A Microscopic Modeling Method of MRE Based on ECC Lattice Structure

Abstract

The present invention relates to a microscopic modeling method of MRE based on the ECC lattice structure, belonging to the field of microscopic structure mechanism analysis and dynamic mechanical property research of intelligent material MRE. A cubic particle network based on the ECC lattice is proposed, where magnetic particles are located at the junctions and connected to elastic springs. By using Langevin dynamics, the control equations of particle motion are derived to calculate the relaxation spectra related to the motion of particles in parallel and perpendicular directions with respect to the applied magnetic field, so as to determine the storage modulus and loss modulus of MRE. Establishing the motion control equations based on the ECC lattice microstructure overcomes the limitations of the phenomenological model and lays a working foundation for the subsequent research on the dynamic magneto-mechanical characteristics of materials. The present invention can more accurately explain the changes in the dynamic mechanical properties of magnetorheological elastomers (MRE) from the mechanism, and the multifunctional MRE with magneto-control characteristics has great potential in the development of new technologies in various industrial fields such as automotive, aerospace, civil, and biomedical.

Description

Technical Field

[0001] The present invention belongs to the field of microstructure mechanism analysis and dynamic mechanical property research of intelligent material MRE. Specifically, a motion control equation is established based on the ECC lattice microstructure, laying a working foundation for the subsequent research on the dynamic magneto-mechanical characteristics of the material. Background Art

[0002] The multifunctional magnetorheological elastomer (MRE) with magneto-control characteristics has great potential in the development of new technologies in various industrial fields such as automotive, aerospace, civil, and biomedical.

[0003] Regarding the model construction of composite material MRE, in 2020, the paper "Experimental characterization and viscoelastic modeling of isotropic and anisotropic magnetorheological elastomers" published by T.H. Nam et al. in "Polymer Testing" used the construction method of the phenomenological model. For example, the Maxwell, Bouc-Wen, and Voigt elements or parametric mathematical formulas were used to represent the viscoelastic characteristics of the material. The disadvantages of such models are that they do not have universality, have more model parameters, and for different experimental environments or operating conditions, the model parameters need to be recalibrated; in addition, most of the model parameters do not have clear physical meanings and cannot explain the dynamic characteristics of MRE from the mechanism.

[0004] Another example is the paper "A nonlinear magnetorheological elastomer model based on fractional viscoelasticity magnetic dipole interactions and adaptive smooth Coulomb friction" published by X.B. Nguyen et al. in "Mechanical Systems and Signal Processing" in 2020. To overcome the limitations of the phenomenological model, a method of combining the microscopic dipole model with the fractional viscoelastic model was adopted, but this model still cannot well explain the changes in the dynamic mechanical properties of MRE from the mechanism.

[0005] To study the dynamic magneto-mechanical properties of MREs. For this purpose, a dynamic physical model considering the microstructure of MREs was developed to accurately predict the frequency and field-related linear viscoelastic characteristics of the material. Summary of the Invention

[0006] Technical problem to be solved

[0007] To develop a model that can more accurately explain the changes in the dynamic mechanical properties of MRE from a mechanism, the present invention provides a cubic particle network based on an ECC (Edge-centered cubic) lattice, where magnetic particles are located at the joints and connected to elastic springs. Using Langevin dynamics, the control equations of particle motion are derived to calculate the relaxation spectra related to the motion of particles in the parallel and perpendicular directions with respect to an applied magnetic field, laying a foundation for determining the storage modulus and loss modulus of MRE.

[0008] Technical solution

[0009] A microscopic modeling method for MRE based on an ECC lattice structure, characterized by the following steps:

[0010] 1) Use software to construct an ECC unit particle structure network ( Figure 2 ), clarify the particle distribution in the unit cube. Among them, the side length relationship of the isotropic cube model unit is In the model, the sphere represents a ferromagnetic particle with a radius of r.

[0011] 2) Relate the internal structure of the magnetorheological elastomer to the idealized assumptions of the ECC microstructure. The ferromagnetic particles in the composite material are represented by Figure 3 the spheres 1 and 2 in Figure 3 , and the elastic filling in the material is represented by x the spring 3 in y . In the isotropic case, the elastic coefficient relationship is K z = K0.

[0012] 3) Establish the motion control equations for the proposed ECC lattice. Brownian dynamics is used to describe the motion of particles 1 and 2, and the motion equations of the particles are expressed using Lagrange equations. Among them, the kinetic energy T(r n ) of the nth particle can be expressed as:

[0013]

[0014] In the formula, m p is the particle mass, r n is the position vector of the particle, is the time derivative of the position vector. The energy dissipation terms E diss , E diss2 of the nth particle of the same type as particle 1 and the particle of the same type as particle 2 are respectively:

[0015]

[0016]

[0017] In the formula, ζ jun is the friction coefficient of the same type of particles as particle 2, and ζ0 is the friction coefficient of the same type of particles as particle 1. The relationship between the two is ζ jun = 3ζ0.

[0018] The elastic potential energy of the nth particle is U elast (r n ), the magnetic potential energy is U magn (r n ), and the total potential energy is U(r n ). The relationship among the three is:

[0019] U(r n ) = U elast (r n ) + U magn (r n )

[0020] Using the extended Euler-Lagrange equation, the model relationship is:

[0021]

[0022] L = T(r n ) - U(r n )

[0023] In the formula, is the generalized force related to the motion of the nth particle. The common generalized forces are mainly divided into the Brownian force and the external load

[0024] 4) Simplify the Euler-Lagrange equation in step 3), and the motion equation of the particle can be obtained as:

[0025]

[0026]

[0027]

[0028]

[0029]

[0029] 4.1) The first derivative of the magnetic potential energy function is:

[0030]

[0031] In the formula, δr n = r n - r n′ is the small change around the average value of the position vector, is the gradient vector of the magnetic energy function at the average position of the particle, is the Hessian matrix.

[0032] After simplification, the first derivative of the total magnetic potential energy is:

[0033]

[0034] In the formula, is obtained by taking the second derivative of the Hessian matrix U magn

[0035] 4.2) The first derivative term of the total elastic potential energy between particles under the ECC lattice is:

[0036]

[0037] Among them, the nth particle is connected to the adjacent n ′ th particle by a spring, and C nn′ is the connectivity parameter.

[0038] 4.3) The first derivative term of the total energy dissipation between particles based on the ECC lattice is:

[0039]

[0040] 4.4) The particle motion equation can be written in the following form:

[0041]

[0042] 5) Transform the motion control equation obtained in step 4.4):

[0043] 5.1) Convert the variable from Cartesian coordinates to normal coordinates. The expression form of the energy loss term in the normal coordinate system is:

[0044]

[0045] In the formula, is the time derivative of the normal coordinate;

[0046] 5.2) There are six particles 2 and six particles 1 in the cubic grid around the nth particle 1 based on the ECC lattice. Then the spring force term in the formula of step 4) can be simplified to:

[0047] ​

[0048] 5.3) After replacing the normal mode, the terms related to magnetism are in the form of:

[0049]

[0050] 5.4) Transform the motion control equation into a homogeneous equation:

[0051]

[0052] 6) For the simplified result of step 5.4), further obtain the eigenvalues and relaxation times. Among them, the approximate solution of the eigenvalue is:

[0053]

[0054]

[0055]

[0056] The relaxation times related to the motion of particles parallel and perpendicular to the magnetic field direction are:

[0057]

[0058] Among them, τ0 is the minimum relaxation time related to isotropic particles when the magnetic flux density is 0.

[0059] Furthermore, step 4) is specifically:

[0060] For the result in step 4.1), the analysis process of the Hessian matrix is:

[0061] When the magnetic potential energy gradient is in equilibrium, The Hessian matrix is also composed of three parts. The first part is the contribution of the interaction between particles of the same type as particle 1, the second part is provided by the interaction between particles of the same type as particle 2, and the third part is provided by the interaction between particles of the same type as particle 1 and particles of the same type as particle 2:

[0062]

[0063] In the equilibrium state, the function expression of the updated Hessian matrix is:

[0064]

[0065] Among them, μ0 is the relative magnetic permeability, particle polarization The critical value of the highest magnetization of isotropic particles G0 is the shear modulus without magnetic field, φ is the CIP volume fraction in MRE, and the newly introduced vector k = n′ - n.

[0066] The analysis process of the first - order derivative term of the total elastic potential energy between particles in step 4.2) is as follows:

[0067] The first - order derivative term of the total elastic potential energy between particles in the ECC lattice is mainly composed of two types: the first - order derivative term of the elastic potential energy of the interaction between particles of the same type as particle 1 and the first - order derivative term of the elastic potential energy of the interaction between particles of the same type as particle 1 and particles of the same type as particle 2. And the relationship between the two is:

[0068]

[0069]

[0070] Among them, r n1 is the spatial vector of particles of the same type as particle 1; r n2′ is the spatial vector of particles of the same type as particle 2.

[0071] Furthermore, in step 5), specifically:

[0072] In step 5.1), to solve the motion equation and find the relaxation time of particle motion, through Fourier transform, the Fourier transform formula is:

[0073]

[0074] In the formula, θ=(θ x ,θ y ,θ z ) is the phase - shift vector, and its value range is [0,π], Q(θ) is the normal coordinate, and nθ is the scalar product of the vectors.

[0075] In step 5.2), to simplify the formula, the Euler formula is introduced as:

[0076]

[0077] Furthermore, in step 6), the specific simplification process is:

[0078]

[0079] Without considering the imaginary term, the definition is introduced That is, when the magnetic flux density is 0, it is the minimum relaxation time related to isotropic particles.

[0080] The expression is simpler and more intuitive, and in a compact form: The matrix can be written as Among them, is the third - order identity matrix; λ0(θ) is the defined function, matrix is in the form of:

[0081]

[0082] Assume that the matrix The motion in one direction is independent of the motion in other directions, then the diagonal elements of the matrix are:

[0083]

[0084] The approximate solution of the eigenvalue λ is:

[0085]

[0086] Beneficial effects

[0087] A microscopic modeling method of MRE based on ECC lattice structure provided by the present invention has the following advantages:

[0088] 1. The present invention proposes a microscopic modeling method of MRE based on ECC lattice structure. The model established by this method can more accurately explain the dynamic mechanical properties of MRE compared with the prior art;

[0089] 2. The present invention expands the microscopic structure modeling of the magnetodynamics of MREs. For the new structure of particles in the ECC lattice network, the motion equations of particles are updated, paving the way for the constitutive modeling of composite materials;

[0090] 3. Based on the modeling of the ECC lattice particle network, calculate the relaxation spectra related to the motion of particles in parallel and perpendicular directions with respect to the applied magnetic field, laying a foundation for determining the storage modulus and loss modulus of MRE. Description of the drawings

[0091] The drawings are only for the purpose of illustrating specific embodiments and are not considered to be a limitation of the present invention. Throughout the drawings, the same reference signs denote the same components.

[0092] Figure 1 is the process flow chart for the constitutive modeling of MRE based on ECC microstructure;

[0093] Figure 2 is the schematic diagram of the geometric model of the isotropic ECC lattice unit structure;

[0094] Figure 3 is the schematic diagram of the idealized assumption of the ECC microstructure of the composite material MRE;

[0095] Figure 4 is the schematic diagram of the geometric model of the internal structure of MRE under the 8-lattice network;

[0096] Figure 5 Schematic diagram of the geometric model of the internal structure of MRE based on the idealized assumptions of ECC. Specific implementation mode

[0097] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0098] A microscopic modeling method of MRE based on the ECC lattice structure provided by the present invention specifically comprises the following steps:

[0099] 1) Use software to construct the ECC unit particle structure network ( Figure 2 ), and clarify the particle distribution in the unit cube. Among them, the side length relationship of the cube model unit is In the model, the sphere represents the ferromagnetic particle, and its radius is r;

[0100] 2) Relate the dynamic magneto-mechanical properties of the composite material MRE to the idealized assumptions of the ECC microstructure. In MRE, the ferromagnetic particles are represented by Figure 3 the spheres 1 and 2 in Figure 3 the elastic material of the material is represented by x the spring 3 in y The elastic coefficient relationship is K z = K

[0101] 3) Establish the motion control equation for the proposed ECC lattice. Brownian dynamics is used to describe the motion of particles 1 and 2, and the motion equation of the particles is represented by the Lagrangian equation. Among them, the kinetic energy T(r n ) of the nth particle can be expressed as:

[0102]

[0103] In the formula, m p is the particle mass, r n is the position vector of the particle, is the time derivative of the position vector.

[0104] The energy dissipation terms E diss1 , E diss2 of the nth particle of the same type as particle 1 and the particle of the same type as particle 2 are respectively:

[0105]

[0106]

[0107] In the formula known from Figure 3 , ζ jun is the friction coefficient of the same type of particles as particle 2, and ζ0 is the friction coefficient of the same type of particles as particle 1. The relationship between the two is ζ jun = 3ζ0.

[0108] The elastic potential energy of the nth particle is U elast (r n ), the magnetic potential energy is U magn (r n ), and the total potential energy is U(r n ). The relationship among the three is:

[0109] U(r n ) = U elast (r n ) + U magn (r n )

[0110] Using the extended Euler - Lagrange equation, the model relationship is:

[0111]

[0112] L = T(r n ) - U(r n )

[0113] In the formula, is the generalized force related to the motion of the nth particle. The common generalized forces are mainly divided into the Brownian force and the external load applied to the particle due to the dynamic mechanical load

[0114] 4) Simplify the Euler - Lagrange equation in step 3). Among them, the first - order derivative of the magnetic potential energy function is:

[0115]

[0116] In the formula, δr n = r n - r n′ is the small change around the average value of the position vector, is the gradient vector of the magnetic energy function at the average position of the particle, is the Hessian matrix. When the magnetic potential gradient is in the equilibrium state, After simplification, the first - order derivative of the total magnetic potential energy is:

[0117]

[0118] In the formula, Obtained from the Hessian matrix U magn by taking the second derivative, where M is the particle polarization, and M * is a parameter related to the particle polarization, and k is a newly introduced vector.

[0119] The first derivative term of the total elastic potential energy between particles under the ECC lattice is:

[0120]

[0121] Among them, the nth particle is connected to the adjacent n'th particle by a spring, and C nn′ is the connectivity parameter.

[0122] The first derivative term based on the total energy dissipation between particles under the ECC lattice is:

[0123]

[0124] The particle motion equation can be written in the following form:

[0125]

[0126] 5) For the motion control equation finally obtained in step 4), through the conversion from Cartesian coordinates to normal coordinates.

[0127] The expression form of the energy loss term in the normal coordinate system is:

[0128]

[0129] In the formula, is the time derivative of the normal coordinate;

[0130] In the cubic grid around the nth particle 1 based on the ECC lattice, there are six particles 2 and six particles 1. The spring force term in the formula of step 4) can be simplified to:

[0131]

[0132] The term related to the magnetic force, after replacing the normal mode, is in the form of:

[0133]

[0134] Convert the motion control equation into a homogeneous equation:

[0135]

[0136] 6) For the simplified result of step 5), further obtain the eigenvalues and relaxation times. Among them, the approximate solution of the eigenvalues is:

[0137]

[0138]

[0139]

[0140] The relaxation times related to the motion of particles parallel and perpendicular to the magnetic field direction are as follows:

[0141]

[0142] where τ0 is the minimum relaxation time related to isotropic particles when the magnetic flux density is 0.

[0143] As described above, only the specific embodiments of the present invention are provided, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention can easily think of various equivalent modifications or substitutions, and these modifications or substitutions should be covered within the protection scope of the present invention.

Claims

1. A microscopic modeling method of MRE based on the ECC lattice structure, characterized in that The steps are as follows: Step 1: Use the 3D modeling software UG to construct a network model of the ECC unit particle structure. The model is a cube structure, and ferromagnetic particles are distributed on each side of the cube. Among them, the side length relationship of the isotropic cube model unit is The radius of the ferromagnetic particle is r; Step 2: Relate the internal structure of the composite MRE to the idealized assumptions of the ECC microstructure, i.e., magnetic particles are located at the joints and connected to elastic springs, and the relationship of the elastic coefficients is K x = K y = K z = K0; Step 3: Establish the motion control equation for the proposed ECC lattice: The motion of two adjacent particles 1 and 2 is described by Brownian dynamics, and the motion equation of the particles is expressed using the Lagrangian equation; among them, the kinetic energy T(r n ) can be expressed as: where m p is the particle mass, r n is the position vector of the particle, is the time derivative of the position vector; The energy dissipation terms E of the nth particle of the same type as particle 1 or of the same type as particle 2 diss1 and E diss2 are respectively: where ζ jun is the friction coefficient of particles of the same type as particle 2, and ζ0 is the friction coefficient of particles of the same type as particle 1; The elastic potential energy of the nth particle is U elast (r n ), the magnetic potential energy is U magn (r n ), the total potential energy is U(r n ), and the relationship among the three is as follows: U(r n ) = U elast (r n ) + U magn (r n ) Using the extended Euler - Lagrange equation, the model relationship is: L = T(r n ) - U(r n ) wherein, is the generalized force related to the motion of the nth particle, and E diss is the total energy dissipation between particles; Step 4: Simplify the Euler - Lagrange equation in Step 3; where, the first - order derivative of the magnetic potential energy function is: where, δr n is the minute change around the average value of the position vector, is the gradient vector of the magnetic energy function at the average position of the particle, is the Hessian moment, r n′ is the position vector of the particle; When the magnetic potential energy gradient is in an equilibrium state, 4.1) After simplification, the first - order derivative of the total magnetic potential energy is: In the formula, obtained by taking the second derivative of the Hessian matrix U magn where M is the particle polarization, and M * is a parameter related to the particle polarization, and k is a newly introduced vector; 4.2) The first - order derivative term of the total elastic potential energy between particles under the ECC lattice is: Among them, the nth particle and the adjacent n'th particle are connected by a spring, and C nn′ is the connectivity parameter, is the r n components in the x, y, and z directions; 4.3) The first - order derivative term of the total energy dissipation between particles under the ECC lattice is: 4.4) The motion equation can be written in the following form: Step 5: Simplify the motion control equation obtained in Step 4.4), and perform the transformation from Cartesian coordinates to normal coordinates; 5.1) The expression form of the energy loss term in the normal coordinate system is: wherein, is the time derivative of the normal coordinate; 5.2) Based on the fact that there are six particles 2 and six particles 1 in the cubic grid around the n - th particle 1 of the ECC lattice, the spring force term in Equation 4.2) can be simplified to: 5.3) After replacing the normal mode, the term related to the magnetic force is in the form of: where θ = (θ x , θ y , θ z ). Since the eigenvalue is affected by the phase shift vector θ, it is defined that the components of the phase shift satisfy θ x = θ y = θ z = θ; 5.4) Convert the motion control equation into a homogeneous equation: Step 6: For the simplified result of Step 5.4), further obtain the eigenvalues and relaxation times; where, the approximate solution of the eigenvalues is: The relaxation times related to the particle motion parallel and perpendicular to the magnetic field direction are: Where, τ0 is the minimum relaxation time related to isotropic particles when the magnetic flux density is 0.

2. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In the said step 3, the friction coefficient ζ ju The relationship between n and ζ0 is: ζ jun = 3ζ0.

3. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In the said step 3, the motion equation of the particle can be derived as and the inertial term is not considered.

4. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In the said step 4.1, the Hessian matrix U magn obtain the matrix by taking the second derivative is: In the formula, k is the newly introduced vector, which here refers to the spatial vector pointing from the n′ - th particle to the n - th particle when assuming the unit lattice side length is 1, expressed as k = n′ - n.

5. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In step 4.1, M * can be expressed as: In the formula, μ0 is the relative magnetic permeability, G0 is the shear modulus without magnetic field, and φ is the CIP volume fraction in the MRE.

6. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In Step 5.1, to solve the motion equation and obtain the relaxation time of particle motion, the variables are transformed from Cartesian coordinates to normal coordinates through Fourier transform, and the Fourier transform is: where θ = (θ x , θ y , θ z ) is the phase shift vector, the value range is [0, π], Q(θ) is the normal coordinate, and nθ is the scalar product of the vectors.

7. The MRE microscopic modeling method based on the ECC lattice structure according to claim 1, characterized in that: In Step 5.2, to simplify the formula, Euler's formula is introduced:

8. A computer system, characterized in thatIncluding: One or more processors, a computer - readable storage medium for storing one or more programs, where, when the one or more programs are executed by the one or more processors, the one or more processors implement the method described in Claim 1.

9. A computer-readable storage medium, characterized in that Stored with computer - executable instructions, the instructions are used to implement the method described in Claim 1 when executed.

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