Construction method of dielectric elastomer electro-mechanical coupling three-dimensional fractional order constitutive model

Abstract

The invention discloses a construction method of a dielectric elastomer force-electricity coupling three-dimensional fractional order constitutive model. The construction method comprises the following steps: S1, establishing a force-electricity coupling fractional order tensor constitutive relation under a constant strain rate condition; s2, considering the thickness change of the dielectric elastomer in a uniaxial deformation process, and correcting a force-electricity coupling fractional order tensor constitutive relation; s3, calculating the uniaxial deformation response of the whole stage under the condition of introducing cyclic load; s4, carrying out numerical processing on the model, and determining model parameters through fitting strain rate related force-electricity coupling experiment data; and S5, establishing a relation criterion between the model parameters and the strain rate, and constructing a strain rate-related force-electric coupling three-dimensional fractional order constitutive model. The invention provides a strain-rate-related three-dimensional fractional order constitutive model which is convenient to apply, clear in physical concept and capable of meeting the precision requirement aiming at the DE force-electricity coupling cyclic deformation behavior, so that the defects of an integer order model and a one-dimensional fractional order model in the process of describing the DE force-electricity coupling cyclic deformation behavior are overcome.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of dielectric elastomer force-electric coupling deformation behavior modeling, and particularly relates to a method for constructing a dielectric elastomer force-electric coupling three-dimensional fractional order constitutive model. BACKGROUND

[0002] As a typical electroactive polymer (EAP), dielectric elastomer (DE) has high flexibility and significant electro-actuation ability, and is an important intelligent soft material. The typical structure of DE is composed of a DE film and flexible electrodes coated on both sides. Under the action of an external voltage, the thickness of the film decreases and the planar area expands due to electrostatic attraction, and the DE material can exhibit obvious force-electric coupling deformation behavior under mechanical load. DE material has the advantages of large strain, fast response, high energy density, and light weight, and is widely used in soft robots, artificial muscles, sensors, actuators, energy harvesting devices and other fields. In recent years, its unique force-electric coupling mechanism and application potential have attracted widespread attention from academia and industry.

[0003] However, although the DE material system and preparation technology have made continuous progress, its practical application still faces key problems such as fracture failure, dielectric breakdown, electromechanical instability and energy loss. The root cause of these problems is that the deformation mechanism of DE material under multi-field coupling conditions has not been fully revealed, and the lack of relevant theoretical understanding has seriously restricted experimental research and performance optimization under extreme working conditions. Therefore, in-depth exploration of the physical nature of the force-electric coupling behavior of DE material and the establishment of a constitutive model that can accurately characterize its mechanical and electrical response not only have important theoretical significance, but also are a key link to promote the engineering application of this type of material.

[0004] At present, many studies have used integer order constitutive models to describe the mechanical behavior of DE material, but such models often have complex structure, numerous parameters, and limited accuracy in describing the actual deformation behavior. On the other hand, although fractional order constitutive models have shown potential in describing material viscoelasticity and rate-dependent characteristics, existing models are mostly limited to one-dimensional cases and have not effectively considered the thickness direction changes of DE under the combined action of voltage and mechanical load and its corresponding three-dimensional force-electric coupling response. Therefore, it is urgent to construct a fractional order constitutive model that can accurately describe the strain rate-dependent characteristics of dielectric elastomer, has simple parameters and is suitable for three-dimensional deformation behavior under force-electric coupling, to fill the existing theoretical gap and support its further application in high-performance intelligent devices. SUMMARY

[0005] In order to overcome the defects of the prior art, a method for constructing a dielectric elastomer force-electric coupling three-dimensional fractional order constitutive model is provided, and for the force-electric coupling cyclic deformation behavior of DE, a strain rate related three-dimensional fractional order constitutive model which is convenient to apply, has clear physical concept and can meet the accuracy requirement is provided, so as to solve the deficiencies of the integer order model and the one-dimensional fractional order model in describing the force-electric coupling cyclic deformation behavior of DE.

[0006] In order to achieve the above object, the present application provides the following technical scheme.

[0007] A method for constructing a dielectric elastomer force-electric coupling three-dimensional fractional order constitutive model, comprising the following steps:

[0008] S1: based on the dielectric elastomer force-electric coupling deformation mechanism, the force-electric coupling fractional order tensor constitutive relationship under the condition of constant strain rate is established;

[0009] S2: considering the influence of the thickness change of the dielectric elastomer on the Maxwell stress in the uniaxial deformation process, the force-electric coupling fractional order tensor constitutive relationship established in step S1 is corrected;

[0010] S3: based on the force-electric coupling fractional order tensor constitutive relationship corrected in step S2, the uniaxial deformation response in the whole stage under cyclic load is calculated;

[0011] S4: the model is numerically processed, and the model parameters are determined by fitting the strain rate related force-electric coupling experimental data;

[0012] S5: according to the correlation between the model parameters and the strain rate, the relationship criterion between the model parameters and the strain rate is established, and the strain rate related force-electric coupling three-dimensional fractional order constitutive model is constructed.

[0013] Preferably, in the S1,

[0014] Based on the dielectric elastomer force-electric coupling deformation mechanism, the initial force-electric coupling tensor constitutive relationship is constructed, as shown in formula (1):

[0015] (1);

[0016] Wherein, σ is the total Cauchy stress, σ vs is the viscoelastic stress tensor, σ Maxwell is the Maxwell stress tensor, which is defined as follows:

[0017] (2);

[0018] Wherein, ϵ0 is the vacuum permittivity, ϵ r is the relative dielectric constant of the material, e is the direction electric field intensity, and I is the unit tensor;

[0019] The viscoelastic mechanical behavior of the dielectric elastomer is characterized by an Abel stick, and σ vs represents formula (3) as shown:

[0020] (3);

[0021] In formula (3), η is a viscosity coefficient, t indicates the time corresponding to the present configuration, α is a fractional order parameter, and 0 < α < 1, D {α} is a fractional derivative of the deformation velocity tensor;

[0022] D {α} is defined as:

[0023] (4);

[0024] In formula (4), Γ( ) represents a gamma function, τ is a time corresponding to a configuration in a deformation history, and τ > 0, T represents a transpose of a matrix, and

[0025] F τ represents a relative deformation gradient tensor at τ, as shown in formula (5):

[0026] (5);

[0027] D represents a strain rate tensor, as shown in formula (6):

[0028] (6);

[0029] In formula (5) and formula (6), F is a deformation gradient tensor.

[0030] Preferably, in S2,

[0031] During uniaxial deformation, considering the influence of the thickness change of the dielectric elastomer on the Maxwell stress, formula (2) is modified as shown in formula (7):

[0032] (7);

[0033] Where d0 is the initial thickness of the dielectric elastomer material, λ3 is the elongation in the thickness change direction of the dielectric elastomer material, represents a directional excitation voltage;

[0034] Substituting formula (3) and formula (7) into formula (1), the total Cauchy stress tensor in the viscoelasticity and electric coupling fractional order constitutive relationship of the dielectric elastomer is finally represented as:

[0035] (8).

[0036] Preferably, in S3,

[0037] The direction of maximum piezoelectric coupling load is defined as the principal deformation direction. The evolution of the elongation λ in the principal deformation direction with time is expressed as:

[0038] (9);

[0039] In formula (9), the strain rate c remains constant during loading and unloading, t1 is the termination time of loading, t2 is the termination time of unloading, and t2 = 2t1 is satisfied; λ a represents the elongation in the principal deformation direction during the loading stage, and λ b represents the elongation in the principal deformation direction during the unloading stage.

[0040] When 0≤t<t1, the dielectric elastomer is in the loading stage, and the total Cauchy stress tensor is expressed by formula (8);

[0041] When t1≤t<t2, the dielectric elastomer is in the unloading recovery stage, and the expression of the total Cauchy stress tensor is shown in formula (10):

[0042] (10);

[0043] If the applied voltage value is u, formula (7) is analytically expressed as:

[0044] (11);

[0045] In the formula, u represents the vector voltage value in the thickness direction, so u=(0, 0, u), e i is a unit directional vector, i=1, 2 and 3;

[0046] For uniaxial deformation, the deformation gradient tensor of the dielectric elastomer is defined as:

[0047] (12);

[0048] Substitute formula (12) into formula (5) and formula (6) to calculate the relative deformation gradient tensor and the deformation velocity tensor D:

[0049] (13);

[0050] (14);

[0051] When 0≤t<t1, substitute formula (13) and formula (14) into formula (4), and then substitute formula (11) into formula (9) to obtain the specific Cauchy stress tensor expression during the loading stage:

[0052] (15);

[0053] When t1≤t<t2, the specific Cauchy stress tensor expression of the unloading stage is obtained by substituting formula (13) and formula (14) into formula (4) and then substituting formula (11) into formula (10) as follows:

[0054] (16);

[0055] For uniaxial deformation, the relationship between λ3 and λ is λ3=λ -1 / 2 , define η1 and α1 as the model parameters of the loading stage, and η2 and α2 as the model parameters of the unloading recovery stage, then based on formula (15) and (16), the Cauchy stress in the uniaxial tensile direction during the entire deformation stage is represented as:

[0056] (17);

[0057] The relationship between the nominal stress P and the Cauchy stress σ satisfies P=JσF -T , J=det(F), then the nominal stress P in the main deformation direction of the dielectric elastomer is represented as:

[0058] (18).

[0059] Preferably, in the S4, the non-elementary integral term contained in formula (18) is processed by using the hypergeometric function and the numerical integral method, and the model parameter values corresponding to multiple sets of strain rate dependent force-electric coupling experimental data are identified by combining the nonlinear least squares fitting.

[0060] Preferably, in the S5, the relationship between the model parameters identified in the S4 and the strain rate is defined as follows:

[0061] (19);

[0062] (20);

[0063] In the formula, p i and q i are constants, and n is the number of different strain rates;

[0064] Based on formula (18), the strain rate dependent force-electric coupling three-dimensional fractional order constitutive model in the uniaxial tensile direction is defined as follows:

[0065] (21).

[0066] Beneficial effects: This invention discloses a method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling, which has the following advantages:

[0067] (1) The present invention constructs a complete tensor constitutive model of dielectric elastomer (DE) mechanical-electric coupling cyclic deformation, which takes into account the influence of the thickness change of dielectric elastomer (DE) during deformation, and includes two components: mechanical field and electric field. This enables the model to adapt to two different load conditions: pure mechanical and mechanical-electric coupling. The model has clear physical meaning, simplified parameters, and high fitting accuracy.

[0068] (2) This invention establishes a relationship criterion between the parameters and strain rate of a three-dimensional fractional model, and the constructed model has a certain data prediction capability. Attached Figure Description

[0069] Figure 1 This is a flowchart summarizing the model construction method of the present invention;

[0070] Figure 2 This is the force-electric coupling deformation diagram of DE in step S1;

[0071] Figure 3 It is Maxwell stress σ Maxwell Graph showing the relationship between elongation λ3 in the thickness direction and its variation;

[0072] Figure 4 This is the stress-elongation relationship diagram of the DE constant strain rate cyclic loading response;

[0073] Figure 5 This is the elongation-time relationship of the DE constant strain rate cyclic loading response;

[0074] Figure 6 It is c=0.1s -1 A comparison of the two numerical calculation methods when α1=0.4;

[0075] Figure 7 It is c=0.05s -1 Verification diagram of the two numerical calculation methods when α1=0.3;

[0076] Figure 8 This is the application of the stress-elongation curve description of the strain rate correlation model of the VHB4905 type DE electromechanical coupling field when u=4000V;

[0077] Figure 9 This describes the relationship between the parameters η1 and α1 of the VHB4905 type DE loading stage model and c when u=4000V.

[0078] Figure 10This describes the relationship between the parameters η2 and α2 of the VHB4905 DE unloading stage model and c when u=4000V.

[0079] Figure 11 This is the data prediction of the strain rate correlation model under the VHB4905 type DE electromechanical coupling field when u=6000V;

[0080] Figure 12 It is an application of the stress-elongation curve description of the VHB4910 type DE strain rate correlation model under pure mechanical load;

[0081] Figure 13 This describes the relationship between the VHB4910 DE loading stage model parameters η1 and α1 and c under pure mechanical load.

[0082] Figure 14 This describes the relationship between the model parameters η2 and α2 and c of the VHB4910 DE unloading stage under pure mechanical load.

[0083] Figure 15 It is c=0.025s -1 A comparison diagram of the descriptions of the three-dimensional fractional-order model and the existing integer-order model;

[0084] Figure 16 It is c=0.1s -1 A comparison diagram of the descriptions of the three-dimensional fractional-order model and the existing integer-order model;

[0085] Figure 17 It is c=0.2s -1 A comparison diagram of the descriptions of the three-dimensional fractional-order model and the existing integer-order model;

[0086] Figure 18 It is λ max =150%, c=0.05s -1 A comparison diagram of the description of the three-dimensional model and the existing one-dimensional fractional-order model;

[0087] Figure 19 It is λ max =200%, c=0.05s -1 A comparison diagram of the description of the three-dimensional model and the existing one-dimensional fractional-order model;

[0088] Figure 20 It is λ max =250%, c=0.01s -1 A comparison diagram of the description of the three-dimensional model and the existing one-dimensional fractional-order model;

[0089] Figure 21 It is λ max =300%, c=0.01s -1A comparison diagram of the description of the three-dimensional model and the existing one-dimensional fractional-order model. Detailed Implementation

[0090] The present invention will now be described with reference to the accompanying drawings, and these improvements and modifications should also be considered within the scope of protection of the present invention.

[0091] Example 1

[0092] For dielectric elastomers (DE), such as Figure 1 As shown in Example 1, a three-dimensional fractional-order constitutive model of electromechanical coupling considering strain rate dependence is constructed. This model takes into account the influence of the thickness change of the electromechanical elastic body (DE) during deformation and can be used to describe the electromechanical coupling cyclic deformation behavior of the DE under different strain rates. The specific method is as follows:

[0093] S1: Analysis as follows Figure 2 The electromechanical coupling deformation mechanism of the dielectric elastic body (DE) shown is based on Maxwell's stress theory. The initial electromechanical coupling tensor constitutive relation is constructed as shown in Equation (1):

[0094] (1);

[0095] Where σ is the total Cauchy stress, σ vs Let σ be the viscoelastic stress tensor. Maxwell Let Maxwell's stress tensor be defined as follows:

[0096] (2);

[0097] Where ϵ0 is the vacuum permittivity, ϵ r Let be the relative permittivity of the material, e be the directional electric field intensity, and I be the unit tensor;

[0098] Considering the ability of a single Abel sticky pot to capture both elastic and viscous behavior, the viscoelastic mechanical behavior of a dielectric elastomer can be characterized by an Abel sticky pot element. Based on the fractional tensor derivative theory, σ in equation (1) vs It can be directly expressed as being proportional to the fractional derivative of the strain rate tensor, as shown in formula (3):

[0099] (3);

[0100] Where η is the viscosity coefficient, t refers to the time corresponding to the current configuration, α is a fractional-order parameter, and 0 < α < 1 (when 0 < α < 1, it can describe viscoelasticity between pure elasticity and viscosity); D {α} Let be the fractional derivative of the deformation velocity tensor.

[0101] Based on the memory characteristics of materials and the fractional short memory theory, D from time 0 to t... {α} Defined as:

[0102] (4);

[0103] Where Γ() represents the gamma function, τ refers to the time corresponding to a certain configuration in the deformation process and τ>0, T represents the transpose of the matrix, and F τ The relative deformation gradient tensor at time τ is shown in equation (5):

[0104] (5);

[0105] D represents the strain rate tensor, as shown in formula (6):

[0106] (6);

[0107] In equations (5) and (6), F is the deformation gradient tensor.

[0108] S2: Analysis as follows Figure 2 As shown in the electromechanical coupling deformation mechanism of the dielectric elastomer (DE), the thickness of the dielectric elastomer (DE) changes slowly during the electromechanical coupling deformation process. From the electric field strength e=u / d, where u represents the directional excitation voltage, and d under the thin flexible electrode can be considered as the instantaneous thickness of the material, the electric field strength changes continuously at each moment. This directly affects the Maxwell stress, and thus indirectly affects the total Cauchy stress. The instantaneous thickness of the dielectric elastomer can be quantitatively expressed by d=λ3d0, where d0 is the initial thickness of the dielectric elastomer material, and λ3 is the length in the direction of thickness change of the dielectric elastomer material. Therefore, formula (2) can be modified as follows:

[0109] (7);

[0110] Substituting equations (3) and (7) into equation (1), the total Cauchy stress tensor in the fractional-order constitutive relation of viscoelastic electro-coupling of dielectric elastomer is finally expressed as:

[0111] (8).

[0112] σ Maxwell The relationship between λ3 and λ3 is as follows: Figure 3 As shown, if thickness variation is neglected, the magnitude of Maxwell stress is independent of λ3. Figure 3It is always a horizontal straight line. Considering the thickness change, it can be clearly seen that at different voltages, as d decreases, during the process where λ3 gradually decreases from 1, the Maxwell stress gradually increases. Moreover, the greater the excitation voltage, the more obvious the increase amplitude, indicating that the influence of thickness change on the Maxwell stress cannot be ignored.

[0113] S3: Dielectric elastomers are in a working condition environment of cyclic loading and unloading for a long time in engineering applications (i.e., cyclic load), such as Figure 4 shown, and their loading path and unloading path show significant asymmetric mechanical responses. As Figure 5 shown is the elongation-time relationship of one-time constant strain rate loading and unloading. In this Example 1, the direction with the largest force-electric coupling load (transverse tensile direction) is defined as the main deformation direction, and the evolution relationship of the elongation λ in the main deformation direction during one-time loading and unloading with time is expressed as:

[0114] (9);

[0115] where the strain rate c remains constant during the loading and unloading process. As Figure 5 shown, it is the absolute value of the straight line slope in it. t1 is the termination moment of loading, t2 is the termination moment of unloading, and t2 = 2t1 is satisfied; λ a represents the elongation in the loading stage in the main deformation direction, and λ b represents the elongation in the unloading stage in the main deformation direction.

[0116] It can be seen from Equation (9) that the variation relationships of the elongation with time in the loading stage and the unloading stage are different.

[0117] When 0 ≤ t < t1, the dielectric elastomer (DE) is in the loading stage, and the total Cauchy stress tensor can be expressed by Equation (8).

[0118] When t1 ≤ t < t2, the dielectric elastomer is in the unloading and recovery stage. When it is loaded to the maximum deformation position and all loads are removed, considering the asymmetric mechanical responses of the two stages and the continuity of the mechanical responses throughout the loading and unloading process, in this Example 1, based on Equation (8), the expression of the total Cauchy stress tensor of the dielectric elastomer (DE) in the unloading and recovery stage is further derived as follows:

[0119] (10);

[0120] According to the classical approximation theory of thin-layer dielectrics, when the direction of the excitation voltage is perpendicular to the dielectric elastomer (DE) film and there is no voltage outside the film. If the applied voltage value is u, then Equation (7) can be analytically expressed as:

[0121] (11);

[0122] where \(u\) represents the vector voltage value in the thickness direction, so \(u=(0, 0, u)\), \(e\) i is the unit vector in the direction, \(i = 1, 2\) and \(3\);

[0123] For uniaxial deformation, the dielectric elastomer (DE) can be regarded as an incompressible material, and its deformation gradient tensor is defined as:

[0124] (12);

[0125] Substituting Equation (12) into Equations (5) and (6), the relative deformation gradient tensor and the deformation velocity tensor \(D\) can be calculated respectively:

[0126] (13);

[0127] (14);

[0128] When \(0\leq t\lt t_1\), substituting Equations (13) and (14) into Equation (4) and then together with Equation (11) into Equation (9), the specific expression of the Cauchy stress tensor in the loading stage is:

[0129] (15);

[0130] When \(t_1\leq t\lt t_2\), substituting Equations (13) and (14) into Equation (4) and then together with Equation (11) into Equation (10), the specific expression of the Cauchy stress tensor in the unloading stage is:

[0131] (16);

[0132] In the uniaxial deformation condition, the load mainly acts on the transverse tensile direction. According to the basic assumption of the free boundary condition, the Cauchy stress components in the other two directions can be approximately zero. Therefore, for uniaxial deformation, the relationship between \(\lambda_3\) and \(\lambda\) can be approximately \(\lambda_3=\lambda\) -1 / 2 . In addition, damage is inevitable during the loading and unloading process of DE, so the model parameter values in the two stages are not the same. Define \(\eta_1\) and \(\alpha_1\) as the model parameters in the loading stage, and \(\eta_2\) and \(\alpha_2\) as the model parameters in the unloading and recovery stage. Then based on Equations (15) and (16), the Cauchy stress in the uniaxial tensile direction during the entire deformation stage is expressed as:

[0133] (17);

[0134] In the subsequent steps of identifying model parameters and evaluating practicality, in this Example 1, the relationship between the nominal stress and elongation is selected as the key characterization index. The conversion relationship between the nominal stress \(P\) and the Cauchy stress \(\sigma\) satisfies \(P = J\sigma F\) -T(J = det(F), the dielectric elastomer can be regarded as an incompressible material. The product of the transverse, longitudinal, and axial elongations is 1, so J = det(F) = 1. Then the nominal stress P in the principal deformation direction of the dielectric elastomer is expressed as:

[0135] (18);

[0136] Physically, both the loading and unloading stages in Equation (18) contain two essentially different stress contributions, namely the mechanical response term of the mechanical load and the Maxwell stress term generated by the electric field excitation. Through the synergistic coupling effect, these two types of stress components jointly determine the complex nonlinear deformation behavior of the DE in the force-electric coupling field.

[0137] S4: Equation (18) contains complex integral terms that cannot be directly solved. The accuracy of dealing with such non-elementary integral problems directly determines the description accuracy of the model. In this Example 1, taking the loading stage as an example, when 0 ≤ t < t1, the non-elementary integral term M in Equation (18) is expressed as:

[0138] (19);

[0139] Next, the accuracy of the numerical calculation of Equation (19) is verified through two numerical processing methods: the Gaussian hypergeometric function and Gauss-Jacobi numerical integration.

[0140] The first is to use the Gaussian hypergeometric function for numerical processing:

[0141] Equation (19) can be analytically expressed by the following hypergeometric function:

[0142] (20);

[0143] In the formula, 2F1[ ] represents the symbol of the Gaussian hypergeometric function, which can be defined by the following power series:

[0144] (21);

[0145] In the formula, a and b are the top parameters, d is the bottom parameter, z is the function variable, and k is the number of terms in the power series expansion.

[0146] At this time, Equation (19) can be directly analytically expressed by Equation (20), and the value of the hypergeometric function can be directly calculated using the Hypergeom function. To facilitate the display of the numerical calculation results, two sets of typical parameter values (c = 0.1s -1 , α1 = 0.4 and c = 0.05s -1 , α1 = 0.3) are taken, and the visualization results of the analytical values of the hypergeometric function between 0 and 20 are plotted in Figure 6 and Figure 7 .

[0147] The second method uses Gauss-Jacobi numerical integration to directly approximate equation (19). Since equation (19) exhibits an algebraic singularity at the integration upper limit τ = t, a variable substitution of v = τ / t is performed to avoid increasing numerical computation errors. N Gauss-Jacobi nodes and their corresponding weights are selected, and the corresponding Gaussian quadrature formula is as follows:

[0148] (twenty two);

[0149] In the formula, v i Let w represent the i-th node. i This indicates the corresponding weight.

[0150] Similarly, the results of Gauss-Jacobi numerical integration are visualized on... Figure 6 and Figure 7 From. Figures 6-7 As can be seen from the curves, the Hypergeom function and the Gauss-Jacobi numerical integral curve almost overlap, indicating that the numerical calculation results of the two methods are very close, which also verifies the accuracy of the numerical calculation of equation (19) in this embodiment 1.

[0151] In this embodiment 1, Gauss-Jacobi numerical integration is used for numerical processing.

[0152] like Figures 8-10 As shown, this is a set of strain rate-related electromechanical coupling experimental data for DE at u=4000V. The model parameter values ​​corresponding to the set of experimental data were identified by combining the above numerical processing with nonlinear least squares fitting. The specific parameter values ​​are listed in Table 1.

[0153] Table 1. Parameters of the strain rate-dependent electromechanical coupling model when u=4000V

[0154]

[0155] S5: Analyzing the parameters in Table 1, the viscosity coefficient decreases with increasing strain rate, which is consistent with the characteristics of viscoelastic materials. It is worth noting that the fractional-order parameters change relatively little with strain rate, exhibiting low strain rate sensitivity. To reduce the number of built-in parameters in the model and improve parameter identification efficiency, the fractional-order parameters are averaged. Therefore, the relationship between model parameters and strain rate is defined as follows:

[0156] (twenty three);

[0157] (twenty four);

[0158] In the formula, pi and q i All are constants, and n is the number of different strain rates.

[0159] Based on equation (18), the three-dimensional fractional constitutive model of strain rate-dependent electromechanical coupling in the uniaxial tensile direction is defined as follows:

[0160] (25);

[0161] Using equation (24), the average values ​​of the fractional-order parameters during the loading and unloading stages were obtained as α1 = 0.3634 and α2 = 0.6395, respectively. The relationship between the viscosity coefficient and strain rate for the two stages was plotted on [the graph]. Figure 9 and Figure 10 The built-in parameters in the viscosity coefficient are further identified by fitting equation (23), as shown in Table 2. Substituting the above parameter values ​​into the model proposed by equation (25), the theoretical stress-elongation response curve is obtained.

[0162] like Figure 8 As shown, the fitting results of multiple strain rate-related electromechanical coupling experiments of VHB4905 DE at u=4000V are presented. The numerical simulation and experimental data of the three different strain rates show good agreement.

[0163] Table 2 Built-in parameters related to viscosity coefficient at u=4000V

[0164]

[0165] To further improve the model's ability to predict force-electric coupling data and apply it to degradation under purely mechanical loads, this invention compares the three-dimensional fractional-order model provided in Example 1 with existing integer-order models, highlighting the superior accuracy and parameter simplification of the proposed model. Furthermore, a comparison between the proposed three-dimensional model and a one-dimensional fractional-order model demonstrates the superiority of the proposed model in handling large DE cyclic deformations. The applications and comparisons of these models fully illustrate the practicality of this invention.

[0166] like Figure 11 The figures show multiple sets of strain rate-dependent electromechanical coupling experimental data and model prediction results for the VHB4905 DE at u=6000V. The electromechanical coupling cyclic deformation of the DE at u=6000V exhibits similar characteristics to the data at u=4000V. Keeping the electromechanical coupling parameter values ​​at u=4000V unchanged in Table 1, the u value in the proposed model is directly changed from 4000V to 6000V. The strain rate-dependent electromechanical coupling model proposed in this invention shows good prediction performance for the experimental data at u=6000V.

[0167] like Figure 12As shown, this is the experimental data and model fitting results of the viscoelastic deformation of VHB4910 DE under pure mechanical load with multiple strain rate correlations. For pure mechanical load, u=0 in equation (25), and the relationship between the model parameters and the strain rate still satisfies equations (23) and (24). The parameter values ​​of the model under pure mechanical load identified by fitting are listed in Table 3, and the built-in parameters related to the viscosity coefficient obtained by fitting equation (24) are listed in Table 4. The average values ​​of the fractional-order parameters obtained by equation (23) are α1=0.2995 and α2=0.5937, respectively. The relationship between the viscosity coefficient and the strain rate is plotted in Table 4. Figure 13 and Figure 14 middle.

[0168] Table 3 Viscosity coefficient and fractional-order parameters under pure mechanical load

[0169]

[0170] Table 4. Built-in parameters related to viscosity coefficient under pure mechanical load.

[0171]

[0172] like Figures 15-17 As shown, this is a comparison of the fitting results between the three-dimensional fractional-order model proposed in Example 1 and the existing integer-order model. Table 5 shows the coefficients of determination for the two models. R 2 Comparative calculations show that the fitting accuracy of the three-dimensional fractional-order model proposed in Example 1 is significantly better than that of the existing integer-order model.

[0173] Table 5. R-values ​​of the three-dimensional fractional-order model in Example 1 and the existing integer-order model 2 Compare

[0174]

[0175] like Figures 18-21 The figure shown is a comparison of the fitting results between the three-dimensional fractional-order model of Embodiment 1 and the existing one-dimensional fractional-order model. Table 6 shows the coefficients of determination R for the two models. 2 The calculation comparison shows that the three-dimensional fractional-order model of this embodiment 1 can adapt to the unloading and recovery characteristics of DE under large deformation.

[0176] Table 6. R-values ​​of the three-dimensional fractional model and the existing one-dimensional fractional model 2 Compare

[0177]

[0178] like Figures 18-21 As shown, these are all practical evaluations of the model in describing the complex deformation behavior of DE, proving the rationality and superiority of the three-dimensional fractional constitutive model constructed in Example 1 in describing the strain rate-related mechanical-electric coupling cyclic deformation behavior of DE.

[0179] The above description is merely an illustration of the present invention and represents a preferred embodiment. It should be noted that those skilled in the art can make various improvements and modifications without departing from 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 three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling, characterized in that, It includes the following steps: S1: Based on the electro-mechanical coupling deformation mechanism of dielectric elastomers, establish the electro-mechanical coupling fractional-order tensor constitutive relation under the condition of constant strain rate; S2: Considering the influence of the thickness change of the dielectric elastomer on the Maxwell stress during the uniaxial deformation process, correct the electro-mechanical coupling fractional-order tensor constitutive relation constructed in step S1; S3: Based on the electro-mechanical coupling fractional-order tensor constitutive relation corrected in step S2, introduce the under cyclic loading, and calculate the uniaxial deformation response during the whole stage; S4: Numerically process the model, and determine the model parameters by fitting the electro-mechanical coupling experimental data related to the strain rate; S5: According to the correlation between the model parameters and the strain rate, establish the relationship criterion between the model parameters and the strain rate, and construct the electro-mechanical coupling three-dimensional fractional-order constitutive model related to the strain rate.

2. The method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling according to claim 1, characterized in that, In the above S1, Based on the electro-mechanical coupling deformation mechanism of dielectric elastomers, construct the initial electro-mechanical coupling tensor constitutive relation as shown in formula (1): （1）； Where σ is the total Cauchy stress, σ vs Let σ be the viscoelastic stress tensor. Maxwell Let Maxwell's stress tensor be defined as follows: （2）； Where ϵ0 is the vacuum permittivity, ϵ r Let be the relative permittivity of the material, e be the directional electric field intensity, and I be the unit tensor; The viscoelastic mechanical behavior of dielectric elastomers is characterized using Abel stick pot, where σ in equation (1) vs The expression is shown in equation (3): （3）； In equation (3), η is the viscosity coefficient, t refers to the time corresponding to the current configuration, α is a fractional-order parameter, and 0 < α < 1, D {α} The fractional derivative of the deformation velocity tensor; Wherein, D from time 0 to time t {α} Defined as: （4）； In formula (4), Γ( ) represents the gamma function, τ is the moment corresponding to a certain configuration in the deformation process, and τ>0, T represents the transpose of the matrix, where, F τ The relative deformation gradient tensor at time τ is shown in equation (5): （5）； D represents the strain rate tensor as shown in formula (6): （6）； In formulas (5) and (6), F is the deformation gradient tensor.

3. The method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling according to claim 2, characterized in that, In the above S2, During the uniaxial deformation process, considering the influence of the thickness change of the dielectric elastomer on the Maxwell stress, correct formula (2) as shown in formula (7): （7）； Where d0 is the initial thickness of the dielectric elastomer material, and λ3 is the elongation of the dielectric elastomer material in the direction of thickness variation. Indicates the directional excitation voltage; Substitute formulas (3) and (7) into formula (1), and the total Cauchy stress tensor in the viscoelastic electro-mechanical coupling fractional-order constitutive relation of the dielectric elastomer is finally expressed as: （8）。 4. The method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling according to claim 3, characterized in that, In the above S3, Define the direction with the maximum electro-mechanical coupling load as the main deformation direction. The evolution relationship of the elongation λ in the main deformation direction during one loading and unloading is expressed separately according to the loading and unloading stages as: （9）； In equation (9), the strain rate c remains constant during loading and unloading, t1 is the end time of loading, t2 is the end time of unloading, and t2 = 2t1 is satisfied; λ a λ represents the elongation during the loading stage in the principal deformation direction. b This indicates the elongation during the unloading stage in the main deformation direction; When 0≤t<t1, the dielectric elastomer is in the loading stage, and the total Cauchy stress tensor is expressed by formula (8); When t1≤t<t2, the dielectric elastomer is in the unloading and recovery stage, and the expression of the total Cauchy stress tensor is as shown in formula (10): （10）； If the applied voltage value is u, then formula (7) is analytically expressed as: （11）； In the formula, u represents the vector voltage value in the thickness direction, then u = (0, 0, u), e i Let i be a unit vector with direction, i = 1, 2, and 3; For uniaxial deformation, the deformation gradient tensor of the dielectric elastomer is defined as: (12)； Substituting formula (12) into formulas (5) and (6), the relative deformation gradient tensor is calculated respectively. and deformation velocity tensor D: （13）； （14）； When 0≤t<t1, substitute formulas (13) and (14) into formula (4), and then substitute them together with formula (11) into formula (9). The specific expression of the Cauchy stress tensor in the loading stage is: （15）； When t1≤t<t2, substitute formulas (13) and (14) into formula (4), and then substitute them together with formula (11) into formula (10). The specific expression of the Cauchy stress tensor in the unloading stage is: （16）； For uniaxial deformation, the relationship between λ3 and λ is λ3 = λ -1 / 2 Define η1 and α1 as model parameters for the loading stage, and η2 and α2 as model parameters for the unloading and recovery stage. Based on equations (15) and (16), the Cauchy stress in the uniaxial tensile direction throughout the deformation stage is expressed as: （17）； Adopt the relationship between the nominal stress P and the elongation as the key characterization index. The conversion relationship between the nominal stress P and the Cauchy stress σ satisfies P=JσF -T If J = det(F), then the nominal stress P in the principal deformation direction of the dielectric elastic body is expressed as: （18）。 5. The method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling according to claim 4, characterized in that, In the above S4, use the hypergeometric function and numerical integration method to jointly process the non-elementary integral terms included in formula (18), and combine the nonlinear least squares method to fit and identify the model parameter values corresponding to multiple groups of electro-mechanical coupling experimental data related to the strain rate.

6. The method for constructing a three-dimensional fractional-order constitutive model of a dielectric elastic body with electromechanical coupling according to claim 5, characterized in that, In the above S5, the relationship criterion between the model parameters identified by fitting in S4 and the strain rate is defined as follows: (19) ； （20）； In the formula, p i and q i All are constants, and n is the number of different strain rates; Based on equation (18), the three-dimensional fractional constitutive model of strain rate-dependent electromechanical coupling in the uniaxial tensile direction is defined as follows: （21）。

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