Prediction and shape control method and device for magneto-induced bending deformation of multilayer hard magnetic soft material plate and medium

Abstract

The invention discloses a prediction and shape control method and device for magnetic bending deformation of a multi-layer hard magnetic soft material plate and a medium, and the method comprises the steps: building an energy density functional of a magnetic multi-layer plate, and deducing a control equation set of the magnetic multi-layer plate through a variational method; establishing a plate equation according to the control equation set, and solving the plate equation to obtain a deformation prediction and control scheme; and obtaining a target curve parameter equation, and substituting the target curve parameter equation into the control scheme to obtain corresponding parameters. According to the scheme provided by the invention, prediction and shape control of bending deformation of the multi-layer hard magnetic soft material plate are realized, and compared with other control modes, the thin plate can quickly generate a target curve under the action of the magnetic field. The accuracy and agility of the control mode have important application prospects and values in the fields of flexible electronics, soft robots and the like. The method can be widely applied to the technical field of prediction and control of bending deformation of hard magnetic soft materials and design and development of flexible intelligent devices.

Description

Technical Field

[0001] The present invention relates to the technical field of prediction and control of bending deformation of hard magnetic soft materials and design and development of flexible intelligent devices, and in particular to a method, device and medium for prediction and shape control of magnetic bending deformation of a multilayer hard magnetic soft material plate. Background Art

[0002] Magnetic soft materials are composed of micrometer-scale or nanometer-scale soft matrices and magnetic particles, which can undergo rapid and reversible deformation under the action of a magnetic field. Among them, hard magnetic soft materials (HMSM) have attracted extensive research interest due to their good deformation and response capabilities. The magnetic particles in HMSM have high coercivity (such as NdFeB and AlNiCr), which enables HMSM to maintain high remanence even in the absence of an external magnetic field. Under the action of a magnetic field or other stimulation, HMSM can exhibit complex deformation and has good shape control capabilities. It is an ideal material for the design and manufacture of new soft actuators, soft robots and flexible electronic devices. In particular, using technologies such as 3D printing and topology optimization, the residual magnetization field distribution in HMSM samples can be artificially controlled, so that HMSM can achieve diverse functions.

[0003] In the past few decades, researchers have been committed to studying the magnetomechanical properties of HMSM. In terms of experiments, Stepanov et al. studied the viscoelastic deformation behavior of HMSM synthesized by filling NdFeB particles in silicone rubber. The study found that hard magnetic particles can cause irrecoverable residual strain even in the absence of a magnetic field. Koo et al. prepared hard magnetic magnetorheological elastomers and tested their bending deformation under the action of 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. studied the instability of HMSM plates induced by magnetic fields after inhomogeneous magnetization. Pal and Sitti designed and fabricated a magnetically responsive bistable beam structure using HMSM and studied the state transition of this structure 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 continuum model for hard magnetic soft materials, in which the magnetic free energy per unit volume can be expressed as ( is the total deformation gradient tensor, M is the magnetization density vector, and B is the magnetic induction intensity of the external magnetic field). This model is in good agreement with the test results of the slender HMSM structure under bending deformation. Mukherjee et al. developed a deformation guidance framework for hard magnetic elastomers with isotropic and incompressible properties. In response to this finding, Yan et al. proposed a modified magnetic potential energy function, mainly by using the rotation tensor replace By adopting model, Zhang et al. studied the buckling problem of bistable hard magnetoelastomers. 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. In addition, Dorfmann and Ogden established a modified model to ensure the symmetry of the Cauchy stress tensor by introducing an additional term in the magnetoelastic energy. In order to study some special experimental characteristics of HMSM (such as viscoelastic effects, microstructural rearrangement of particles, etc.), the researchers also proposed a viscoelastic model. Based on the theoretical model, Liu et al. developed a numerical framework to perform finite element simulations of the complex deformation of HMSM samples.

[0005] In practical applications, magnetic soft devices are usually designed in the form of beams, plates or shells to achieve the goals of fast response and easy operation. In order to meet these requirements, some 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 bending. Yan et al. reduced the magnetic energy in three-dimensional form to one dimension and introduced a shell model to study the buckling deformation of axisymmetric shells. Yang et al. established a finite strain shell model and implanted it into the finite element program. The model can quantitatively predict the deformation and morphological evolution of HMSM shell structures.

[0006] In addition to single-layer structures, flexible functional devices are usually designed in the form of multilayer boards, in which different boards have different geometric parameters, materials and magnetization strengths. In order to study the magnetomechanical properties of these HMSM devices, it is necessary to establish a theory (model) of HMSM multilayer boards within a limited strain range. This theory (model) can predict the bending deformation of multilayer boards under different parameters, and can also determine the magnetization vector distribution inside the thin plate soft material sample for any target curve (satisfying the necessary strain conditions), thereby guiding the multilayer board sample to generate the target curve. The solution to this problem is of great significance for the design of magnetic soft robots and flexible sensors, but there is still a lack of relevant technical solutions. Summary of the invention

[0007] In order to solve at least one of the technical problems existing in the prior art to a certain extent, the purpose of the present invention is to provide a method, device and medium for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate.

[0008] The first technical solution adopted by the present invention is:

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

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

[0011] Establish plate equations according to the control equations, and solve the plate equations to obtain deformation prediction and control solutions;

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

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

[0014]

[0015] In the formula, x k represents the position vector of the current configuration of the k-th plate; X k represents the position vector of the reference configuration of the k-th plate; represents the deformation gradient tensor of the k-th plate; represents the transpose of the inverse of the k-th plate deformation gradient tensor; h k represents the thickness of the kth layer; represents the elastic energy of the kth layer; μ 0 Represents vacuum magnetic permeability (μ 0 =4π×107 H / m); M k represents the magnetization vector of the kth plate; H a represents the external magnetic field vector; χ represents the magnetic potential energy; Gradχ represents the gradient of χ; It describes the change in volume after the kth plate is deformed. If This means that the volume of the kth plate is the same before and after deformation; q - ,q + and q represent the action on the lower surface of the first layer The upper surface of the nth layer and side surface External force; p k Appears as a Lagrangian operator, As a constraint item for the incompressibility of volume; 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 position coordinate of the board in the out-of-plane direction (thickness direction); r represents the in-plane direction vector; V represents the volume element of the entire multilayer board.

[0016] Furthermore, the control equations of the magnetic multilayer plate are derived by the variational method, including:

[0017] By varying the magnetic potential energy χ, we obtain the governing equations of the magnetic field;

[0018] The position vector x of the current configuration of the k-th plate k By solving the variational equations, we can obtain the governing equations of the magnetic multilayer board sample;

[0019] It is assumed 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 sample.

[0020] Furthermore, the simplified governing equations of the magnetic multilayer board sample are expressed as:

[0021]

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

[0023] According to the equilibrium equation and boundary conditions established in equation (1), and M k Expand the series about Z, thus eliminating one variable and establishing the two-dimensional plate equation:

[0024]

[0025] In the formula, Z i It represents the order of Z after the series expansion along the Z direction, It represents the order of the nominal stress of the kth plate after series expansion along the Z direction, represents the order of the deformation gradient of the kth plate after series expansion along the Z direction, represents the order of the magnetization vector of the kth plate after series expansion along the Z direction, O(Z 4 ) indicates that the series is expanded along the Z direction to Z 4 An infinitesimal quantity.

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

[0027]

[0028] in and They are respectively expressed as follows:

[0029]

[0030] In the formula, represents the Nabla operator within the surface; Defined as nominal stress.

[0031] Furthermore, the multi-layer hard magnetic soft material plate is a magnetic double-layer plate material; the deformation prediction and control scheme obtained according to the plate equation includes:

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

[0033]

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

[0035] A2. Based on the plane strain assumption, assuming that the magnetic double-layer plate does not deform along the Y-axis direction, the magnetization vector M of each layer 1 and M 2 、External magnetic field B a And the external force Q is decomposed into the following form:

[0036]

[0037] In the formula, {MA1 (X), M A2 (X), B A , Q A} respectively represent the magnitude of the corresponding vector, {M θ1 (X), M θ2 (X), B θ , Q θ} represents the direction of the corresponding vector; E 1 represents the unit vector in the horizontal direction, and k represents the unit vector in the vertical direction;

[0038] A3. Expand the current configuration position parameters in series:

[0039]

[0040] Now further simplifying equation (2) yields only two unknowns: and The two equations of and It reflects the position 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. For the bending deformation of the magnetic double-layer plate, further and Written in the following form, the unknown quantity in the plate equation is converted to f 0 (X) and g 0 (X):

[0042]

[0043] In the formula, f 0 (X) represents the axial elongation of the bottom surface of the magnetic double-layer plate, g 0 (X) represents the bending angle of the lower bottom surface at position X;

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

[0045] C 1 =C 0 ,C 2 =αC 0 ,h 1 =b 1 h=(1-β)h,h 2 =b 2 h=βh (7)

[0046] By making 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. In order to avoid the cumbersome calculation caused by dimension, the following dimensionless calculation is performed:

[0048]

[0049] A7, because the calculation result of step A6 is B A M A and Q A The value is much smaller than C 0 , so we can assume that a small quantity ∈ is k 1 , k 2 and f 0 (X) Further asymptotic simplification:

[0050] k 1 =∈K 1 ,k 2 =∈K 2 ,f 0 (X)=1+∈Δf 0 (X) (9)

[0051] Where Δf 0 It represents the strain value at X after the double-layer plate is bent;

[0052] A8. Based on the assumptions and processing in steps A1-A7, substitute into the plate equation (2) to obtain the following equation:

[0053]

[0054]

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

[0056] When given B A 、M A , C 0 After , g(X) and h, further calculation of formula (11) is performed to obtain the magnetization vector angle M θ The expression is:

[0057]

[0058] A9. Substitute the 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 this time a complete prediction model for the magnetic double-layer plate after bending deformation is established.

[0059] Furthermore, the obtaining of the target curve parameter equation and substituting the target curve parameter equation into the control scheme includes:

[0060] Given the parametric equation of the target curve Further calculation of g 0 (X):

[0061]

[0062] G 0 After substituting (X) into formula (10), we can further obtain Δf 0 , if Δf is satisfied 0 →0, then the angle M of the magnetization vector required to generate the target curve is solved according to formula (11): θ (X); Based on the solution results, numerical methods are further used to verify whether the target configuration can be generated. At this time, 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 further comprises the step of verifying the equation result:

[0064] Select the calculation method (full field or reduced field) for the magnetic potential energy in the Magnetic Field (No Current) module;

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

[0066] Select elastic and magnetic constitutive models for thin plates and external air;

[0067] Compute the bending deformation of a thin plate due to mechanical-magnetic coupling.

[0068] The second technical solution adopted by the present invention is:

[0069] An electronic device comprises 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 are loaded and executed by the processor to implement the method for predicting and controlling the magneto-induced bending deformation of a multilayer hard magnetic soft material plate as described above.

[0070] The third technical solution adopted by the present invention is:

[0071] A computer-readable storage medium stores 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 implement a method for predicting and controlling the magneto-induced bending deformation of a multilayer hard magnetic soft material plate as described above.

[0072] The fourth technical solution adopted by the present invention is:

[0073] A computer program product or a 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 the processor executes the computer instructions, so that the computer device executes the above method.

[0074] The beneficial effects of the present invention are as follows: the scheme proposed by the present invention realizes the prediction and shape control of the bending deformation of the multi-layer hard magnetic soft material plate, and compared with other control methods, the effect of the magnetic field can make the thin plate quickly generate the target curve. The accuracy and agility of this control method have important application prospects and value in the fields of flexible electronics and soft robots. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the embodiments of the present invention or the drawings of related technical solutions in the prior art are introduced below. It should be understood that the drawings introduced below are only for the convenience of clearly describing 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 work.

[0076] Figure 1 Flow chart of the bending deformation control scheme of the magnetic double-layer plate.

[0077] Figure 2 Schematic diagram of the boundary constraints of the magnetic double-layer plate in numerical simulation.

[0078] Figure 3 Schematic diagram of the bending deformation prediction of magnetic double-layer plates under different thickness parameters.

[0079] Figure 4 Schematic diagram of the bending deformation prediction of the 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 between the numerical simulation results of the magnetic double-layer plate and the target curve 1.

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

[0083] Figure 8 It is a comparison chart between the numerical simulation results of the magnetic double-layer plate and the target curve 3. DETAILED DESCRIPTION

[0084] The embodiments of the present invention are described in detail below, and examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and are not to be construed as limitations of the present invention. For the step numbers in the following embodiments, they are only provided for the convenience of explanation, and the order between the steps is not limited in any way, and the execution order of each step in the embodiment can be adaptively adjusted according to the understanding of those skilled in the art.

[0085] In the description of the present invention, it should be understood that descriptions involving orientations, such as up, down, front, back, left, right, etc., and orientations or positional relationships indicated are based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as a limitation on the present invention.

[0086] In the description of the present invention, "several" means one or more, "more" means more than two, "greater than", "less than", "exceed" etc. are understood as not including the number itself, and "above", "below", "within" etc. are understood as including the number itself. If there is a description of "first" or "second", it is only used for the purpose of distinguishing the technical features, and cannot be understood as indicating or implying the relative importance or implicitly indicating the number of the indicated technical features or implicitly indicating the order of the indicated technical features.

[0087] In the description of the present invention, unless otherwise clearly defined, terms such as setting, installing, connecting, etc. should be understood in a broad sense, and technicians in the relevant technical field can reasonably determine the specific meanings of the above terms in the present invention based on 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 magnetic bending deformation of a multilayer hard magnetic soft material plate. This method can predict the bending deformation of a magnetic soft material plate driven by an external magnetic field, and guide the selection of magnetization parameters according to the target configuration to achieve shape control. Therefore, it has important application prospects and value in the fields of flexible electronics and soft robots. The method specifically includes the following steps:

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

[0091]

[0092] where x k represents the position vector of the current configuration of the k-th plate; X k represents the position vector of the reference configuration of the k-th plate; represents the deformation gradient tensor of the kth plate; h k represents the thickness of the kth layer; represents the elastic energy of the kth layer; μ 0 Represents vacuum magnetic permeability (μ 0 =4π×10 7 H / m); M k represents the magnetization vector of the kth plate; H a represents the external magnetic field vector; χ represents the magnetic potential energy; It describes the change in volume after the kth plate is deformed. If This means that the volume of the kth plate is the same before and after deformation; q - ,q + and q represent the action on the lower surface of the first layer The upper surface of the nth layer and side surface External force; p k Appears as a Lagrangian operator, As a constraint term for volume incompressibility.

[0093] S2. Derive the control equations of the magnetic multilayer board by variational method.

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

[0095] S21. By varying χ, we obtain the magnetic field control equations.

[0096]

[0097] Among them B L Represents the magnetic induction intensity vector of the magnetic plate sample or the external space; double brackets It indicates the size of the mutation corresponding to the interface between the magnetic multilayer board sample and the air.

[0098] S22, x k The control equations of the magnetic multilayer board sample are obtained by solving the variational equations.

[0099]

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

[0101] S23, due to the magnetic plate sample in the object exciting the magnetic field H s Size and sample magnetization intensity M k Therefore, it is assumed that the excitation magnetic field caused by the magnetization of the object is much smaller than the external magnetic field, and the control equations of the plate sample can be simplified as follows:

[0102]

[0103] S3. According to the equilibrium equation and boundary conditions established in equation (4), we can and M k The series expansion about Z allows us to eliminate one variable and thus establish the two-dimensional plate equation (5).

[0104]

[0105] After the magnetization vectors of each layer of the magnetic multilayer plate and the external magnetic field vector are given, the magneto-induced bending deformation of the magnetic multilayer plate can be predicted by the asymptotic solution solved 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 expressed as follows

[0109]

[0110] S5. Solving the plate equation and proposing a method for predicting and controlling bending deformation In this embodiment, n=2, that is, a double-layer plate is taken as an example for research.

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

[0112]

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

[0114] S52, based on the plane strain assumption, assuming that the magnetic double-layer plate does not deform along the Y-axis direction, the magnetization vector M of each layer 1 and M 2 、External magnetic field B a And the external force Q can be decomposed into the following form:

[0115]

[0116] Where {M A1 (X), M A2 (X), B A , Q A} respectively represent the magnitude of the corresponding vector, {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 expand the current configuration position parameters in series:

[0118]

[0119] At this point, further simplifying equation (5) yields the equation containing only two unknowns: and The two equations of and It reflects the position 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. For the bending deformation of the magnetic double-layer plate, the and Written in the following form, the unknown quantity in the plate equation is converted to f 0 (X) and g 0 (X):

[0121]

[0122] where f 0 (X) represents the axial elongation of the bottom surface of the magnetic double-layer plate, g 0 (X) represents the bending angle of the bottom surface at position X.

[0123] S55. In order to systematically study the influence of the material parameters and thickness of the magnetic double-layer plate on the overall bending deformation, we further give the following equivalent form:

[0124] C 1 =C 0 ,C 2=αC 0 ,h 1 =b 1 h=(1-β)h,h 2 =b 2 h=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. In order to avoid the cumbersome calculation caused by dimension, the following dimensionless calculation is performed.

[0127]

[0128] S57, since the calculation result of step S56 is B A M A and Q A The value is much smaller than C 0 , so we can assume that a small quantity ∈ is k 1 , k 2 and f 0 (X) Perform further asymptotic simplification.

[0129] k 1 =∈K 1 ,k 2 =∈K 2 ,f 0 (X)=1+∈Δf 0 (X) (13)

[0130] Where Δf 0 It represents the strain value at X after the double-layer plate is bent.

[0131] S58. Based on the assumptions and processing of steps S51-S57, substitute into the plate equation (5), and the equation can be obtained as follows:

[0132]

[0133]

[0134] Where E αβ It is a quantity used to characterize bending stiffness; g 0,X Indicates g 0 (X) the first derivative of X; g 0,XX Indicates g 0 (X) The second derivative of X.

[0135]

[0136] When given B A 、M A , C0 After , g(X) and h, further calculation of formula (15) can obtain M θ expression.

[0137]

[0138] where k = B A M A / C 0 .

[0139] S59. Substituting the results of formulas (13)-(16) into formula (10), we can 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 this time a complete prediction model for the magnetic double-layer plate after bending deformation is established.

[0140] S6. Parametric equation of given target curve We can further calculate g 0 (X):

[0141]

[0142] G 0 Substituting (X) into formula (14) we can further obtain Δf 0 , if Δf is satisfied 0 →0, the angle M of the magnetization vector required to generate the target curve can be solved according to formula (15): θ (X). Based on the solution results, a numerical method is further used to verify whether the target configuration can be generated. At this point, a complete shape control method for the bending deformation of the magnetic double-layer plate is 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 (full field or reduced field) of the magnetic potential energy in the magnetic field (no current) module;

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

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

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

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

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

[0151] For the sake of demonstration, this embodiment takes a magnetic double-layer plate as an example. The solution includes predicting the bending deformation of the plate after given parameters such as the material, thickness and magnetization vector of the magnetic double-layer plate. At the same time, after a target curve is given, the required magnetization angle of the magnetic double-layer plate can be calculated, thereby achieving the bending deformation shape control of the plate.

[0152] This embodiment provides a method for predicting and controlling the shape of a multilayer hard magnetic soft material plate's magneto-bending deformation, which specifically includes the following steps:

[0153] 1) Set the material parameters according to the experimental conditions (the thickness of the double-layer board h 1 and h 2 、Material parameters C 1 and C 2 and magnetic field parameters (external magnetic field size B A , external magnetic field angle B θ , the magnetization amplitude of the thin plate sample M A )wait.

[0154] 2) According to the solved control equations, set the body force on the magnetic double-layer plate and the surface force at the boundary. The constraint setting at the sample boundary is as follows: Figure 2 shown.

[0155]

[0156] 3) According to 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. At the same time, the parameters can also be implanted into the finite element software COMSOL Multiphysics to calculate and obtain numerical results. In this way, it is predicted that the magnetic double-layer plate has different thickness ratios (β=h 2 / h 1 ) and material parameter ratio (α=C 2 / C 1 )Bending deformation under the action of magnetic coupling.

[0157] a) Different thickness ratio (β)

[0158]

[0159] b) Ratio of different materials (α)

[0160]

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

[0162]

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

[0164]

[0165] 6) Calculate the magnetization vector angle M θ The distribution of (k = B A M A / C 0 )

[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) The bending deformation of the magnetic double-layer plate is verified by numerical calculation (here we assume that the thickness of the double-layer plate is the same as the material, that is, h 1 =h 2 , C 1 =C 2 ). In the numerical calculation, the initial configuration of the thin plate is selected as Ω r =1m×0.3m×0.04m, the external magnetic field (air) model configuration is selected as Ω r =10m×10m×10m, the numerical calculation steps are as follows:

[0174] 7.1) Select the "Solid Mechanics" and "Magnetic Field (No Current)" modules of the commercial finite element software COMSOL Multiphysics for simulation. In the "Solid Mechanics" module, set the elastic-hyperelastic constitutive model of the sample to the incompressible Neo-Hookean constitutive model, where C 0 = 0.1MPa, and set the constitutive model of the external magnetic field model area to the air constitutive model built into the software. The boundary conditions of the thin plate sample are set as follows: Figure 2 As shown. In addition, since the present invention is only for bending deformation, the thin plate is assumed to be plane strain in the numerical calculation, that is, the displacement along the Y axis is constrained. According to the derived boundary conditions, the body force and surface force of the magnetic field on the sample are applied to the sample. In this module, the quadratic coincidence edge point unit 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 area to B = μ(H + M), where μ = 4π×10 -7 is the vacuum permeability, the magnetization vector M of the sample area is set by formula (13), and the magnetization vector M of the external area is 0. Finally, select the "magnetic flux density" and "zero magnetic scalar potential" options to specify the magnitude of the external magnetic field magnetic induction intensity and the magnetic field boundary conditions. In this module, the magnetic potential energy order is set to quadratic.

[0176] The numerical simulation results of the bending deformation of the magnetic double-layer plate samples under different material and thickness parameters are shown in the figure. Figure 3 , Figure 4 As shown in the figure, it can be seen that the simulation results are very consistent with the calculated results, verifying the accuracy of the bending deformation prediction. Figure 5 , Figure 6 , Figure 7 and Figure 8 As shown, it can be seen that the deformed thin plate is very consistent with the target curve, which verifies the correctness of the solution and setting of the magnetization vector.

[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 complete theoretical plan.

[0179] The theoretical model established by the present invention does not require a priori assumptions about displacement or stress, and can obtain the control equations of the multilayer plate by the variational method, and establish a two-dimensional plate equation. Further, under the assumption of plane strain, the asymptotic analysis method is used to solve the configuration of the magnetic multilayer plate (taking a double-layer plate as an example) after shape deformation, thereby establishing a prediction model for the bending deformation of the magnetic multilayer plate under the action of a magnetic field. In addition, the required magnetization vector can also be obtained by giving a target curve, solving the bending angle of the curve and then substituting it into the plate equation to guide the shape control of the magnetic multilayer plate.

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

[0181] The technical solution proposed in the present invention can predict the bending deformation of the magnetic multilayer plate in the magnetic field, and can guide the setting of the magnetization vector angle in the plate to achieve the overall bending deformation into an arbitrary curve. Taking Neo-Hookean hyperelastic material as an example, Figure 3 and Figure 4 A comparison between the asymptotic and numerical solutions for the prediction of the bending deformation of magnetic bilayer plates with different thickness-to-material ratios in a magnetic field is shown. Figure 6-Figure 8The paper shows that the required magnetization vector is calculated after a given target curve, and then the bending deformation and target configuration obtained at this time are verified by numerical methods. The results also show that this scheme can accurately control the bending deformation of the superelastic thin plate.

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

[0183] The equation for predicting and controlling the bending deformation of the magnetic multilayer plate established by the present invention has a very simple form. On the one hand, it helps to expand the application and facilitate the efficient design and manufacture of magnetic soft robots. On the other hand, the conditions that affect the bending deformation of the magnetic multilayer plate can also be understood from the form of the equation, which provides a deeper insight into the magneto-induced deformation behavior of the multilayer plate.

[0184] Example 2

[0185] An embodiment of the present invention further provides an electronic device, the electronic device comprising a processor and a memory, the memory storing 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 being loaded and executed by the processor to implement the following Figure 1 A method for predicting and controlling the magneto-induced bending deformation of a multilayer hard magnetic soft material plate is shown.

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

[0187] The processor may include one or more processing cores. The processor uses various interfaces and lines to connect the various parts of the entire server, and executes various functions of the server and processes data by running or executing instructions, programs, code sets or instruction sets stored in the memory, and calling data stored in the memory. Optionally, the processor can be implemented in at least one hardware form of digital signal processing (DSP), field programmable gate array (FPGA), and programmable logic array (PLA). The processor can integrate one or a combination of a central processing unit (CPU) and a modem. Among them, the CPU mainly processes the operating system and application programs; the modem is used to process wireless communications. It can be understood that the above-mentioned modem may not be integrated into the processor, but implemented separately through a chip.

[0188] Since the electronic device is an electronic device corresponding to the method for predicting and controlling the magneto-induced bending deformation of a multilayer hard magnetic soft material plate in an embodiment of the present invention, and the principle of solving the problem by the electronic device is similar to that of the method, the implementation of the electronic device can refer to the implementation process of the above-mentioned method embodiment, and the repeated parts will not be repeated.

[0189] Example 3

[0190] The embodiment of the present invention further provides a computer-readable storage medium, wherein the storage medium stores at least one instruction, at least one program, code set or instruction set, and the at least one instruction, the at least one program, the code set or instruction set is loaded and executed by a processor to implement the following Figure 1 A method for predicting and controlling the shape of a multilayer hard magnetic soft material plate by magneto-induced bending deformation is shown.

[0191] Those skilled in the art can understand that all or part of the steps in the various methods of the above embodiments can be completed by instructing related hardware through a program, and the program can be stored in a computer-readable storage medium, and the storage medium includes a read-only memory (ROM), a random access memory (RAM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), a one-time programmable read-only memory (OTPROM), an electronically erasable rewritable read-only memory (EEPROM), a compact disc (CD-ROM) or other optical disc storage, magnetic disk storage, magnetic tape storage, or any other computer-readable medium that can be used to carry or store data.

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

[0193] Example 4

[0194] In some possible implementations, various aspects of the method of the embodiment of the present invention can also be implemented in the form of a program product, which includes a program code. When the program product is run on a computer device, the program code is used to enable the computer device to execute the steps of the method for predicting and controlling the magnetic bending deformation of a multilayer hard magnetic soft material plate according to various exemplary embodiments of the present application described above in this specification. Among them, the executable computer program code or "code" for executing each embodiment can be written in a high-level programming language such as C, C++, C#, Smalltalk, Java, JavaScript, VisualBasic, structured query language (e.g., Transact-SQL), Perl, or in various other programming languages.

[0195] It should be understood that the various parts of the present invention can be implemented by hardware, software, firmware or a combination thereof. In the above-mentioned embodiments, a plurality of steps or methods can be implemented by software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented by hardware, as in another embodiment, it can be implemented by any one of the following technologies known in the art or their combination: a discrete logic circuit having a logic gate circuit for implementing a logic function for a data signal, a dedicated integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0196] In the description of this specification, the description with reference to the terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art may combine and combine the different embodiments or examples described in this specification and the features of the different embodiments or examples, without contradiction.

[0197] The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable ordinary technicians in the field to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. Any equivalent changes or modifications made based on the essence of the content of the present invention should be included in the protection scope of the present invention.

[0198] The source of the project results of this patent application is: Analytical research and experimental verification of multi-field coupled mechanical behavior of magnetic plate-shell soft material samples, 2022A1515010653, Guangdong Natural Science Foundation-General Project.

Claims

1. A method for predicting and controlling the shape of a multilayer hard magnetic soft material plate by magneto-bending deformation, characterized in that: The following steps are involved: The energy density functional of the magnetic multilayer plate is established, and the governing equations of the magnetic multilayer plate are derived by the variational method; Establish plate equations according to the control equations, and solve the plate equations to obtain deformation prediction and control solutions; Obtain the target curve parameter equation, substitute the target curve parameter equation into the control scheme, and obtain the corresponding parameters.

2. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 1, characterized in that: The energy density functional Ψ(χ,x k ,p k ;X k ) is: In the formula, x k represents the position vector of the current configuration of the k-th plate; X k represents the position vector of the reference configuration of the k-th plate; represents the deformation gradient tensor of the k-th plate; represents the transpose of the inverse of the k-th plate deformation gradient tensor; h k represents the thickness of the kth layer; represents the elastic energy of the kth plate; μ0 represents the vacuum magnetic permeability; M k represents the magnetization vector of the kth plate; H a represents the external magnetic field vector; χ represents the magnetic potential energy; Gradχ represents the gradient of χ; It describes the volume change of the kth plate before and after deformation. This means that the volume of the kth plate is the same before and after deformation; q - ,q + and q represent the action on the lower surface of the first layer The upper surface of the nth layer and side surfaces External force; p k Appears as a Lagrangian operator, As a constraint item for volume incompressibility; x represents the position vector under the current configuration; n represents the layer number of the nth layer; Z represents the out-of-plane position coordinate of the plate; r represents the in-plane position vector of the plate; and V represents the volume element of the multilayer plate.

3. A method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 1 or 2, It is characterized in that The control equations of the magnetic multilayer plate are derived by the variational method, including: By varying the magnetic potential energy χ, we obtain the governing equations of the magnetic field; The position vector x of the current configuration of the k-th plate k By solving the variational equations, we can obtain the governing equations of the magnetic multilayer board sample; It is assumed 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 sample.

4. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 3, characterized in that: The simplified governing equations of the magnetic multilayer board sample are expressed as: In the formula, Defined as nominal stress; T represents transposition; k represents the unit vector perpendicular to the in-plane direction; N represents the unit vector perpendicular to the side surface; Indicates the lower bottom surface of the first layer of board; Represents the upper surface of the nth layer of board; represents the side surface of the kth layer of the multilayer board; Represents the interface between plates; x k represents the position vector of the kth layer; r k Represents the position vector within the k-th layer of the board; According to the equilibrium equation and boundary conditions established in equation (1), and M k Expand the series about Z, thus eliminating one variable and establishing the two-dimensional plate equation: In the formula, Z i It represents the order of Z after the series expansion along the Z direction, It represents the order of the nominal stress of the kth plate after series expansion along the Z direction, represents the order of the deformation gradient of the kth plate after series expansion along the Z direction, represents the order of the magnetization vector of the kth plate after series expansion along the Z direction, O(Z 4 ) indicates that the series is expanded along the Z direction to Z 4 An infinitesimal quantity.

5. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 2, characterized in that: The expression of the plate equation is as follows: in and They are respectively expressed as follows: In the formula, represents the Nabla operator within the surface; Defined as nominal stress.

6. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 4, characterized in that: The multi-layer hard magnetic soft material plate is a magnetic double-layer plate material; the deformation prediction and control scheme obtained according to the plate equation includes: A1. Assume that the material constitutive model of the double-layer plate is the incompressible neo-Hookean constitutive model; In the formula, C k It represents the material parameters; A2. Based on the plane strain assumption, assuming that the magnetic double-layer plate does not deform along the Y-axis direction, the magnetization vectors M1 and M2 of each layer, and the external magnetic field B a And the external force Q is decomposed into the following form: In the formula, {M A1 (X), M A2 (X), B A , Q A } respectively represent the magnitude of the corresponding vector, {M θ1 (X), M θ2 (X), B θ , Q θ } represents the direction of the corresponding vector; E1 represents the unit vector in the horizontal direction, and k represents the unit vector in the vertical direction; A3. Expand the current configuration position parameters in series: Now further simplifying equation (2) yields only two unknowns: and The two equations of and It reflects the position parameters of the bottom surface of the magnetic double-layer plate after bending deformation, and other quantities can be obtained from and express; A4. For the bending deformation of the magnetic double-layer plate, further and Written in the following form, the unknown quantities in the plate equation are converted to f0(X) and g0(X): 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; A5. In order to systematically study the influence of the material parameters and thickness of the magnetic double-layer plate on the overall bending deformation, the following equivalent form is further given: C1=C0, C2=αC0, h1=b1h=(1-β)h, h2=b2h=βh (7) By making 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; A6. In order to avoid the cumbersome calculation caused by dimension, the following dimensionless calculation is performed: A7, because the calculation result of step A6 is B A M A and Q A The value of is much smaller than C0, so we can assume a small amount ∈ to further asymptotically simplify k1, k2 and f0(X): k1=∈K1,k2=∈K2,f0(X)=1+∈Δf0(X) (9) Where Δf0 represents the strain value at point X after the double-layer plate is bent and deformed; A8. Based on the assumptions and processing in steps A1-A7, substitute into the plate equation (2) to obtain the following equation: In the formula, E αβ It is a quantity used to characterize bending stiffness; g 0,X represents the first-order derivative of g0(X) with respect to X; g 0,XX represents the second-order derivative of g0(X) with respect to X; When given B A 、M A After C0, g(X) and h, the formula (11) is further calculated to obtain the magnetization vector angle M θ The expression is: 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. At this time, a complete prediction model of the magnetic double-layer plate after bending deformation is established.

7. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 6, characterized in that: The step of obtaining a target curve parameter equation and substituting the target curve parameter equation into a control scheme includes: Given the parametric equation of the target curve Further calculation of g0(X): Substituting g0(X) into formula (10), we can further obtain Δf0. If Δf0→0 is satisfied, the angle M of the magnetization vector required to generate the target curve is solved according to formula (11): θ (X); Based on the solution results, numerical methods are further used to verify whether the target configuration can be generated. At this time, a complete shape control scheme for the bending deformation of the magnetic double-layer plate is established.

8. The method for predicting and controlling the magneto-induced bending deformation of a multi-layer hard magnetic soft material plate according to claim 1, characterized in that: The prediction and shape control method also includes the step of verifying the equation results: Select the calculation method of magnetic potential energy in the magnetic field module; Establish the geometric model of the thin plate and the external air; Select elastic and magnetic constitutive models for thin plates and external air; Compute the bending deformation of a thin plate due to mechanical-magnetic coupling.

9. 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 described in any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that: The storage medium 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 according to any one of claims 1 to 8.

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