings
FIG. 1(a) is a graph of engineering stress versus engineering strain obtained from a constant strain rate tensile test conducted on a test specimen in accordance with an embodiment of the present invention; FIG. 1(b) is an engineering stress-engineering strain curve of a sample according to an embodiment of the present invention before stress softening and strain localization occurs during a tensile test;
FIG. 2 shows a sample of an embodiment of the invention at a strain rate of 10-3s-1Axial strain-time curves and transverse strain-time curves measured under the conditions;
FIG. 3 is a transverse strain-axial strain curve of a sample according to an embodiment of the present invention;
FIG. 4 is a true stress-true strain curve for samples of an embodiment of the present invention at different strain rates;
FIG. 5 is a linear relationship of yield stress at different strain rates during a tensile test for a sample according to an embodiment of the present invention;
FIG. 6 is a graph of instantaneous compliance versus true strain for a sample at different strain rates and model fitting according to an embodiment of the present invention;
FIG. 7 shows an embodiment of the present inventionInitial modulus of elasticity E of the test specimen of (2)0A curve that varies with strain rate;
FIG. 8 is a plot of yield strain versus strain rate for a sample 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 according to the present invention.
Detailed Description
In the prior art, the properties of pipes made of polyethylene are generally evaluated by the stress-strain behavior measured by tensile tests at different strain rates. However, when the tensile test is to be conducted under a relatively low strain rate condition, it is necessary to conduct the test for a long time, and when the strain rate is low
Figure BDA0003299349350000061
When the tensile test is carried out under the condition, the tensile test needs to be carried out for more than ten hours, and the strain rate is lower than 10-5s-1When the tensile test is carried out under the condition, a large amount of time and cost are consumed. Therefore, the applicant establishes a hyperbolic constitutive model of polyethylene by performing tensile tests on polyethylene to be tested under different strain rates and extrapolating a stress-strain constitutive model related to the rates to correlate the strain rates with the stress-strain relationship of the polyethylene to be tested. Therefore, by substituting any given strain rate into the constructed polyethylene hyperbolic constitutive model, the yield strain of the polyethylene to be tested can be predicted, and the performance of the polyethylene pipe at different strain rates can be predicted.
The construction process of the polyethylene hyperbolic constitutive model is further explained by the description of the experimental process.
The PE100 buried polyethylene gas pipeline is obtained, and is processed into a sample through numerical control milling, the model of the pipe is SDR11/Dn315 multiplied by 28.6, the brand of the raw material is P6006, and the PE buried polyethylene gas pipeline is produced by Hebei plastic pipeline manufacturing Limited liability company. The specimen size meets the requirements of ISO 527-2: 2012.
The samples were then subjected to constant strain rate tensile testing by a CSS44020 electronic tensile tester (manufactured by chandelian tester institute). Test at room temperature (23 ℃) and relative humidity 50% RH stripsThe method is carried out under the condition that a sample is stretched and deformed at a constant strain rate, the axial strain is measured by an extensometer, and the engineering strain rates applied to the sample are respectively 5 multiplied by 10-2s-1、10-2s-1、10-3s-1、10-4s-1And 10-5s-1. In order to measure the poisson's ratio of the sample, the axial and transverse strains during tensile deformation are measured simultaneously using Digital Image Correlation (DIC) techniques to obtain the poisson's ratio of the sample.
Referring to fig. 1(a), fig. 1(a) is a graph of engineering stress-engineering strain obtained by a constant strain rate tensile test on a sample. As can be seen from 1(a), when the strain increases to a certain value, stress softening and strain localization occur due to necking, and the subsequent engineering stress-engineering strain curve does not truly reflect the constitutive properties of the material. Therefore, applicants retained only the engineering stress-engineering strain curve prior to necking for subsequent analysis, as shown in fig. 1(b), which is the engineering stress-engineering strain curve prior to stress softening and strain localization of the specimen during the tensile test.
Referring to FIG. 2, FIG. 2 shows a sample with a strain rate of 10-3s-1Axial strain-time curves and transverse strain-time curves measured under the conditions. As can be seen from fig. 2, the axial strain measured by the extensometer is consistent with that measured by DIC techniques, so in other constant strain rate tensile tests, the applicant measured the axial strain using only the extensometer for stress-strain analysis.
The poisson's ratio of polyethylene can be measured as the ratio of transverse strain to axial strain, plotting transverse strain against axial strain in fig. 2, resulting in an axial strain-time curve and a transverse strain-time curve as shown in fig. 3. As can be seen from fig. 3, during the stretching process, the transverse strain of the sample changes in proportion to the axial strain, and the data is linearly fitted, so that the poisson ratio of the material is 0.456 according to the fitting slope.
As is clear from FIGS. 1(a) and 1(b), the strain rate is 10-5s-1～5×10-2s-1Under the condition that the strain experienced by the test specimen before neck-in yield reaches0.07-0.17, the deformation is large, so the applicant converts the test data corresponding to the engineering stress-engineering strain curve before yielding in the constant strain rate uniaxial tensile test shown in fig. 1(b) through an equation (1) and an equation (2) to obtain the true stress-true strain curve of the sample under different strain rates as shown in fig. 4.
εtrue＝ln(1+ε) (1)
σtrue＝σ(1+ε)2μ (2)
In the formula, epsilontrueIs true strain, ε is engineering strain, σtrueIs true stress, σ is engineering stress, and μ is Poisson's ratio.
As can be seen from fig. 1(a), the elongation process of the sample shows a neck-in yield behavior, and the peak stress before necking (i.e., the inflection point) is referred to as yield stress, and the corresponding strain is referred to as yield strain, which are all related to the load strain rate. Referring to FIG. 5, FIG. 5 is a linear relationship of yield stress of the sample under different strain rates during the tensile test, which shows the true yield stress σyIncreases linearly with increasing logarithmic strain rate. This strain rate dependence is manifested in many polymeric materials, in strain rates<0.1s-1Under the quasi-static loading condition, the Eying model (Ailin model) is satisfied.
The Eying model is firstly used for describing the chemical reaction rate, but is applied to the viscoelastic-plastic mechanical analysis of the high polymer material soon after being proposed, the viscoelasticity is considered as a thermal activation rate process, and the yield stress and the logarithmic strain rate at plastic yield satisfy a linear function relation, namely
Figure BDA0003299349350000081
In the formula (I), the compound is shown in the specification,
Figure BDA0003299349350000082
for reference viscoplastic strain rate, kBIs 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 parameterAnd (4) material parameters.
Fitting the experimental data of fig. 5 according to equation (3) yields the yield stress versus strain rate relationship shown in fig. 5:
Figure BDA0003299349350000083
in the formula, σyIn order to be able to obtain a yield stress,
Figure BDA0003299349350000084
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 three-axis mechanical response of consolidation and non-drainage of rock and soil media, Kondner and Duncan and the like provide a rate-dependent stress-strain hyperbolic constitutive model characterized by hyperbolic function, which is as follows:
Figure BDA0003299349350000085
in the formula, σ1-σ3Is the principal stress difference, epsilon is the axial strain, and a and b are the strain rate related material parameters, respectively.
Further, Merry and Scott apply equation (5) to strain rate dependent mechanical behavior analysis of polymeric materials. Suleiman et al further developed a focal-method hyperbolic constitutive model considering the linear correlation of parameters a and b.
For uniaxial tensile or compressive mechanical behavior, equation (5) reduces to:
Figure BDA0003299349350000086
wherein epsilon is axial strain,
Figure BDA0003299349350000087
is the strain rate.
From equation (6), when the axial strain ε tends to be infinite, the ultimate stress of the material is obtained
Figure BDA0003299349350000091
The ultimate stress is not present for a real material, so applicants set the failure stress of the material to be
Figure BDA0003299349350000092
Wherein R isf<And 1 is a failure material parameter.
Further, the equation (6) is differentiated to obtain the tangent modulus of the model as:
Figure BDA0003299349350000093
wherein epsilon is axial strain,
Figure BDA0003299349350000094
for strain rate, a and b are the strain rate related material parameters, respectively.
From equation (7), when the axial strain ε tends to be 0, the initial elastic modulus of the sample
Figure BDA0003299349350000095
As previously described, applicants have analyzed the uniaxial tensile stress-strain behavior of the test specimens using true stress and true strain. Taking into account the parameters
Figure BDA0003299349350000096
And
Figure BDA0003299349350000097
according to the above-derived ultimate strain of the material
Figure BDA0003299349350000098
And initial modulus of elasticity of the material
Figure BDA0003299349350000099
The hyperbolic constitutive model given in equation (6) can be written as:
Figure BDA00032993493500000910
in the formula, σtrueIs true stress,. epsilontrueTo yield strain, σfTo failure stress, E0In order to be the initial modulus of elasticity,
Figure BDA00032993493500000911
is the strain rate, RfIs a failure material parameter.
Since polyethylene yield means failure in engineering, applicant has established
Figure BDA00032993493500000912
Corresponding strain to failure epsilonfIs epsilonf＝εyThen a rate dependent hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain can be derived from equation (8):
Figure BDA00032993493500000913
in the formula, σtureIs true stress,. epsilontrueIs true strain, σyTo yield stress, E0In order to be the initial modulus of elasticity,
Figure BDA00032993493500000914
is the strain rate;
and, a yield strain model expressed by yield stress and initial elastic modulus:
Figure BDA00032993493500000915
in the formula, epsilonyTo yield strain, σyTo yield stress, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000101
is the strain rate, RfIs a failure material parameter;
further, equation (9) may be rewritten as:
Figure BDA0003299349350000102
in the formula, epsilontrueIs true strain, σtureIs true stress, σyTo yield stress, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000103
is the strain rate, RfIs a failure material parameter.
Applicants relate the ratio of true strain to true stress (. epsilon.) in equation (11)true/σtrue) Defined as 'instantaneous compliance', the true stress-true strain curve is converted into instantaneous compliance (epsilon)true/σtrue) True strain epsilontrueAnd (4) obtaining the instantaneous compliance-true strain curve of the sample under different strain rates as shown in the figure 6. As can be seen in fig. 6, at 10-5s-1To 5X 10-2s-1Within the range of strain rates, the instantaneous compliance and true strain satisfy a substantially linear relationship and can therefore be described by the model given in equation (11).
Equation (4) is substituted for equation (11), and the data in FIG. 6 are linearly fitted using equation (11), thereby determining the initial modulus of elasticity of the test specimens at different strain rates
Figure BDA0003299349350000104
And failure material parameter RfAs shown in table 1 below. As can be seen from Table 1, the failure material parameter RfBeing constant, the fitting results are shown as a solid line in fig. 8.
TABLE 1 model parameters E0And Rf
Figure BDA0003299349350000105
Referring to FIG. 7, FIG. 7 shows the initial modulus E of elasticity of the sample0Curve of variation with strain rate. As can be seen from fig. 7, in the log-log coordinate system, the initial elastic modulus increases linearly with the increase of the strain rate, and therefore, can be described by a power law model, and the relationship between the initial elastic modulus and the strain rate is:
Figure BDA0003299349350000106
in the formula, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000107
is the strain rate.
Further, equation (12) may be expressed as:
Figure BDA0003299349350000108
in the formula, E0In order to be the initial modulus of elasticity,
Figure BDA0003299349350000109
and C is a third material parameter, and D is a fourth material parameter.
The failure material parameter RfWhen equation (10) is substituted for 0.9, equation (4) and equation (12), a hyperbolic constitutive model of the polyethylene sample is obtained:
Figure BDA0003299349350000111
wherein epsilonyIn order to be able to yield to the strain,
Figure BDA0003299349350000112
is the strain rate.
Referring to fig. 8, fig. 8 is a graph of yield strain versus strain rate for a sample according to an embodiment of the present invention. The solid line in the graph is a calculated value of yield strain calculated by equation (13), and the scatter point in the graph is at 5 × 10-2s-1、10-2s-1、10-3s-1、10-4s-1And 10-5s-1Yield strain test value measured by tensile test at 8 strain rate. As can be seen from fig. 8, for any given strain rate, the calculated yield strain calculated by the polyethylene hyperbolic constitutive model of equation (13) is compared with the test value obtained by the tensile test, and the calculated model value matches well with the test value.
A parameter RfSubstituting equation (9) with 0.9, equation (4), and equation (12), the rate-dependent hyperbolic constitutive model represented by strain rate and true stress-true strain is:
Figure BDA0003299349350000113
referring to fig. 6, the solid line in the graph is the calculated true stress value obtained by equation (14), and the scatter in the graph is the test true stress value converted from the engineering stress measured by the tensile test, and it can be seen from fig. 6 that the calculated model value and the test value also agree well.
In conclusion, the prediction result of the polyethylene hyperbolic constitutive model is reliable.
In practical application, any strain rate can be substituted into the hyperbolic constitutive model of equation (13) to calculate a calculated yield strain value, and the calculated yield strain 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. Or converting engineering strain measured by a tensile test of a sample at any strain rate into true strain, substituting the true strain into equation (14), calculating a true stress calculated value, and comparing the true stress calculated value with a true stress test value converted from the engineering stress measured by the tensile test to judge the reliability of the prediction result of the polyethylene hyperbolic constitutive model. It should be noted that, the user can set the acceptable range of the error between the calculated value and the test value as required, and if the error is within the acceptable range, the prediction result of the hyperbolic constitutive model of polyethylene is reliable.
Referring to fig. 9, fig. 9 is a schematic flow chart summarizing the method for constructing the hyperbolic constitutive model of polyethylene according to the present invention based on the above-mentioned whole experimental process, wherein the method comprises the following steps:
s1: setting failure stress according to rate-dependent stress-strain hyperbolic constitutive model
Figure BDA0003299349350000114
Wherein R isfIs a parameter of the failed material, and Rf<1 and setting the failure stress equal to the yield stress, i.e.
Figure BDA0003299349350000121
Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula derivation:
Figure BDA0003299349350000122
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σyIn order to be able to obtain a yield stress,
Figure BDA0003299349350000123
is the strain rate, RfIs a failure material parameter;
based on polyethylene having a true stress at yield equal to the yield stress, i.e.
Figure BDA0003299349350000124
Will be provided with
Figure BDA0003299349350000125
Substituting the yield stress, the initial elastic modulus and the rate correlation hyperbolic constitutive model expressed by the true stress-true strain into the yield strain model expressed by the yield stress and the initial elastic modulus:
Figure BDA0003299349350000126
in the formula (I), the compound is shown in the specification,
Figure BDA0003299349350000127
is a relation between yield stress and strain rate,
Figure BDA0003299349350000128
is a power law relation of initial elastic modulus and strain rate,
Figure BDA0003299349350000129
is the strain rate, RfIs a failure material parameter;
performing a tensile test on 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 relational expression of yield stress and 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 of 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 represented by the true stress-true strain obtained in the step S1 and the relation of the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: substituting the yield stress-strain rate relation obtained in the step S1, the power law relation of 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:
Figure BDA00032993493500001210
in the formula (I), the compound is shown in the specification,
Figure BDA00032993493500001211
represents the yield stress σyI.e. by
Figure BDA00032993493500001212
The material is a relational expression of yield stress and strain rate of polyethylene, wherein A is a first material parameter, and B is a second material parameter;
Figure BDA00032993493500001213
denotes the primary modulus of elasticity E0I.e. by
Figure BDA00032993493500001214
The power law relation between the initial elastic modulus and the strain rate of the polyethylene is shown, C is a third material parameter, and D is a fourth material parameter; rfIs a parameter of the failed material, and Rf<1。
Further, the construction method further comprises the following step of verifying the reliability of the polyethylene hyperbolic constitutive model:
s4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a calculated value of yield strain, comparing the 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, judging that the prediction result of the polyethylene hyperbolic constitutive model is reliable.
Or, 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 RfSubstituting the obtained ratio correlation hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain into the step S1 to obtain a pass strain ratio and a true strainRate-related hyperbolic constitutive model of force-true strain representation:
Figure BDA0003299349350000131
in the formula, σtrueIs true stress,. epsilontrueIn order to be able to yield to the strain,
Figure BDA0003299349350000132
for the strain rate, A is a first material parameter, B is a second material parameter, RfIs a failure material parameter, C is a third material parameter, and D is a fourth material parameter;
and then substituting any given strain rate into the hyperbolic constitutive model related to the rate represented by the yield stress, the initial elastic modulus and the true stress-true strain to calculate a true stress calculation value, and then comparing the true stress calculation value with a true stress test value converted from the engineering stress measured by a tensile test to judge the reliability of the prediction result of the polyethylene hyperbolic constitutive model.
Based on the obtained polyethylene hyperbolic constitutive model reflecting the relation between yield strain and strain rate 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 of the high-density polyethylene pipe material, such as yield strength, the initial elastic modulus, the yield strain and the like under different strain rate conditions can be predicted.
It should be noted that the method for constructing the hyperbolic constitutive model of polyethylene according to the present invention is also applicable to other viscoelastic plastic materials, and the steps of the method for constructing the hyperbolic constitutive model of polyethylene are the same as those of the embodiment of the present invention, except that some parameters need to be re-fitted and determined according to the test results, including:
(1) when the method is used for constructing a hyperbola constitutive model of other viscoelastic-plastic materials, the value of the first material parameter a and the value of the second material parameter B in the equation (3) are different from those of the embodiment, and the first material parameter a and the second material parameter B of the viscoelastic-plastic material to be measured need to be obtained by satisfying a linear function relationship between the yield stress and the logarithmic strain rate of the material.
(2) Initial modulus of elasticity E in equation (8) when used for the construction of a hyperbolic constitutive model of other viscoelastic-plastic materials0And failure material parameter RfThe value of (A) is different from that of the specific embodiment, and the initial elastic modulus E needs to be determined through the instantaneous compliance-true strain curve and model fitting of the material under different strain rates0And failure material parameter Rf。
(3) Equation (12) initial modulus of elasticity E when used for the construction of a hyperbolic constitutive model of other viscoelastic-plastic materials0And strain rate
Figure BDA0003299349350000141
Unlike the present embodiment, the relationship (E) is determined by the initial modulus of elasticity E for the material0The initial elastic modulus E is deduced from the results obtained by fitting the curve of variation with strain rate0And strain rate
Figure BDA0003299349350000143
The empirical relationship of (2).
From the above (1), (2) and (3), when the method is used for constructing a hyperbola constitutive model of other viscoelastic-plastic materials, the hyperbola constitutive model of equation (13) is:
Figure BDA0003299349350000142
compared with the prior art, the method has the advantages that the correlation law among the yield stress, the initial elastic modulus model and the strain rate of the polyethylene is determined, the stress-strain relation related to the rate is described on the basis of 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 properties of the polyethylene under different strain rates, particularly under the condition of extremely low strain rate, are predicted.
The present invention is not limited to the above-described embodiments, and various modifications and variations of the present invention are intended to be included within the scope of the claims and the equivalent technology of the present invention if they do not depart from the spirit and scope of the present invention.

Claims (10)

1. A method for constructing a polyethylene hyperbolic constitutive model is characterized by comprising the following steps:
s1: setting failure stress according to rate-dependent stress-strain hyperbolic constitutive model
Figure FDA0003299349340000011
Wherein R isfIs a parameter of the failed material, and Rf< 1 and set the failure stress equal to the yield stress, i.e.
Figure FDA0003299349340000012
Obtaining a rate-related hyperbolic constitutive model expressed by yield stress, initial elastic modulus and true stress-true strain through calculation and formula derivation:
Figure FDA0003299349340000013
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σyIn order to be able to obtain a yield stress,
Figure FDA0003299349340000014
is the strain rate, RfIs a failure material parameter;
based on polyethylene having a true stress at yield equal to the yield stress, i.e.
Figure FDA0003299349340000015
Will be provided with
Figure FDA0003299349340000016
Substituting the yield stress, the initial elastic modulus and the rate correlation hyperbolic constitutive model expressed by the true stress-true strain into the yield strain model expressed by the yield stress and the initial elastic modulus:
Figure FDA0003299349340000017
in the formula (I), the compound is shown in the specification,
Figure FDA0003299349340000018
is a relation between yield stress and strain rate,
Figure FDA0003299349340000019
is a power law relation of initial elastic modulus and strain rate,
Figure FDA00032993493400000110
is the strain rate, RfIs a failure material parameter;
performing a tensile test on 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 relational expression of yield stress and 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 of 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 represented by the true stress-true strain obtained in the step S1 and the relation of the yield stress and the strain rate obtained in the step S1, and determining the value of the failure material parameter;
s3: and substituting the yield stress-strain rate relation obtained in the step S1, the initial elastic modulus-strain rate power law relation obtained in the step S2 and the value of the failure material parameter into the yield strain model represented by the yield stress and the initial elastic modulus obtained in the step S1 to obtain a polyethylene hyperbolic constitutive model represented by the yield strain and the strain rate.
2. The method for constructing the polyethylene hyperbolic constitutive model according to claim 1, wherein:
in step S1, the construction process of the rate-dependent hyperbolic constitutive model represented by the yield stress, the initial elastic modulus, and the true stress-true strain is as follows:
firstly, the following rate-dependent stress-strain hyperbolic constitutive model is prepared:
Figure FDA0003299349340000021
simplified processing is carried out to obtain a simplified rate-dependent stress-strain hyperbolic constitutive model:
Figure FDA0003299349340000022
wherein epsilon is axial strain,
Figure FDA0003299349340000023
for strain rate, a and b are respectively strain rate related material parameters;
the epsilon tends to be infinite to obtain the ultimate stress of the polyethylene
Figure FDA0003299349340000024
Setting the failure stress of the polyethylene to be tested as
Figure FDA0003299349340000025
RfIs a parameter of the failed material, and Rf＜1；
And carrying out differential operation on the simplified rate-related stress-strain hyperbolic constitutive model to obtain the tangent modulus of the simplified rate-related stress-strain hyperbolic constitutive model:
Figure FDA0003299349340000026
driving epsilon to 0 to obtain the initial elastic modulus of the polyethylene
Figure FDA0003299349340000027
The simplified rate-dependent stress-strain hyperbolic constitutive model can be written as:
Figure FDA0003299349340000028
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σfIn order to be a failure stress,
Figure FDA0003299349340000029
is the strain rate, RfIs a failure material parameter;
stress at failure
Figure FDA00032993493400000210
Deriving from said simplified rate-dependent stress-strain hyperbolic constitutive model a rate-dependent hyperbolic constitutive model represented by yield stress, initial modulus of elasticity and true stress-true strain:
Figure FDA00032993493400000211
in the formula, σtrueIs true stress,. epsilontrueIs true strain, E0Is the initial modulus of elasticity, σyIn order to be able to obtain a yield stress,
Figure FDA00032993493400000212
is the strain rate, RfIs a failure material parameter.
3. The method for constructing the polyethylene hyperbolic constitutive model according to claim 1, wherein:
in step S1, the yield stress-strain ratio relation is constructed by:
measuring Poisson's ratio of polyethylene to be measured, and converting engineering stress and engineering strain obtained by performing a tensile test on the polyethylene to be measured under different strain rates into true stress and true strain through the following equations to obtain true stress-true strain curves under different strain rates:
εtrue＝ln(1+ε)
σtrue＝σ(1+ε)2μ
in the formula, epsilontrueIs true strain, ε is engineering strain, σtrueIs true stress, σ is engineering stress, μ is poisson's ratio;
then obtaining the yield stress of the polyethylene to be tested under different strain rates according to the true stress-true strain curve under different strain rates;
according to an Erying model, a linear function relation of yield stress and logarithmic strain rate is deduced:
Figure FDA0003299349340000031
in the formula (I), the compound is shown in the specification,
Figure FDA0003299349340000032
for reference viscoplastic strain rate, kBIs 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 fitting the yield stress of the polyethylene to be tested under different strain rates to determine the value of the first material parameter and the value of the second material parameter in a linear function relation of the yield stress and the logarithmic strain rate, so as to obtain a relation of the yield stress and the strain rate.
4. The method for constructing a polyethylene hyperbolic constitutive model according to claim 3, characterized in that:
in step S2, the relationship between the yield stress and the strain rate is:
Figure FDA0003299349340000033
in the formula, σyIn order to be able to obtain a yield stress,
Figure FDA0003299349340000034
is the strain rate.
5. The method for constructing the polyethylene hyperbolic constitutive model according to claim 1, wherein:
in step S2, the process of constructing the relationship between initial modulus of elasticity and strain rate and the process of determining the failure material parameter are:
defining the ratio of true strain to true stress as the instantaneous compliance, and then correlating the yield stress, initial modulus of elasticity, and true stress-true strain rate obtained in step S1 with a hyperbolic constitutive model:
Figure FDA0003299349340000035
conversion to instantaneous compliance versus true strain:
Figure FDA0003299349340000036
in the formula, epsilontrueIs true strain, σtrueIs true stress, σyTo yield stress, E0In order to be the initial modulus of elasticity,
Figure FDA0003299349340000041
is the strain rate, RfIs a failure material parameter;
substituting the yield stress relation obtained in the step S1 with the strain rate change and the real stress and the real strain obtained in the step S1 under different strain rates into the relation between 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 according to the law that the initial elastic modulus changes along with different strain rates, constructing and obtaining a power law relational expression of the initial elastic modulus and the strain rate:
Figure FDA0003299349340000042
in the formula, E0In order to be the initial modulus of elasticity,
Figure FDA0003299349340000043
is the strain rate, C is the third material parameter, D is the fourth material parameter; and C and D are determined according to the rule that the initial elastic modulus changes along with the strain rate.
6. The method for constructing a polyethylene hyperbolic constitutive model according to claim 5, characterized in that:
in step S2, the power law relation between the initial elastic modulus and the strain rate is:
Figure FDA0003299349340000044
in the formula, E0In order to be the initial modulus of elasticity,
Figure FDA0003299349340000045
is strain ofAnd (4) rate.
7. The method for constructing the polyethylene hyperbolic constitutive model according to claim 1, wherein:
in step S3, the hyperbolic constitutive model of the polyethylene reflecting the relationship between yield strain and strain rate is:
Figure FDA0003299349340000046
in the formula (I), the compound is shown in the specification,
Figure FDA00032993493400000412
represents the yield stress σyI.e. by
Figure FDA0003299349340000047
The material is a relational expression of yield stress and strain rate of polyethylene, wherein A is a first material parameter, and B is a second material parameter;
Figure FDA0003299349340000048
denotes the primary modulus of elasticity E0I.e. by
Figure FDA0003299349340000049
The power law relation between the initial elastic modulus and the strain rate of the polyethylene is shown, C is a third material parameter, and D is a fourth material parameter; rfIs a parameter of the failed material, and Rf＜1。
8. The polyethylene hyperbolic constitutive model of claim 7, characterized in that:
the polyethylene hyperbolic constitutive model reflecting the relation between yield strain and strain rate is as follows:
Figure FDA00032993493400000410
wherein epsilonyIn order to be able to yield to the strain,
Figure FDA00032993493400000411
is the strain rate.
9. The method for constructing the polyethylene hyperbolic constitutive model according to claim 1, wherein:
further comprising step S4: substituting any given strain rate into the polyethylene hyperbolic constitutive model to calculate a calculated value of yield strain, comparing the 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, judging that the prediction result of the polyethylene hyperbolic constitutive model is reliable;
or, 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 RfSubstituting the value of (D) into the yield stress, initial elastic modulus and rate-related hyperbolic constitutive model represented by true stress-true strain obtained in step S1 to obtain a rate-related hyperbolic constitutive model represented by strain rate and true stress-true strain, substituting any given strain rate into the rate-related hyperbolic constitutive model represented by yield stress, initial elastic modulus and true stress-true strain to calculate a true stress calculation value, comparing the true stress calculation value with a true stress test value obtained by converting engineering stress measured by a tensile test, and judging the reliability of the polyethylene hyperbolic constitutive model prediction result.
10. The application of the polyethylene hyperbolic constitutive model constructed according to the construction method of any one of claims 1 to 9, wherein the application comprises the following steps:
and predicting the yield stress, the initial elastic modulus and the yield strain of the polyethylene by substituting any strain rate into the polyethylene hyperbolic constitutive model.
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