A method for analyzing nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure

Abstract

The application discloses a method for analyzing nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure. The method considers the strain hardening effect and viscoelasticity of the dielectric elastomer material, establishes a dynamic model of the dielectric elastomer actuator structure from linear damping assumption and considering material viscoelasticity respectively, calculates the equilibrium deformation and inherent frequency of micro-vibration near the equilibrium state, studies the nonlinear dynamic response of the structure under alternating voltage load, explores the influence law of the constitutive parameters on the nonlinear dynamic response, and analyzes the internal relationship between material viscoelasticity and structure vibration damping. In addition, considering compressibility, the stress wave propagation characteristics in the finite deformation state are studied, the separation phenomenon of transverse wave and longitudinal wave is analyzed, and the influence of the constitutive parameters and strain hardening effect on the stress wave propagation characteristics is explored. The application can provide a theoretical basis for the fine design of the dielectric elastomer actuator and the control of stress wave propagation by adjusting the electric field.

Description

Technical Field

[0001] This invention relates to the fields of nonlinear dynamics and stress wave propagation analysis. It focuses on the nonlinear dynamic behavior caused by the electro-mechanical coupling working environment of dielectric elastomers, material constitutive relations, and large geometric deformation characteristics. In particular, it relates to the analysis of stress wave propagation characteristics of structures when considering strain hardening and material viscoelasticity. It provides a nonlinear dynamics and stress wave propagation characteristic analysis method for dielectric elastomer actuator structures, providing a theoretical basis for the design of flexible actuators based on dielectric elastomers. Background Technology

[0002] Dielectric elastomers are active soft materials whose deformation is caused by an electric field. This electric field significantly alters the size and shape of the dielectric elastomer, leading to its reputation as an electroactive polymer. When a dielectric elastomer film is sandwiched between two flexible electrodes, applying a voltage through the electrodes generates an electric field along its thickness. Opposite charges accumulate on the upper and lower surfaces of the film. Under the influence of the electric field, the dielectric elastomer film contracts in the thickness direction and expands in the lateral direction. Even under relatively small voltage excitation, the dielectric elastomer film can produce significant deformation, with a driven strain greater than 100%, and a fast response speed. This overcomes the shortcomings of shape memory alloys and electroactive ceramic actuators, such as slow response speed, fragility, low driven strain capacity, and unpredictable motion. Therefore, dielectric elastomers are used in the design and manufacture of various novel actuators.

[0003] Studying the static and dynamic responses of dielectric elastomer actuator structures under excitation is the first and most important step in fully utilizing their structural advantages. Dielectric elastomer actuators operate in a force-electric coupling environment, exhibiting constitutive nonlinearity and electroinduced large deformation characteristics, which poses a significant challenge to accurate dynamic modeling of dielectric elastomers. Simultaneously, quantitative analysis and numerical calculation of nonlinear problems are relatively difficult. Therefore, in-depth research into the static and dynamic characteristics of dielectric elastomers under excitation is of great scientific significance. With the rapid development of dielectric elastomers in various engineering fields, a large number of theoretical and experimental studies have emerged both domestically and internationally. However, limited by the large deformation and failure issues of dielectric elastomers, their applications remain largely in the laboratory stage, and commercial applications face significant challenges. Further quantitative research on the dynamic characteristics of typical dielectric elastomer actuator structures is needed to guide the design and application of dielectric elastomer actuators.

[0004] Most studies focus on the static or quasi-static properties of dielectric elastomers, neglecting viscoelasticity or simplifying it to linear damping without considering strain hardening effects. To fill this gap, this invention establishes a dynamic model of a dielectric elastomer considering viscoelasticity and compares the results with those under the linear damping assumption to analyze the effectiveness of the linear damping assumption. It then employs the Gent model, which considers strain hardening effects, to analyze the stress wave propagation characteristics of the dielectric elastomer. The method proposed in this invention provides a theoretical basis for the design of dielectric elastomer actuators and for controlling stress wave propagation through electric fields. Summary of the Invention

[0005] To overcome the shortcomings of existing technologies, this invention provides a nonlinear dynamics and stress wave propagation characteristic analysis method for dielectric elastomer actuator structures. It fully considers the viscoelasticity and strain hardening effects of dielectric elastomer materials, providing a theoretical basis for the design of flexible actuators based on dielectric elastomers. The design results obtained are more consistent with real-world conditions and have stronger engineering applicability.

[0006] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0007] A nonlinear dynamics and stress wave propagation characteristic analysis method for dielectric elastomer actuator structures is proposed for the design of flexible actuators based on dielectric elastomers. The implementation steps are as follows:

[0008] Step 1: Based on the principle of virtual work, derive the dynamic control equations for the planar actuator of the dielectric elastic body;

[0009] Step 2: Calculate the equilibrium deformation and the natural frequency of the small-amplitude vibration around the equilibrium position using the small perturbation method;

[0010] Step 3: Solve the nonlinear vibration response of the dielectric elastomer planar actuator under alternating voltage load;

[0011] Step 4: Derive explicit expressions for the generalized acoustic tensor and stress wave phase velocity based on the theory of finite deformation material dynamics;

[0012] Step 5: Solve for the stress wave propagation characteristics (wave velocity and transverse / longitudinal wave separation angle) of the dielectric elastic planar actuator under a unidirectional electric field.

[0013] Furthermore, the first step of obtaining the dynamic control equations based on the principle of virtual work specifically includes:

[0014] Consider a planar actuator structure made of dielectric elastomer. Before deformation, its dimensions are L1, L2, L3. After being subjected to external tensile forces P1 and P2 in the XY plane and a voltage φ in the Z direction, the dielectric elastomer film expands in the XY plane, increasing its area, while its thickness decreases in the Z direction. Let the dimensions after deformation be l1, l2, l3. Define the system's state variables as the elongation ratios λ1 = l1 / L1, λ2 = l2 / L2, and λ3 = l3 / L3, which are time-dependent.

[0015] According to Gauss's law of electrostatics, the relationship between the charge and voltage in a dielectric elastomer actuator can be obtained as follows:

[0016]

[0017] Where Φ represents the voltage in the Z direction and ε represents the dielectric constant.

[0018] Taking its partial derivative yields the expression for δQ:

[0019]

[0020] When a small amount of charge passes through the dielectric elastic membrane, the voltage does work ΦδQ, and the external force does work P1δl1 and P2δl2, respectively. The inertial forces per unit length in the x and y directions are ρL2L3x, respectively. 2 (d 2 λ1 / dt 2 ),ρL1L3y 2 (d 2 λ2 / dt 2 ), where ρ is the density of the dielectric elastomer material. The linear damping forces are -cx(dλ1 / dt) and -cy(dλ2 / dt), respectively, where c is the linear damping coefficient. Integrating over elements of length δλ1dx and δλ2dx in the x and y directions respectively, the work done by the inertial force and damping force is obtained as follows:

[0021]

[0022] According to the principle of virtual work, the change in free energy of a dielectric elastic film is equal to the work done by voltage, external force, inertial force, and damping, that is:

[0023]

[0024] Where ρ is the density of the dielectric elastomer material, and c is the linear damping coefficient.

[0025] The free energy density of a dielectric elastic body is equal to the sum of the elastic strain energy density and the electrostatic energy density. Using the Gent hyperelastic strain energy model and the linear polarization assumption, the total free energy density can be expressed as:

[0026]

[0027] Where μ represents the shear modulus, J lim denoted by , where represents the strain limit constant, and D represents the electric displacement.

[0028] Substituting equation (2) into equation (4), we get:

[0029]

[0030] Among them, the independent variables are δλ1 and δλ2;

[0031]

[0032]

[0033] Substituting equation (5) into equations (7) and (8), we get:

[0034]

[0035]

[0036] This refers to the nonlinear dynamic equations of a planar actuator made of dielectric elastomer material under linear damping. To simplify the analysis without loss of generality, we consider the biaxial deformation case of actuators made of incompressible dielectric elastomer material, i.e., P1 = P2 = P, λ1 = λ2 = λ, λ3 = λ. -2 L1 = L2 = L, after dimensionless transformation, equations (9) and (10) are simplified to the dimensionless dynamic control equations with elongation λ as the state variable under the assumption of linear damping:

[0037]

[0038]

[0039] Among them, dimensionless time Dimensionless damping Dimensionless voltage Dimensionless mechanical force This represents the strain hardening constant.

[0040] Considering viscoelasticity, the application of the principle of virtual work is the same, except that the linear damping term is subtracted from equations (7) and (8), resulting in:

[0041]

[0042]

[0043] Using the Gent elastic strain energy model that considers viscoelasticity, the total free energy expression of the system is:

[0044]

[0045] Where, μ α ,μ β Indicates shear modulus, ξ represents the strain hardening constant, and ξ1 and ξ2 represent the damping ratios.

[0046] Substituting equation (15) into equations (13) and (14), we get:

[0047]

[0048]

[0049] The damper is modeled as a Newtonian fluid, and its deformation follows the following evolution law:

[0050]

[0051]

[0052] Where η represents the damper viscosity.

[0053] To simplify the analysis without loss of generality, we consider biaxial deformation cases such as incompressible dielectric elastic planar actuators, i.e., P1=P2=P, λ1=λ2=λ, λ3=λ -2 L1=L2=L, after dimensionless transformation, equations (16)(17)(18)(19) are simplified to the dimensionless dynamic governing equations considering the viscoelasticity of the material, with the elongation λ and damping ξ as state variables:

[0054]

[0055]

[0056]

[0057] Where T v Indicates the viscoelastic relaxation time. Represents the strain hardening constant, a dimensionless quantity:

[0058]

[0059] t v =η / μ β

[0060] Furthermore, the equilibrium deformation and natural frequency obtained in the second step using the small perturbation method specifically include:

[0061] For the linear damping assumption, the equilibrium deformation λ eq satisfy:

[0062]

[0063] Where, λ eq J represents the elongation at equilibrium. lim Represents the strain hardening constant. This represents the aforementioned dimensionless system parameter.

[0064] The dimensionless natural frequency of the small-amplitude oscillations near the equilibrium state can be obtained using the small perturbation method. The expression:

[0065]

[0066] Therefore, for the linear damping assumption, the deformation of the equilibrium state can be calculated according to equation (23), and the natural frequency of the system can be calculated by substituting it into equation (24).

[0067] Considering the viscoelasticity of the material, the equilibrium deformation λ eq satisfy:

[0068]

[0069] The natural frequency can be obtained using the small perturbation method. The expression:

[0070]

[0071]

[0072] Furthermore, the third step of solving the nonlinear vibration characteristics under alternating voltage load specifically includes:

[0073] A dielectric elastomer planar actuator vibrates near its equilibrium state when subjected to a small disturbance under constant internal air pressure and alternating voltage. The alternating voltage consists of DC and sinusoidal voltages.

[0074] φ(t)=φ dc +φ ac sin(ωt) (28)

[0075] Where φ dc DC voltage magnitude, φ ac ω is the amplitude of the sinusoidal voltage, and ω is the excitation frequency. This is obtained after dimensionless conversion.

[0076] Substituting equation (28) into equations (20)-(22), we obtain the governing equations for the viscoelastic Gent model:

[0077]

[0078]

[0079] Substituting equation (28) into equation (11), we obtain the governing equation under the linear damping assumption:

[0080]

[0081] Furthermore, the explicit expressions for the generalized acoustic tensor and stress wave phase velocity in the fourth step specifically include:

[0082] To analyze the finite deformation state of dielectric elastomer materials, a deformation gradient tensor is introduced. in and These are the position vectors after deformation / current configuration and before deformation / reference configuration, respectively. Let the gradient and dyadic operators be represented respectively, and introduce the strain energy function. The independent variables are the deformation gradient F and the electric displacement vector under the reference configuration. The first Piola-Kirchhoff stress tensor and electric field intensity in the reference configuration can be written as:

[0083]

[0084] The corresponding quantities between the current configuration and the reference configuration are related:

[0085] T = J -1 P0·F T

[0086] Linearizing the constitutive formula (32), we have:

[0087]

[0088] Where δ represents the incremental change. K0 is the nominal electroelastic modulus tensor, defined as:

[0089]

[0090] Next, we consider small-amplitude motions superimposed on a finite deformation state, and equation (34) is rewritten in incremental configuration form:

[0091]

[0092] Where, δP=J -1 δP0·F T , δH=δF·F -1

[0093] K = JF -T·K0·F -1 (37)

[0094] Here, the superscript (2134) indicates that the tensor isomers are isomers.

[0095] The linearized equations of motion and the Euler form of Maxwell's equations are as follows:

[0096]

[0097] in, This represents the incremental displacement, where ρ = ρ0 / J is the material density after deformation, and ρ0 is the initial material density before deformation.

[0098] Next, we find the plane wave solution for equation (38). According to solid wave theory, a plane wave can be expressed as:

[0099]

[0100] Where c is the phase velocity and t represents time. It is a unit vector along the direction of propagation. Let f be a unit vector along the direction of motion, and g be a quadratically differentiable function. It is a unit vector of electric displacement. The direction of motion of a shear wave is orthogonal to the direction of propagation, therefore... A longitudinal wave (pressure wave) is a wave whose direction of motion is the same as its direction of propagation, therefore it has...

[0101] Substituting equations (36) and (39) into equation (38), we get:

[0102]

[0103] This is an eigenvalue problem of a tensor, where the direction of the wave's displacement is the direction of the eigenvector. Here, ρ represents the density of the propagation medium, and A is the generalized sound tensor, which defines the propagation conditions of plane waves in nonlinear electroelastic materials and has an arbitrary strain energy function. The generalized sound tensor of an electroelastic material has the following form:

[0104]

[0105] in, Indicates perpendicular to The projection on the surface, tr() represents the tensor trace operator.

[0106] Neglecting compressibility, longitudinal wave propagation does not occur in dielectric elastomers. Therefore, to analyze transverse and longitudinal waves, we consider the energy function of a compressible dielectric elastomer in the following form:

[0107]

[0108] in, It is a purely elastic strain energy function, and various purely elastic strain energy models can be used, such as Neo-Hookean, Gent, etc. ε is the dielectric constant of the material in the reference configuration before deformation, and γ is... i These are dimensionless parameters that satisfy γ0 + γ1 + γ2 = 1. For an ideal dielectric elastic body model, we have γ0 = γ2 = 0, γ1 = 1. The generalized acoustic tensor simplifies to:

[0109] A = Q elas (43)

[0110] Given a purely elastic strain energy function ψ elas After the formality, Q elas Q represents the purely elastic contribution. elas The derivation process is as follows:

[0111]

[0112]

[0113]

[0114] This is the expression for the generalized acoustic tensor.

[0115] Furthermore, the fifth step of solving for the stress wave propagation characteristics (wave velocity and transverse / longitudinal wave separation angle) under a unidirectional electric field specifically includes:

[0116] Consider the propagation of stress waves caused by electrostatic deformation in a dielectric elastic body under a unidirectional uniform electric field. In the principal direction... Apply voltage, i.e. Dielectric elastomer in a plane Extending upwards, the voltage-induced deformation gradient can be expressed as:

[0117]

[0118] For an incompressible ideal dielectric elastic body, the elongation can be written as λ(D)=(1+D) 2 ) -1 / 3 , for For highly compressible materials, the above elongation expression also holds approximately true within the range of deformation and voltage considered.

[0119] Next, specific pure elastic strain energy models—the Neo-Hookean model and the Gent model considering strain hardening effects—are used to derive the explicit expression of the generalized acoustic tensor in order to analyze the propagation of stress waves in dielectric elastic bodies.

[0120] (1) Neo-Hookean model

[0121] The compressible Neo-Hookean model has the following elastic strain energy functions:

[0122]

[0123] Where I1=trC,C=F·F T Let μ denote the first invariant of the right Cauchy-Green tensor, K denote the shear modulus, and J = detF. det represents taking the determinant. Substituting equation (48) into equations (44) to (46),

[0124]

[0125]

[0126] in, Representative at Directional projection, Represents perpendicular to Projection onto the plane in the direction of the wave. Review the propagation conditions for plane waves:

[0127]

[0128] For any wave propagation direction The generalized acoustic tensor A has three characteristic directions, one of which is related to... Collinear, the other two are located perpendicular to On the plane, perpendicular to The former consists of one longitudinal wave, while the latter consists of two transverse waves.

[0129] Let the wave propagation angle be ? Propagation direction vector Explicit expressions for the wave velocities of transverse and longitudinal waves can be obtained. The magnitude of the transverse wave velocity is:

[0130]

[0131] The magnitude of the longitudinal wave velocity is:

[0132]

[0133] (2) Gent model

[0134] Considering the strain hardening effect of the material, the elastic strain energy function of the compressible Gent model is:

[0135]

[0136] Where I1=trB,B=F·F T Let μ denote the first invariant of the left Cauchy-Green tensor, μ denote the shear modulus, K denote the bulk modulus, and J denote the first invariant of the left Cauchy-Green tensor. m Let J represent the strain hardening constant, J = detF. Substituting equation (54) into equations (44)-(46), we get:

[0137]

[0138]

[0139]

[0140] in, ξ=J m -I1+3.

[0141] Beneficial effects:

[0142] The advantages of this invention compared to existing technologies lie in the fact that the structural dynamics model and stress wave theoretical analysis framework of the dielectric elastomer actuator established in this invention consider the viscoelasticity and strain hardening effects of the material, thus overcoming and improving the limitations of traditional dielectric elastomer material dynamics research that does not consider strain hardening effects and adopts linear damping assumptions. On the one hand, it performs numerical simulation of the parametric vibration response of the structure under alternating voltage excitation, and gives the influence law of constitutive parameters such as relaxation time and strain hardening constant on the structural vibration response, and analyzes the relationship between material viscoelasticity and structural vibration damping. On the other hand, it studies the influence of strain hardening effect on dynamic response and stress wave propagation characteristics, providing a theoretical basis for the design of dielectric elastomer actuators and the control of stress wave propagation and separation. Attached Figure Description

[0143] Figure 1 This is a schematic diagram of the initial and energized state of a dielectric elastomer planar actuator structure.

[0144] Figure 2 It is the amplitude-frequency response curve under alternating voltage excitation;

[0145] Figure 3Figure 1 shows the harmonic response at different relaxation times; Figure 2 shows the harmonic response at a relaxation time of 5s, Figure 3 shows the harmonic response at a relaxation time of 10s, Figure 4 shows the harmonic response at a relaxation time of 50s, Figure 5 shows the harmonic response at a relaxation time of 100s, Figure 6 shows the harmonic response at a relaxation time of 200s, and Figure 7 shows the harmonic response at a relaxation time of 400s.

[0146] Figure 4 Figure 1 shows the phase plane diagrams and Poincaré maps of the harmonic response at different relaxation times. Figure 2 shows the phase plane diagram and Poincaré map for a relaxation time of 5 s, Figure 3 shows the phase plane diagram and Poincaré map for a relaxation time of 10 s, Figure 4 shows the phase plane diagram and Poincaré map for a relaxation time of 50 s, Figure 5 shows the phase plane diagram and Poincaré map for a relaxation time of 100 s, Figure 6 shows the phase plane diagram and Poincaré map for a relaxation time of 200 s, and Figure 7 shows the phase plane diagram and Poincaré map for a relaxation time of 400 s.

[0147] Figure 5 It is the amplitude-frequency response under the linear damping assumption;

[0148] Figure 6(a) shows the time-domain response results of the superharmonic wave;

[0149] Figure 6(b) shows the harmonic time-domain response results;

[0150] Figure 6(c) shows the time-domain response results of the subharmonics;

[0151] Figure 7(a) shows the change of transverse and longitudinal wave separation angles with electric displacement in the compressible Neo-Hookean model;

[0152] Figure 7(b) shows the change of the shear wave and longitudinal wave separation angle of the compressible Neo-Hookean model with the incident angle;

[0153] Figure 7(c) shows the change of the shear wave and longitudinal wave separation angle of the compressible Neo-Hookean model as a function of the compressibility coefficient;

[0154] Figure 8(a) shows the change of the shear wave and longitudinal wave separation angle as a function of electric displacement in the compressible Gent model;

[0155] Figure 8(b) shows the change of the shear wave and longitudinal wave separation angle of the compressible Gent model as a function of the incident angle;

[0156] Figure 8(c) shows the change of the shear wave and longitudinal wave separation angle of the compressible Gent model as a function of the compressibility coefficient;

[0157] Figure 9(a) shows the phase velocity distribution of longitudinal waves with an electric displacement of 2 for the compressible Gent model and the compressible Neo-Hookean model.

[0158] Figure 9(b) shows the phase velocity distribution of the compressible Gent model and the compressible Neo-Hookean model in a longitudinal wave with an electric displacement of 4.

[0159] Figure 9(c) shows the phase velocity distribution of the compressible Gent model and the compressible Neo-Hookean model in a transverse wave with an electric displacement of 2.

[0160] Figure 9(d) shows the phase velocity distribution of the compressible Gent model and the compressible Neo-Hookean model in a transverse wave with an electric displacement of 4.

[0161] Figure 10 This is a flowchart illustrating the analysis of the planar actuator structure of the dielectric elastomer according to the present invention;

[0162] Figure 11(a) is a schematic diagram of the propagation of the first plane wave;

[0163] Figure 11(b) is a schematic diagram of the propagation of the second plane wave;

[0164] Figure 11(c) is a schematic diagram of the propagation of the third plane wave. Detailed Implementation

[0165] 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.

[0166] like Figure 10 As shown, this invention proposes a method for analyzing the nonlinear dynamics and stress wave propagation characteristics of dielectric elastomer actuator structures, comprising the following steps:

[0167] Step (1) derives the dynamic control equations for the dielectric elastomer planar actuator based on the principle of virtual work. Considering the planar actuator structure made of dielectric elastomer material, such as... Figure 1 As shown, the original dimensions are L1, L2, L3. After being subjected to external tensile forces P1 and P2 in the XY plane and a voltage φ in the Z direction, the dielectric elastomer film expands in the XY plane, increasing its area, while its thickness decreases in the Z direction. Let the dimensions after deformation be l1, l2, l3, and define the system's state variables as the elongation ratios λ1 = l1 / L1, λ2 = l2 / L2, and λ3 = l3 / L3, which are time-dependent.

[0168] When a small amount of charge passes through the dielectric elastic membrane, the voltage does work ΦδQ, and the external force does work P1δl1 and P2δl2, respectively. The inertial forces per unit length in the x and y directions are ρL2L3x, respectively. 2 (d 2 λ1 / dt 2 ),ρL1L3y 2 (d 2 λ2 / dt 2 ), where ρ is the density of the dielectric elastomer material. The linear damping forces are -cx(dλ1 / dt) and -cy(dλ2 / dt), respectively, where c is the linear damping coefficient. Integrating over elements of length δλ1dx and δλ2dx in the x and y directions respectively, the work done by the inertial force and damping force is obtained as follows:

[0169]

[0170] According to the principle of virtual work, the change in the free energy of a dielectric elastomer film is equal to the work done by voltage, external force, inertial force, and damping. The free energy density of a dielectric elastomer is equal to the sum of the elastic strain energy density and the electrostatic energy density. Using the Gent hyperelastic strain energy model, the nonlinear dynamic equations of the dielectric elastomer planar actuator under linear damping are obtained.

[0171] To simplify the analysis without loss of generality, we consider biaxial deformation cases such as actuators made of incompressible dielectric elastomer materials, i.e., P1=P2=P, λ1=λ2=λ, λ3=λ -2 L1 = L2 = L, thus obtaining the dimensionless governing equations of motion:

[0172]

[0173]

[0174] Considering the viscoelasticity of the material, the damper is modeled as a Newtonian fluid, using the Gent elastic strain energy model that considers viscoelasticity. To simplify the analysis without loss of generality, biaxial deformation cases such as incompressible dielectric elastic planar actuators are considered, i.e., P1=P2=P, λ1=λ2=λ, λ3=λ. -2 L1 = L2 = L, which is then dimensionless, resulting in the dimensionless governing equations of motion:

[0175]

[0176]

[0177]

[0178] Among them, T v Indicates the viscoelastic relaxation time. Represents the strain hardening constant, a dimensionless quantity:

[0179]

[0180] t v =η / μ β

[0181] Step (2) uses the small perturbation method to calculate the equilibrium deformation and the natural frequency of the small-amplitude oscillation around the equilibrium position. For the linear damping assumption, the equilibrium deformation λ... eq satisfy:

[0182]

[0183] Where, λ eq J represents the deformation of the equilibrium state. lim Represents the strain hardening constant. This represents the aforementioned dimensionless system parameter.

[0184] The dimensionless natural frequency of the small-amplitude oscillations near the equilibrium state can be obtained using the small perturbation method. The expression is:

[0185]

[0186] Therefore, under the linear damping assumption, the natural frequency of the system can be calculated by substituting the equilibrium deformation obtained from the calculation into the expression.

[0187] Considering the viscoelasticity of the material, the equilibrium deformation λ eq satisfy:

[0188]

[0189] The natural frequency can be obtained using the small perturbation method. The expression is:

[0190]

[0191]

[0192] Step (3) Solve for the nonlinear vibration response of the dielectric elastomer planar actuator under alternating voltage load. The dielectric elastomer planar actuator vibrates near its equilibrium state when subjected to a small disturbance under constant internal air pressure and alternating voltage. The alternating voltage consists of DC and sinusoidal voltages:

[0193] φ(t)=φ dc +φ ac sin(ωt) (28)

[0194] Where φ dc DC voltage magnitude, φac ω is the amplitude of the sinusoidal voltage, and ω is the excitation frequency. This is obtained after dimensionless conversion.

[0195] Substituting the voltage expression into the dimensionless dynamic equation, we obtain the governing equations for the viscoelastic Gent model:

[0196]

[0197]

[0198] Substituting equation (28) into equation (11), we obtain the governing equation under the linear damping assumption:

[0199]

[0200] Step (4) derives explicit expressions for the generalized acoustic tensor and stress wave phase velocity based on the theory of finite deformation material dynamics. To analyze the finite deformation state of dielectric elastic materials, the deformation gradient tensor is introduced. in and These are the position vectors after deformation / current configuration and before deformation / reference configuration, respectively. Let the gradient and dyadic operators be represented respectively, and introduce the strain energy function. The independent variables are the deformation gradient F and the electric displacement vector under the reference configuration. The first Piola-Kirchhoff stress tensor and electric field intensity in the reference configuration can be written as:

[0201]

[0202] The corresponding quantities between the current configuration and the reference configuration are related:

[0203] T = J -1 P0·F T

[0204] The linearized constitutive model has:

[0205]

[0206] Where δ represents the incremental change. K0 is the nominal electroelastic modulus tensor, defined as:

[0207]

[0208] Next, we consider the small-amplitude motions superimposed on a finite deformation state, which can be rewritten in an incremental configuration form:

[0209]

[0210] Where, δP=J -1 δP0·F T , δH=δF·F -1

[0211] K = JF -T ·K0·F -1 (37)

[0212] Here, the superscript (2134) indicates that the tensor isomers are isomers.

[0213] The linearized equations of motion and the Euler form of Maxwell's equations are as follows:

[0214]

[0215] in, This represents the incremental displacement, where ρ = ρ0 / J is the material density after deformation, and ρ0 is the initial material density before deformation.

[0216] Next, we find the plane wave solution for equation (38). According to solid wave theory, a plane wave can be expressed as:

[0217]

[0218] Where c is the phase velocity and t represents time. It is a unit vector along the direction of propagation. Let f be a unit vector along the direction of motion, and g be a quadratically differentiable function. It is a unit vector of electric displacement. The direction of motion of a shear wave is orthogonal to the direction of propagation, therefore... A longitudinal wave (pressure wave) is a wave whose direction of motion is the same as its direction of propagation, therefore it has...

[0219] Combining the above equations, we get:

[0220]

[0221] This is an eigenvalue problem of a tensor, where the direction of the wave's displacement is the direction of the eigenvector. Here, ρ represents the density of the propagation medium, and A is the generalized sound tensor, which defines the propagation conditions of plane waves in nonlinear electroelastic materials and has an arbitrary strain energy function. The generalized sound tensor of an electroelastic material has the following form:

[0222]

[0223] in, Indicates perpendicular to The projection on the surface, Represents the tensor trace operator;

[0224] Neglecting compressibility, longitudinal wave propagation does not occur in dielectric elastomers. Therefore, to analyze transverse and longitudinal waves, consider the energy function of a compressible dielectric elastomer in the following form:

[0225]

[0226] in, It is a purely elastic strain energy function, and various purely elastic strain energy models can be used, such as Neo-Hookean, Gent, etc. ε is the dielectric constant of the material in the reference configuration before deformation, and γ is... i These are dimensionless parameters, satisfying γ0 + γ1 + γ2 = 1, i = 0, 1, 2; for the ideal dielectric elastic body model, we have γ0 = γ2 = 0, γ1 = 1, and the generalized acoustic tensor simplifies to:

[0227] A = Q elas (43)

[0228] Given a purely elastic strain energy function ψ elas After the formality, Q elas Q represents the purely elastic contribution. elas The derivation process is as follows:

[0229]

[0230]

[0231]

[0232] This is the expression for the generalized acoustic tensor.

[0233] Step (5) Solve for the stress wave propagation characteristics of the dielectric elastic body planar actuator under a unidirectional electric field. Consider the stress wave propagation caused by electrostatic deformation of the dielectric elastic body under a unidirectional uniform electric field. In the principal direction... Apply voltage, i.e. Dielectric elastomer in a plane Extending upwards, the voltage-induced deformation gradient can be expressed as:

[0234]

[0235] For an incompressible ideal dielectric elastic body, the elongation can be written as λ(D)=(1+D) 2 ) -1 / 3 , for For highly compressible materials, the above elongation expression also holds approximately true within the range of deformation and voltage considered.

[0236] The compressible Neo-Hookean model has the following elastic strain energy functions:

[0237]

[0238] Where I1=trC,C=F·F T Let represent the first invariant of the right Cauchy-Green tensor, μ represent the shear modulus, K represent the bulk modulus, and J = detF. det represents taking the determinant. Solving the above equations simultaneously, we have:

[0239]

[0240]

[0241] in, Representative at Directional projection, Represents perpendicular to Projection onto the plane in the direction of the wave. Review the propagation conditions for plane waves:

[0242]

[0243] For any wave propagation direction The generalized acoustic tensor A has three characteristic directions, one of which is related to... Collinear, the other two are located perpendicular to On the plane, perpendicular to The former consists of one longitudinal wave, while the latter consists of two transverse waves.

[0244] Let the wave propagation angle be... Propagation direction vector Explicit expressions for the wave velocities of transverse and longitudinal waves can be obtained. The magnitude of the transverse wave velocity is:

[0245]

[0246] The magnitude of the longitudinal wave velocity is:

[0247]

[0248] For the compressible Gent model, considering the strain hardening effect of the material, the elastic strain energy function of the compressible Gent model is:

[0249]

[0250] Where I1=trB,B=F·F TLet μ denote the first invariant of the left Cauchy-Green tensor, μ denote the shear modulus, K denote the bulk modulus, and J denote the first invariant of the left Cauchy-Green tensor. m Let J represent the strain hardening constant, J = detF. Combining the above equations, we have:

[0251]

[0252]

[0253]

[0254] in, ξ=J m -I1+3.

[0255] Review the propagation conditions of plane waves:

[0256]

[0257] As can be seen from equation (56), the expression for the generalized acoustic tensor matrix A obtained by applying the Gent model is quite complex, and solving its eigenvalue problem requires knowledge of the wave propagation direction. We will discuss the different scenarios.

[0258] See Figure 11(a) for a schematic diagram of Case 1.

[0259] when Not on any main plane When above, it can be proven that the characteristic direction is not the same as Collinear, and not perpendicular There are neither transverse waves nor longitudinal waves.

[40] .

[0260] The schematic diagram for scenario 2 is shown in Figure 11(b).

[0261] when On the main plane When this is done, it can be proven that a characteristic direction is perpendicular to... And with Collinearity means that there exists a pure transverse wave, and the other two characteristic directions are collinear with each other. It is at a certain angle (≠90°), that is, there are quasi-transverse waves and quasi-longitudinal waves.

[0262] Let's assume the direction of wave propagation. Explicit expressions for the wave velocities of pure transverse waves, quasi-longitudinal waves, and quasi-transverse waves can be obtained. The derivation process and the wave velocity expressions for quasi-longitudinal and quasi-transverse waves are given in the appendix. Only the magnitude of the wave velocity of pure transverse waves is given here:

[0263]

[0264] The schematic diagram for scenario 3 is shown in Figure 11(c).

[0265] when On the main plane When this is done, it can be proven that a characteristic direction is... Collinear, the other two characteristic directions are the same as Vertical means that there is one pure transverse wave and two pure longitudinal waves.

[0266] Let's assume the direction of wave propagation. The magnitude of the pure transverse wave velocity can be obtained as:

[0267]

[0268] The magnitude of the pure longitudinal wave velocity is:

[0269]

[0270] Example:

[0271] To gain a fuller understanding of the features of this invention and its applicability to engineering practice, this invention first studies the creep and relaxation phenomena of a planar actuator structure of a dielectric elastomer under quasi-static conditions to verify the effectiveness of the established viscoelastic dielectric elastomer constitutive model. Subsequently, numerical simulations are performed to calculate the response of the planar actuator under alternating voltage and the stress wave propagation characteristics under a unidirectional electric field.

[0272] Take the following parameters: T v =10 / 50 / 100, the equilibrium deformation λ is calculated. eq The magnitude is 2.4613, and the natural frequency of the small-amplitude oscillation near the equilibrium state is... The value is 1.24. Initial conditions are set near equilibrium: λ0 = ξ0 = 2.46. Numerical simulations were performed using the ode solver in MATLAB to analyze the nonlinear vibration response characteristics of a plane actuator considering viscoelastic effects under alternating voltage.

[0273] from Figure 2 As can be seen from this, considering the viscoelasticity of the material, the natural frequency of the planar actuator structure is time-dependent and also resonates at multiple excitation frequencies (harmonic resonance, superharmonic resonance, and subharmonic resonance are shown in Figures 6(a), (b), and (c)). Figure 3 Figures (a) to (f) and Figure 4 Figures (a) to (f) show that as the relaxation time increases, the viscoelastic effect intensifies. This viscoelastic effect reduces the resonant frequency of the planar actuator structure, increases the maximum amplitude, and increases the time it takes for the structure to reach steady-state vibration. The beat vibration phenomenon during steady-state vibration also becomes more pronounced. Figure 5It can be seen that, comparing the dynamic response under the linear damping assumption, the next harmonic dynamic response under the linear damping assumption differs significantly from that considering the viscoelastic case. Figures 7(a)-7(c) and 8(a)-8(c) show that the electric field magnitude, wave propagation direction, and compressibility affect the separation angle between the transverse and longitudinal waves. Furthermore, the strain hardening effect alters the direction of the separation angle, and it is no longer affected by changes in the compressibility coefficient. Figures 9(a)-9(d) show that the strain hardening effect significantly increases the wave phase velocity and enhances its dependence on the propagation direction.

[0274] The parts of this invention not described in detail are well-known to those skilled in the art.

[0275] 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 with parameter uncertainties. All technical solutions formed by equivalent transformation or equivalent substitution fall within the scope of protection of the present invention.

Claims

1. A method for analyzing the nonlinear dynamics and stress wave propagation characteristics of dielectric elastomer actuator structures, used in the design of dielectric elastomer actuator structures, characterized in that... Includes the following steps: Step 1: Based on the principle of virtual work, derive the dynamic control equations for the planar actuator of the dielectric elastic body; To simplify the analysis without loss of generality, we consider biaxial deformation cases such as incompressible dielectric elastomer planar actuators, i.e. , Dimensionless transformation is performed, taking into account the viscoelasticity of the material, using the elongation rate. and damping rate Dimensionless dynamic governing equations for state variables: (20) (21) (22) in, Indicates strain, strain hardening constant, superscript Indicates a superelastic spring, superscript This represents a time-dependent viscoelastic spring, where the two springs have different strain hardening constants. Let P be the external tensile force in the XY plane; All are elongation. Voltage in the Z direction. Represents dimensionless mechanical force. Represents dimensionless voltage. Indicates shear modulus, Indicates the viscoelastic relaxation time. Represents the strain hardening constant, a dimensionless quantity: in, These are the dimensions before deformation. The density of the dielectric elastomer material, Indicates dimensionless time; The shear modulus represents the viscoelastic portion. The shear modulus represents the hyperelastic portion; This represents the dimensionless viscoelastic relaxation time. This represents the viscosity coefficient of the damper. Indicates the dielectric constant of the material; Step 2: Calculate the equilibrium deformation and the natural frequency of the small-amplitude vibration around the equilibrium position using the small perturbation method; Step 3: Solve the nonlinear vibration response of the dielectric elastomer planar actuator under alternating voltage load; Step 4: Derive explicit expressions for the generalized acoustic tensor and stress wave phase velocity based on the theory of finite deformation material dynamics; given the purely elastic strain energy function. After the formalities, This represents the purely elastic contribution. The derivation process is as follows: (44) (45) (46) That is, the expression for the generalized sound tensor; in, Represents the elastic modulus tensor under the reference configuration. Represents the elastic strain energy function. Represents the deformation gradient tensor. This represents the elastic modulus tensor under the current configuration. The Jacobian determinant representing the deformation gradient, The superscript represents the transpose of the deformation gradient tensor. Isomer labeling of fourth-order tensors This represents the purely elastic contribution of the generalized vocal tensor. Indicates a characteristic direction unit vector; Step 5: Solve for the stress wave propagation characteristics of the dielectric elastomer planar actuator under a unidirectional electric field, including wave velocity and transverse and longitudinal wave separation angles.

2. The method for analyzing the nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure according to claim 1, characterized in that: The first step specifically includes: Considering the dielectric elastomer planar actuator structure, the dimensions before deformation are: Subjected to external tensile force in the XY plane After a voltage φ is applied in the Z direction, the dielectric elastomer film expands in the XY plane, increasing its area, while its thickness decreases in the Z direction; let the dimensions after deformation be... Define the system's state variable as the elongation. , , It is related to time; According to Gauss's law of electrostatics, the relationship between the charge and voltage in a dielectric elastic material actuator is as follows: (1) Taking the partial derivative with respect to it yields expression: (2) When a small amount of charge passes through the dielectric elastomer film, the voltage does work. external force doing work The inertial forces per unit length element in the x and y directions are respectively ,in, Let be the density of the dielectric elastomer material, and be the linear damping forces respectively. ,in, The linear damping coefficient is given by the length in the x and y directions, respectively. Integrating the units, the work done by the inertial force and the damping force are obtained as follows: (3) According to the principle of virtual work, the change in free energy of a dielectric elastic film is equal to the work done by voltage, external force, inertial force, and damping, that is: (4) in, Indicates the linear damping coefficient; The free energy density of a dielectric elastic body is equal to the sum of the elastic strain energy density and the electrostatic energy density. Using the Gent hyperelastic strain energy model and the linear polarization assumption, the total free energy density is expressed as: (5) in, Indicates shear modulus, D represents the strain hardening constant, and D represents the electric displacement. Substituting equation (2) into equation (4), we get: (6) Among them, the independent variables are , ; (7) (8) Substituting equation (5) into equations (7) and (8), we get: (9) (10) That is, the nonlinear dynamic equation of the dielectric elastomer planar actuator considering linear damping; To simplify the analysis without loss of generality, we consider biaxial deformation cases such as actuators made of incompressible dielectric elastomer materials, i.e. , , After dimensionless transformation, equations (9) and (10) are simplified to the following: considering the linear damping assumption, with the elongation ratio Dimensionless dynamic governing equations for state variables: (11) (12) Among them, dimensionless time Dimensionless damping Dimensionless voltage Dimensionless mechanical force , Indicates elongation; Considering viscoelasticity, the application of the principle of virtual work is the same, except that the linear damping term is subtracted from equations (7) and (8), resulting in: (13) (14) Using the Gent elastic strain energy model that considers viscoelasticity, the total free energy expression of the system is: (15) in, Indicates shear modulus, Represents the strain hardening constant. Indicates the damping ratio; Substituting equation (15) into equations (13) and (14), we get: (16) (17) The damper is modeled as a Newtonian fluid, and its deformation follows the following evolution law: (18) (19)。 3. The method for analyzing the nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure according to claim 2, characterized in that: The second step specifically includes: For the linear damping assumption, equilibrium deformation satisfy: (23) in, Represents the elongation at equilibrium. Represents the strain hardening constant. This represents the aforementioned dimensionless system parameter; The dimensionless natural frequency of the small-amplitude oscillations near the equilibrium state can be obtained using the small perturbation method. The expression: (24) Therefore, for the linear damping assumption, the deformation of the equilibrium state is calculated according to equation (23), and the natural frequency of the system is calculated by substituting it into equation (24). Considering the viscoelasticity of materials and equilibrium deformation satisfy: (25) The natural frequency is obtained using the small perturbation method. The expression: (26) (27)。 4. The method for analyzing the nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure according to claim 3, characterized in that: The third step specifically includes: A dielectric elastomer planar actuator vibrates near its equilibrium state when subjected to a small disturbance under constant internal air pressure and alternating voltage. The alternating voltage consists of DC and sinusoidal voltages. (28) Among them, is DC voltage magnitude, It is the amplitude of the sinusoidal voltage. It is the excitation frequency, obtained after dimensionless conversion. ; Substituting equation (28) into equations (20)-(22), we obtain the governing equations for the viscoelastic Gent model: (29) (30) Substituting equation (28) into equation (11), we obtain the governing equation under the linear damping assumption: (31)。 5. The method for analyzing the nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure according to claim 4, characterized in that: The fourth step specifically includes: To analyze the finite deformation state of dielectric elastomer materials, a deformation gradient tensor is introduced. ,in and These are the position vectors after deformation / current configuration and before deformation / reference configuration, respectively. Let the gradient and dyadic operators be represented respectively, and introduce the strain energy function. The independent variable is the deformation gradient. Electric displacement vector under reference configuration The first Piola-Kirchhoff stress tensor and electric field strength in the reference configuration are written as: (32) The corresponding quantities between the current configuration and the reference configuration are related: (33) Linearizing the constitutive formula (32), we have: (34) in, Represents incremental change. It is the nominal electroelastic modulus tensor, defined as: (35) Considering the small-amplitude motion superimposed on a finite deformation state, equation (34) can be rewritten in incremental configuration form: (36) in, (37) Here, the superscript (2134) indicates that the tensor isomers are isomers; The linearized equations of motion and the Euler form of Maxwell's equations are as follows: (38) in, Indicates incremental displacement. It is the density of the material after deformation. It is the initial density of the material before deformation; Next, we find the plane wave solution for equation (38). According to solid wave theory, the plane wave is expressed as: (39) Where c is the phase velocity and t represents time. It is a unit vector along the direction of propagation. It is a unit vector along the direction of motion. g is a quadratically differentiable function, and g is a continuously differentiable function. It is a unit vector of electric displacement; the direction of transverse wave motion is orthogonal to the direction of propagation, therefore we have A longitudinal wave is a wave whose direction of motion is the same as its direction of propagation, therefore it has ; Substituting equations (36) and (39) into equation (38), we get: (40) It is an eigenvalue problem of a tensor, where the direction of the wave's displacement is the direction of the eigenvector; where, Let A represent the density of the propagation medium, and let A be the generalized sound tensor, which defines the propagation conditions of plane waves in nonlinear electroelastic materials and has an arbitrary strain energy function. The generalized sound tensor of an electroelastic material has the following form: (41) in, Indicates perpendicular to The projection on the surface, , , , Represents the tensor trace operator; Neglecting compressibility, longitudinal wave propagation does not occur in dielectric elastomers. Therefore, to analyze transverse and longitudinal waves, consider the energy function of a compressible dielectric elastomer in the following form: (42) in, It is a purely elastic strain energy function, employing various purely elastic strain energy models, including Neo-Hookean, Gent, and others. It is the dielectric constant of the material in the reference configuration before deformation. It is a dimensionless parameter, i=0,1,2, satisfying For the ideal dielectric elastic body model, we have The generalized sound tensor simplifies to the following form: (43)。 6. The method for analyzing the nonlinear dynamics and stress wave propagation characteristics of a dielectric elastomer actuator structure according to claim 5, characterized in that: The fifth step specifically includes: Consider the propagation of stress waves in a dielectric elastic body undergoing electrostatic deformation under a unidirectional uniform electric field, in the principal direction Apply voltage, i.e. Dielectric elastomer in a plane The upward extension, the voltage-induced deformation gradient is expressed as: (47) For an incompressible ideal dielectric elastic body, the elongation is written as... , ,for For highly compressible materials, the above elongation expression also holds approximately true within the range of deformation and voltage considered. Using specific pure elastic strain energy models: the Neo-Hookean model and the Gent model considering strain hardening effects, an explicit expression for the generalized acoustic tensor is derived to analyze the propagation of stress waves in dielectric elastic bodies. (1) Neo-Hookean model: The compressible Neo-Hookean model has the following elastic strain energy functions: (48) in, Denotes the first invariant of the right Cauchy-Green tensor. Indicates shear modulus, Indicates bulk modulus. , This indicates taking the determinant; substituting equation (48) into equations (44)-(46), (49) (50) in, , Representative at Directional projection, Represents perpendicular to Projection onto the plane in the direction of the wave; review the propagation conditions of plane waves: (51) For any wave propagation direction The generalized sound tensor A has three characteristic directions, one of which is related to... Collinear, the other two are located perpendicular to On the plane, perpendicular to The former consists of one longitudinal wave, while the latter consists of two transverse waves; Let the wave propagation angle be... Propagation direction vector This yields explicit expressions for the wave velocities of transverse and longitudinal waves; the magnitude of the transverse wave velocity is: (52) The magnitude of the longitudinal wave speed is (53) (2) Gent model Considering the strain hardening effect of the material, the elastic strain energy function of the compressible Gent model is: (54) in, Denotes the first invariant of the left Cauchy-Green tensor. Indicates shear modulus, Indicates bulk modulus. Represents the strain hardening constant. Substituting equation (54) into equations (44)-(46), we get: (55) (56) (57) in, .

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