A microstructure design method for poisson's ratio beyond the thermodynamic limit

Abstract

The present application belongs to the technical field of structure design, and discloses a microstructure design method for exceeding the thermodynamic limit of Poisson's ratio, comprising the following steps: (1) arranging self-contacting gaps for the design domain of the microstructure to be designed; (2) applying relative displacement constraints to the four boundaries of the design domain based on orthotropic anisotropy; dividing the design domain into a plurality of quadrilateral grids, and setting a density variable for each grid; (3) determining an interpolation function of the hyperelastic material model parameters and the density variable based on the density variable, and then constructing a design model; (4) determining the target Poisson's ratio and the constraint threshold of the objective function and the constraint function of the design model respectively; (5) performing gradient optimization on the design model, and determining the shape of the microstructure based on the density variable obtained by optimization. The present application does not depend on the linear homogenization theory, and introduces self-contacting gaps in the design and medium, so that the designed microstructure can exhibit different topological configurations under different working conditions.

Description

Technical Field

[0001] This invention belongs to the technical field of structural design, and more specifically, relates to a microstructure design method that achieves a Poisson's ratio exceeding the thermodynamic limit. Background Technology

[0002] It possesses physical properties rarely found in nature, such as negative refractive index, negative dielectric constant, negative thermal expansion, and negative Poisson's ratio. Leveraging these novel physical properties, microstructure design demonstrates exceptional application potential in cutting-edge fields such as weaponry, aerospace, and biomedicine, for applications like stealth cloaks and cardiac stents.

[0003] Poisson's ratio is a typical mechanical parameter describing the deformability of microstructures. In nature, almost all materials possess a positive Poisson's ratio. Under longitudinal compression (tension), materials with a positive Poisson's ratio undergo lateral expansion (contraction), while artificially designed microstructures with a negative Poisson's ratio exhibit lateral contraction (expansion). Benefiting from this negative Poisson's ratio property, microstructures with a negative Poisson's ratio also exhibit superior performance compared to traditional materials in terms of energy absorption, indentation resistance, surface symmetry, shear resistance, and permeability variability. Therefore, designing microstructures with unique Poisson's ratios has high application potential in practical engineering.

[0004] However, traditional microstructure design methods often rely on linear homogenization theory, and the designed microstructures lack topological reconstruction capabilities under different operating conditions, limiting their Poisson's ratio design range to the thermodynamic limit 0 ≤ v. 12 v 21 <1, where v 12 Let v be the Poisson's ratio in the x-direction of the microstructure. 21 Let be the Poisson's ratio in the y-direction of the microstructure. Accordingly, there is a technical need in this field to develop a microstructure design method with a Poisson's ratio exceeding the thermodynamic limit. Summary of the Invention

[0005] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a microstructure design method with a Poisson's ratio exceeding the thermodynamic limit. This method does not rely on linear homogenization theory and introduces self-contact gaps in the design process, enabling the designed microstructure to exhibit different topological configurations under different operating conditions.

[0006] To achieve the above objectives, according to one aspect of the present invention, a microstructure design method for achieving a Poisson's ratio exceeding the thermodynamic limit is provided, the method comprising the following steps:

[0007] (1) Lay out self-contact gaps in the design domain of the microstructure to be designed; wherein, the design domain is square; the self-contact gaps are geometric gaps with self-contact nonlinearity applied.

[0008] (2) Apply relative displacement constraints to the four boundaries of the design domain based on orthogonal anisotropy; at the same time, divide the design domain into multiple quadrilateral grids and set a density variable x for each grid;

[0009] (3) Based on the density variable x, determine the interpolation function between the hyperelastic material model parameters and the density variable, and then construct the design model; wherein, the objective function of the design model is to minimize the equivalent Poisson's ratio v(x) and the target Poisson's ratio v * The square of the difference is constrained by the condition that the volume fraction vol(x) is no greater than the upper limit vol. * The density variable x is constrained within the interval [0,1].

[0010] (4) Determine the objective Poisson's ratio and constraint threshold of the objective function and constraint function of the design model respectively;

[0011] (5) Perform gradient optimization on the design model and determine the shape of the microstructure based on the density variable obtained from the optimization.

[0012] Furthermore, the self-contact gap distribution exhibits rotational symmetry, which includes two types: rotation around the center of the square by 90 degrees and rotation by 180 degrees.

[0013] Furthermore, the relative displacement constraint is u 13 -u 11 =u1、u 21 =u 23 u 24 -u 22 =u2 and u 12 =u 14 , where u 11 and u 21 The displacements in the x and y directions of the left side of the square are u, respectively. 12 and u 22 These represent the x and y displacements of the lower side of the square, respectively. 13 and u 23 The displacements in the x and y directions of the right side of the square are u, respectively. 14 and u 24 U1 and U2 are the displacements of the upper side of the square in the x and y directions, respectively, and the relative displacements of the left and right sides and the top and bottom sides of the square, respectively.

[0014] Furthermore, a unidirectional displacement is assigned to a corner point of the square design domain to drive the deformation of the square design domain; the mesh size does not exceed 1 / 100 of the side length of the square.

[0015] Furthermore, the interpolation function is C″=C′+x 3(CC′), where C is the original material parameter of the hyperelastic material model, C′ is one millionth of C, and C″ is the interpolated result.

[0016] Furthermore, the mathematical expression of the design model is:

[0017] minimize(v(x)-v * ) 2

[0018] subjectto vol(x)≤vol * .

[0019] 0≤x≤1

[0020] Furthermore, the target Poisson ratio includes non-reciprocal Poisson ratio, ultra-large Poisson ratio, and step Poisson ratio, with the constraint threshold being the upper limit of the volume fraction (vol). * .

[0021] Furthermore, the non-reciprocal Poisson ratio satisfies v 12 v 21 <0, an extremely large Poisson's ratio satisfies v 12 v 21 >1, the step Poisson's ratio satisfies v 12 or v 21 The value under tensile conditions is not equal to the value under compressive conditions, where v 12 Let v be the Poisson's ratio in the x-direction of the microstructure. 21 Let be the Poisson's ratio of the microstructure in the y-direction.

[0022] Furthermore, the adjoint method is used to calculate the gradients of the objective function and the constraint function with respect to the density variables, and the density variables are optimized based on the obtained gradients.

[0023] Furthermore, when the density variable x = 1, the corresponding mesh is solid material, and when the density variable x = 0, the corresponding mesh is pores, thereby determining the shape of the microstructure.

[0024] In summary, compared with the prior art, the microstructure design method with Poisson's ratio exceeding the thermodynamic limit provided by the present invention has the following advantages:

[0025] 1. Existing technology relies on linear homogenization theory, which is derived from thermodynamics. Therefore, the Poisson's ratio that existing technology can design is within the thermodynamic limit, i.e., 0 ≤ v. 12 v 21 <1; The present invention is based on applying relative displacement constraints to the design domain using orthogonal anisotropy to achieve periodic boundary conditions, which does not depend on linear homogenization theory. Therefore, the Poisson's ratio that the present invention can achieve is not limited by the thermodynamic limit.

[0026] 2. This invention introduces self-contact gaps into the design domain, which can take into account material, geometric and contact nonlinearities in microstructure design, enabling the designed microstructure to exhibit different topological configurations under different working conditions.

[0027] 3. Supported by different topological configurations, the designed microstructures can exhibit Poisson's ratios exceeding the thermodynamic limit, including non-reciprocal Poisson's ratios, ultra-large Poisson's ratios, and step Poisson's ratios; the non-reciprocal Poisson's ratio satisfies v 12 v 21 <0, an extremely large Poisson's ratio satisfies v 12 v 21 >1, and in addition, the step Poisson's ratio satisfies v 12 The value under tensile conditions is not equal to the value under compressive conditions. Attached Figure Description

[0028] Figure 1 This is a flowchart illustrating a microstructure design method for exceeding the thermodynamic limit of Poisson's ratio provided in Embodiment 1 of the present invention.

[0029] Figure 2 (a), (b), and (c) are schematic diagrams of non-reciprocal Poisson ratio, ultra-large Poisson ratio, and economical Poisson ratio involved in the microstructure design method for exceeding the thermodynamic limit of Poisson ratio provided in Embodiment 2 of the present invention. Detailed Implementation

[0030] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0031] Example 1

[0032] Please see Figure 1 The microstructure design method for achieving a Poisson's ratio exceeding the thermodynamic limit provided in Embodiment 1 of this invention is based on applying relative displacement constraints to the design domain using orthogonal anisotropy, thereby achieving periodic boundary conditions without relying on linear homogenization theory. Introducing self-contact gaps into the design domain allows consideration of material, geometric, and contact nonlinearities in the microstructure design, ensuring the feasibility of achieving a Poisson's ratio exceeding the thermodynamic limit.

[0033] The design method mainly includes the following steps:

[0034] Step 1: Lay out self-contact gaps in the design domain of the microstructure to be designed; wherein, the design domain is square; the self-contact gaps are geometric gaps with applied self-contact nonlinearity.

[0035] Specifically, the self-contact gap distribution has rotational symmetry, which includes two types: rotation around the center of the square by 90 degrees and rotation by 180 degrees.

[0036] Step 2: Apply relative displacement constraints to the four boundaries of the design domain based on orthogonal anisotropy; at the same time, divide the design domain into multiple quadrilateral grids and set a density variable x for each grid.

[0037] Specifically, relative displacement constraints are applied to the four boundaries of the square design domain based on orthogonal anisotropy to achieve periodic boundary conditions. The relative displacement constraints are specifically u 13 -u 11 =u1、u 21 =u 23 u 24 -u 22 =u2 and u 12 =u 14 , where u 11 and u 21 The displacements in the x and y directions of the left side of the square are u, respectively. 12 and u 22 These represent the x and y displacements of the lower side of the square, respectively. 13 and u 23 The displacements in the x and y directions of the right side of the square are u, respectively. 14 and u 24 The x and y directions represent the displacements of the upper side of the square, respectively; u1 and u2 represent the relative displacements of the left and right sides and the top and bottom sides of the square, respectively; and a unidirectional displacement is assigned to a corner point of the square design domain to drive the deformation of the square design domain.

[0038] The design domain is divided into grids. The grid of the design domain is a quadrilateral grid, and the grid size does not exceed 1 / 100 of the side length of the square. A density variable x is set for each grid, with a value range of [0,1].

[0039] Step 3: Determine the interpolation function between the hyperelastic material model parameters and the density variable x, and then construct the design model; wherein, the objective function of the design model is to minimize the equivalent Poisson's ratio v(x) and the target Poisson's ratio v * The square of the difference is constrained by the condition that the volume fraction vol(x) is no greater than the upper limit vol. * The density variable x is constrained within the interval [0,1].

[0040] The interpolation function is C″=C′+x 3(CC′), where C is the original material parameter of the hyperelastic material model, C′ is one millionth of C, and C″ is the interpolated result.

[0041] The mathematical expression for the design model is:

[0042] minimize(v(x)-v * ) 2

[0043] subjectto vol(x)≤vol *

[0044] 0≤x≤1

[0045] The objective function is to minimize the equivalent Poisson ratio v(x) and the target Poisson ratio v. * The square of the difference is constrained by the function that the volume fraction vol(x) is no greater than the upper limit vol. * The density variable x is constrained within the interval [0,1].

[0046] Step four: Determine the objective function and the objective Poisson's ratio and constraint threshold of the design model, respectively.

[0047] The target Poisson ratio includes non-reciprocal Poisson ratio, super-large Poisson ratio, and step Poisson ratio, with the constraint threshold being the upper limit of the volume fraction (vol). * Non-reciprocal Poisson ratios satisfy v 12 v 21 <0, an extremely large Poisson's ratio satisfies v 12 v 21 >1, the step Poisson's ratio satisfies v 12 or v 21 The value under tensile conditions is not equal to the value under compressive conditions, where v 12 Let v be the Poisson's ratio in the x-direction of the microstructure. 21 Let be the Poisson's ratio of the microstructure in the y-direction.

[0048] Step 5: Perform gradient optimization on the design model and determine the shape of the microstructure based on the density variable obtained from the optimization.

[0049] The adjoint method is used to calculate the gradients of the objective function and constraint functions with respect to the density variable, and the density variable values ​​are optimized based on these gradients. When the density variable x = 1, the corresponding mesh represents solid material; when the density variable x = 0, the corresponding mesh represents pores, thus determining the shape of the microstructure.

[0050] Example 2

[0051] Please see Figure 2The microstructure design method for Poisson's ratio exceeding the thermodynamic limit provided in Embodiment 2 of the present invention is used to design non-reciprocal Poisson's ratio, ultra-large Poisson's ratio, and step Poisson's ratio for square microstructure design domains. Figure 2 (a) in the diagram represents a microstructure with a non-reciprocal Poisson ratio, where the Poisson ratio is v. 12 =-0.5 and v 21 =1.4; Figure 2 (b) in the diagram represents a microstructure with an extremely large Poisson ratio, whose Poisson ratio is v. 12 =v 21 =3; Figure 2 (c) in the diagram represents a microstructure with a step Poisson's ratio, where the Poisson's ratio is v under tensile conditions. 12 = -0.8 and v under compression conditions 12 =0.8.

[0052] The method mainly includes the following steps:

[0053] (1) Establish a microstructure design domain, which is a square. Layout self-contact gaps in the design domain. The distribution of self-contact gaps has rotational symmetry, which includes two types: rotation around the center of the square, 90 degrees and 180 degrees. The rotational symmetry of non-reciprocal Poisson ratio corresponds to 180 degrees, while the rotational symmetry of ultra-large Poisson ratio and step Poisson ratio corresponds to 90 degrees.

[0054] (2) Relative displacement constraints are applied to the four boundaries of the orthogonal anisotropic square design domain to achieve periodic boundary conditions. Specifically, the relative displacement constraints are u... 13 -u 11 =u1、u 21 =u 23 u 24 -u 22 =u2 and u 12 =u 14 , where u 11 and u 21 The displacements in the x and y directions of the left side of the square are u, respectively. 12 and u 22 These represent the x and y displacements of the lower side of the square, respectively. 13 and u 23 The displacements in the x and y directions of the right side of the square are u, respectively. 14 and u 24 The x and y directions represent the displacements of the upper side of the square, respectively; u1 and u2 represent the relative displacements of the left and right sides and the top and bottom sides of the square, respectively; and a unidirectional displacement is assigned to a corner point of the square to drive the square to deform.

[0055] (3) Divide the design domain into a grid. The grid of the design domain is a quadrilateral grid. The grid size is 1 / 100 of the side length of the square. Set a density variable x for each grid, with a value range of [0,1].

[0056] (4) Set the interpolation function C″=C′+x between the hyperelastic material model parameters and the density variable x. 3 (CC′), where C is the original material parameter of the hyperelastic material model, C′ is one millionth of C, and C″ is the interpolated result. The objective function and constraint functions are in the following model:

[0057] minimize(v(x)-v * ) 2

[0058] subjectto vol(x)≤vol *

[0059] 0≤x≤1

[0060] The objective function is to minimize the equivalent Poisson ratio v(x) and the target Poisson ratio v. * The square of the difference is constrained by the function that the volume fraction vol(x) is no greater than the upper limit vol. * The density variable x is constrained within the interval [0,1].

[0061] (5) In the model of step (4), the target Poisson ratio v * Setting: The target Poisson's ratio for non-reciprocal Poisson's ratios is v. 12 =-0.5 and v 21 =1.4, the target Poisson's ratio for ultra-large Poisson's ratios is v 12 =v 21 =3. The target Poisson's ratio for a step Poisson's ratio is v under tensile conditions. 12 = -0.8 and v under compression conditions 12 =0.8; Upper limit of volume fraction vol * =0.5; perform gradient optimization on the model in step (4).

[0062] (6) Output the optimized density variable value. When the density variable x = 1, the corresponding mesh is solid material. When the density variable x = 0, the corresponding mesh is pore. This determines the shape of the microstructure.

[0063] This invention provides a microstructure design method for achieving Poisson's ratios exceeding the thermodynamic limit. This method is applicable to the Poisson's ratio design of microstructures. During the design process, relative displacement constraints are applied to the design domain based on orthogonal anisotropy to achieve periodic boundary conditions, independent of linear homogenization theory. Simultaneously, self-contact gaps are introduced into the design domain, enabling consideration of material, geometric, and contact nonlinearities in the microstructure design. Furthermore, the designed microstructures can exhibit different topological configurations under different operating conditions, thereby achieving Poisson's ratios exceeding the thermodynamic limit, including non-reciprocal Poisson's ratios, ultra-large Poisson's ratios, and step Poisson's ratios. Non-reciprocal Poisson's ratios satisfy v 12 v 21 <0, an extremely large Poisson's ratio satisfies v 12 v 21 >1, and in addition, the step Poisson's ratio satisfies v 12 or v 21 The value under tensile conditions is not equal to the value under compressive conditions.

[0064] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A microstructure design method with a Poisson's ratio exceeding the thermodynamic limit, characterized in that, The method includes the following steps: (1) Layout self-contact gaps in the design domain of the microstructure to be designed; wherein, the design domain is square; the self-contact gaps are geometric gaps with self-contact nonlinearity applied; (2) Apply relative displacement constraints to the four boundaries of the design domain based on orthogonal anisotropy; at the same time, divide the design domain into multiple quadrilateral grids and set a density variable for each grid. d ; (3) Based on density variables d The interpolation function for the parameters and density variable of the hyperelastic material model is determined, and then a design model is constructed; wherein, the objective function of the design model is to minimize the equivalent Poisson's ratio. v ( d ) and the target Poisson ratio The square of the difference, with the constraint being the volume fraction. vol ( d (Not greater than the upper limit) Density variables d The constraint is within the interval [0, 1]. (4) Determine the objective Poisson's ratio and constraint threshold of the objective function and constraint function of the design model respectively; (5) Perform gradient optimization on the design model and determine the shape of the microstructure based on the density variable obtained from the optimization; The self-contact gap distribution exhibits rotational symmetry, which includes two types: rotation around the center of the square by 90 degrees and rotation by 180 degrees. Relative displacement constraint is , , and ,in, and The left side of the square x and y Directional displacement, and These are the bottom sides of the square. x and y Directional displacement, and The right side of the square x and y Directional displacement, and The top side of the square x and y Directional displacement, and These represent the relative displacements of the left and right sides and the top and bottom sides of the square, respectively. Specify a unidirectional displacement for a corner point of a square design domain to drive the deformation of the square design domain; the mesh size does not exceed 1 / 100 of the side length of the square. The target Poisson ratio includes non-reciprocal Poisson ratio, super-large Poisson ratio, and step Poisson ratio, with the constraint threshold being the upper limit of the volume fraction. Non-reciprocal Poisson ratios satisfy The ultra-large Poisson ratio satisfies The step Poisson ratio satisfies or The value under tensile conditions is not equal to the value under compressive conditions, where For microstructure in x Poisson's ratio in direction For microstructure in y Poisson's ratio in direction; When the density variable d=1, the corresponding mesh is solid material; when the density variable d=0, the corresponding mesh is pores, thus determining the shape of the microstructure.

2. The microstructure design method with Poisson's ratio exceeding the thermodynamic limit as described in claim 1, characterized in that: The interpolation function is ,in C These are the original material parameters for the hyperelastic material model. One millionth of C This is the result after interpolation.

3. The microstructure design method with Poisson's ratio exceeding the thermodynamic limit as described in claim 2, characterized in that: The mathematical expression for the design model is: 。 4. The microstructure design method with Poisson's ratio exceeding the thermodynamic limit as described in any one of claims 1-3, characterized in that: The adjoint method is used to calculate the gradients of the objective function and constraint function with respect to the density variables, and the density variables are optimized based on the obtained gradients.

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