Method for controlling shape of hyperelastic plane strain plate in gravity environment and related equipment

Abstract

The embodiment of the invention provides a method for controlling the shape of a hyperelastic plane strain plate in a gravity environment and related equipment, and belongs to the technical field of flexible materials and intelligent structures. The method comprises the following steps: firstly, establishing a hyperelastic plate control equation considering physical strength based on a finite strain theory, and deducing a dimensionality reduction plate equation through asymptotic analysis; secondly, innovatively analyzing and inversely solving an explicit expression of a growth function required for realizing any given target curved surface, and establishing an analytic relationship among a growth field, target shape geometric characteristics and physical power; and finally, preparing a pre-strain sample by designing a mold corresponding to the target configuration, and physically realizing a growth field. The gravity is converted into a design variable from an interference factor, the influence of the gravity is compensated through growth deformation, so that the plane strain plate structure with one side clamped and supported accurately keeps the expected shape under the gravity, and the method has the advantages of being high in control precision, high in design efficiency and easy to implement and can be applied to the fields of soft robots, space deployable structures and the like.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of flexible materials and smart structures, and particularly relates to a method for shape control of a super-elastic plane strain plate under gravity and related equipment. BACKGROUND

[0002] In nature, organisms can evolve from simple initial morphologies to complex morphologies with different functions through internal non-uniform growth (i.e., "morphogenesis"). In recent years, inspired by the biological morphogenesis process, the design of an internal "growth field" to achieve a target shape has become an important development direction. The core idea is to pre-design a non-uniform growth field (such as local expansion or contraction) in a soft material, so as to drive the soft material structure to actively and controllably deform to a pre-set shape. This technology has shown broad application prospects in the fields of soft robots, intelligent actuators, biological tissue engineering, and deployable structures.

[0003] However, the existing shape programming theories and methods have the following common limitations: 1) Ignoring the influence of body force (gravity, centrifugal inertia, acceleration field). This assumption is still valid for small-size, light-weight, and high-stiffness samples, but in large-scale or engineering scenarios requiring horizontal cantilevers, the deflection caused by gravity cannot be ignored.

[0004] 2) Not considering gravity as a design variable. Most existing solutions only consider gravity as an external factor that needs to be avoided. Therefore, additional compensation (thickening, stiffening, external support) is needed, which increases the additional mass, energy consumption, and complexity.

[0005] 3) Lack of analytical results of the inverse problem of sample deformation under growth-gravity induction. Existing numerical optimization schemes are often computationally expensive, sensitive to initial values, and difficult to give explicit relationships between geometry-material-loading parameters and the required growth field, which is not conducive to rapid design and control.

[0006] In engineering practice, if the local non-uniform expansion / contraction (equivalent growth field) of the sample can be set to deform the component to the expected shape under gravity, the dependence on additional reinforcement structures, energy-consuming closed-loop control systems, and excessive safety redundancy design can be reduced. Therefore, there is an urgent need for a growth-driven shape control technology based on the finite strain plate theory, which establishes an explicit analytical relationship between the growth function, the geometric characteristics of the target shape, and the distribution of body forces (especially gravity), establishes a shape control scheme, and proposes a corresponding experimental preparation method. SUMMARY

[0007] The main purpose of the embodiments of the present application is to provide a method for realizing shape control of a super-elastic planar strain plate in a gravity environment based on growth deformation and related equipment. By pre-designing a non-uniform growth field (or pre-strain field) in the material, the super-elastic plate can actively deform and accurately maintain the predetermined target shape under the action of its own gravity and other body forces.

[0008] To achieve the above purpose, one aspect of the embodiments of the present application provides a method for shape control of a super-elastic planar strain plate in a gravity environment, which comprises the following steps: Theoretical modeling: based on the finite strain plate theory, the control equation of the super-elastic plate considering the action of body force is established; Multiplicative decomposition of total deformation gradient is introduced , wherein is the growth tensor, is the elastic deformation tensor; The growth tensor is introduced in the reference configuration only in the principal direction, and its expression is:

[0009] , wherein is a local scaling control function, is a curvature control function; is the coordinate in the thickness direction, is the base vector of the rectangular coordinate system, denotes the tensor product operation; Analytical inverse solution: the control equation of the super-elastic plate is reduced in dimension and analyzed asymptotically, and the analytical expression of the growth function and required to realize a given target surface is derived, which contains the body force term related to gravity; Calculation and implementation: according to the set target shape, geometric size and body force distribution, the specific and function distribution is calculated, and the growth field distribution is realized in the super-elastic plate material through the preparation process, so that the plate can actively deform and stabilize in the target shape after being released in the gravity environment.

[0010] In some embodiments, the analytical expression of the growth function and is:

[0011]

[0012] , wherein denotes the first-order derivative of the rectangular coordinate , a parametric equation of the target shape (surface), a first derivative of a second derivative of a second derivative of a normal vector of the target surface, are the first and second fundamental form coefficients of a unit volume force applied in the plate where denotes the material density, denotes the gravitational acceleration, is half of the total thickness of the plate, is the length of the plate, is a dimensionless small parameter used to characterize the effect of gravity on the deformation of the sample, is half of the shear modulus, is a gravity-related correction term.

[0013] In some embodiments, the function is obtained by integration of

[0014] and determined by integration of the boundary conditions

[0015] In some embodiments, the target shape is a planar curve and satisfies the left-end clamped boundary condition, expressed as:

[0016] where , is the component of the parametric equation, is the Z component of the parametric equation; is the length of the plate; In some embodiments, the constitutive model of the hyperelastic plate is an isotropic incompressible neo-Hookean model, whose elastic strain energy function is:

[0017] where denotes the determinant of the growth tensor, is half of the shear modulus, denotes the trace operation, is the transpose.

[0018] ​​​​​​In some embodiments, the growth field distribution is achieved in the super-elastic sheet material by a fabrication process, including: According to the calculated post-growth configuration, a corresponding mold is designed and manufactured; An uncured super-elastic material precursor liquid is poured into the mold; After curing, the mold is removed to directly obtain a sample having a non-uniform pre-strain distribution corresponding to the growth field distribution.

[0019] In some embodiments, after the steps of calculation and implementation, a numerical verification step is further included, which specifically includes: A three-dimensional finite element model of the super-elastic sheet is established; The growth function is introduced into the model in the form of a state variable, and the evolution process thereof from a unit tensor to a target value is defined; The deformation process of the sheet under the body force is simulated, and the final equilibrium shape is compared with the target shape.

[0020] To achieve the above object, another aspect of the embodiments of the present application provides an electronic device, which comprises a memory and a processor, the memory stores a computer program, and the processor implements the above method when executing the computer program.

[0021] To achieve the above object, another aspect of the embodiments of the present application provides a computer readable storage medium, which stores a computer program, and the computer program implements the above method when executed by a processor.

[0022] To achieve the above object, another aspect of the embodiments of the present application provides a computer program product, which comprises a computer program, and the computer program implements the above method when executed by a processor The embodiments of the present application at least have the following beneficial effects: the present application provides a method for shape control of a super-elastic plane strain sheet in a gravity environment, an electronic device, a storage medium and a program product, an explicit mathematical relationship between a growth field, a target geometry and a body force is established through theoretical analysis and deduction, the growth distribution inside the material can be efficiently and accurately designed, and the structure can automatically reach and maintain the desired target shape under the action of gravity after release. The present application can realize lightweight and high-precision shape control of large-size flexible structures, the analytical formula provided improves the design efficiency, and has a wide application prospect in the fields of soft robots, large deployable structures (such as antennas, solar sails), biological tissue engineering and precision optical device forming. BRIEF DESCRIPTION OF DRAWINGS

[0023] Figure 1is a schematic diagram of deformation of a plane strain plate under body force and growth field in the embodiment of the present application; Figure 2 is a flow chart of shape control of a super-elastic plane strain plate in a gravity environment based on growth deformation in the embodiment of the present application; Figure 3 is a schematic diagram of experimental verification in the embodiment of the present application; Figure 4 is a schematic diagram when the target curved surface is set as a flat plate in the embodiment of the present application; Figure 5 is a schematic diagram when the target curved surface is set as a downward bending semicircle in the embodiment of the present application; Figure 6 is a schematic diagram of a method of shape control of a super-elastic plane strain plate in a gravity environment provided in the embodiment of the present application; Figure 7 is a step diagram of a method of shape control of a super-elastic plane strain plate in a gravity environment provided in the embodiment of the present application; Figure 8 is a schematic diagram of a hardware structure of an electronic device provided in the embodiment of the present application. DETAILED DESCRIPTION

[0024] In order to make the objects, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not intended to limit the present application. When the following description refers to the accompanying drawings, the same numbers in different drawings represent the same or similar elements unless otherwise indicated. The implementation modes described in the following exemplary embodiments do not represent all implementation modes consistent with the embodiments of the present application, but are only examples of devices and methods consistent with some aspects of the embodiments of the present application as detailed in the appended claims.

[0025] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the present application belongs. The terms used herein are only for the purpose of describing the embodiments of the present application and are not intended to limit the present application.

[0026] Before the embodiments of the present application are described in detail, first, some nouns and terms involved in the embodiments of the present application are described, and the nouns and terms involved in the embodiments of the present application are applicable to the following explanations.

[0027] 1) Super-elastic plate: Super-elastic plate refers to a plate-shaped structure composed of super-elastic materials (such as silicone, hydrogel, biological tissue, etc.). Super-elastic materials are a special class of materials that can almost completely recover to their original shape after experiencing very large elastic deformation, with minimal energy dissipation during the process. The stress-strain relationship of such materials is nonlinear, and the mechanical behavior is usually described by the strain energy density function.

[0028] 2) Plane strain: Plane strain is a hypothesis used to simplify the mechanical analysis of three-dimensional objects. Under the plane strain hypothesis, the strain components of the object in certain directions are considered to be zero or negligible.

[0029] 3) Clamped constraint: Clamped constraint is a boundary condition of a structure, which means that one or more boundaries of the structure are completely fixed and cannot move or rotate.

[0030] Inspired by biological morphogenesis in nature, pre-programming a non-uniform growth field inside the material during the design stage to drive soft intelligent structures to deform to a pre-set target shape has become a hot research topic. The core of this technology is to offset or utilize the deformation caused by external loads through the design of growth tensors.

[0031] However, existing shape programming methods have obvious limitations: First, most studies ignore the influence of body forces (such as gravity), which is only applicable to small-scale or high-stiffness structures. For large-scale, low-stiffness flexible cantilever structures, the deformation caused by gravity cannot be ignored and even plays a dominant role. Second, existing methods usually regard gravity as a purely negative interference factor, which needs to be passively compensated by increasing the stiffness of the structure or adding supports, which leads to an increase in the redundant weight and complexity of the structure. Third, existing techniques lack analytical solution methods for shape control inverse problems in a gravity environment, and are heavily dependent on numerical optimization algorithms with high computational cost and sensitivity to initial values, making it difficult to establish clear design criteria and hindering its engineering application.

[0032] Therefore, the embodiments of the present application provide a method for shape control by compensating for the influence of gravity through growth deformation, electronic devices, storage media and program products. This scheme is based on the theory of finite strain plates, studies super-elastic plane strain plates under the action of body forces (such as gravity), and establishes an explicit analytical relationship between the geometric characteristics of the target shape, the distribution of body forces, and the internal growth function to calculate the required growth field distribution. The growth field is linearly distributed in the thickness direction, while controlling the local stretching and bending of the plate. Physically, this method pre-fabricates a sample with a specific initial bending morphology through mold design, which will deform accurately to the target shape under the action of gravity.

[0033] The embodiment of the application provides a method for shape control of a super-elastic plane strain plate in a gravity environment, and relates to the technical field of flexible materials and intelligent structures. The method for shape control of the super-elastic plane strain plate in the gravity environment can be applied to a terminal, can also be applied to a server, and can be software running in the terminal or the server. In some embodiments, the terminal can be a smart phone, a tablet computer, a notebook computer, a desktop computer, a smart speaker, a smart watch, a vehicle-mounted terminal and the like, but is not limited thereto; the server end can be configured as a stand-alone physical server, can also be configured as a server cluster or a distributed system formed by multiple physical servers, can also be configured as a cloud server providing basic cloud computing services such as cloud service, cloud database, cloud computing, cloud function, cloud storage, network service, cloud communication, middleware service, domain name service, security service, CDN and big data and artificial intelligence platform, and the server can also be a node server in a blockchain network; and the software can be an application of the method for shape control of the super-elastic plane strain plate in the gravity environment and the like, but is not limited to the above forms.

[0034] The application can be used in many general or special computer system environments or configurations. For example: personal computers, server computers, handheld devices or portable devices, tablet devices, multi-processor systems, microprocessor-based systems, set-top boxes, programmable consumer electronics, network PCs, minicomputers, mainframe computers, distributed computing environments including any of the above systems or devices, and the like. The application can be described in the general context of computer-executable instructions executed by a computer, such as a program module. Generally, program modules include routines, programs, objects, components, data structures, and the like that perform specific tasks or implement specific abstract data types. The application can also be practiced in a distributed computing environment in which tasks are performed by remote processing devices connected by a communication network. In a distributed computing environment, program modules can be located in local and remote computer storage media, including storage devices.

[0035] As shown in Figure 6 and Figure 7 , the embodiment provides a method for shape control of a super-elastic plane strain plate in a gravity environment based on growth deformation, including the following steps: S1, theoretical modeling and control equation establishment.

[0036] The research object of the embodiment is a super-elastic plate with constant thickness in a three-dimensional Euclidean space, and the deformation process is as shown in Figure 1 First, a rectangular plate with a length L, a width W and a thickness 2h is taken as a reference configuration, and a rectangular coordinate system is established. A unit volume body force is applied in the plate, where denotes the material density, denotes the gravitational acceleration. The growth effect is decomposed by multiplication with the total deformation gradient is introduced, where is the growth tensor, is the elastic deformation tensor. To satisfy the shape control requirement of the plane strain plate, only the non-uniform growth in the X direction is introduced and is linearly distributed along the thickness Z: where is responsible for local stretching, controls the curvature. The material is considered as an isotropic incompressible neo-Hookean model, whose elastic strain energy is , denotes the determinant of the growth tensor, is half of the shear modulus, and tr denotes the trace operation. Since the elastic response of soft materials is usually incompressible (e.g., biological soft tissues, polymer gels), the elastic strain tensor should satisfy the incompressibility constraint , denotes the determinant of the elastic deformation tensor. The nominal stress is derived from the elastic strain energy where p is the hydrostatic pressure. The three-dimensional stress equilibrium equation, the incompressibility constraint, and the cantilever boundary condition constitute a nonlinear three-dimensional control system containing unknown functions x and p: (1) To realize the dimension reduction, the method of thickness direction Taylor series expansion and truncation is adopted to derive the plate equation which is asymptotically consistent with the three-dimensional weak form: (2) where and are the average stress and body force (gravity) along the plate thickness, and subscripts 11 and 13 denote the stress components.

[0037] S2, Derivation and analytical inverse solution of the plate equation.

[0038] To facilitate the solution, the dimensionless small parameter is introduced in this embodiment to represent the effect of gravity on the deformation of the sample. For the uniform body force in the direction, The expression of the nominal stress is substituted into the plate equation (2) and the boundary conditions in (1) are satisfied, and the terms with respect to and are retained, so that the growth function distribution for any target surface can be obtained: (3) where denotes the first derivative with respect to X, E and L are the first and second fundamental form coefficients of the surface (4) cannot be written explicitly, but its first derivative can be expressed as (5) The analytical expression of can be obtained by integrating with respect to X, and the integration constant is determined by the boundary condition at which is free end. For the target shape with complex parametric equation, numerical integration method should be used.

[0039] S3, growth function calculation and implementation.

[0040] Based on the above analytical results, this embodiment proposes an implementation process of programming the cantilever plate to the target shape by growth deformation under the conditions of given body force load and target surface, as shown in the schematic diagram Figure 2 . S31, define the geometric size, body force and target shape. Set the length L, width W, thickness 2h and body force distribution of the sample. Give the parametric equation of the bottom surface . To meet the left end fixed condition, the tangential direction needs to be consistent with the clamping direction, for example, for horizontal fixed support.

[0041] S32, calculate the growth function expression. Calculate the geometric quantities E and L according to formula (4). First, calculate by formula (5), then integrate with respect to X and apply the free end condition to obtain . If the explicit expression of cannot be obtained, numerical integration method should be used. Substitute the obtained into formula (3) to obtain and required to achieve the target shape under the given body force.

[0042] When , formula (3) will degenerate to the growth function without body force, and based on the growth function, any target surface can also be generated. It should be emphasized that although the term of about is usually small, the term of about is relatively large, especially for slender and soft plates.

[0043] S4, verification and test.​​​

[0044] To verify the correctness of the theoretical design, both numerical simulation and physical experiment can be used. Numerically, a model is built in a finite element software (e.g. ABAQUS) with the growth tensor as a state variable and its evolution from the identity tensor to the target value is defined. The deformation of the plate under gravity is simulated and the result is compared with the target shape. Physically, a prepared sample is fixed at one end and then released to deform freely under its own weight. The final equilibrium shape is recorded by optical measurement and compared with the theoretical target shape to evaluate the control accuracy.

[0045] Exemplarily, the numerical verification is as follows: the growth deformation simulation is implemented in a three-dimensional finite element model. The growth function is input as a state variable and linearly changes in the analysis step: (time t increases from 0 to 1), where is the identity tensor, thereby achieving a smooth transition from the initial to the fully grown state. The simulation result of the plate is compared with the target shape.

[0046] In some embodiments, referring to Figure 3 , the experimental verification includes the following steps S41-S44: S41, material characterization. The super-elastic material used in the experiment is silicone rubber. To characterize the material performance, the two components of the silicone rubber are mixed according to the specified mass ratio, poured into a mold to form a cylindrical sample with a diameter of 30 mm and a height of 20 mm, and cured at room temperature for 6 hours. After curing, the material density is obtained by weighing. Uniaxial compression test is performed under the assumption of incompressible neo-Hookean to obtain the shear modulus.

[0047] S42, theoretical design. Based on the material parameters obtained in step S41, repeat step 2 to calculate the required and . The growth function is input into the finite element model, and the geometric configuration after "full growth" can be obtained by numerical simulation.

[0048] S43, sample preparation. According to the geometric configuration after growth obtained in step S42, a mold is designed and a corresponding mold is made using 3D printing technology. Pour the prepared silicone liquid into the mold, and after curing at room temperature, remove the mold to obtain a silicone plate with a specific curvature.

[0049] S44, loading and comparison. One end (X=0) of the prepared curved silicone plate is fixed and clamped, and after release, the free section deforms under its own weight until it reaches a stable equilibrium. The side view is collected and compared with the target shape to evaluate the accuracy of the theoretical prediction.

[0050] In the experiments, the growth field is physically realized by a non-uniform pre-strain field. The specimen is first directly molded in the stress-free growth configuration, and then the body force (gravity) is applied. This indirect approach avoids inducing growth during the loading process.

[0051] The order and path of applying growth and body force (e.g. gravity) affect the final equilibrium state the system reaches, especially in thin plates where multiple equilibrium bifurcations exist and instability can occur. In practice, by adjusting the evolution process of the external loading and growth field, one can guide the system to evolve along the desired equilibrium branch and avoid unexpected snap-through instability.

[0052] Three basic paths are considered in this embodiment: (i) apply body force first and then growth; (ii) apply growth first and then body force; (iii) apply body force and growth simultaneously. For thicker or stiffer plates, the difference between these three paths can be negligible; but for thin / soft plates, path (ii) is often more robust when the body force effect is not negligible. Path (i) can be more advantageous when the direction of realizing the target curvature is aligned with the body force induced bending. However, if the shape programming needs to be performed "against" the body force, path (i) is more sensitive and prone to premature buckling.

[0053] With the loading strategy described above, the final configuration can usually stably approach the desired target shape. If there is still a significant deviation or instability, the loading path can be further fine-tuned by adjusting the relative proportion of the body force and growth evolution rate.

[0054] In the following, the scheme of the embodiments of the present application is described in detail and explained in combination with the drawings and specific application examples.

[0055] To verify the theoretical framework of this embodiment, this embodiment designs six experimental working conditions, as shown in Table 1. These working conditions are used to investigate the influence of several key factors on the shape programming accuracy and effect. We study two target shapes: a horizontal flat plate and a downward bending semicircle. Two different silicone elastomers (Ecoflex 00-30 and Dragon Skin 20) are selected to evaluate the influence of material properties. In addition, samples with thicknesses of 10 mm and 5 mm are made to explore the effect of structural stiffness.

[0056] Table 1 Six experimental settings: target shape, material and sample thickness

[0057] First, examples 1-3 are shown, where we choose a flat plate as the target surface, which includes the following steps: Step A1: Define the geometric dimensions, body force and target shape. Set the length L of the sample to be 100 mm, the width W to be 50 mm, and the thickness 2h to be 5 or 10 mm. The body force distribution Because the gravitational acceleration at the Earth's surface is approximately The force per unit mass of the sample is The parametric equations of the base surface are given. .

[0058] Step A2: Calculate the growth function expression. Calculate the geometric quantities E and L according to equation (4). First, calculate using equation (5). Then integrate over X and apply the free-end condition. get The obtained Substituting into equation (3) yields the result required to achieve the target shape under a given physical force. and as follows: (6) Where the unit and range of values ​​for X are: m.

[0059] Step A3: Numerical Verification. Simulate growth deformation in a 3D finite element model. Input the growth function as a state variable and allow it to change linearly within the analysis step: (Time t increases from 0 to 1), where As a unit tensor, this allows for a smooth transition from initial to full growth (e.g. Figure 4 (as shown in the third column).

[0060] Step A4: Experimental verification, the specific steps are as follows: Step A4.1: Material Characterization. The superelastic material used in the experiment was Ecoflex 00-30 and Dragon Skin 20 silicone rubber manufactured by Smooth-On. To characterize the material properties, the two-component liquid was mixed at the specified mass ratio, poured into a mold to form cylindrical samples with a diameter of 30 mm and a height of 20 mm, and cured at room temperature for 6 hours. The material density was obtained by weighing after curing. The shear modulus was obtained by uniaxial compression tests under the incompressible neo-Hookean assumption. For Ecoflex 00-30, the measured average density... for Its density is slightly higher than that of water. Material parameters were obtained from uniaxial compression tests. (Half of the shear modulus) is For DragonSkin 20, the average density is... The corresponding material parameters for .

[0061] Step A4.2: Theoretical design. With the material parameters obtained from Step A4.1, repeat Step 2 to calculate the required With . The dimensionless parameter is about in Ecoflex material and about in Dragon Skin material. Subsequently, a finite element model with the same geometry and material parameters as the experiment is built in ABAQUS software, and the boundary conditions are set to remove the rigid body motion. The growth function is input into the finite element model, and the fully grown geometry configuration can be obtained by numerical simulation.

[0062] Step A4.3: Sample preparation. Design the mold according to the grown geometry configuration obtained from Step A4.2, and use 3D printing technology to make the corresponding mold. Pour the prepared Ecoflex or Dragon Skin liquid into the mold, and after solidification at room temperature, the silicone plate with specific curvature is obtained (e.g. Figure 4 as shown in the first column).

[0063] Step A4.4: Loading and comparison. Fix one end (X=0) of the prepared curved silicone plate with a clamp, and after releasing, the free section deforms under the action of gravity until it reaches a stable equilibrium. Since gravity cannot be applied gradually, a quasi-static loading scheme is used to simulate the smooth loading process. First, apply a vertical support force to balance the self-weight of the free section, and then gradually remove it at a low rate until the plate loses support completely (e.g. Figure 4 as shown in the second column). Compare the experimental results with Figure 4 the numerical simulation in the third column, and the red solid line is the target shape.

[0064] Overall, the sample shape measured by the experiment is close to the target shape, indicating that the sample can effectively offset the effect of gravity by applying the growth field and return to the flat plate configuration.

[0065] Examples 4-6 are shown below, where we choose a semicircle as the target curved surface, which includes the following steps: Step B1: Define the geometric dimensions, body force, and target shape. Set the length L = 100 mm, width W = 50 mm, and thickness 2h = 5 or 10 mm of the sample. The body force distribution is due to the acceleration of gravity on the surface of the Earth, which is about , and the force on the unit mass of the sample is . The parametric equation of the bottom surface is given by .

[0066] Step B2: Calculate the growth function expression. Calculate the geometric quantity E and L according to formula (4). First, calculate from formula (5), and then integrate X and apply the free end condition Obtained . The obtained is substituted into equation (3) to obtain the and as follows: (7) where the unit and value range of X is m, The expression of is as follows (8) Step B3: Numerical verification. Implement the growth deformation simulation in a three-dimensional finite element model. Input the growth function as a state variable and linearly change it in the analysis step: (time t increases from 0 to 1), where is the unit tensor, thereby achieving a smooth transition from initial to full growth (as shown in the third column). Figure 5

[0067] Step B4: Experimental verification, the specific steps are as follows: Step B4.1: Material characterization. The super-elastic material used in the experiment is Ecoflex 00-30 and Dragon Skin 20 silicone rubber produced by Smooth-On Company. To characterize the material performance, mix the two components of the liquid according to the specified mass ratio, pour into the mold to make a cylindrical sample with a diameter of 30 mm and a height of 20 mm, and cure at room temperature for 6 hours. After curing, the material density is obtained by weighing. Uniaxial compression test is carried out under the assumption of incompressible neo-Hookean to obtain the shear modulus. For Ecoflex 00-30, the average density measured is , slightly higher than the density of water. The uniaxial compression test obtains the material parameter (half of the shear modulus) as . For Dragon Skin 20, the average density is , and the corresponding material parameter is .

[0068] Step B4.2: Theoretical design. Combine the material parameters obtained in step B4.1 to repeat step 2 to calculate the required and . The dimensionless parameter is about in Ecoflex material, and about in Dragon Skin material.Subsequently, a finite element model with the same geometry and material parameters as the experiment was established in ABAQUS software, and the boundary conditions were set to remove the rigid body motion. The growth function was input into the finite element model, and the geometry after full growth was obtained by numerical simulation.

[0069] Step B4.3: sample preparation. The mold was designed according to the geometry after growth obtained in step B4.2, and the corresponding mold was made by 3D printing technology. The prepared Ecoflex or Dragon Skin liquid was poured into the mold, and after curing at room temperature, the silicone plate with specific curvature was obtained (such as Figure 5 as shown in the first column).

[0070] Step B4.4: loading and comparison. One end (X=0) of the prepared curved silicone plate was fixed and clamped, and after release, the free section deformed under the action of gravity until stable equilibrium was reached. Since gravity cannot be applied gradually, a quasi-static loading scheme was used to simulate the smooth loading process. First, a vertical support force was applied to balance the self-weight of the free section, and then it was gradually removed at a low rate until the plate completely lost support (such as Figure 5 as shown in the second column). The experimental results were compared with Figure 5 the numerical simulation in the third column, and the red solid line was the target shape.

[0071] The sample shape measured in the experiment was close to the target shape. It can be predicted that when the target shape (curved upward) is opposite to the natural downward bending trend induced by gravity, the growth field at the left end of the plate needs to provide a larger gradient in the thickness direction to generate sufficient bending moment to offset the gravity load. On the contrary, when the target shape (curved downward) is consistent with the bending mode dominated by gravity, the body force will cooperate with the shape programming, and the required growth field gradient will be significantly reduced. The results show the potential of body force as a means to achieve target morphological transformation, i.e. body force can be regarded as a design variable rather than an external disturbance.

[0072] In summary, the method of the present embodiment has at least the following advantages and beneficial effects compared to the prior art: 1) turning harm into benefit. Gravity is transformed from an interference factor that causes deformation into a predictable and usable design parameter, achieving active compensation for gravity deformation.

[0073] 2) high precision and lightweight. Without the need to increase additional support or material stiffness, lightweight and high-precision shape control of large-size flexible structures can be achieved.

[0074] 3) high design efficiency. A systematic design process based on analytical formulas is provided, greatly reducing the cost and time of traditional reliance on numerical optimization algorithms or physical trial and error.

[0075] 4) Wide application. It can be used to manufacture flexible mechanical arms that can keep straight under gravity, accurately shaped deployable antenna reflectors, etc.

[0076] 5) The growth function calculation formula (3) established in this embodiment has a very simple form. On the one hand, it is helpful for popularization and application, and provides convenience for efficient design and manufacture of intelligent soft instruments. On the other hand, the analytical result can also provide a more profound insight into the growth behavior of certain soft biological tissues in nature.

[0077] The embodiment of the present application further provides an electronic device, which comprises a memory and a processor. The memory stores a computer program, and the processor implements the method described above when executing the computer program. The electronic device can be any intelligent terminal, such as a tablet computer or a vehicle-mounted computer.

[0078] It can be understood that the contents in the method embodiments described above are applicable to the device embodiments, the device embodiments specifically implement the functions of the method embodiments described above, and achieve the same beneficial effects as the method embodiments described above.

[0079] Please refer to Figure 8 , Figure 8 The hardware structure of the electronic device of another embodiment is illustrated, which comprises: The processor 801 can be implemented in the form of a general-purpose CPU (Central Processing Unit), a microprocessor, an ASIC (Application Specific Integrated Circuit), or one or more integrated circuits, and is used to execute related programs to implement the technical solutions provided by the embodiments of the present application. The memory 802 can be implemented in the form of a ROM (Read Only Memory), a static storage device, a dynamic storage device, or a RAM (Random Access Memory). The memory 802 can store an operating system and other application programs. When the technical solutions provided by the embodiments of the present application are implemented by software or firmware, the related program codes are stored in the memory 802 and called and executed by the processor 801 to implement the method described above. The input / output interface 803 is used to realize information input and output. The communication interface 804 is used to realize the communication interaction between the device and other devices. The communication can be realized in a wired manner (for example, USB, network cable, etc.) or in a wireless manner (for example, mobile network, WIFI, Bluetooth, etc.). A bus 805 transmits information between various components (for example, the processor 801, the memory 802, the input / output interface 803, and the communication interface 804) of the device. The processor 801, the memory 802, the input / output interface 803, and the communication interface 804 are communicatively connected to each other within the device through the bus 805.

[0080] The embodiment of the present application further provides a computer readable storage medium, which stores a computer program, and the computer program is executed by a processor to implement the method.

[0081] It can be understood that the contents in the above method embodiments are all applicable to the present storage medium embodiment, the present storage medium embodiment specifically implements the functions of the above method embodiments, and achieves the same beneficial effects as the above method embodiments.

[0082] The memory is a non-transitory computer readable storage medium, and can be used to store non-transitory software programs and non-transitory computer executable programs. In addition, the memory can include a high-speed random access memory, and can also include a non-transitory memory, for example, at least one magnetic disk storage device, a flash memory device, or other non-transitory solid-state memory device. In some embodiments, the memory can optionally include a memory remotely arranged relative to the processor, and the remote memory can be connected to the processor through a network. Examples of the above network include but are not limited to the Internet, an intranet, a local area network, a mobile communication network, and a combination thereof.

[0083] The embodiment of the present application further provides a computer program product, which includes a computer program, and the computer program is executed by a processor to implement the method.

[0084] It can be understood that the contents in the above method embodiments are all applicable to the present program product embodiment, the present program product embodiment specifically implements the functions of the above method embodiments, and achieves the same beneficial effects as the above method embodiments. The executable computer program code or "code" for executing various embodiments can be written in a high-level programming language such as C, C++, Python, Smalltalk, Java, JavaScript, Visual Basic, Structured Query Language (for example, Transact-SQL), Perl, or in various other programming languages.

[0085] The embodiments described in the specification are for more clearly illustrating the technical solutions of the embodiments of the present application, and do not constitute a limitation on the technical solutions provided by the embodiments of the present application. Those skilled in the art can know that, with the evolution of technology and the appearance of new application scenarios, the technical solutions provided by the embodiments of the present application are also applicable to similar technical problems.

[0086] Those skilled in the art can understand that the technical solutions shown in the figures do not constitute a limitation on the embodiments of the present application, and can include more or fewer steps than shown in the figures, or combine certain steps, or different steps.

[0087] The device embodiments described above are only schematic, and the units described as separate components can or can not be physically separate, i.e., can be located in one place, or can be distributed on multiple network units. Part or all of the modules can be selected according to actual needs to achieve the purpose of the embodiments of the present application.

[0088] Those skilled in the art can understand that all or some of the steps in the above disclosed method, the functional modules / units in the system and the device can be implemented as software, firmware, hardware and their appropriate combinations.

[0089] The terms "first", "second", "third", "fourth" and the like (if any) in the specification and above-described drawings of the present application are used to distinguish similar objects, and do not necessarily have to be used to describe a specific order or sequence. It should be understood that the data thus used can be interchanged under appropriate circumstances, so that the embodiments of the present application described herein can be implemented in an order other than that illustrated or described herein. In addition, the terms "include" and "have" and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device that includes a series of steps or units does not have to be limited to those steps or units clearly listed, but can include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

[0090] It should be understood that, in the application, "at least one" refers to one or more, and "multiple" refers to two or more. "And / or" is used to describe the association relationship of the associated objects, which means that there can be three relationships, for example, "A and / or B" can represent three cases of only A, only B, and A and B existing at the same time, wherein A and B can be singular or plural. The character " / " generally represents an "or" relationship between the associated objects before and after it. "At least one of the following" or similar expressions means any combination of these items, including any combination of single or multiple items. For example, at least one of a, b or c can represent a, b, c, "a and b", "a and c", "b and c", or "a and b and c", wherein a, b, and c can be single or multiple.

[0091] In several embodiments provided in the application, it should be understood that the disclosed devices and methods can be implemented in other ways. For example, the device embodiments described above are only illustrative, for example, the division of the above units is only a logical function division, and actual implementation can have another division manner, for example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. In addition, the coupling or direct coupling or communication connection between the displayed or discussed units can be indirect coupling or communication connection through some interfaces, devices or units, which can be electrical, mechanical or other forms.

[0092] The units described above as separate components can or can not be physically separated, and the components shown as units can or can not be physical units, that is, they can be located in one place, or they can be distributed on multiple network units. According to actual needs, some or all of the units can be selected to achieve the purpose of the embodiment scheme.

[0093] In addition, the functional units in each embodiment of the application can be integrated in one processing unit, or each unit can exist physically, or two or more units can be integrated in one unit. The integrated unit can be realized in the form of hardware or in the form of a software functional unit.

[0094] The integrated unit, if implemented in the form of a software function unit and sold or used as an independent product, can be stored in a computer-readable storage medium. Based on such understanding, the technical solutions of the present application essentially or the part that contributes to the prior art or the whole or part of the technical solutions can be embodied in the form of a software product. The computer software product is stored in a storage medium, including a plurality of instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods of the various embodiments of the present application. The aforementioned storage medium includes: a U disk, a mobile hard disk, a read-only memory (ROM), a random access memory (RAM), a magnetic disk or an optical disk, and various program storage media.

[0095] The above-described embodiments are only the preferred embodiments of the present application, and are not intended to limit the scope of the present application. Any changes made in the shape or principle of the present application should be covered within the scope of protection of the present application. The present application is supported by the following fund project: Postdoctoral Innovative Talent Support Plan (No.: BX20240119).

Claims

1. A method of shape control of a super-elastic plane strain plate in a gravitational environment, characterized by, The method comprises the following steps: Theoretical modeling: based on the finite strain plate theory, the super-elastic plate control equation considering the body force is established; Multiplicative decomposition of the total deformation gradient wherein is the growth tensor, is the elastic deformation tensor; The growth tensor In the reference configuration only the non-uniform distribution is introduced in the principal directions, whose expression is: wherein is a local scaling control function, is a curvature control function; is a coordinate in the thickness direction, is a basis vector of the Cartesian coordinate system, denotes a tensor product operation; Analytical inverse solution: By reducing the dimensionality and performing an asymptotic analysis of the governing equations of the super-elastic plate, the analytical expression of the growth function needed to realize a given target surface and is derived, which contains a body force term related to gravity; Calculation and implementation: according to the set target shape, geometric size and body force distribution, the specific and function distribution is calculated, and the growth field distribution is realized in the super-elastic plate material through the preparation process, so that the plate can actively deform and stabilize in the target shape after being released in the gravity environment.

2. The method of claim 1, wherein, The growth function and The analytical expression for where denotes the first-order derivative of the Cartesian coordinates with respect to the parameter is the parametric equation of the target shape (surface), is the first-order derivative of with respect to is the first-order derivative of with respect to is the second-order derivative of with respect to is the normal vector of the target surface, and are the first and second fundamental form coefficients of respectively, is the unit volume force applied in the plate where denotes the material density, denotes the gravitational acceleration, is half of the total thickness of the plate, is the length of the plate, is a dimensionless small parameter used to characterize the effect of gravity on the deformation of the sample, is half of the shear modulus, is a correction term related to gravity.

3. The method of claim 2, wherein, Function By integration we obtain: and by boundary conditions are determined by integration.

4. The method of claim 1, wherein, The expression of the target shape is: wherein , is a Z component of the parametric equation; is a Y component of the parametric equation, is a Z component of the parametric equation; is a length of the plate; And satisfies the left end fixed boundary condition.

5. The method of claim 1, wherein, The constitutive model of the super-elastic plate is an isotropic incompressible neo-Hookean model, and the elastic strain energy function thereof is: wherein denotes the determinant of the growth tensor, is half the shear modulus, denotes the trace operation, is the transpose.

6. The method of claim 1, wherein, The growth field distribution is realized in the super-elastic plate material through a preparation process, which comprises: According to the calculated post-growth configuration, a corresponding mold is designed and manufactured; Unsolidified super-elastic material precursor liquid is poured into the mold; After solidification, demolding is performed, and a sample directly having a non-uniform pre-strain distribution corresponding to the growth field distribution is obtained.

7. The method of claim 1, wherein, After the calculation and implementation steps, a numerical verification step is further included, and the numerical verification step specifically comprises: A three-dimensional finite element model of the super-elastic plate is established; The growth function is introduced in the form of a state variable and its evolution from the unit tensor to the target value is defined; The deformation process of the plate under the action of the body force is simulated, and the final equilibrium shape is compared with the target shape.

8. An electronic device, comprising: The electronic device comprises a memory and a processor, the memory stores a computer program, and the processor implements the method of any one of claims 1 to 7 when executing the computer program.

9. A computer-readable storage medium storing a computer program, the computer-readable storage medium comprising: The computer program is executed by the processor to implement the method of any one of claims 1 to 7.

10. A computer program product comprising a computer program, characterized in that, The computer program is executed by the processor to implement the method of any one of claims 1 to 7.

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