A method for constructing a mechanical model of high polymer rheological behavior considering temperature effect

Abstract

The application discloses a method for constructing a mechanical model of high polymer rheological behavior considering temperature effect, and specific steps are as follows: S1, a temperature-dependent fractional derivative constitutive model is established according to the change of high polymer viscoelastic properties in a creep or relaxation process; S2, a linear relationship between a fractional order alpha of the temperature-dependent fractional derivative constitutive model and temperature T is constructed; S3, a function relationship between material parameters E of the temperature-dependent fractional derivative constitutive model and temperature T is constructed; and S4, the change of mechanical properties of the high polymer in the creep or relaxation process at different temperatures is described by using the established temperature-dependent fractional derivative constitutive model, and model parameters are determined by fitting experimental data. α The application provides a temperature-dependent fractional derivative constitutive model which is convenient to apply, clear in physical concept and capable of meeting accuracy requirements, so as to solve the problem that there is no simple and high-precision theoretical model for describing the temperature-related rheological behavior of high polymers.

Description

Technical Field

[0001] This invention belongs to the field of polymer mechanical behavior modeling technology, specifically relating to a method for constructing a mechanical model of polymer rheological behavior that considers temperature effects. Background Technology

[0002] Polymer materials, due to their outstanding properties, have been widely used in industry and agriculture, and are still being further studied to explore their potential applications in advanced technologies such as microelectronics, aerospace, and soft robotics. Among these, the mechanical properties of polymers, especially their mechanical behavior considering temperature effects, are crucial for promoting their widespread application. The deformation response of polymers at different temperatures indicates that creep and relaxation are their main failure mechanisms; therefore, the exploration and research of the rheological behavior of polymers has attracted widespread interest.

[0003] Currently, creep or relaxation experimental results for various polymer materials have been widely reported, such as high-density polyethylene (HDPE), polypropylene (PP), polycarbonate (PC), polymethyl methacrylate (PMMA), and polyethylene terephthalate (PET). Experimental results show that both creep and relaxation responses exhibit nonlinearity and strong temperature dependence. In this context, constitutive models are needed to describe temperature-sensitive mechanical behavior. However, existing constitutive models present complex mathematical forms when describing temperature-dependent creep or relaxation behavior. Therefore, it is essential to construct an effective, simple, and accurate model to characterize the nonlinear evolution of temperature-dependent creep strain or relaxation stress. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention proposes a mechanical model construction method for polymer rheological behavior that considers temperature effects. It provides a temperature-dependent fractional derivative constitutive model that is easy to apply, has clear physical concepts, and meets the accuracy requirements, thereby solving the problem of lacking a simple and accurate theoretical model to describe the temperature-related rheological behavior of polymers.

[0005] The main technical solution adopted in this invention is as follows:

[0006] A method for constructing a mechanical model of polymer rheological behavior considering temperature effects, the specific steps of which are as follows:

[0007] S1: Based on the changes in the viscoelastic properties of polymers during creep or relaxation, establish a temperature-dependent fractional derivative constitutive model;

[0008] S2: Constructing a linear relationship between the fractional order α and temperature T of a temperature-dependent fractional derivative constitutive model;

[0009] S3: Constructing the material parameters Eθ of the temperature-dependent fractional derivative constitutive model α The functional relationship with temperature T;

[0010] S4: The established temperature-dependent fractional derivative constitutive model is used to describe the changes in mechanical properties of polymers during creep or relaxation at different temperatures. The model parameters are determined by fitting experimental data.

[0011] Preferably, the temperature-dependent fractional derivative constitutive model constructed in S1 is shown in equation (1):

[0012]

[0013] In equation (1), T is temperature, E is elastic modulus, θ is relaxation time, t is loading time, σ(T,t) and ε(T,t) are stress and strain at temperatures T and t, respectively, α(T) is the order of the fractional derivative at temperature T, and d α / dt α Let be the symbol for the fractional derivative, which is defined as:

[0014]

[0015] In equation (2), f(·) is an arbitrary function, f′(·) represents the first derivative, τ is the integration variable, and Γ(·) is the gamma function, which is specifically defined as follows:

[0016]

[0017] Here, Re(α) represents the real part of the complex number α.

[0018] Preferably, when the external force is a constant control stress σ0, the formula (1) is substituted to describe the creep deformation of the polymer considering the temperature effect, as shown in formula (4):

[0019]

[0020] Where J(T,t) represents the creep compliance at temperature T and time t.

[0021] Preferably, when the external force is a constant control stress σ0, the corresponding polymer creep deformation at a known specific temperature T0 is as shown in formula (5):

[0022] Substituting T = T0 into formula (4) and performing logarithmic transformation, we obtain formula (5):

[0023]

[0024] Preferably, when the external force is a constant control strain ε0, the formula (1) is substituted to describe the relaxation behavior of the polymer considering the temperature effect, as shown in formula (6):

[0025]

[0026] Where G(T,t) is the relaxation modulus at temperature T and time t.

[0027] Preferably, when the external force is a constant control strain ε0, the corresponding polymer relaxation behavior at a known specific temperature T0 is as shown in formula (7):

[0028] Substituting T = T0 into formula (6) and performing logarithmic transformation, we obtain formula (7):

[0029]

[0030] Preferably, in S2, the linear relationship between the fractional order α of the temperature-dependent fractional derivative constitutive model and the temperature T is shown in Equation (8):

[0031] α(T)=kT+b,0<α<1 (8);

[0032] Where k and b are fixed parameters.

[0033] Preferably, in S3, the material parameter Eθ of the temperature-dependent fractional derivative constitutive model... α The functional relationship with temperature T is shown in formula (9):

[0034] ln(Eθ α )=-λT+v (9);

[0035] Where λ and v are fixed parameters.

[0036] Preferably, the specific method for determining the model parameters of the fractional derivative constitutive model in S4, which describes the creep deformation of polymers considering temperature effects, is as follows:

[0037] First, the experimental data of polymer creep deformation at different temperatures after logarithmic processing are fitted using formula (5) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer creep deformation at different temperatures are fitted using formula (4) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

[0038] Preferably, the specific method for determining the model parameters of the fractional derivative constitutive model in S4, which describes the relaxation behavior of polymers considering temperature effects, is as follows:

[0039] First, the experimental data of polymer relaxation behavior at different temperatures after logarithmic processing are fitted using formula (7) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer relaxation behavior at different temperatures are fitted using formula (6) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

[0040] Beneficial effects: This invention proposes a method for constructing a mechanical model of polymer rheological behavior that considers temperature effects, which has the following advantages:

[0041] (1) Compared with traditional constitutive models, the mechanical model constructed in this invention can be used to characterize the nonlinear evolution of creep strain or relaxation stress that considers temperature dependence. The model is simple in form, easy to apply, has high fitting accuracy, and the physical meaning of the model parameters is clear.

[0042] (2) The mechanical properties of the viscoelastic behavior of polymers can be characterized by fractional order α. Attached Figure Description

[0043] Figure 1 This is a flowchart illustrating the method for constructing the temperature-dependent fractional derivative constitutive model of the present invention.

[0044] Figure 2 This is a graph showing the experimental data of polymer creep at a specific temperature (45°C) in Example 1;

[0045] Figure 3 for Figure 2 The diagram shows the logarithmic representation of the polymer creep experimental data.

[0046] Figure 4 The fractional order α and material parameter Eθ describe the creep response of the polymer at different temperatures in Example 1. α Relationship with temperature T;

[0047] Figure 5 This is a diagram illustrating the temperature-dependent polymer creep deformation effect of the model in Example 1.

[0048] Figure 6 To verify the applicability of the model to temperature-dependent creep deformation of polymers;

[0049] Figure 7This is a graph showing the polymer relaxation experiment data at a specific temperature (5°C) in Example 2;

[0050] Figure 8 for Figure 7 The diagram shows the logarithmic representation of the polymer relaxation experiment data.

[0051] Figure 9 The fractional order α and material parameter Eθ are used to describe the relaxation behavior of the polymer at different temperatures in Example 2. α Relationship with temperature T;

[0052] Figure 10 This is a graph showing the effect of temperature-dependent relaxation behavior represented by the model in Example 2;

[0053] Figure 11 This serves to verify the applicability of the model to the temperature-dependent relaxation behavior of polymers. Detailed Implementation

[0054] To enable those skilled in the art to better understand the technical solutions in this application, the technical solutions in the embodiments of this application are clearly and completely described below. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of this application.

[0055] Example 1

[0056] A method for constructing a mechanical model of polymer rheological behavior that considers temperature effects, such as Figure 1 As shown, the specific steps for characterizing the creep deformation of polymers at different temperatures are as follows:

[0057] S1: Based on the changes in the viscoelastic properties of polymers during creep deformation, a temperature-dependent fractional derivative constitutive model for creep deformation is established. The specific steps are as follows:

[0058] S1-1: The temperature-dependent fractional derivative constitutive model is shown in equation (1):

[0059]

[0060] In equation (1), T is temperature, E is elastic modulus, θ is relaxation time, t is loading time, σ(T,t) and ε(T,t) are stress and strain at temperatures T and t, respectively, α(T) is the order of the fractional derivative at temperature T, and d α / dt α Let be the symbol for the fractional derivative, which is defined as:

[0061]

[0062] In equation (2), f(·) is an arbitrary function, f′(·) represents the first derivative, τ is the integration variable, and Γ(·) is the gamma function, which is specifically defined as follows:

[0063]

[0064] Here, Re(α) represents the real part of the complex number α.

[0065] S1-2: When the external force is a constant control stress σ0, it can be substituted into formula (1) to describe the creep deformation of polymers considering temperature effects, as shown in formula (4):

[0066]

[0067] Where J(T,t) represents the creep compliance at temperature T and time t.

[0068] S1-3: When the external force is a constant control stress σ0, the corresponding polymer creep deformation at a known specific temperature T0 is shown in formula (5):

[0069] Substituting T = T0 into formula (4) and performing logarithmic transformation, we obtain formula (5):

[0070]

[0071] S2: The linear relationship between the fractional order α of the temperature-dependent fractional derivative constitutive model and temperature T is shown in Equation (8):

[0072] α(T)=kT+b,0<α<1 (8);

[0073] Where k and b are fixed parameters.

[0074] S3: Material parameters Eθ of the temperature-dependent fractional derivative constitutive model α The functional relationship with temperature T is shown in formula (9):

[0075] ln(Eθ α )=-λT+v (9);

[0076] Where λ and v are fixed parameters.

[0077] S4: The specific method for determining the model parameters of the fractional derivative constitutive model used to describe the creep deformation of polymers considering temperature effects is as follows:

[0078] First, the experimental data of polymer creep deformation at different temperatures after logarithmic processing are fitted using formula (5) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer creep deformation at different temperatures are fitted using formula (4) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

[0079] Based on the mechanical model construction method for polymer creep behavior considering temperature effects provided in Example 1, a temperature-dependent fractional derivative constitutive model of creep behavior is constructed using experimental data on the creep behavior of polymers (bagasse-based high-density polyethylene composites), as follows:

[0080] like Figure 2 The figure shows the relationship between polymer creep strain and time at a certain temperature (45℃). The logarithm of the experimental data is shown below. Figure 3 As shown, the relationship between ln(ε) and ln(t) can be approximated as a straight line. Therefore, the value of the fractional order α can be obtained by fitting the ln(ε)-ln(t) data using formula (5), i.e., α = 0.1909. Following the above method, the fractional orders of four different temperatures (45℃, 55℃, 75℃, 85℃) were obtained respectively. Then, formula (8) was used to fit and determine the parameters k and b. The fitting results are shown below. Figure 4 As shown. Subsequently, based on the known α value and formula (4), experimental data on polymer creep behavior at different temperatures were fitted to obtain the material parameter Eθ corresponding to different temperatures (45℃, 55℃, 75℃, 85℃). α and ln(Eθ) α And according to formula (9), the parameters λ and v are determined by fitting, and the fitting result is as follows: Figure 4 As shown.

[0081] For the creep behavior of the polymer (bagasse-based high-density polyethylene composite), based on the parameter values ​​determined above, formulas (8) and (9) are substituted into formula (4) of the fractional derivative constitutive model to characterize the temperature-dependent creep behavior of the polymer. The characterization effect is as follows: Figure 5 As shown in the figure, the creep response of the polymer at 65℃ predicted by the fractional derivative constitutive model (the dashed line represents the fractional derivative constitutive model prediction) agrees well with the experimental data.

[0082] Based on the mechanical model construction method for polymer creep behavior considering temperature effects provided in Example 1, the applicability of the constructed temperature-dependent fractional derivative constitutive model of creep deformation in the temperature-related creep behavior of polymers is verified as follows:

[0083] like Figure 6 The figure shows the fitting results of the temperature-dependent creep mechanical behavior of the polymer material (polypropylene), and the parameters involved in the model are shown in Table 1.

[0084] Table 1. Values ​​of other physical parameters for polypropylene creep deformation.

[0085]

[0086] from Figure 6 As can be seen from the data, the fractional derivative constitutive model constructed according to Example 1 can well describe the experimental data of polypropylene, which demonstrates the applicability of the fractional derivative constitutive model to temperature-dependent creep deformation of polymer materials.

[0087] Example 2

[0088] A method for constructing a mechanical model of polymer rheological behavior that considers temperature effects, such as Figure 1 As shown, the specific steps for describing and characterizing the relaxation behavior of polymers at different temperatures are as follows:

[0089] S1: Based on the changes in the viscoelastic properties of the polymer during relaxation, a fractional derivative constitutive model of the temperature-dependent relaxation behavior is established. The specific steps are as follows:

[0090]

[0091] In equation (1), T is temperature, E is elastic modulus, θ is relaxation time, t is loading time, σ(T,t) and ε(T,t) are stress and strain at temperatures T and t, respectively, α(T) is the order of the fractional derivative at temperature T, and d α / dt α Let be the symbol for the fractional derivative, which is defined as:

[0092]

[0093] In equation (2), f(·) is an arbitrary function, f′(·) represents the first derivative, τ is the integration variable, and Γ(·) is the gamma function, which is specifically defined as follows:

[0094]

[0095] Here, Re(α) represents the real part of the complex number α.

[0096] S1-2: When the external force is a constant control strain ε0, substituting into equation (1) can be used to describe the relaxation behavior of polymers considering temperature effects, as shown in the formula:

[0097]

[0098] Where G(T,t) is the relaxation modulus at temperature T and time t.

[0099] S1-3: When the external force is a constant control strain ε0, the corresponding polymer relaxation behavior at a known specific temperature T0 is as shown in formula (7):

[0100] Substituting T = T0 into formula (6) and performing logarithmic transformation, we obtain formula (7):

[0101]

[0102] S2: Construct the linear relationship between the fractional order α and temperature T of the temperature-dependent fractional derivative constitutive model, as follows:

[0103] α(T)=kT+b,0<α<1 (8);

[0104] Where k and b are fixed parameters.

[0105] S3: Constructing the material parameters Eθ of the temperature-dependent fractional derivative constitutive model α The functional relationship with temperature T is shown in formula (9):

[0106] ln(Eθ α )=-λT+v (9);

[0107] Where λ and v are fixed parameters.

[0108] S4: The specific method for determining the model parameters of the fractional derivative constitutive model used to describe the relaxation behavior of polymers considering temperature effects is as follows:

[0109] First, the experimental data of polymer relaxation behavior at different temperatures after logarithmic processing are fitted using formula (7) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer relaxation behavior at different temperatures are fitted using formula (6) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

[0110] Based on the mechanical model construction method for polymer relaxation behavior considering temperature effects provided in Example 2, a temperature-dependent fractional derivative constitutive model for creep deformation is constructed using experimental data on the relaxation behavior of polymers (high-density polyethylene / linen felt composite materials), as follows:

[0111] like Figure 7As shown, this represents the relationship between the relaxation modulus of a polymer and time at a certain temperature (5℃). The logarithm of the experimental data is then presented as follows. Figure 8 As shown, the relationship between ln(G) and ln(t) can be approximated as a straight line. Therefore, the value of the fractional order α can be obtained from formula (7) and The value α is obtained by fitting ln(G)—ln(t) data, i.e., α = 0.07708. Following the above method, the fractional order of six temperature groups (5℃, 10℃, 15℃, 20℃, 25℃, 30℃) is obtained respectively. Then, the parameters k and b are determined by fitting using formula (8), and the fitting results are as follows: Figure 9 As shown. Subsequently, based on the known α value and formula (4), experimental data on the relaxation behavior of polymers at different temperatures were fitted to obtain the corresponding material parameters Eθ for different temperatures (5℃, 10℃, 15℃, 20℃, 25℃, 30℃). α and ln(Eθ) α And according to formula (9), the parameters λ and v are determined by fitting, and the fitting result is as follows: Figure 9 As shown.

[0112] For the relaxation behavior of the polymer (high-density polyethylene / linen felt composite), based on the parameter values ​​determined above, formulas (8) and (9) are substituted into the fractional derivative constitutive model formula (6) to characterize the temperature-dependent relaxation behavior of the polymer. The characterization effect is as follows: Figure 10 As shown in the figure, the relaxation response of the polymer at 35℃ and 40℃ predicted by the fractional derivative constitutive model (the dashed line represents the fractional derivative constitutive model prediction) is in good agreement with the experimental data.

[0113] Based on the mechanical model construction method for polymer relaxation behavior considering temperature effects provided in Example 2, the applicability of the constructed temperature-dependent fractional derivative constitutive model of relaxation behavior in the temperature-related relaxation behavior of polymers is verified as follows:

[0114] like Figure 11 As shown, the fitting results for the temperature-dependent relaxation mechanical behavior of the polymer material (tetrafunctional epoxy resin) are presented. The experimental data corresponding to low temperature (T≤(70+273.15)K) and high temperature (T>(70+273.15)K) are fitted separately, and the parameters involved in the model are shown in Table 2. (Note: Parameter subscript 1 represents the parameter value corresponding to low temperature, and subscript 2 represents the parameter value corresponding to high temperature.)

[0115] Table 2. Values ​​of other physical parameters for the relaxation behavior of tetrafunctional epoxy resins.

[0116]

[0117] from Figure 11As can be seen from the data, the fractional derivative constitutive model constructed according to Example 2 can well describe the experimental data of the tetrafunctional epoxy resin, which demonstrates the applicability of the fractional derivative constitutive model to the temperature-dependent relaxation behavior of polymer materials.

[0118] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for constructing a mechanical model of polymer rheological behavior considering temperature effects, characterized in that, The specific steps are as follows: S1: Based on the changes in the viscoelastic properties of polymers during creep or relaxation, a temperature-dependent fractional derivative constitutive model is established; wherein, the temperature-dependent fractional derivative constitutive model is shown in equation (1): （1）； In equation (1), T is temperature, E is elastic modulus, θ is relaxation time, t is loading time, σ(T, t) and ε(T, t) are stress and strain at temperatures T and t, respectively, and α(T) is the order of the fractional derivative at temperature T. Let be the symbol for the fractional derivative, which is defined as: （2）； In formula (2) For any function, Let τ denote the first derivative, and τ be the integration variable. The gamma function is defined as follows: （3）； in, Represented as complex number The real part; S2: Construct the linear relationship between the fractional order α and temperature T of the temperature-dependent fractional derivative constitutive model, where the linear relationship between the fractional order α and temperature T of the temperature-dependent fractional derivative constitutive model is shown in formula (8): （8）； Where k and b are fixed parameters; S3: Material parameters for constructing a temperature-dependent fractional derivative constitutive model The functional relationship with temperature T, where the material parameters of the temperature-dependent fractional derivative constitutive model are... The functional relationship with temperature T is shown in formula (9): （9）； Where λ and v are fixed parameters; S4: The established temperature-dependent fractional derivative constitutive model is used to describe the changes in mechanical properties of polymers during creep or relaxation at different temperatures. The model parameters are determined by fitting experimental data.

2. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 1, characterized in that, When the external force is a constant control stress σ0, the formula (1) is used to describe the creep deformation of polymers considering temperature effects, as shown in formula (4): （4）； Where J(T, t) represents the creep compliance at temperature T and time t.

3. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 2, characterized in that, When the external force is a constant control stress σ0, the corresponding polymer creep deformation at a known specific temperature T0 is shown in formula (5): Substituting T=T0 into formula (4) and performing logarithmic transformation, we obtain formula (5): （5）。 4. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 1, characterized in that, When the external force is a constant control strain ε0, the formula (1) is used to describe the relaxation behavior of polymers considering temperature effects, as shown in formula (6): （6）； Where G(T, t) is the relaxation modulus at temperature T and time t.

5. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 4, characterized in that, When the external force is a constant control strain ε0, the corresponding polymer relaxation behavior at a known specific temperature T0 is as shown in equation (7): Substituting T=T0 into formula (6) and performing logarithmic transformation, we obtain formula (7): （7）。 6. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 3, characterized in that, The specific method for determining the model parameters in S4, which describes the fractional derivative constitutive model considering temperature effects in polymer creep deformation, is as follows: First, the experimental data of polymer creep deformation at different temperatures after logarithmic processing are fitted using formula (5) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer creep deformation at different temperatures are fitted using formula (4) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

7. The method for constructing a mechanical model of polymer rheological behavior considering temperature effects according to claim 5, characterized in that, The specific method for determining the model parameters of the fractional derivative constitutive model in S4, which describes the relaxation behavior of polymers considering temperature effects, is as follows: First, the experimental data of polymer relaxation behavior at different temperatures after logarithmic processing are fitted using formula (7) to obtain the fractional order corresponding to different temperatures. Then, the linear relationship between the fractional order and temperature is determined according to formula (8). Subsequently, the experimental data of polymer relaxation behavior at different temperatures are fitted using formula (6) to obtain the material parameter Eθ corresponding to different temperatures. α And determine the material parameter Eθ according to formula (9). α The functional relationship with temperature.

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