Numerical simulation method of viscoelastic nonlinear dielectric elastomer constitutive model

Abstract

The application discloses a numerical simulation method of a viscoelastic nonlinear dielectric elastomer constitutive model, constructs a free energy function of a dielectric elastomer considering a strain hardening effect; obtains a super-elastic constitutive model of the dielectric elastomer according to the free energy function of the dielectric elastomer; based on the obtained super-elastic constitutive model, introduces viscoelasticity of the dielectric elastomer material, and constructs a visco-hyper-elastic constitutive model of the dielectric elastomer; based on the constructed visco-hyper-elastic constitutive model, finite element simulation is carried out on a driver or a sensor structure with the dielectric elastomer as the material to carry out numerical simulation, and electromechanical stability and dynamic response of the dielectric elastomer structure are obtained, thereby laying a theoretical foundation for structure design of the dielectric elastomer. The application provides a new method for accurately calculating critical instability voltage of a complex dielectric elastomer structure, and also provides a possibility for accurately simulating dynamic behaviors of the dielectric elastomer.

Description

Technical Field

[0001] This invention relates to the field of constitutive model construction and numerical simulation of dielectric elastomers, and particularly to a numerical simulation method for a nonlinear constitutive model of a viscoelastic dielectric elastomer based on the finite element method when considering strain hardening effect and viscoelasticity under large deformation. This provides important theoretical basis and engineering application value for the complex structural design of dielectric elastomer materials and the determination of critical instability voltage. Background Technology

[0002] Dielectric elastomers are a typical type of electroactive soft material. Due to their unique properties such as large deformation, light weight, flexibility, and excellent chemical and biocompatibility, dielectric elastomers have become the most popular actuators in the field of soft materials. These unique characteristics have facilitated the application of dielectric elastomer actuators, which are mainly used in artificial muscles and soft robots, active control of vibration and noise, resonators, and other fields.

[0003] Despite the numerous advantages mentioned above, dielectric elastomers have not yet seen large-scale practical applications. The main limitations are: insufficient electroinduced deformation at low voltages to meet practical requirements, and unpredictable instability of complex structures at high voltages. Therefore, enabling dielectric elastomers to possess sufficient deformation while satisfying electromechanical stability has become a key research focus. However, dielectric elastomers exhibit significant geometric and material nonlinearities under large deformation strain conditions. Furthermore, in most operating environments, dielectric elastomers must consider not only the electric field force but also their own strain and external mechanical forces; therefore, the analysis of dielectric elastomers must also consider the coupling of multiple physics fields. Calculating analytical solutions for complex dielectric elastomer structures exhibiting both nonlinearity and electromechanical coupling becomes extremely difficult.

[0004] In recent years, with the rapid development of finite element numerical simulation, numerical methods for solving practical problems using simulation software have become increasingly mature. For complex mechanical problems, we can fully utilize currently mature commercial finite element analysis software, select appropriate materials, construct corresponding models and boundary conditions, and obtain simulation solutions under corresponding conditions through finite element methods. This allows us to obtain physical phenomena that cannot be observed intuitively, increasing our understanding of the mechanical behavior of complex structures. For this reason, finite element analysis has become an indispensable tool for R&D departments. Therefore, if finite element analysis can be applied to dielectric elastomers with complex mechanisms, it will be possible to predict the critical voltage of dielectric elastomers under instability conditions, which will have enormous engineering application value for the structural design of dielectric elastomers.

[0005] To date, finite element method (FEM) software does not consider strain hardening and viscoelasticity in dielectric elastomers. Without considering strain hardening, it is impossible to accurately calculate the critical buckling voltage that satisfies electromechanical stability under large deformations in dielectric elastomer structures, thus negating the application value of FEM in precise design guidance. Conversely, neglecting viscoelasticity introduces defects into the finite element dynamics analysis of dielectric elastomers. Summary of the Invention

[0006] The technical problem to be solved by this invention is to overcome the shortcomings of the prior art and provide a numerical simulation method for the constitutive model of viscoelastic nonlinear dielectric elastomers. This method fills the gap in the field of finite element analysis of viscoelastic dielectric elastomers that considers strain hardening effects and viscoelastic dielectric elastomer materials. It provides a new method for accurately calculating the critical instability voltage of complex dielectric elastomer structures and also provides the possibility for accurately simulating the dynamic behavior of dielectric elastomers.

[0007] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0008] A numerical simulation method for a viscoelastic nonlinear dielectric elastomer constitutive model is provided for structural design using dielectric elastomers as raw materials for actuators or sensors. The implementation steps are as follows:

[0009] Step 1: Construct the free energy function of the dielectric elastic body considering strain hardening effect;

[0010] Step 2: Obtain the hyperelastic constitutive model of the dielectric elastic body based on the free energy function of the dielectric elastic body;

[0011] Step 3: Based on the hyperelastic constitutive model obtained in Step 2, the viscoelasticity of dielectric elastomer materials is introduced to construct a visco-hyperelastic constitutive model of dielectric elastomers;

[0012] Step 4: Based on the viscoelastic-hyperelastic constitutive model constructed in Step 3, finite element simulation is performed on the actuator or sensor structure using dielectric elastomer as the material to obtain the electromechanical stability and dynamic response of the dielectric elastomer structure, laying a theoretical foundation for the structural design of dielectric elastomers.

[0013] Furthermore, in the first step, the free energy function of the dielectric elastic body considering the strain hardening effect is constructed as follows:

[0014]

[0015] Where: F is the deformation gradient tensor of the material, E i Let F be the nominal electric field in each direction under the electric field of the material, H be the inverse of the deformation gradient tensor F, ε be the dielectric constant of the material, and J be the third invariant of the deformation gradient tensor F, i.e., the determinant satisfies J = det(F). For isochoric gradient tensor, For isochoric gradient tensor The first invariant satisfies Let μ be the isochoric right Cauchy-Green deformation tensor, μ be the initial shear modulus of the material, D1 be the incompressibility parameter of the material, and J be the incompressibility parameter of the material. m It is the limiting chain limit value of the material, also known as the maximum average tensile parameter.

[0016] Furthermore, in the second step, the hyperelastic constitutive model includes the following hyperelastic stress-strain relationship and hyperelastic Jacobian matrix:

[0017] The hyperelastic stress-strain relationship is as follows:

[0018]

[0019] Where: σ is the stress tensor, F is the material deformation gradient tensor, and J is the third invariant of the deformation gradient tensor F, i.e., the determinant satisfies J = det(F). For isochoric gradient tensor, Let I be the isochoric left-hand Cauchy-Green deformation tensor, I be the second-order unit tensor, μ be the initial shear modulus of the material, D1 be the incompressibility parameter of the material, and J be the incompressibility parameter of the material. m This is the limiting chain constraint value of the material, also known as the maximum average tensile parameter;

[0020] The hyperelastic Jacobian matrix c is:

[0021]

[0022] in: Let i be the tensor union product symbol, and let i be a fourth-order tensor satisfying i ijkl =I ij I kl .

[0023] Furthermore, in the third step, constructing the viscoelastic constitutive model includes the viscoelastic stress-strain relationship and the viscoelastic Jacobian matrix.

[0024] The viscoelastic stress-strain relationship is as follows:

[0025]

[0026]

[0027]

[0028] in: I is a second-order unit tensor, σ D (t) and σ H(t) represents the deviatoric stress and hydrostatic pressure components of the stress tensor σ at time t, considering the viscoelastic effect. and These represent the deviatoric stress and hydrostatic pressure components of the Kirchhoff stress σ at time t, excluding viscoelastic effects, where i is the i-th branch, N is the total number of branches, and τ... i Let g be the relaxation time corresponding to the sticky pot in the i-th branch. i and k i These are the shear and volume relaxation moduli relative to the i-th branch. These are all viscoelastic parameters of the dielectric elastomer material, and t' is the time variable in the integration. Let be the isochoric deformation gradient tensor of the material at time t. Let be the isochoric deformation gradient tensor of the state at time t relative to the state at time t-t';

[0029] Jacobian matrix of the constitutive model of visco-hyperelastic dielectric elastomer as follows:

[0030]

[0031]

[0032] Where: Δt is the step size of the finite element analysis. Let g be the viscoelastic factor, c be the hyperelastic Jacobian matrix, and g be the hyperelasticity factor. ∞ It represents the relative shear modulus as time approaches infinity.

[0033] Furthermore, in the fourth step, the numerical simulation process of the actuator or sensor structure using finite element software based on dielectric elastomer material specifically includes:

[0034] (1) Determine the input parameters of the viscoelastic dielectric elastomer material and give the material properties according to the actual material;

[0035] (2) To realize finite element analysis, complete the finite element discretization of the viscous-hyperelastic constitutive model in the third step;

[0036] (3) Construct a corresponding geometric model based on the actual structure of the dielectric elastomer, and set boundary conditions and external electric field loads;

[0037] (4) Using the viscoelastic-hyperelastic constitutive model of dielectric elastomer, the viscoelastic-hyperelastic Jacobian matrix update and the viscoelastic-hyperelastic Jacobian matrix stress relationship update are completed to complete the finite element simulation;

[0038] (5) Based on the finite element simulation results, obtain the stress-strain cloud diagram and stress-strain variation curve with external electric field under the corresponding external conditions. Based on the cloud diagram and dynamic curve, obtain the electromechanical stability and dynamic response of the dielectric elastomer structure.

[0039] The advantages of this invention compared to existing technologies are:

[0040] (1) Current research methods for dielectric elastomer actuators mainly consist of experiments and theoretical analysis. Experiments require a significant amount of time and effort, while theoretical analysis is mostly limited to simple configurations and is difficult to apply to complex structures in practical applications. This invention addresses the lack of constitutive models for dielectric elastomer materials in current commercial finite element software, filling and improving the gap in finite element analysis of dielectric elastomer materials that consider strain hardening effects and viscoelasticity. The constitutive model of the dielectric elastomer constructed for finite element analysis considers both the influence of material viscoelasticity on the constitutive relationship and its significant strain hardening characteristics under large deformation. This allows for a more accurate description of the material properties of dielectric elastomers, increases the understanding of the complex structural behavior of dielectric elastomers, and has significant engineering application value for the refined design of dielectric elastomer structures.

[0041] (2) The present invention constructs a free energy function that can more accurately describe dielectric elastomer materials, takes into account the obvious strain hardening phenomenon that occurs when dielectric elastomers undergo large deformation in experiments, and selects the Gent model with obvious strain hardening effect as the tensile strain energy of dielectric elastomer materials. It is more accurate than other hyperelastic models in describing large deformation conditions such as electromechanical instability of dielectric elastomers.

[0042] (3) This invention constructs a hyperelastic constitutive model for dielectric elastomers that considers strain hardening effects. Compared with existing constitutive models, it can more accurately describe the stress-strain relationship of dielectric elastomer structures under large deformations.

[0043] (4) This invention constructs a viscoelastic-hyperelastic constitutive model for dielectric elastomers. Existing hyperelastic constitutive models cannot characterize phenomena such as creep and relaxation that occur in dielectric elastomer experiments. In the technical solution of this invention, based on the theory of large deformation viscoelasticity, a viscoelastic-hyperelastic constitutive model for dielectric elastomers is constructed through mathematical means such as tensor analysis, providing a theoretical basis for realizing numerical simulation of viscoelastic dielectric elastomer materials.

[0044] (5) This invention provides viscoelastic dielectric elastomer materials for the material property library in general finite element software. Currently, there are no material properties in general finite element software that consider the strain hardening effect and viscoelasticity of dielectric elastomers, which will lead to the inability to accurately calculate the electromechanical stability and dynamic response of dielectric elastomer structures.

[0045] (6) This invention enables finite element analysis of actuator or sensor structures using dielectric elastomers as materials. Considering practical engineering problems, the design process of dielectric elastomer actuators often requires extensive experimentation, while theoretical solutions for dielectric elastomers are mostly only applicable to simple configurations and cannot provide guidance for practical engineering problems. Numerical simulation of complex structures using methods such as finite element analysis can greatly accelerate the research process, narrow the experimental scope, thereby accelerating the R&D cycle and enhancing engineering applicability. Attached Figure Description

[0046] Figure 1 This is a flowchart illustrating the implementation of the method of the present invention;

[0047] Figure 2 This is a schematic diagram of the initial state of a planar dielectric elastomer.

[0048] Figure 3 This is a schematic diagram of the current state of a planar dielectric elastomer;

[0049] Figure 4 Viscoelastic model of dielectric elastomers;

[0050] Figure 5 The figures show the displacement contour maps in the x and y directions under the critical state with prestress. The left figure shows the displacement contour map in the x direction, and the right figure shows the displacement contour map in the y direction.

[0051] Figure 6 For comparison of finite element results and theoretical results;

[0052] Figure 7 The graphs show the elongation ratio as a function of time under sinusoidal voltage at various frequencies. (a) is the elongation ratio as a function of time when ωτ=10, (b) is the elongation ratio as a function of time when ωτ=1, and (c) is the elongation ratio as a function of time when ωτ=0.1. Here, ω is the frequency of the applied sinusoidal voltage, and τ is the relaxation time of the dielectric elastomer. Detailed Implementation

[0053] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the protection scope of the present invention.

[0054] like Figure 1 As shown, the present invention provides a numerical simulation method for a constitutive model of a viscoelastic nonlinear dielectric elastic body, comprising the following steps:

[0055] Step 1: Construct the free energy function of the dielectric elastic body considering strain hardening effect;

[0056] The free energy function is determined based on the assumption of an ideal dielectric elastic body. In the ideal dielectric elastic body model, the free energy function of the dielectric elastic body is... The contributions include both tensile and polarization components, namely elastic energy U(F) and electric field energy. Free energy can be expressed as:

[0057]

[0058] In the above formula, F is the deformation gradient tensor of the material. Let be the actual electric field under the electric field of the material. To establish the constitutive relation of the dielectric elastic body, appropriate elastic strain energy and electric field energy need to be selected. First, consider the construction of the electric field energy density function. For the electric field energy density function, the dielectric elastic body material is regarded as a capacitor structure, and the electric field energy function is expressed as follows:

[0059]

[0060] Where detF is the determinant of the material's deformation gradient tensor F. Let ε be the actual electric field after material deformation, and ε be the dielectric constant. Considering that the actual electric field after deformation is difficult to control using an initial voltage, for subsequent UMAT subroutine writing and finite element calculations, the input parameters for the electric force component of the material are designed as the initial electric field under no deformation conditions (hereinafter referred to as the nominal electric field). The nominal electric field will consist of three components: x, y, and z. From the physical meaning, the component form of the nominal electric field E of the material is Ex... i for:

[0061]

[0062] The actual electric field of the material component form for:

[0063]

[0064] Among them is H ik Substituting this into the component form of the inverse of the gradient tensor F, we get:

[0065]

[0066] In the above formula, i, m, and l are tensor indices, and i, m, and l can independently take any number from 1, 2, and 3.

[0067] To describe the strain hardening effect of dielectric elastomers, the Gent model is chosen as the elastic energy component of dielectric elastomers. The strain energy function of the Gent model is as follows:

[0068]

[0069] Where μ is the initial shear modulus of the material, D1 is the incompressibility parameter of the material, and J m This is the limiting chain limit value of the material, also known as the maximum average tensile parameter. All of these are intrinsic parameters of the material.

[0070] And J = det(F), For the isotropic right Cauchy-Green deformation tensor, Let F be the isochoric deformation gradient tensor.

[0071] In summary, the free energy function of a dielectric elastic body considering strain hardening effect can be expressed as:

[0072]

[0073] Step 2: Obtain the hyperelastic constitutive model of the dielectric elastic body based on the free energy function of the dielectric elastic body;

[0074] Differentiating the free energy function of the dielectric elastic body obtained in the first step, we can obtain the result in the actual electric field. Components of the true stress σ of the dielectric elastomer material under action ij satisfy:

[0075]

[0076] i, j, m are tensor indices, where i, j, m can independently take any number from 1, 2, 3, where δ ij The symbol is Kronecker. Here we have obtained the Maxwell electric force of the dielectric elastomer under the action of an electric field. Next, we will construct the constitutive model of the elastic strain energy of the dielectric elastomer material.

[0077] The Piola-Kirchhoff stress S of type 2 and the material free energy function W have the following relationship:

[0078]

[0079] C = F T F is a right Cauchy-Green tensor, which can be known from the differential operation between the tensor and its invariants;

[0080] If I1 = trace(C) is the trace of the right Cauchy-Green tensor, then... I is a second-order unit tensor;

[0081] If J 2 =det(C), has Since C is a symmetric tensor, then we have

[0082] Therefore:

[0083]

[0084]

[0085] Then the Piola-Kirchhoff stress S of type 2 can be obtained as:

[0086]

[0087] Further calculations yield the Cauchy stress σ as follows:

[0088]

[0089] After simplification, we have:

[0090]

[0091] b = FF T For the left Cauchy-Green tensor, For the left-inclusive Cauchy-Green tensor.

[0092] The Jacobian matrix c has the following relationship with the Piola-Kirchhoff stress of the second kind:

[0093]

[0094] in: Let i be the tensor union product symbol, and let i be a fourth-order tensor component satisfying i ijkl =I ij I kl I ij I kl All are component forms of the second-order unit tensor I, and the table below i,j,k,l are the sub-indices of the tensor, which can all independently take any number from 1, 2, 3.

[0095] In summary, the hyperelastic constitutive model includes the following hyperelastic stress-strain relationship and hyperelastic Jacobian matrix:

[0096] The hyperelastic stress-strain relationship is as follows:

[0097]

[0098] The hyperelastic Jacobian matrix c is:

[0099]

[0100] Step 3: Based on the hyperelastic constitutive model obtained in Step 2, the viscoelasticity of dielectric elastomer materials is introduced to construct a visco-hyperelastic constitutive model of dielectric elastomers;

[0101] Assume the instantaneous response of the material obeys the hyperelastic constitutive equation:

[0102]

[0103] For compressible materials, and These are the deviatoric stress and hydrostatic pressure components of the instantaneous Kirchhoff stress σ0(t), respectively. J = det(F) is the "distorted" deformation gradient tensor related to the deformation gradient F, and J = det(F) is the determinant of the deformation gradient tensor F.

[0104] Under the reference configuration of the hyperelastic material, the following system of equations was obtained using genetic integrals and standard advance operators under the current configuration:

[0105]

[0106]

[0107] in: I is a second-order unit tensor, σ D (t) and σ H (t) represents the deviatoric stress and hydrostatic pressure components of the stress tensor σ at time t, considering the viscoelastic effect. and These are the deviatoric stress and hydrostatic pressure components of the Kirchhoff stress σ at time t, excluding viscoelastic effects, respectively, where t' is the time variable in the integral. Let be the isochoric deformation gradient tensor of the material at time t. for The inverse matrix, for The transpose of the inverse matrix, Let Gt be the isochoric deformation gradient tensor of the state at time t relative to the state at time t-t', where G0 and K0 are the initial shear modulus and bulk modulus, and G(t) and K(t) are the time-dependent shear and bulk relaxation moduli. and These are the derivatives of G(t) and K(t) with respect to time t, respectively.

[0108]

[0109]

[0110] Let i be the i-th branch, where g i and k ig is the relative modulus of the shear modulus and bulk modulus on the i-th branch. ∞ k is the relative shear modulus as time approaches infinity. ∞ Let be the relative bulk modulus as time approaches infinity. It satisfies [condition missing]. Here we assume that the relaxation time is the same, that is... The number of branches that vary with time is also the same, i.e., N. G =N K =N.

[0111] The constitutive equations of hyperelasticity theory after time-domain generalization are:

[0112]

[0113]

[0114] For the Jacobian matrix of the material, considering the effect of viscoelasticity, the entire Jacobian matrix will be multiplied by a viscoelastic factor. For viscoelastic factor have:

[0115]

[0116] Therefore, the Jacobian matrix of the constitutive model material after considering the viscoelastic effect is:

[0117]

[0118] Where: Δt is the step size of the finite element analysis. Let be the viscoelastic factor, and c be the hyperelastic Jacobian matrix.

[0119] In summary, constructing a viscoelastic constitutive model includes the viscoelastic stress-strain relationship and the viscoelastic Jacobian matrix.

[0120] The viscoelastic stress-strain relationship is as follows:

[0121]

[0122]

[0123]

[0124] Jacobian matrix of the constitutive model of visco-hyperelastic dielectric elastomer as follows:

[0125]

[0126]

[0127] Step 4: Based on the viscoelastic-hyperelastic constitutive model constructed in Step 3, finite element simulation is performed on the actuator or sensor structure using dielectric elastomer as material to obtain the electromechanical stability and dynamic response of the dielectric elastomer structure, laying a theoretical foundation for the structural design of dielectric elastomers.

[0128] The first three material parameters are those of the Gent model itself: initial shear modulus, incompressibility parameter, and limit chain constraint value. The fourth parameter is the dielectric constant of the dielectric elastomer, all of which are material parameters of the model itself. Since the electric field cannot be applied to the model in the UMAT subroutine, the temperature field magnitude is defined as the total nominal electric field magnitude during the programming process. props(5), props(6), and props(7) are set as the projections of the unit direction vector of the nominal electric field in the x, y, and z directions, respectively. A nominal electric field is applied to the dielectric elastomer in this way.

[0129] Meanwhile, considering the impact of pre-strain on the stability of dielectric elastomer structures during the programming process, three input variables corresponding to the three principal directions in the deformation gradient tensor were added to the material parameter input to increase applicability. That is, λ1, λ2, and λ3 in the initial deformation gradient tensor F0 are the pre-stretch coefficients in the x, y, and z directions, respectively.

[0130]

[0131] For the viscoelastic part of the dielectric elastomer material, a two-splitter viscoelastic constitutive model is selected, so the viscoelastic parameters include three parameters: g1, k1, and relaxation time τ.

[0132] After determining the input parameters, the corresponding stress matrix and Jacobian matrix can be obtained by using the material input parameters and deformation parameters such as the material deformation gradient tensor as independent variables, according to the corresponding equations in the constitutive model. The user subroutine is then compiled and its correctness is verified.

[0133] The user subroutine is called to update the viscoelastic-hyperelastic Jacobian matrix and the stress relationship of the viscoelastic-hyperelastic Jacobian matrix, and to complete the finite element simulation. Based on the finite element simulation results, the stress-strain contour map and the stress-strain curve with respect to the external electric field are obtained. According to the contour map and the dynamic curve, the electromechanical stability and dynamic response of the dielectric elastomer structure are obtained.

[0134] Test case

[0135] To gain a fuller understanding of the characteristics of this invention and its applicability to practical engineering, this invention verifies the proposed finite element-based numerical simulation method using ABAQUS and its UMAT subroutine as examples, focusing on the electromechanical stability and dynamic response under sinusoidal voltage in quasi-static problems. Subsequently, to further validate the proposed finite element-based numerical simulation method for dielectric elastomers, this invention performs two numerical simulation examples for a dielectric elastomer plate: one for a quasi-static problem and the other for a dynamic problem.

[0136] In the embodiment of the quasi-static problem, all physical quantities are expressed as normalized parameters, and the normalized nominal electric field strength is: Where E0 is the nominal electric field strength, μ is the initial shear modulus of the material, and ε is the dielectric constant of the material. Only the limiting chain constraint value J of the material needs to be given as the material parameters. m =120, and the electric field-stretch rate curve can be obtained. A 50mm × 50mm × 1mm dielectric elastomer plate is selected as the research object, with prestress applied to both sides of the plate. In this case, σ represents the initial prestress of the material, a nominal electric field is applied, and the direction is parallel to the z-axis, for finite element analysis. The finite element analysis results are as follows: Figure 5 As shown, Figure 5 The figures show the displacement contour maps in the x and y directions of a dielectric elastomer plate under prestressed critical state. The left figure shows the displacement contour map in the x direction, and the right figure shows the displacement contour map in the y direction. Figure 6 This paper presents a comparison of finite element simulation and theoretical solutions under prestressed conditions. The theoretical and numerical simulation results are in near-perfect agreement, demonstrating the high accuracy of the numerical simulation method used in this invention to determine the critical voltage under instability conditions in dielectric elastomer structures. The correctness of the subroutine and the accuracy and feasibility of using this numerical simulation method to analyze and calculate the electromechanical stability of dielectric elastomer actuators are verified. This method can be used to verify whether the designed dielectric elastomer actuator structure can be used under normal operating voltage, providing guidance for specific structural designs.

[0137] In the embodiment of the dynamic problem, the present invention uses the same parameters as the quasi-static problem. For a dielectric elastic body undergoing in-plane deformation, in order to study sinusoidal voltages of different frequencies, the present invention solves the free vibration problem with different frequency values ​​ωτ = 10, ωτ = 1, and ωτ = 0.1. The results of the dynamic response are as follows: Figure 7As shown in (a), (b), and (c), (a) is the elongation curve of the dielectric elastomer versus time when ωτ = 10, (b) is the elongation curve of the dielectric elastomer versus time when ωτ = 1, and (c) is the elongation curve of the dielectric elastomer versus time when ωτ = 0.1, where ω is the frequency of the applied sinusoidal voltage and τ is the relaxation time of the dielectric elastomer. It can be seen that as the relaxation time decreases, the influence of damping on the dynamic response becomes increasingly significant. This conforms to the theoretical results regarding the dynamic response curve of the dielectric elastomer under sinusoidal voltage. The embodiments verify the feasibility and accuracy of this invention in analyzing the dynamic response of dielectric elastomers, and have significant engineering application value for the refined design of dielectric elastomer structures.

[0138] The parts of this invention not described in detail are well-known to those skilled in the art.

[0139] The above are merely specific steps of the present invention and do not constitute any limitation on the scope of protection of the present invention; it can be extended to the field of quasi-static and dynamic analysis of viscoelastic dielectric elastomers including numerical simulation. All technical solutions formed by equivalent transformation or equivalent substitution fall within the scope of protection of the present invention.

Claims

1. A numerical simulation method for a constitutive model of a viscoelastic nonlinear dielectric elastomer, used for structural design using dielectric elastomers as raw materials for actuators or sensors, characterized in that, The implementation steps are as follows: Step 1: Construct the free energy function of the dielectric elastic body considering strain hardening effect; In the first step, the free energy function of the dielectric elastic body considering the strain hardening effect is constructed as follows: (1) in: For the material's true electric field, Let be the deformation gradient tensor of the material. The nominal electric field in each direction under the electric field of the material. For the deformation gradient tensor The reverse, Let be the dielectric constant of the material. For isochoric gradient tensor, For the isotropic right Cauchy-Green tensor The first invariant satisfies , For the isotropic right Cauchy-Green tensor, For the deformation gradient tensor The third invariant, i.e., the determinant, satisfies , The initial shear modulus of the material. The incompressible parameter of the material, The limiting chain limit value of the material, also known as the maximum average tensile parameter, i, m, l are tensor indices, which can be taken independently as any number from 1, 2, 3; Step 2: Obtain the hyperelastic constitutive model of the dielectric elastic body based on the free energy function of the dielectric elastic body; Step 3: Based on the hyperelastic constitutive model obtained in Step 2, the viscoelasticity of dielectric elastomer materials is introduced to construct a visco-hyperelastic constitutive model of dielectric elastomers; In the third step, constructing the viscoelastic constitutive model includes the viscoelastic stress-strain relationship and the viscoelastic Jacobian matrix. The viscoelastic stress-strain relationship is as follows: (4) (5) (6) in: , It is a second-order unit tensor. and These are Kirchhoff stresses. The deviatoric stress and hydrostatic pressure components considering the viscoelastic effect at time t. and These are Kirchhoff stresses. The deviatoric stress and hydrostatic pressure components at time t, without considering the viscoelastic effect, are represented by z, which is the z-th branch. This represents the total number of branches. Let z be the relaxation time corresponding to the sticky pot in the z-th branch. and It is the shear and volume relaxation modulus relative to the z-th branch. For time variables in integration, For materials in The isochoric gradient tensor at time t. for The state at any given moment relative to The isochoric deformation gradient tensor of the state at time step; Jacobian matrix of the constitutive model of visco-hyperelastic dielectric elastomer as follows: (7) (8) in: This is the step size for finite element analysis. It is the viscoelastic factor. It is a hyperelastic Jacobian matrix. The relative shear modulus as time approaches infinity; Step 4: Based on the viscoelastic-hyperelastic constitutive model constructed in Step 3, numerical simulation of the actuator or sensor structure using dielectric elastomer as material is performed using finite element simulation. The electromechanical stability and dynamic response of the dielectric elastomer structure are obtained, laying a theoretical foundation for the structural design of dielectric elastomers.

2. The numerical simulation method for a constitutive model of a viscoelastic nonlinear dielectric elastic body according to claim 1, characterized in that: In the second step, the hyperelastic constitutive model includes the following hyperelastic stress-strain relationship and hyperelastic Jacobian matrix: The hyperelastic stress-strain relationship is as follows: (2) in: For stress tensor, Let be the deformation gradient tensor of the material. For the isotropic right Cauchy-Green tensor, For the deformation gradient tensor The third invariant, i.e., the determinant, satisfies , For isochoric gradient tensor, For an isotropic left-handed Cauchy-Green tensor, It is a second-order unit tensor. The initial shear modulus of the material. The incompressible parameter of the material, This is the limiting chain constraint value of the material, also known as the maximum average tensile parameter; Hyperelastic Jacobian Matrix for: (3) in: The symbol for tensor union product is... For a fourth-order tensor to satisfy j and k are tensor indices, which can be taken independently as any number from 1, 2, and 3.

3. The numerical simulation method for a constitutive model of a viscoelastic nonlinear dielectric elastic body according to claim 1, characterized in that: In the fourth step, the numerical simulation process of the actuator or sensor structure using dielectric elastomer as the material based on finite element simulation is specifically as follows: (1) Determine the input parameters of the viscoelastic dielectric elastomer material and give the material properties according to the actual material; (2) To realize finite element analysis, complete the finite element discretization of the viscous-hyperelastic constitutive model in the third step; (3) Construct the corresponding geometric model based on the actual structure of the dielectric elastomer, and set the boundary conditions and external electric field loads; (4) Using the viscoelastic-hyperelastic constitutive model of dielectric elastomer, the viscoelastic-hyperelastic Jacobian matrix update and the viscoelastic-hyperelastic Jacobian matrix stress relationship update are completed to complete the finite element simulation; (5) Based on the finite element simulation results, obtain the stress-strain cloud diagram and stress-strain variation curve with external electric field under the corresponding external conditions. Based on the cloud diagram and dynamic curve, obtain the electromechanical stability and dynamic response of the dielectric elastomer structure.

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