Method, device and medium for predicting and shape controlling of magnetic bending deformation of multi-layer hard magnetic soft material plate

Abstract

The application discloses a kind of multilayer hard magnetic soft material plate magnetic bending deformation prediction and shape control method, equipment and medium, wherein the method comprises: the energy density functional of magnetic multilayer plate is established, and the control equation set of magnetic multilayer plate is derived by variational method;Plate equation is established according to the control equation set, and the deformation prediction and control scheme are obtained by solving plate equation;Obtain target curve parameter equation, substitute target curve parameter equation into control scheme, obtain corresponding parameters.The scheme proposed in the application realizes the prediction and shape control of multilayer hard magnetic soft material plate bending deformation, and compared with other control modes, the action of magnetic field can make the thin plate quickly generate target curve.The accuracy and agility of this control mode have important application prospect and value in the field of flexible electronics and soft robot.The application can be widely applied in the technical field of hard magnetic soft material bending deformation prediction and control and flexible intelligent device design and development.

Description

Technical Field

[0001] This invention relates to the technical field of predicting and controlling the bending deformation of hard magnetic soft materials and designing and developing flexible intelligent devices, and particularly to a method, device and medium for predicting and controlling the magneto-induced bending deformation of multilayer hard magnetic soft material plates. Background Technology

[0002] Magnetic soft materials consist of a soft matrix and magnetic particles at the micrometer or nanometer scale, and can undergo rapid and reversible deformation under the influence of a magnetic field. Hard magnetic soft materials (HMSMs) have attracted widespread research interest due to their excellent deformation and responsiveness. The magnetic particles in HMSMs possess high coercivity (such as neodymium iron boron and AlNiCr), which allows HMSMs to maintain high remanence even in the absence of an external magnetic field. Under the influence of a magnetic field or other stimuli, HMSMs can exhibit complex deformations and possess excellent shape control capabilities, making them ideal materials for designing and fabricating novel soft actuators, soft robots, and flexible electronic devices. In particular, by utilizing techniques such as 3D printing and topology optimization, the distribution of residual magnetization field in HMSM samples can be artificially controlled, thereby enabling HMSMs to achieve diverse functions.

[0003] For decades, researchers have been dedicated to studying the magnetomechanical properties of hard magnetic polymers (HMSMs). Experimentally, Stepanov et al. investigated the viscoelastic deformation behavior of HMSMs synthesized by filling silicone rubber with NdFeB particles. Their study found that even in the absence of a magnetic field, the hard magnetic particles induce irreversible residual strain. Koo et al. prepared hard magnetic magnetorheological elastomers and tested their bending deformation under a magnetic field. Zhao et al. conducted magnetically driven bending tests on HMSM beams with different thickness-to-length ratios. Manish et al. measured the magnetomechanical response of hard magnetic hydrogels containing different mass fractions of magnetic particles. Chen et al. investigated the instability of non-uniformly magnetized HMSM plates under magnetic field-induced instability. Pal and Sitti designed and fabricated magnetically responsive bistable beam structures using HMSMs and studied the state transitions of these structures under a magnetic field.

[0004] In terms of theoretical modeling, Dorfmann and Ogden established some commonly used frameworks for magnetically active soft materials. Zhao et al. proposed a finite-strain continuous model for hard magnetic soft materials, in which the magnetic free energy per unit volume can be expressed as... ( (where M is the total deformation gradient tensor, B is the magnetization density vector, and B is the magnetic flux density of the applied magnetic field). This model agrees very well with the test results of slender HMSM structures under bending deformation. Mukherjee et al. developed a deformation guidance framework for isotropic and incompressible hard magnetic elastomers. Based on this finding, Yan et al. proposed a modified magnetic potential energy function, primarily using a rotation tensor. replace By adopting a based Zhang et al. investigated the buckling problem of bistable hard magnetoelastics using their model. Further analysis by Danas and Reis showed that the fully dissipative model established by Mukherjee et al. can be simplified to the model of Yan et al., but not to the model of Zhao et al. Furthermore, Dorfmann and Ogden developed a modified model, ensuring the symmetry of the Cauchy stress tensor by introducing an additional term into the magnetoelastic properties. To study some unique experimental characteristics of HMSMs (such as viscoelastic effects and microstructural rearrangement of particles), researchers also proposed a viscoelastic model. Based on the theoretical model, Liu et al. developed a numerical framework for finite element simulation of the complex deformation of HMSM samples.

[0005] In practical applications, magnetic soft devices are typically designed as beams, plates, or shells to achieve fast response and ease of operation. To meet these requirements, several beam, plate, or shell theories / models for describing HMSM samples have been reported in the literature. Wang et al. proposed a nonlinear theory to predict the deflection of slender HMSM beams under magnetic field. Chen and Wang analyzed the large deformation of HMSM beams under magnetic field buckling. Yan et al. reduced the magnetic energy from three dimensions to one dimension and introduced it into a shell model to study the buckling deformation of axisymmetric shells. Yang et al. established a finite strain shell model and incorporated it into a finite element program, which can quantitatively predict the deformation and morphological evolution of HMSM shell structures.

[0006] Besides single-layer structures, flexible functional devices are typically designed as multilayer plates, with different plates possessing different geometric parameters, materials, and magnetization. To investigate the magnetomechanical properties of these HMSM devices, a theoretical model of HMSM multilayer plates within a finite strain range is needed. This model can predict the bending deformation of multilayer plates under different parameters and, similarly, determine the magnetization vector distribution within thin-plate soft material samples for any target curve (satisfying the necessary strain conditions), thereby guiding the generation of the target curve in the multilayer plate sample. Solving this problem is of great significance for the design of magnetic soft robots and flexible sensors, but relevant technical solutions are currently lacking. Summary of the Invention

[0007] In order to at least partially solve one of the technical problems existing in the prior art, the present invention aims to provide a method, device and medium for predicting and controlling the magnetostrictive bending deformation of multilayer hard magnetic soft material plates.

[0008] The first technical solution adopted in this invention is:

[0009] A method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate includes the following steps:

[0010] The energy density functional of magnetic multilayer plates is established, and the governing equations of magnetic multilayer plates are derived by variational method.

[0011] Establish plate equations based on the set of control equations, and solve the plate equations to obtain deformation prediction and control schemes;

[0012] Obtain the target curve parametric equation, substitute the target curve parametric equation into the control scheme, and obtain the corresponding parameters.

[0013] Furthermore, the energy density functional Ψ(χ,x) k ,p k ;X k The expression for ) is:

[0014]

[0015] In the formula, x k X represents the position vector of the current configuration of the k-th layer plate; k The position vector representing the reference configuration of the k-th layer plate; Let represent the deformation gradient tensor of the k-th layer plate; h represents the transpose of the inverse of the deformation gradient tensor of the k-th layer plate; k This represents the thickness of the k-th layer. μ represents the elastic energy of the k-th layer; μ0 represents the free magnetic permeability (μ0 = 4π × 10⁻⁶). 7 H / m); Mk H represents the magnetization vector of the k-th layer; a χ represents the external magnetic field vector; χ represents the magnetic potential energy; Gradχ represents the gradient of χ. This describes the change in volume of the k-th layer after deformation. This indicates that the volume of the k-th layer is the same before and after deformation; q - q + q and q represent the action on the lower surface of the first plate, respectively. The surface of the nth layer and side surface External force; p k Appearing as a Lagrange operator As an incompressible constraint term; x represents the position vector of the side surface of the multilayer board; n represents the layer number of the multilayer board (top layer); Z represents the out-of-plane (thickness direction) position coordinate of the board; r represents the in-plane direction vector; V represents the volume element of the entire multilayer board.

[0016] Furthermore, the derivation of the governing equations for the magnetic multilayer plate using the variational method includes:

[0017] By taking the variational equation for the magnetic potential energy χ, the system of magnetic field control equations is obtained;

[0018] The position vector x of the current configuration of the k-th layer plate k By taking the variational equations, the governing equations for the magnetic multilayer plate sample are obtained.

[0019] We assume that the excitation magnetic field caused by the magnetization of the object is much smaller than the external magnetic field in order to simplify the governing equations of the magnetic multilayer plate sample.

[0020] Furthermore, the simplified governing equations for the magnetic multilayer plate sample are expressed as follows:

[0021]

[0022] In the formula, Defined as nominal stress; T represents transpose; k represents a unit vector perpendicular to the in-plane direction (thickness direction); N represents a unit vector perpendicular to the side surface; This indicates the bottom surface of the first layer of board; This represents the upper surface of the nth layer. This refers to the side surface of a multilayer board; Indicates the interface between plates; x k r represents the position vector of the k-th layer plate; k This represents the position vector within the surface of the k-th layer.

[0023] Based on the equilibrium equation and boundary conditions established in equation (1), for and Mk Expanding Z using a series, we can eliminate one variable and thus establish the two-dimensional plate equation:

[0024]

[0025] In the formula, Z i This indicates the order of Z after a series expansion along the Z direction. This indicates the order of the nominal stress in the k-th layer of the plate after series expansion along the Z direction. This represents the order of the deformation gradient of the k-th layer after series expansion along the Z direction. O(Z) represents the order of the magnetization vector of the k-th layer after series expansion along the Z direction. 4 ) indicates a series expansion along the Z direction to Z. 4 An infinitesimal quantity.

[0026] Furthermore, the expression for the plate equation is as follows:

[0027]

[0028] in and They are represented as follows:

[0029]

[0030] In the formula, Represents the Nabla operator within a face; Defined as nominal stress.

[0031] Furthermore, the multilayer hard magnetic soft material plate is a magnetic double-layer plate material; the step of obtaining the deformation prediction and control scheme based on the plate equation includes:

[0032] A1. Assume that the constitutive model of the double-layer plate material is an incompressible neo-Hookean constitutive model;

[0033]

[0034] In the formula, C k This represents material parameters;

[0035] A2. Based on the plane strain assumption, assuming the magnetic double-layer plate does not deform along the Y-axis, the magnetization vectors M1 and M2 of each layer, and the external magnetic field B... a The external force Q can be decomposed into the following form:

[0036]

[0037] In the formula, {M A1 (X), M A2 (X), B AQ A} represent the magnitudes of the corresponding vectors, {M θ1 (X), M θ2 (X), B θ Q θ} represents the direction of the corresponding vector; E1 represents the horizontal unit vector, and k represents the vertical unit vector;

[0038] A3. Perform a series expansion on the current configuration position parameters:

[0039]

[0040] Further simplification of equation (2) yields an equation containing only two unknowns. and The two equations, and and This reflects the positional parameters of the bottom surface of the magnetic double-layer plate after bending deformation, and other quantities can be obtained from... and express;

[0041] A4. Regarding the bending deformation of the magnetic double-layer plate, further... and Rewritten in the following form, the unknowns in the plate equation are transformed into f0(X) and g0(X):

[0042]

[0043] In the formula, f0(X) represents the axial elongation of the bottom surface of the magnetic double-layer plate, and g0(X) represents the bending angle of the bottom surface at X.

[0044] A5. To systematically study the influence of the material parameters and thickness of the magnetic double-layer plates on the overall bending deformation, the following equivalent form is further given:

[0045] C1=C0, C2=αC0, h1=b1h=(1-β)h, h2=b2h=βh (7)

[0046] With this assumption, only α and β are needed to reflect the difference in thickness and material parameters between the two layers of the magnetic double-layer plate;

[0047] A6. To avoid the cumbersome calculations caused by dimensions, the following dimensionless calculations are performed:

[0048]

[0049] A7. Because of B in the calculation result of step A6 A M A and Q AThe value of is much smaller than C0, so we can assume a small quantity ∈ to further asymptotically simplify k1, k2, and f0(X):

[0050] k1=∈K1,k2=∈K2,f0(X)=1+∈Δf0(X) (9)

[0051] In the formula, Δf0 represents the strain value at point X after the double-layer plate undergoes bending deformation;

[0052] A8. Based on the assumptions and treatments in steps A1-A7, substituting them into the plate equation (2), we can obtain the following equation:

[0053]

[0054]

[0055] In the formula, E αβ It is a quantity used to characterize bending stiffness; g 0,X G represents the first derivative of g0(X) with respect to X; g 0,xx Let g0(X) be the second derivative of g0(X) with respect to X.

[0056] Given B A M A After calculating C0, g(X) and h, further calculations are performed on formula (11) to obtain the magnetization vector angle M. θ The expression:

[0057]

[0058] A9. Substitute the calculation results of formulas (9)-(12) into formula (6) to obtain the asymptotic solution of the current configuration of the bottom surface of the magnetic double-layer plate. (i.e., the morphology after bending deformation), at which point a complete predictive model of the magnetic double-layer plate after bending deformation is established.

[0059] Further, obtaining the target curve parametric equation and substituting it into the control scheme includes:

[0060] Given the parametric equation of the target curve Further calculation of g0(X):

[0061]

[0062] Substituting g0(X) into formula (10), we can further obtain Δf0. If Δf0→0 is satisfied, then we can solve for the angle M of the magnetization vector required to generate the target curve according to formula (11). θ (X); Based on the solution results, we further verify whether the target configuration can be generated through numerical methods. At this point, a complete shape control scheme for the bending deformation of the magnetic double-layer plate is established.

[0063] Furthermore, the prediction and shape control method also includes a step of verifying the equation results:

[0064] Choose the calculation method for magnetic potential energy in the magnetic field (no current) module (full field or reduced field);

[0065] Establish a geometric model of the thin plate and the external air;

[0066] Choose the elastic and magnetic constitutive models for the thin plate and external air;

[0067] Calculate the bending deformation of a thin plate under force-magnetic coupling.

[0068] The second technical solution adopted in this invention is:

[0069] An electronic device includes a processor and a memory, wherein the memory stores at least one instruction, at least one program, a code set, or an instruction set, and the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by the processor to realize a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate as described above.

[0070] The third technical solution adopted in this invention is:

[0071] A computer-readable storage medium storing at least one instruction, at least one program, code set, or instruction set, wherein the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by a processor to implement a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate as described above.

[0072] The fourth technical solution adopted in this invention is:

[0073] A computer program product or computer program includes computer instructions stored in a computer-readable storage medium. A processor of a computer device can read the computer instructions from the computer-readable storage medium and execute the computer instructions to cause the computer device to perform the method described above.

[0074] The beneficial effects of this invention are: the proposed solution achieves prediction and shape control of bending deformation in multilayer hard magnetic soft material plates, and compared with other control methods, the magnetic field can enable the thin plate to quickly generate the target curve. The accuracy and agility of this control method have significant application prospects and value in fields such as flexible electronics and soft robotics. Attached Figure Description

[0075] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following description is provided with accompanying drawings of the relevant technical solutions in the embodiments of the present invention or the prior art. It should be understood that the accompanying drawings described below are only for the purpose of clearly illustrating some embodiments of the technical solutions of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0076] Figure 1 This is a flowchart of a scheme for controlling the bending deformation of a magnetic double-layer plate.

[0077] Figure 2 This is a schematic diagram of the boundary constraints of a magnetic double-layer plate in a numerical simulation.

[0078] Figure 3 This is a schematic diagram illustrating the prediction of bending deformation of a magnetic double-layer plate under different thickness parameters.

[0079] Figure 4 This is a schematic diagram illustrating the prediction of bending deformation of a magnetic double-layer plate under different material parameters.

[0080] Figure 5 M corresponding to the three target curves θ picture.

[0081] Figure 6 This is a comparison chart of the numerical simulation results of the magnetic double-layer plate and the target curve 1.

[0082] Figure 7 This is a comparison chart of the numerical simulation results of the magnetic double-layer plate and the target curve 2.

[0083] Figure 8 This is a comparison chart of the numerical simulation results of the magnetic double-layer plate and the target curve 3. Detailed Implementation

[0084] The embodiments of the present invention are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention. The step numbers in the following embodiments are set only for ease of explanation, and there is no limitation on the order between the steps. The execution order of each step in the embodiments can be adaptively adjusted according to the understanding of those skilled in the art.

[0085] In the description of this invention, it should be understood that the orientation descriptions, such as up, down, front, back, left, right, etc., are based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limiting this invention.

[0086] In the description of this invention, "several" means one or more, "more than" means two or more, "greater than," "less than," and "exceeding" are understood to exclude the stated number, while "above," "below," and "within" are understood to include the stated number. The use of "first" and "second" in the description is merely for distinguishing technical features and should not be construed as indicating or implying relative importance, or implicitly indicating the number of indicated technical features, or implicitly indicating the order of the indicated technical features.

[0087] In the description of this invention, unless otherwise explicitly defined, terms such as "set up," "install," and "connect" should be interpreted broadly, and those skilled in the art can reasonably determine the specific meaning of the above terms in this invention in conjunction with the specific content of the technical solution.

[0088] Example 1

[0089] like Figure 1 As shown, this embodiment provides a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate. This method can predict the bending deformation of the magnetic soft material plate under the drive of an external magnetic field, and guide the selection of magnetization parameters based on the target configuration to achieve shape control. Therefore, it has important application prospects and value in fields such as flexible electronics and soft robotics. The method specifically includes the following steps:

[0090] S1. Establish the energy density functional Ψ(χ,x) of a magnetic multilayer plate. k ,p k ;X k ).

[0091]

[0092] Where x k X represents the position vector of the current configuration of the k-th layer plate; k The position vector representing the reference configuration of the k-th layer plate; h represents the deformation gradient tensor of the k-th layer plate. k This represents the thickness of the k-th layer. μ represents the elastic energy of the k-th layer; μ0 represents the free magnetic permeability (μ0 = 4π × 10⁻⁶). 7 H / m); M k H represents the magnetization vector of the k-th layer; a χ represents the external magnetic field vector; χ represents the magnetic potential energy. This describes the change in volume of the k-th layer after deformation. This indicates that the volume of the k-th layer is the same before and after deformation; q - q + q and q represent the action on the lower surface of the first plate, respectively. The surface of the nth layer and side surface External force; p k Appearing as a Lagrange operator As a constraint term for incompressible volume.

[0093] S2. Derive the governing equations of the magnetic multilayer plate using the variational method.

[0094] As an optional implementation, step S2 includes the following steps:

[0095] S21. The set of magnetic field control equations is obtained by taking variational measures over χ.

[0096]

[0097] Among them B L Represents the magnetic flux density vector of the magnetic plate sample or the external space; double brackets This indicates the magnitude of the abrupt change in the corresponding quantity at the interface between the magnetic multilayer plate sample and air.

[0098] S22, regarding x k The governing equations for the magnetic multilayer plate sample are obtained by variational methods.

[0099]

[0100] in Defined as nominal stress; H s Represents the magnetic field generated by an object; <H s > represents the average value of the abrupt change in the magnetic field at the interface between the magnetic plate sample and the air.

[0101] S23, Due to the magnetic plate sample in the object generating a magnetic field H s Size and sample magnetization M k Therefore, assuming that the excitation magnetic field caused by the magnetization of the object is much smaller than the external magnetic field, the governing equations for the plate sample can be simplified as follows:

[0102]

[0103] S3. Based on the equilibrium equation and boundary conditions established in equation (4), we can... and M k We can perform a series expansion on Z, which makes it easier for us to eliminate a variable and then establish the two-dimensional plate equation (5).

[0104]

[0105] Given the magnetization vectors of each layer of the magnetic multilayer plate and the external magnetic field vector, the magneto-induced bending deformation of the magnetic multilayer plate can be predicted by the asymptotic solution obtained in 5) or the numerical method given in 10).

[0106] S4. Combining the simplified control equations (4)-(5), the following two-dimensional plate equation is derived:

[0107]

[0108] in and They are respectively represented as follows

[0109]

[0110] S5. Solution of plate equations and proposal of methods for predicting and controlling bending deformation. In this embodiment, n=2 is taken, that is, a double-layer plate is used as an example for research.

[0111] S51. Assume that the constitutive model of the double-layer plate material is an incompressible neo-Hookean constitutive model.

[0112]

[0113] Where C k (k=1,2) represents the material parameters.

[0114] S52. Based on the plane strain assumption, assuming the magnetic double-layer plate does not deform along the Y-axis, the magnetization vectors M1 and M2 of each layer, and the external magnetic field B... a The external force Q can be decomposed into the following form:

[0115]

[0116] Where {M A1 (X), M A2 (X), B A Q A} represent the magnitudes of the corresponding vectors, respectively, {M θ1 (X), M θ2 (X), B θ Q θ} indicates the direction of the corresponding vector.

[0117] S53. Based on the above assumptions, in order to further solve the problem, we perform a series expansion of the positional parameters of the current configuration:

[0118]

[0119] Further simplification of equation (5) yields an equation containing only two unknowns. and The two equations, and and This reflects the positional parameters of the bottom surface of the magnetic double-layer plate after bending deformation, and other quantities can be obtained from... and express.

[0120] S54. Regarding the bending deformation of magnetic double-layer plates, further... and Rewritten in the following form, the unknowns in the plate equation are transformed into f0(X) and g0(X):

[0121]

[0122] Where f0(X) represents the axial elongation of the bottom surface of the magnetic double-layer plate, and g0(X) represents the bending angle of the bottom surface at point X.

[0123] S55. To systematically study the influence of the material parameters and thickness of the magnetic double-layer plates on the overall bending deformation, we further present the following equivalent form:

[0124] C1=C0, C2=αC0, h1=b1h=(1-β)h, h2=b2h=βh, (11)

[0125] With this assumption, only α and β are needed to reflect the difference in thickness and material parameters between the two layers of the magnetic double-layer plate.

[0126] S56. To avoid the cumbersome calculations caused by dimensions, the following dimensionless calculations are performed.

[0127]

[0128] S57. Due to B in the calculation result of step S56 A M A and Q A The value of is much smaller than C0, so we can assume a small quantity ∈ to further asymptotically simplify k1, k2 and f0(X).

[0129] k1=∈K1,k2=∈K2,f0(X)=1+∈Δf0(X) (13)

[0130] Where Δf0 represents the strain value at point X after the double-layer plate undergoes bending deformation.

[0131] S58. Based on the assumptions and processing in steps S51-S57, substituting into the plate equation (5), we can obtain the following equation:

[0132]

[0133]

[0134] Where E αβ It is a quantity used to characterize bending stiffness; g 0,X G represents the first derivative of g0(X) with respect to X; g 0,XX Let g0(X) be the second derivative of g0(X) with respect to X.

[0135]

[0136] Given B A M A After calculating C0, g(X) and h, further calculations of formula (15) can yield M. θ expression.

[0137]

[0138] Where k = B A M A / C0.

[0139] S59. Substituting the calculation results of formulas (13)-(16) into formula (10), the asymptotic solution of the current configuration of the bottom surface of the magnetic double-layer plate can be obtained. (i.e., the morphology after bending deformation), at which point a complete predictive model of the magnetic double-layer plate after bending deformation is established.

[0140] S6. Given the parametric equation of the target curve g0(X) can be further calculated:

[0141]

[0142] Substituting g0(X) into formula (14) yields Δf0. If Δf0→0 is satisfied, then the angle M of the magnetization vector required to generate the target curve can be calculated using formula (15). θ (X). Based on the solution results, the ability to generate the target configuration was further verified through numerical methods. At this point, a complete shape control method for the bending deformation of the magnetic double-layer plate was established.

[0143] S7. Use the "Solid Mechanics" and "Magnetic Field (No Current)" modules of the commercial finite element software Comsol to verify the equation results.

[0144] In some embodiments, step S7 includes the following steps:

[0145] S71. Select the calculation method for magnetic potential energy in the magnetic field (no current) module (full field or reduced field);

[0146] S72. Establish a geometric model of the thin plate and the external air;

[0147] S73. Select the elastic and magnetic constitutive models for the thin plate and the external air model;

[0148] S74. Input the boundary conditions and basic material parameters;

[0149] S75. Calculate the bending deformation of a thin plate under force-magnetic coupling.

[0150] The above method will be explained in detail below with reference to the accompanying drawings and specific embodiments.

[0151] For ease of demonstration, this embodiment uses a magnetic double-layer plate as an example. The solution includes predicting the bending deformation of the plate given parameters such as the material, thickness, and magnetization vector of the magnetic double-layer plate. Simultaneously, given a target curve, the required magnetization angle of the magnetic double-layer plate can be calculated, thereby achieving shape control of the plate's bending deformation.

[0152] This embodiment provides a method for predicting and controlling the magnetostrictive bending deformation of multilayer hard magnetic soft material plates, specifically including the following steps:

[0153] 1) Set the material parameters (thickness h1 and h2 of the double-layer plate, material parameters C1 and C2) and magnetic field parameters (magnitude B of the external magnetic field) according to the experimental conditions. A external magnetic field angle B θ Magnetization amplitude M of thin plate sample A )wait.

[0154] 2) Based on the obtained governing equations, set the body forces and surface forces at the boundaries of the magnetic double-layer plate. The constraint settings at the sample boundaries are as follows: Figure 2 As shown.

[0155]

[0156] 3) Following the settings in 1) and 2), the position vector (asymptotic solution) of the magnetic double-layer plate after bending deformation can be obtained through asymptotic analysis and other methods. Simultaneously, the parameters can be incorporated into the finite element software COMSOL Multiphysics for calculation to obtain numerical results. This allows for the prediction of the bending deformation of the magnetic double-layer plate under force-magnetic coupling at different thickness ratios (β = h2 / h1) and material parameter ratios (α = C2 / C1).

[0157] a) Different thickness ratios (β)

[0158]

[0159] b) Different material ratios (α)

[0160]

[0161] 4) Given the target curve parameter coordinate equation r(X):

[0162]

[0163] 5) Calculate the curvature angle of the target curve:

[0164]

[0165] 6) Calculate the magnetization vector angle M θ Distribution (k = B) A M A / C0)

[0166]

[0167] a) Target curve 1 (0≤X≤1)

[0168]

[0169] b) Target curve 2 (0≤X≤1)

[0170]

[0171] c) Target curve 3 (0≤X≤1)

[0172]

[0173] 7) Verify the bending deformation of the magnetic double-layer plate through numerical calculations (here we assume the thickness and material of the double-layer plate are the same, i.e., h1 = h2, C1 = C2). In the numerical calculations, the initial configuration of the thin plate is chosen as Ω. r = 1m × 0.3m × 0.04m, the configuration of the external magnetic field (air) model is selected as Ω. r =10m×10m×10m, the numerical calculation steps are as follows:

[0174] 7.1) The "Solid Mechanics" and "Magnetic Field (No Current)" modules of the commercial finite element software COMSOL Multiphysics were selected for simulation. In the "Solid Mechanics" module, the hyperelastic constitutive model of the sample was set to the incompressible Neo-Hookean constitutive model, where C0 = 0.1 MPa. The constitutive model of the external magnetic field was set to the software's built-in air constitutive model. The boundary conditions for the thin plate sample were set as follows: Figure 2 As shown. Furthermore, since this invention only addresses bending deformation, the thin plate is assumed to be subjected to plane strain in the numerical calculations, meaning the displacement along the Y-axis is constrained. Based on the derived boundary conditions, the body force and surface force exerted by the magnetic field on the sample are applied. In this module, a quadratic coincident edge element is selected to simulate the bending deformation of the magnetic thin plate.

[0175] 7.2) In the "Magnetic Field (No Current)" module, first set the "Full Field" option to calculate the magnetic potential energy of the magnetic field. Set the magnetic constitutive model of the sample and the external region as B = μ(H + M), where μ = 4π × 10⁻⁶. -7 The magnetization vector M of the sample region is set according to formula (13), while the magnetization vector M of the external region is 0. Finally, the "magnetic flux density" and "zero magnetic scalar potential" options are selected to specify the magnitude of the magnetic induction intensity of the external magnetic field and the magnetic field boundary conditions. In this module, the magnetic potential energy order is set to second-order.

[0176] Numerical simulation results of bending deformation of magnetic double-layer plate samples under different material and thickness parameters are as follows: Figure 3 , Figure 4 As shown in the figure, the simulation results and the calculation results are in excellent agreement, verifying the accuracy of the bending deformation prediction. The numerical simulation results for the shape control of the magnetic double-layer plate sample are as follows: Figure 5 , Figure 6 , Figure 7 and Figure 8 As shown, the deformed thin plate matches the target curve very well, verifying the correctness of the magnetization vector solution and setting.

[0177] In summary, compared with the prior art, the method of the present invention has at least the following advantages and beneficial effects:

[0178] (1) A well-developed theoretical scheme.

[0179] The theoretical model established in this invention does not require prior assumptions about displacement or stress. It can obtain the governing equations for multilayer plates using variational methods and establish two-dimensional plate equations. Furthermore, under the assumption of plane strain, asymptotic analysis is used to solve for the deformed configuration of the magnetic multilayer plate (taking a double-layer plate as an example), thus establishing a predictive model for the bending deformation of the magnetic multilayer plate under a magnetic field. In addition, given a target curve, the required magnetization vector can be obtained by solving for the bending angle of the curve and substituting it into the plate equations, which can then guide the shape control of the magnetic multilayer plate.

[0180] (2) Precise prediction and shape control of the growth of two-dimensional hyperelastic thin plates.

[0181] The technical solution proposed in this invention can predict the bending deformation of magnetic multilayer boards in a magnetic field and guide the setting of the magnetization vector angle in the board, achieving an arbitrary curve for the overall bending deformation. Taking Neo-Hookean hyperelastic material as an example... Figure 3 and Figure 4 The asymptotic and numerical solutions for predicting the bending deformation of magnetic double-layer plates with different thickness and material ratios in a magnetic field are presented. Figures 6-8The paper demonstrates how the required magnetization vector is calculated based on a given target curve, and then uses numerical methods to verify the obtained bending deformation against the target configuration. The results also show that this method can precisely control the bending deformation of hyperelastic thin plates.

[0182] (3) Simple and easy-to-use analytical formulas for shape design

[0183] The equations for predicting and controlling the bending deformation of magnetic multilayer boards established in this invention have a very simple form. On the one hand, this helps to expand applications and facilitates the efficient design and manufacturing of magnetic soft robots. On the other hand, the form of the equations also allows us to understand the conditions that affect the bending deformation of magnetic multilayer boards, providing a deeper understanding of the magnetostrictive behavior of multilayer boards.

[0184] Example 2

[0185] This invention also provides an electronic device, which includes a processor and a memory. The memory stores at least one instruction, at least one program, a code set, or an instruction set. The at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by the processor to achieve the following: Figure 1 This paper presents a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate.

[0186] It is understood that the memory may include random access memory (RAM) or read-only memory. Optionally, the memory may include non-transitory computer-readable storage medium. The memory can be used to store instructions, programs, code, code sets, or instruction sets. The memory may include a stored program area and a stored data area, wherein the stored program area may store instructions for implementing an operating system, instructions for at least one function, instructions for implementing the various method embodiments described above, etc.; the stored data area may store data created according to the use of the server, etc.

[0187] A processor may include one or more processing cores. The processor connects to various parts of the server via various interfaces and lines, executing instructions, programs, code sets, or instruction sets stored in memory, and accessing data stored in memory to perform various server functions and process data. Optionally, the processor may be implemented using at least one of the following hardware forms: Digital Signal Processing (DSP), Field-Programmable Gate Array (FPGA), and Programmable Logic Array (PLA). The processor may integrate one or more of the following: Central Processing Unit (CPU) and Modem. The CPU primarily handles the operating system and applications; the modem handles wireless communication. It is understood that the modem may also be implemented as a separate chip without being integrated into the processor.

[0188] Since this electronic device corresponds to the method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to an embodiment of the present invention, and the principle of solving the problem by this electronic device is similar to that of the method, the implementation of this electronic device can refer to the implementation process of the above method embodiment, and the repeated parts will not be described again.

[0189] Example 3

[0190] This invention also provides a computer-readable storage medium storing at least one instruction, at least one program, a code set, or an instruction set, wherein the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by a processor to achieve the following: Figure 1 This paper presents a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate.

[0191] Those skilled in the art will understand that all or part of the steps in the various methods of the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, including read-only memory (ROM), random access memory (RAM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), one-time programmable read-only memory (OTPROM), electrically-Erasable Programmable Read-Only Memory (EEPROM), compact disc read-only memory (CD-ROM) or other optical disc storage, disk storage, magnetic tape storage, or any other computer-readable medium capable of carrying or storing data.

[0192] Since this storage medium is the storage medium corresponding to the method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to an embodiment of the present invention, and the principle of solving the problem by this storage medium is similar to that of this method, the implementation of this storage medium can refer to the implementation process of the above method embodiment, and the repeated parts will not be described again.

[0193] Example 4

[0194] In some possible implementations, various aspects of the methods of the embodiments of the present invention can also be implemented as a program product comprising program code that, when run on a computer device, causes the computer device to perform the steps of a method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to various exemplary embodiments of the present application, as described above. The executable computer program code or "code" used to perform the various embodiments can be written in high-level programming languages ​​such as C, C++, C#, Smalltalk, Java, JavaScript, Visual Basic, Structured Query Language (e.g., Transact-SQL), Perl, or in various other programming languages.

[0195] It should be understood that various parts of the present invention can be implemented in hardware, software, firmware, or a combination thereof. In the above embodiments, multiple steps or methods can be implemented in software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (PGAs), field-programmable gate arrays (FPGAs), etc.

[0196] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0197] The above embodiments are merely illustrative of the technical concept and features of the present invention, and are intended to enable those skilled in the art to understand the content of the present invention and implement it accordingly. They should not be construed as limiting the scope of protection of the present invention. All equivalent changes or modifications made based on the essence of the content of the present invention should be covered within the scope of protection of the present invention.

[0198] The project for which this patent application was submitted is: Analytical Study and Experimental Verification of Multi-Field Coupled Mechanical Behavior of Magnetic Plate-Shell Soft Material Samples, 2022A1515010653, Guangdong Provincial Natural Science Foundation - General Project.

Claims

1. A method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate, characterized in that, Includes the following steps: The energy density functional of a magnetic multilayer plate is established, and the governing equations of the magnetic multilayer plate are derived by variational method. Establish plate equations based on the set of control equations, and solve the plate equations to obtain deformation prediction and control schemes; Obtain the target curve parametric equation, substitute the target curve parametric equation into the control scheme, and obtain the corresponding parameters; The energy density functional The expression is: In the formula, Indicates the first The position vector of the current configuration of the shelf; Indicates the first The position vector of the reference configuration of the shelf; Indicates the first The deformation gradient tensor of the plate; Indicates the first The transpose of the inverse of the plate deformation gradient tensor; Indicates the first The thickness of the shelf; Indicates the first The elastic properties of the laminate; Indicates the permeability of free space; Indicates the first The magnetization vector of the plate; Represents the external magnetic field vector; Represents magnetic potential energy; express The gradient; The description is the first The change in volume of the laminate before and after deformation, if Then it means the first The volume of the shelf remains the same before and after deformation. , and These respectively represent the action on the lower surface of the first layer plate. , No. Top surface of the shelf and side surface External forces; Appearing as a Lagrange operator As a constraint term for incompressible volume; Represents the position vector under the current configuration; Indicates the first Sheet layer number; Indicates the out-of-plane position coordinates of the plate; Represents the in-plane position vector of the plate; Volume elements representing multilayer boards; The expression for the plate equation is as follows: in , and They are represented as follows: In the formula, Represents the Nabla operator within a plane; Defined as nominal stress; The multilayer hard magnetic soft material board is a magnetic double-layer board material; The method of solving the plate equations to obtain deformation prediction and control schemes includes: A1. Assume that the constitutive model of the double-layer plate material is an incompressible neo-Hookean constitutive model; （3） In the formula, This represents material parameters; A2. Based on the plane strain assumption, assuming the magnetic double-layer plate does not deform along the Y-axis, the magnetization vector of each layer is... and External magnetic field and external forces Decomposed into the following form: （4） In the formula, { , , , } represent the magnitudes of the corresponding vectors, { , , , } indicates the direction of the corresponding vector; Represents a unit vector in the horizontal direction. Represents a unit vector in the vertical direction; A3. Perform a series expansion on the current configuration position parameters: （5） At this point, further simplification of equation (2) yields an equation containing only two unknowns. and The two equations, and and This reflects the positional parameters of the bottom surface of the magnetic double-layer plate after bending deformation, and other quantities can be obtained from... and express; A4. Regarding the bending deformation of the magnetic double-layer plate, further... and When written in the following form, the unknowns in the plate equation are transformed into... and : （6） In the formula, This represents the axial elongation of the bottom surface of the magnetic double-layer plate. Indicates the bottom surface is The bending angle at that point; A5. To systematically study the influence of the material parameters and thickness of the magnetic double-layer plates on the overall bending deformation, the following equivalent form is further given: （7） Based on this assumption, it is only necessary to... and This can reflect the differences in thickness and material parameters between the two layers of the magnetic double-layer plate; A6. To avoid the cumbersome calculations caused by dimensions, the following dimensionless calculations are performed: （8） A7. Due to the calculation results in step A6 and The value is much smaller than Therefore, we can assume a small amount right , and Further asymptotic simplification: （9） In the formula, This represents the strain value at point X after the double-layer plate undergoes bending deformation; A8. Based on the assumptions and treatments in steps A1-A7, substituting them into the plate equation (2), we can obtain the following equation: （10） （11） In the formula, It is a quantity used to characterize bending stiffness; express right X The first derivative; express right X The second derivative; When given , , , and Then, the magnetization vector angle is obtained by further calculating formula (11). The expression: （12） A9. Substitute the calculation results of formula (9)-(12) into formula (6) to obtain the asymptotic solution of the current configuration of the bottom surface of the magnetic double-layer plate. At this time, a complete prediction model of the magnetic double-layer plate after bending deformation is established. The step of obtaining the target curve parameter equation and substituting the target curve parameter equation into the control scheme includes: Given the parametric equation of the target curve Further calculations : Will Substituting into formula (10), we further obtain If satisfied Then, according to formula (12), the angle of the magnetization vector required to generate the target curve can be calculated. Based on the solution results, numerical methods were used to verify whether the target configuration could be generated. At this point, a complete shape control scheme for the bending deformation of the magnetic double-layer plate was established.

2. The method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to claim 1, characterized in that, The derivation of the governing equations for the magnetic multilayer plate using the variational method includes: magnetic potential energy By taking the variational equations, we obtain the magnetic field control equations. For the The position vector of the current configuration of the shelf By taking the variational equations, the governing equations for the magnetic multilayer plate sample are obtained. We assume that the excitation magnetic field caused by the magnetization of the object is much smaller than the external magnetic field in order to simplify the governing equations of the magnetic multilayer plate sample.

3. The method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to claim 2, characterized in that, The simplified governing equations for the magnetic multilayer plate sample are expressed as follows: （1） In the formula, Defined as nominal stress; Indicates transpose; Represents a unit vector perpendicular to and within the plane; Represents a unit vector perpendicular to the side surface; This indicates the bottom surface of the first layer of board; Indicates the first n The upper surface of the shelf; Indicates the multilayer board. The side surface of the layer; Indicates the interface between two boards; Indicates the first k The position vector of the shelf; Indicates the first k Position vector within the layer surface; Based on the equilibrium equation and boundary conditions established in equation (1), for , and Expanding Z using a series, we can eliminate one variable and thus establish the two-dimensional plate equation: （2） In the formula, Indicates along Z After performing a series expansion in the direction Z The order of Indicates along Z After performing a series expansion in the direction, the first k The order of the nominal stress in the laminate. Indicates along Z After performing a series expansion in the direction, the first k The order of the deformation gradient of the laminate. Indicates along Z After performing a series expansion in the direction, the first k The order of the magnetization vector of the laminate. Indicates along Z Expand the direction into a series to An infinitesimal quantity.

4. The method for predicting and controlling the magnetostrictive bending deformation of a multilayer hard magnetic soft material plate according to claim 1, characterized in that, The prediction and shape control method also includes a step of verifying the equation results: Choose the method for calculating magnetic potential energy in the magnetic field module; Establish a geometric model of the thin plate and the external air; Choose the elastic and magnetic constitutive models for the thin plate and external air; Calculate the bending deformation of a thin plate under force-magnetic coupling.

5. An electronic device, characterized in that, The electronic device includes a processor and a memory, wherein the memory stores at least one instruction, at least one program, a code set, or an instruction set, and the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by the processor to implement the method as described in any one of claims 1 to 4.

6. A computer-readable storage medium, characterized in that, The storage medium stores at least one instruction, at least one program, code set, or instruction set, wherein the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by a processor to implement the method as described in any one of claims 1 to 4.

Images