WO2021139287A1 - 负泊松比结构的非线性本构关系分析方法、系统及装置 - Google Patents

负泊松比结构的非线性本构关系分析方法、系统及装置 Download PDF

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WO2021139287A1
WO2021139287A1 PCT/CN2020/120393 CN2020120393W WO2021139287A1 WO 2021139287 A1 WO2021139287 A1 WO 2021139287A1 CN 2020120393 W CN2020120393 W CN 2020120393W WO 2021139287 A1 WO2021139287 A1 WO 2021139287A1
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correction factor
load
cell
ratio
constitutive relationship
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PCT/CN2020/120393
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French (fr)
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蓝林华
黄泽彬
汪大洋
孙静
张永山
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广州大学
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation

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  • the invention relates to the field of materials science and technology, in particular to a method, system and device for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure.
  • Negative Poisson's ratio refers to the transverse expansion of the material within the elastic range when it is stretched; while the transverse contraction of the material occurs when it is compressed.
  • Cell The smallest structural unit in a multicellular structure. For example, in a honeycomb structure, a unit surrounded by each small hole is called a cell.
  • Bending moment a kind of internal moment on the section of a stressed member.
  • the honeycomb structure is a good lightweight structure due to its high specific strength, high specific rigidity, good heat and sound insulation performance and strong energy absorption.
  • the microscopic structure of the honeycomb is different, the macroscopic mechanical and physical properties are also different.
  • the equivalent material parameters of the honeycomb structure can be adjusted by changing the geometry and material parameters of the honeycomb structure.
  • the concave honeycomb structure exhibits a very strong negative Poisson's ratio (NPR) effect.
  • Negative Poisson's ratio materials have the characteristics of becoming narrower during stretching and wider during extrusion. With the development of modern industrial technology, negative Poisson's ratio materials are used in aviation, aerospace, and machinery due to their special mechanical properties. And transportation and other engineering fields have huge potential.
  • Negative Poisson's ratio materials have appeared for more than 100 years, but it was not until 1987 that Lakes et al. invented the first artificial negative Poisson's ratio foam that attracted widespread attention.
  • the shortcomings of low stiffness limit the application of negative Poisson materials in structures.
  • researchers have done a lot of work on the stiffness of negative Poisson's ratio materials.
  • Zied et al. improved the in-plane stiffness of the concave honeycomb structure by implanting two different substructures.
  • Lu and Fu et al. used the finite element technology to increase the thickness of the cell vertical plate through the concave honeycomb structure, and also obtained the effect of improving the rigidity of the structure in the plane.
  • the concave honeycomb structure is usually regarded as an anisotropic material with periodic characteristics.
  • the complex geometrical form and anisotropic mechanical properties make the honeycomb structure more effective.
  • Experiment design and analysis are time-consuming, laborious and uneconomical.
  • computing science it has become a reality to use general finite element programs for structural modeling and mechanical simulation of large and complex structures.
  • refined models will greatly increase the difficulty of model calculations and significantly reduce computational efficiency.
  • the complex and diverse structure and boundary conditions also make the application of the model have certain limitations.
  • the purpose of the present invention is to at least to some extent solve one of the technical problems in the prior art. For this reason, the purpose of the present invention is to provide a non-linear constitutive with better calculation accuracy and a negative Poisson's ratio structure. Relationship analysis methods, systems and devices.
  • An object of the embodiments of the present invention is to provide a method for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure, which includes the following steps:
  • the correction factors include: Young's modulus correction factor and Poisson's ratio correction factor.
  • the method for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure may also have the following additional technical features:
  • the step of performing deformation analysis on the cell element specifically includes: obtaining the bending moment and the ultimate bending moment at the end point of the inclined wall plate of the cell; comparing the bending moment with the ultimate bending moment , Determining the deformation stage of the cell; the deformation stage includes an elastic deformation stage and a plastic deformation stage.
  • the Young's modulus correction factor includes a Young's modulus correction factor in the horizontal coordinate direction and a Young's modulus correction factor in the vertical coordinate direction;
  • the Poisson's ratio correction factor includes Poisson's ratio in the horizontal coordinate direction Correction factor and Poisson's ratio correction factor in the vertical coordinate direction.
  • the step of obtaining the correction factor according to the result of the deformation analysis specifically includes: establishing the governing equation of the semi-inclined wall plate of the cell; converting the governing equation into a dimensionless equation, and calculating the The dimension equation is simplified; the dimensionless load and the projection rate of the inclined wall plate are obtained according to the simplified dimensionless equation; the correction factor is obtained according to the dimensionless load and the projection rate of the inclined wall plate.
  • the step of obtaining a correction factor according to the dimensionless load and the projection rate of the inclined wall plate specifically includes the following steps: obtaining the geometric characteristics of the cell and the Young's modulus of the concave honeycomb structure material; Combining the geometric feature, the Young's modulus of the material, the non-dimensional load and the projection rate of the inclined wall plate to obtain a correction factor.
  • the projection rate includes a projection rate in the horizontal coordinate direction and a projection rate in the vertical coordinate direction.
  • the step of establishing the control equation of the semi-inclined wall panel of the cell includes at least one of the following steps: when the load is a tensile load, establishing a first control equation for the elastic deformation of the semi-inclined wall panel; When the load is a tensile load, the second governing equation is established for the plastic deformation of the semi-inclined wall; when the load is a compressive load, the third governing equation is established for the elastic deformation of the semi-inclined wall; when the load is a compressive load, the second governing equation is established for the semi-inclined wall.
  • the plastic deformation of the inclined wall plate establishes the fourth governing equation.
  • the load includes the load borne by the cell element in the horizontal coordinate direction and the load borne by the cell element in the vertical coordinate direction.
  • an embodiment of the present invention proposes a system for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure, including:
  • the target acquisition unit is used to acquire the cells of the concave honeycomb structure
  • a deformation analysis unit for performing deformation analysis on the cell for performing deformation analysis on the cell
  • the core processing unit is configured to obtain a correction factor according to the result of the deformation analysis and establish the nonlinear constitutive relationship of the negative Poisson's ratio structure according to the correction factor;
  • the output unit is used to simulate and output the structure according to the nonlinear constitutive relationship.
  • an embodiment of the present invention provides a device for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure, including:
  • At least one processor At least one processor
  • At least one memory for storing at least one program
  • the at least one processor When the at least one program is executed by the at least one processor, the at least one processor is caused to implement the analysis method of the nonlinear constitutive relationship of the negative Poisson's ratio structure.
  • the technical solution provided by the present invention first conducts deformation analysis on the representative cell of the honeycomb structure, and obtains the correction factor of the nonlinear constitutive relationship, and further establishes the nonlinear constitutive relationship to design It has better calculation accuracy and scope of application compared with the numerical methods and test methods of the prior art; in scenarios with special requirements for materials or structures, the design, production and manufacturing process are more Convenient and further shorten the product design cycle.
  • FIG. 1 is a schematic flowchart of a specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 2 is a schematic diagram of the concave honeycomb structure and the cell structure in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 3 is a diagram of the overall force when the concave honeycomb structure is subjected to a tensile load in the y direction in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 4 is a schematic diagram of the deformation of the inclined wall plate in the concave honeycomb structure subjected to tensile load in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 5 is an analysis diagram of the elastic deformation of the inclined wall plate in the concave honeycomb structure under tensile load in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 6 is an analysis diagram of the plastic deformation analysis of the inclined wall plate in the concave honeycomb structure under tensile load in the specific embodiment of the analysis method of the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 7 is a diagram of the overall force when the concave honeycomb structure is subjected to a compressive load in the y direction in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 8 is a schematic diagram of the deformation of the inclined wall plate in the concave honeycomb structure under compression load in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 9 is an analysis diagram of the elastic deformation of the inclined wall plate in the concave honeycomb structure under compressive load in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • Fig. 10 is an analysis diagram of the plastic deformation analysis of the inclined wall plate in the concave honeycomb structure under compressive load in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • 11 is a comparison diagram of the results of the theoretical model and the finite element model of the Almansch strain and the dimensionless Euler stress in the specific embodiment of the analysis method of the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 12 is a graph showing the relationship between k 1 and ⁇ in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different angles ⁇ ;
  • FIG. 13 is a graph showing the relationship between k 2 and ⁇ in a specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different angles ⁇ ;
  • FIG. 14 is a graph showing the relationship between m 1 and ⁇ in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different angles ⁇ ;
  • FIG. 15 is a graph showing the relationship between m 2 and ⁇ in a specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different angles ⁇ ;
  • 16 is a graph showing the relationship between k 1 and ⁇ in a specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different slenderness ratios t/l;
  • 17 is a graph showing the relationship between k 2 and ⁇ in specific embodiments of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different slenderness ratios t/l;
  • 19 is a graph showing the relationship between m 2 and ⁇ in specific embodiments of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention at different slenderness ratios t/l;
  • FIG. 20 is a graph showing the relationship between k 1 and ⁇ under different yield strength ⁇ s of honeycomb materials in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 21 is a graph showing the relationship between k 2 and ⁇ under different yield strength ⁇ s of honeycomb materials in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 22 is a graph showing the relationship between m 1 and ⁇ under different yield strength ⁇ s of honeycomb materials in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention
  • FIG. 23 is a graph of the relationship between m 2 and ⁇ under the yield strength ⁇ s of different honeycomb materials in the specific embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure of the present invention.
  • the method for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure described in the embodiment of the present invention mainly includes the following steps:
  • S01 Obtain the cells of the concave honeycomb structure; referring to FIG. 2, the present embodiment regards the inner honeycomb structure as an anisotropic material with periodic characteristics, and selects the representative cell of the honeycomb structure for the next processing, and then reflects Constitutive relationship of the same type of concave honeycomb structure (same shape).
  • S02 Perform deformation analysis on the cell; specifically, it is mainly to judge whether the inclined wall plate of the cell is in an elastic deformation stage or a plastic deformation stage;
  • ⁇ s is the yield strength of the honeycomb material
  • t is the wall thickness of the honeycomb structure
  • b is the depth of the honeycomb structure.
  • the bending moment M B is less than the limit bending moment M U , there is no plastic turning angle ⁇ 0 , and the semi-inclined wall panel is in the elastic deformation stage; if the bending moment M B is greater than the limit bending moment M U , then the plastic turning angle ⁇ 0 appears, semi-inclined
  • the wall panel is in the plastic deformation stage, and when constructing the control equation of the semi-inclined wall panel and calculating the dimensionless load and displacement, the bending moment M B is equal to the limit bending moment M U.
  • the correction factor is obtained according to the results of the deformation analysis; specifically, for the geometric characteristics of the inner honeycomb structure (angle, relative density), the Young's modulus of the material and the actual deformation (through the deformation analysis of the representative cell under load in different directions) Obtain the corresponding equivalent material parameters), and obtain the correction factor that can reflect the nonlinear constitutive relationship of the same type of concave honeycomb structure.
  • the correction factor comprises: a Young's modulus when subjected to loading in the cells of the X-axis direction correction factor k 1 k 2 poise and Poisson's ratio and the correction factor m 1, the Y-axis direction, the Young's modulus to withstand the load correction factor Loose ratio correction factor m 2 ;
  • the non-linear equivalent material parameters of the cell include: the equivalent Young's modulus E 1 and the equivalent Poisson's ratio ⁇ 1 of the representative cell in the X-axis direction and the cell in the Y-axis direction
  • the nonlinear equivalent material parameters of the cell, the nonlinear constitutive relationship correction factor, the relative density R of the structure and the Young’s modulus E of the material The relational equations of s are:
  • ⁇ * is the density of the concave honeycomb material
  • ⁇ 0 is the density of the cell wall
  • h is the length of the honeycomb cell vertical plate
  • t is the wall thickness of the honeycomb structure
  • ⁇ is the hypotenuse of the honeycomb cell
  • Y The included angle of the axis.
  • the relative density R is:
  • ⁇ is the dimensionless load
  • Respectively are the projection rates of the cell swash plate in the X-axis direction and the Y-axis direction when the load is carried in the X-axis direction
  • Respectively are the projection rates of the cell swash plate in the X-axis direction and the Y-axis direction when the load is carried in the Y-axis direction
  • It is the sum of the deformation of the cell vertical plate DB and vertical plate AC in Figure 1 (tension is positive and compression is negative).
  • ⁇ is the dimensionless load
  • Respectively are the projection rates of the cell swash plate in the X-axis direction and the Y-axis direction when the load is carried in the X-axis direction
  • Respectively the projection rates of the cell swash plate in the X-axis direction and the Y-axis direction when the load is loaded in the Y-axis direction.
  • Both can be connected with the corners of the two ends and midpoints of the structural cell swash plate through large deformation analysis. The specific process is as follows:
  • P is the equivalent load applied to the representative cell
  • E s I is the bending stiffness of the honeycomb structure sloping wall section
  • s is the arc coordinate along the beam axis (0 ⁇ s ⁇ 1/ 2)
  • ⁇ is the rotation angle at the arc coordinate s.
  • the governing equation can be further transformed into a dimensionless equation:
  • ⁇ is the turning angle of the midpoint O of the inclined wall panel.
  • the dimensionless load ⁇ is expressed by the first type of elliptic integral F( ⁇ ) ( ⁇ is the angle of rotation at point O);
  • the projection rate of the inclined wall of the honeycomb structure along the X-axis direction is expressed by the first type of elliptic integral F( ⁇ );
  • the projection rate of the inclined wall of the honeycomb structure along the Y axis is expressed by the first type of elliptic integral F( ⁇ ) and the second type of elliptic integral E( ⁇ ).
  • the four combinations of the elastic deformation stage or the plastic deformation stage of the semi-inclined wall panel are respectively analyzed by large deformation and the two ends of the structural cell sloping plate Establish a connection with the corner of the midpoint.
  • the following selects when the honeycomb structure bears the load in the Y-axis direction for detailed description.
  • the solution process is similar to that when the load is applied in the Y-axis direction. Convert the projection rate in the X-axis direction and the Y-axis direction with ⁇ in is transformed into And exchange the projection rate in the X-axis direction and the Y-axis direction, I won't go into details here:
  • E s I is the bending stiffness of the honeycomb structure sloping wall section; I is the section moment of inertia of the honeycomb structure wall; ⁇ is the rotation angle at the arc coordinate S.
  • F( ⁇ ) is the elliptic integral of the first kind
  • E( ⁇ ) is the elliptic integral of the second kind
  • ⁇ is based on the coordinate transformation formula The new variable generated
  • ⁇ is the turning angle of the midpoint of the inclined wall
  • E s I is the bending stiffness of the honeycomb structure sloping wall section; I is the section moment of inertia of the honeycomb structure wall; ⁇ is the rotation angle at the arc coordinate S, and ⁇ 0 is the plastic deformation stage.
  • the plastic turning angle, ⁇ 0 is determined by the bending moment M B at the end of the sloping wall. As described in step S022, the bending moment M B is equal to the limit bending moment M U and the following formula is obtained:
  • ⁇ is the turning angle of the midpoint of the inclined wall; the boundary conditions of the governing equation are:
  • M O represents the bending moment at the midpoint of the slanted wall plate of the cell.
  • Dimensionless load ⁇ the projection rate of the inclined wall plate along the X axis
  • the projection rate on the Y axis are as follows:
  • F( ⁇ ) is the elliptic integral of the first kind
  • E( ⁇ ) is the elliptic integral of the second kind
  • ⁇ ' is based on the coordinate transformation formula Generating a new variable
  • E s I is the bending stiffness of the honeycomb structure sloping wall section; I is the section moment of inertia of the honeycomb structure wall; ⁇ is the rotation angle at the arc coordinate S.
  • F( ⁇ ) is the elliptic integral of the first kind
  • E( ⁇ ) is the elliptic integral of the second kind
  • ⁇ is based on the coordinate transformation formula The new variable generated
  • ⁇ is the turning angle of the midpoint of the inclined wall
  • E s I is the bending stiffness of the honeycomb structure sloping wall section; I is the section moment of inertia of the honeycomb structure wall; ⁇ is the rotation angle at the arc coordinate S, and ⁇ 0 is the plastic deformation stage.
  • the plastic turning angle, ⁇ 0 is determined by the bending moment M B at the end of the sloping wall. As described in step S022, the bending moment M B is equal to the limit bending moment M U and the following formula is obtained:
  • ⁇ is the rotation angle of the midpoint of the inclined wall, the dimensionless load ⁇ , the projection rate of the inclined wall along the X axis And the projection rate on the Y axis They are as follows:
  • F( ⁇ ) is the elliptic integral of the first kind
  • E( ⁇ ) is the elliptic integral of the second kind
  • ⁇ ' is based on the coordinate transformation formula Generating a new variable
  • ⁇ 'x and ⁇ ' y shows the case where the load subjected to the X-axis direction
  • ⁇ "x and ⁇ " y shows the case where the load subjected to the Y-axis direction.
  • FIG. 11 it is a schematic diagram of the results of Almansch strain and dimensionless Euler stress in the theoretical model and the finite element model in the specific embodiment of the present invention.
  • the Almansch strain curve in the theoretical model is roughly the same as the Almansch strain curve in the finite element model, indicating that the concave honeycomb structure in this embodiment is in good agreement with the relatively fine finite element concave honeycomb structure.
  • FIG. 12 are schematic diagrams of the influence of the concave honeycomb structure parameter ⁇ on the honeycomb structure parameters in the embodiment of the present invention.
  • the positive strain represents the tensile load of the honeycomb structure
  • the negative strain represents the honeycomb structure. Compressive load to bear.
  • the Poisson’s ratio correction factor m 1 in the X-axis direction becomes smaller as the cell deformation becomes larger, and the Poisson’s ratio correction factor m 1 in the Y-axis direction becomes smaller.
  • the Poisson's ratio correction factor m 2 becomes larger as the cell deformation becomes larger;
  • FIG. 16 a schematic diagram of the influence of the slenderness ratio t/l of the concave honeycomb structure in the embodiment of the present invention on the structural parameters.
  • the positive strain represents the tensile load borne by the honeycomb structure
  • the negative strain represents the honeycomb structure Compressive load to bear.
  • the Young's modulus correction factor k 2 becomes smaller as the cell deformation becomes larger;
  • the slenderness ratio t/l the larger, the X-axis direction is smaller than a Poisson correction factor m, the Poisson ratio of the Y-axis direction correction factor m 2 no significant change in the Poisson ratio of the X-axis direction correction factor cell element becomes deformed as 1 m Larger and smaller, the Poisson's ratio correction factor m 2 in the Y-axis direction becomes larger as the cell deformation becomes larger;
  • the larger the slenderness ratio t/l the smaller the Young's modulus correction factors k 1 and k 2 in the X-axis direction and the Y-axis direction, and the Young's modulus in the X-axis direction
  • the correction factor k 1 becomes smaller as the cell deformation becomes larger
  • the Young's modulus correction factor k 2 in the Y axis direction becomes larger as the cell deformation becomes larger
  • the larger the slenderness ratio t/l, the X axis direction is larger than the Poisson correction factor m 1, m correction factor Poisson's ratio without significant changes in the Y-axis direction 2
  • the cell element with a Poisson deformation becomes larger correction factor m larger than the X-axis direction
  • the Poisson's ratio correction factor m 2 in the Y-axis direction becomes smaller as the cell deformation becomes larger;
  • FIG. 20 Figure 21, Figure 22 and Figure 23, it is a schematic diagram of the influence of the yield strength ⁇ s of the honeycomb material on the structural parameters in the embodiment of the present invention.
  • the positive strain represents the tensile load borne by the honeycomb structure
  • the negative strain represents the honeycomb structure bears The compressive load.
  • the yield strength ⁇ s of the honeycomb material has no effect on the structural parameters
  • the correction factor k 1 becomes smaller as the cell deformation becomes larger, and the Young's modulus correction factor k 2 in the Y-axis direction becomes larger as the cell deformation becomes larger; the greater the yield strength ⁇ s of the honeycomb material, the X
  • the Poisson's ratio correction factor m 2 becomes smaller as the cell deformation becomes larger.
  • the embodiment of the method for analyzing the nonlinear constitutive relationship of the negative Poisson's ratio structure has the following advantages:
  • the embodiment of the present invention obtains a correction factor that can reflect the non-linear constitutive relationship of the same type of concave honeycomb structure through deformation analysis combined with structural geometric characteristics (angle, relative density) and material Young's modulus, and establishes a correction factor that reflects the same type of internal honeycomb structure.
  • the non-linear constitutive relationship of the concave honeycomb structure, and the elasto-plastic deformation of the material is fully considered in the analysis process. Compared with the numerical method and the test method, it has better calculation accuracy and scope of application;
  • the embodiments of the present invention can design a suitable honeycomb structure conveniently and quickly, making the design, production and manufacturing process more convenient, and further shortening the product design cycle.
  • the embodiment of the present invention proposes a system for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure.
  • the system specifically includes:
  • the target acquisition unit is used to acquire the cells of the concave honeycomb structure
  • a deformation analysis unit for performing deformation analysis on the cell for performing deformation analysis on the cell
  • the core processing unit is configured to obtain a correction factor according to the result of the deformation analysis and establish the nonlinear constitutive relationship of the negative Poisson's ratio structure according to the correction factor;
  • the output unit is used to simulate and output the structure according to the nonlinear constitutive relationship.
  • an embodiment of the present invention provides a device for analyzing the nonlinear constitutive relationship of a negative Poisson's ratio structure, including:
  • At least one processor At least one processor
  • At least one memory for storing at least one program
  • the at least one processor When the at least one program is executed by the at least one processor, the at least one processor is caused to implement the analysis method of the nonlinear constitutive relationship of the negative Poisson's ratio structure.
  • the functions/operations mentioned in the block diagram may occur out of the order mentioned in the operation diagram.
  • two blocks shown in succession may actually be executed substantially simultaneously or the blocks may sometimes be executed in the reverse order.
  • the embodiments presented and described in the flowchart of the present invention are provided by way of example, with the purpose of providing a more comprehensive understanding of the technology. The disclosed method is not limited to the operations and logic flow presented herein. Alternative embodiments are contemplated in which the order of various operations is changed and in which sub-operations described as part of a larger operation are performed independently.
  • the function is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer readable storage medium.
  • the technical solution of the present invention essentially or the part that contributes to the prior art or the part of the technical solution can be embodied in the form of a software product, and the computer software product is stored in a storage medium, including Several instructions are used to make a computer device (which may be a personal computer, a server, or a network device, etc.) execute all or part of the steps of the methods described in the various embodiments of the present invention.
  • the aforementioned storage media include: U disk, mobile hard disk, read-only memory (ROM, Read-Only Memory), random access memory (RAM, Random Access Memory), magnetic disks or optical disks and other media that can store program codes. .
  • each part of the present invention can be implemented by hardware, software, firmware or a combination thereof.
  • multiple steps or methods can be implemented by software or firmware stored in a memory and executed by a suitable instruction execution system.
  • a suitable instruction execution system For example, if it is implemented by hardware, as in another embodiment, it can be implemented by any one or a combination of the following technologies known in the art: Discrete logic circuits, application specific integrated circuits with suitable combinational logic gates, programmable gate array (PGA), field programmable gate array (FPGA), etc.

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Abstract

一种负泊松比结构的非线性本构关系分析方法、系统及装置。该方法通过获取内凹蜂窝结构的胞元（S01）；对所述胞元进行变形分析（S02）；根据变形分析的结果得到修正因子（S03）；根据所述修正因子建立所述内凹蜂窝结构的非线性本构关系（S04）；根据所述非线性本构关系设计和生产对应结构（S05）。通过使用该方法，能够在对材料或结构有殊结需求的场景中，使设计、生产及制作过程更加便捷，进一步缩短产品的设计周期，比现有技术的数值方法和试验方法，具有更好的计算精度和适用范围。

Description

负泊松比结构的非线性本构关系分析方法、系统及装置 技术领域
本发明涉及材料科学技术领域，尤其是一种负泊松比结构非线性本构关系的分析方法、系统及装置。
背景技术
负泊松比：是指受拉伸时,材料在弹性范围内横向发生膨胀；而受压缩时,材料的横向反而发生收缩。
胞元：在多胞结构的最小结构单元，例如在蜂窝结构中，每个小孔洞围成的一个单元称为胞元。
弯矩：受力构件截面上的内力矩的一种。
蜂窝结构由于具有比强度大、比刚度高、隔热隔音性能好以及能量吸收强等特性，是一种良好的轻量化结构。蜂窝的微观结构不同，宏观力学和物理性能也不同。可以通过改变蜂窝结构的几何及材料参数来调整蜂窝结构的等效材料参数，特别地，内凹蜂窝结构体现出了极强的负泊松比(NPR)效应。负泊松比材料具有在拉伸时变得更窄和挤压时变得更宽的特性，伴随着现代工业技术的发展，负泊松比材料由于其特殊的力学性能在航空、航天、机械及交通等各个工程领域有着巨大的潜能。负泊松比材料已经出现100多年，但直到1987年，Lakes等人发明了第一种人造负泊松比泡沫才引起人们的广泛关注。然而因为刚度低的缺点限制了负泊松材料在结构中的应用。近年来研究人员对负泊松比材料的刚度方面做了大量的工作，Zied等人通过植入两种不同子结构来提高内凹蜂窝结构的面内刚度。Lu和Fu等人利用有限元技术通过增加胞元竖板厚度的内凹蜂窝结构，同样得出了提高结构面内刚度的效果。
上述研究大多是基于实验或数值模拟，内凹蜂窝结构通常被认为一种具有周期特性的各向异性材料，在工程设计和分析中，复杂的几何形式和各项异性的力学性能使得蜂窝结构的试验设计与分析费时费力且不经济。随着计算科学的发展，对大型复杂结构采用通用有限元程序进行结构建模及力学仿真已成为现实，然而，精细化的模型将大大增加模型计算难度，使得计算效率显著降低。同时复杂多样的结构及边界条件，也使得模型的应用具有一定的局限性。
发明内容
本发明的目的在于至少一定程度上解决现有技术中存在的技术问题之一，为此，本发明目的在于提供一种具有更好的计算精度和适用范围负泊松比结构的非线性本构关系分析方法、系统及装置。
本发明实施例的一个目的在于提供一种负泊松比结构的非线性本构关系的分析方法，包括以下步骤：
获取内凹蜂窝结构的胞元；
对所述胞元进行变形分析；
根据变形分析的结果得到修正因子；
根据所述修正因子建立所述内凹蜂窝结构的非线性本构关系；
根据所述非线性本构关系设计和生产对应结构；
所述修正因子包括：杨氏模量修正因子和泊松比修正因子。
另外，根据本发明上述实施例的一种负泊松比结构的非线性本构关系的分析方法，还可以具有以下附加的技术特征：
进一步，所述对所述胞元进行变形分析这一步骤，其具体包括：获取所述胞元的斜壁板端点处的弯矩以及极限弯矩；对比所述弯矩与所述极限弯矩，确定所述胞元所处的变形阶段；所述变形阶段包括弹性变形阶段和塑性变形阶段。
进一步，所述杨氏模量修正因子包括水平坐标方向上杨氏模量修正因子以及竖直坐标方向上的杨氏模量修正因子；所述泊松比修正因子包括水平坐标方向上泊松比修正因子以及竖直坐标方向上的泊松比修正因子。
进一步，所述根据变形分析的结果得到修正因子这一步骤，其具体包括：建立所述胞元的半斜壁板的控制方程；将所述控制方程转化为无量纲方程，并对所述无量纲方程进行简化；根据简化后的无量纲方程得到无量纲荷载以及斜壁板的投影率；根据所述无量纲载荷以及斜壁板的投影率得到修正因子。
进一步，所述根据所述无量纲载荷以及斜壁板的投影率得到修正因子这一步骤，具体包括下步骤：获取所述胞元的几何特征以及所述内凹蜂窝结构材料杨氏模量；结合所述几何特征、所述材料杨氏模量、所述无量纲载荷以及斜壁板的投影率得到修正因子。
进一步，所述投影率包括水平坐标方向上的投影率和竖直坐标方向上的投影率。
进一步，所述建立所述胞元的半斜壁板的控制方程这一步骤，其包括以下步骤至少一项：当荷载为拉荷载，针对半斜壁板的弹性变形，建立第一控制方程；当荷载为拉荷载，针对半斜壁板的塑性变形，建立第二控制方程；当荷载为压荷载，针对半斜壁板的弹性变形，建立第三控制方程；当荷载为压荷载，针对半斜壁板的塑性变形，建立第四控制方程。
进一步，所述荷载包括所述胞元在水平坐标方向上承受的荷载以及在所述胞元竖直坐标方向上承受的荷载。
第二方面，本发明实施例提出了一种负泊松比结构的非线性本构关系的分析系统，包括：
目标获取单元，用于获取内凹蜂窝结构的胞元；
变形分析单元，用于对所述胞元进行变形分析；
核心处理单元，用于根据变形分析的结果得到修正因子以及根据所述修正因子建立所述负泊松比结构的非线性本构关系；
输出单元，用于根据所述非线性本构关系对结构进行仿真以及输出。
第三方面，本发明实施例提供了一种负泊松比结构的非线性本构关系的分析装置，包括：
至少一个处理器；
至少一个存储器，用于存储至少一个程序；
当所述至少一个程序被所述至少一个处理器执行时，使得所述至少一个处理器实现所述的负泊松比结构的非线性本构关系的分析方法。
本发明的优点和有益效果：本发明所提供的技术方案先针对蜂窝结构的代表胞元进行了变形分析，并由此得到非线性本构关系的修正因子，进一步建立非线性本构关系从而设计出和合适的蜂窝结构；相比现有技术的数值方法和试验方法，具有更好的计算精度和适用范围；在对材料或结构有殊结需求的场景中，使设计、生产及制作过程更加便捷，进一步缩短产品的设计周期。
附图说明
为了更清楚地说明本发明实施例或者现有技术中的技术方案，下面对本发明实施例或者现有技术中的相关技术方案附图作以下介绍，应当理解的是，下面介绍中的附图仅仅为了方便清晰表述本发明的技术方案中的部分实施例，对于本领域的技术人员来说，在无需付出创造性劳动的前提下，还可以根据这些附图获取到其他附图。
图1为本发明负泊松比结构的非线性本构关系的分析方法具体实施例的流程示意图；
图2为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构以及胞元结构示意图；
图3为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构y方向上承受拉荷载时的整体受力图；
图4为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中斜壁板承受拉荷载变形示意图；
图5为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中的斜壁板承受拉荷载弹性变形分析图；
图6为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中的斜壁板承受拉荷载塑性变形分析图；
图7为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构y方向上承受压荷载时的整体受力图；
图8为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中斜壁板的承受压荷载变形示意图；
图9为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中的斜壁板承受压荷载弹性变形分析图；
图10为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中内凹蜂窝结构中的斜壁板承受压荷载塑性变形分析图；
图11为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中阿尔曼希应变与无量纲欧拉应力在理论模型和有限元模型中的结果比较图；
图12为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同大小的夹角φ下k 1与β的关系曲线图；
图13为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同大小的夹角φ下k 2与β的关系曲线图；
图14为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同大小的夹角φ下m 1与β的关系曲线图；
图15为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同大小的夹角φ下m 2与β的关系曲线图；
图16为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同细长比t/l下，k 1与β的关系曲线图；
图17为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同细长比t/l下，k 2与β的关系曲线图；
图18为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同细长比t/l下，m 1与β的关系曲线图；
图19为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同细长比t/l下，m 2与β的关系曲线图；
图20为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同蜂窝材料屈服强度σ s下，k 1与β的关系曲线图；
图21为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同蜂窝材料屈服强度σ s下，k 2与β的关系曲线图；
图22为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同蜂窝材料屈服强度σ s下，m 1与β的关系曲线图；
图23为本发明负泊松比结构的非线性本构关系的分析方法具体实施例中不同蜂窝材料屈服强度σ s下，m 2与β的关系曲线图。
具体实施方式
下面详细描述本发明的实施例，所述实施例的示例在附图中示出，其中自始至终相同或类似的标号表示相同或类似的元件或具有相同或类似功能的元件。下面通过参考附图描述的实施例是示例性的，仅用于解释本发明，而不能理解为对本发明的限制。对于以下实施例中的步骤编号，其仅为了便于阐述说明而设置，对步骤之间的顺序不做任何限定，实施例中的各步骤的执行顺序均可根据本领域技术人员的理解来进行适应性调整。
下面参照附图详细描述根据本发明实施例提出的负泊松比结构的非线性本构关系的分析方法、系统及装置， 首先将参照附图描述根据本发明实施例提出的负泊松比结构的非线性本构关系的分析方法。
参照图1，本发明实施例中所述的一种负泊松比结构的非线性本构关系的分析方法主要包括以下步骤：
S01：获取内凹蜂窝结构的胞元；参照图2，本实施例将内蜂窝结构视为一种具有周期特性的各向异性材料，并选取蜂窝结构的代表胞元进行下一步处理，进而反映同一类内凹蜂窝结构(相同形状)的本构关系。
S02：对所述胞元进行变形分析；具体的，主要是判断判断出胞元的斜壁板处于弹性变形阶段还是塑性变形阶段；
S021：计算出代表胞元的斜壁板端点处的弯矩M B，以及所能承受的极限弯矩M U：
Figure PCTCN2020120393-appb-000001
式中，σ s是蜂窝材料屈服强度，t是蜂窝结构的壁板厚度，b是蜂窝结构的深度。
S022：判断弯矩M B与极限弯矩M U的大小；
若弯矩M B小于极限弯矩M U，则未出现塑性转角β 0，半斜壁板处于弹性变形阶段；若弯矩M B大于极限弯矩M U，则出现塑性转角β 0，半斜壁板处于塑性变形阶段，并在构建半斜壁板的控制方程和计算无量纲荷载和位移量时，将弯矩M B等于极限弯矩M U。
S03：根据变形分析的结果得到修正因子；具体的，针对内蜂窝结构几何特征(角度、相对密度)、材料杨氏模量及实际变形情况(通过代表胞元在不同方向上承受荷载的变形分析得到相应的等效材料参数)，得到能够反映同一类内凹蜂窝结构非线性本构关系的修正因子。
其中修正因子包括：胞元在X轴方向上承受荷载时的杨氏模量修正因子k 1和泊松比修正因子m 1、在Y轴方向上承受荷载时的杨氏模量修正因子k 2和泊松比修正因子m 2；其中胞元非线性等效材料参数包括：代表胞元在X轴方向上的等效杨氏模量E 1和等效泊松比ν 1以及胞元在Y轴方向上的等效杨氏模量E 2和等效泊松比ν 2；所述胞元非线性等效材料参数、非线性本构关系修正因子、结构的相对密度R与材料杨氏模量E s的关系方程组为：
Figure PCTCN2020120393-appb-000002
式中，结构的相对密度由公式(3)计算得到：
Figure PCTCN2020120393-appb-000003
其中ρ *是内凹蜂窝材料的密度，ρ 0是胞元壁板的密度，h是是蜂窝胞元竖板的长度，t是蜂窝结构的壁板厚度，φ是蜂窝胞元斜边与Y轴方向的夹角。特别地，当
Figure PCTCN2020120393-appb-000004
和h＝l时，相对密度R为：
Figure PCTCN2020120393-appb-000005
根据公式(3)及公式(2)可以得到：
Figure PCTCN2020120393-appb-000006
式中，ζ为无量纲荷载，
Figure PCTCN2020120393-appb-000007
和
Figure PCTCN2020120393-appb-000008
分别为胞元斜板在X轴方向上承受荷载时在X轴方向上和Y轴方向上的投影率，
Figure PCTCN2020120393-appb-000009
和
Figure PCTCN2020120393-appb-000010
分别为胞元斜板在Y轴方向上承受荷载时在X轴方向上和Y轴方向上的投影率，
Figure PCTCN2020120393-appb-000011
为图1中胞元竖板DB和竖板AC的变形之和(受拉为正，受压为负)。
其中，ζ为无量纲荷载，
Figure PCTCN2020120393-appb-000012
和
Figure PCTCN2020120393-appb-000013
分别为胞元斜板在X轴方向上承受荷载时在X轴方向上和Y轴方向上的投影率，
Figure PCTCN2020120393-appb-000014
和
Figure PCTCN2020120393-appb-000015
分别为胞元斜板在Y轴方向上承受荷载时在X轴方向上和Y轴方向上的投影率，均可以通过大变形分析与结构胞元斜板两端及中点的转角建立联系，具体过程如下：
S031：利用对称性，根据梁的弯曲理论建立对应的半斜壁板的控制方程；
具体的，根据梁的弯曲理论，其控制方程为：
Figure PCTCN2020120393-appb-000016
公式(6)中，P是施加在代表胞元上的等效荷载；E sI是蜂窝结构斜壁板截面的抗弯刚度；s为沿梁轴线方向的弧坐标(0≤s≤1/2)，γ是弧坐标s处的转角。
S032：将控制方程转化为无量纲方程，再利用边界条件对无量纲方程进行简化；
具体的，控制方程可进一步转化为无量纲方程：
Figure PCTCN2020120393-appb-000017
公式(7)中，
Figure PCTCN2020120393-appb-000018
为两端铰支工况下压杆稳定的临界荷载，S＝s/l(0≤S≤0.5)为无量纲弧长。无量纲方程两边乘以
Figure PCTCN2020120393-appb-000019
并对无量纲弧坐标S积分，同时利用边界条件M 0＝0后可以化为：
Figure PCTCN2020120393-appb-000020
公式(8)中，β是斜壁板中点O点的转角。
S033：针对于简化后的无量纲方程，通过第一类椭圆积分F(β)对无量纲荷载ζ进行表示(β为O点处转角)；
通过第一类椭圆积分F(β)对蜂窝结构的斜壁板沿X轴方向上的投影率进行表示；
通过第一类椭圆积分F(β)和第二类椭圆积分E(β)对蜂窝结构的斜壁板沿Y轴方向上的投影率进行表示。
更为细化地，根据内蜂窝结构所受载荷为拉负载或压负载，半斜壁板的弹性变形阶段或塑性变形阶段的四种组合情况分别通过大变形分析与结构胞元斜板两端及中点的转角建立联系，下文选取当蜂窝结构在Y轴方向上承受荷载的情况进行详细说明，当X轴方向上承受荷载时，其求解过程与Y轴方向上承受荷载时类似，只需要将X轴方向和Y轴方向上的投影率
Figure PCTCN2020120393-appb-000021
和
Figure PCTCN2020120393-appb-000022
中的φ变换为
Figure PCTCN2020120393-appb-000023
并将X轴方向和Y轴方向上的投影率互换，在此不多过多赘述：
A、参照图3、图4及图5，当荷载为拉荷载时，针对于半斜壁板的弹性变形阶段，所建立的半斜壁板的控制方程为：
Figure PCTCN2020120393-appb-000024
公式(9)中，E sI是蜂窝结构斜壁板截面的抗弯刚度；I是蜂窝结构壁板的截面惯性矩；γ是弧坐标S处的转角。
无量纲荷载ζ、斜壁板沿X轴方向上的投影率
Figure PCTCN2020120393-appb-000025
和Y轴方向上的投影率
Figure PCTCN2020120393-appb-000026
分别如下：
Figure PCTCN2020120393-appb-000027
其中：
Figure PCTCN2020120393-appb-000028
在公式(11)中F(β)是第一类椭圆积分；E(β)是第二类椭圆积分；η是根据坐标变换公式
Figure PCTCN2020120393-appb-000029
产生的新变量；β是斜壁板中点的转角；η 0是当弧坐标S＝0，即弧坐标S处的转角γ＝0时的η值。
B、参照图6，当荷载为拉荷载时，针对于半斜壁板的塑性变形阶段，所建立的半斜壁板的控制方程为：
Figure PCTCN2020120393-appb-000030
公式(12)中，E sI是蜂窝结构斜壁板截面的抗弯刚度；I是蜂窝结构壁板的截面惯性矩；γ是弧坐标S处的转角，β 0是塑性变形阶段所存在的塑性转角，β 0由斜壁板端点处的弯矩M B确定，如步骤S022中所述弯矩M B等于极限弯矩M U进而得到下式：
Figure PCTCN2020120393-appb-000031
公式(13)中，β是斜壁板中点的转角；其控制方程的边界条件是：
Figure PCTCN2020120393-appb-000032
公式(14)中，M O是代表胞元斜壁板中点的弯矩。无量纲荷载ζ、斜壁板沿X轴方向上的投影率
Figure PCTCN2020120393-appb-000033
和Y轴方向上的投影率
Figure PCTCN2020120393-appb-000034
分别如下：
Figure PCTCN2020120393-appb-000035
其中：
Figure PCTCN2020120393-appb-000036
在公式(16)中F(β)是第一类椭圆积分；E(β)是第二类椭圆积分；η'是是根据坐标变换公式
Figure PCTCN2020120393-appb-000037
Figure PCTCN2020120393-appb-000038
产生的新变量；η' 0是当弧坐标S＝0，即弧坐标S处的转角γ＝β 0时的η'值。
C、参照图7、图8以及图9，当荷载为压荷载时，针对于半斜壁板的弹性变形阶段，所建立的半斜壁板的控制方程为：
Figure PCTCN2020120393-appb-000039
公式(17)中，E sI是蜂窝结构斜壁板截面的抗弯刚度；I是蜂窝结构壁板的截面惯性矩；γ是弧坐标S处的转角。
无量纲荷载ζ、斜壁板沿X轴方向上的投影率
Figure PCTCN2020120393-appb-000040
和Y轴方向上的投影率
Figure PCTCN2020120393-appb-000041
分别如下：
Figure PCTCN2020120393-appb-000042
其中：
Figure PCTCN2020120393-appb-000043
在公式(19)中F(β)是第一类椭圆积分；E(β)是第二类椭圆积分；η是根据坐标变换公式
Figure PCTCN2020120393-appb-000044
产生的新变量；β是斜壁板中点的转角；η 0是当弧坐标S＝0，即弧坐标S处的转角γ＝0时的η值。
D、当荷载为压荷载时，针对于半斜壁板的塑性变形阶段，所建立的半斜壁板的控制方程为：
Figure PCTCN2020120393-appb-000045
公式(20)中，E sI是蜂窝结构斜壁板截面的抗弯刚度；I是蜂窝结构壁板的截面惯性矩；γ是弧坐标S处的转角，β 0是塑性变形阶段所存在的塑性转角，β 0由斜壁板端点处的弯矩M B确定，如步骤S022中所述弯矩M B等于极限弯矩M U进而得到下式：
Figure PCTCN2020120393-appb-000046
公式(21)中，β是斜壁板中点的转角，无量纲荷载ζ、斜壁板沿X轴方向上的投影率
Figure PCTCN2020120393-appb-000047
和Y轴方向上的投影率
Figure PCTCN2020120393-appb-000048
分别如下：
Figure PCTCN2020120393-appb-000049
其中；
Figure PCTCN2020120393-appb-000050
在公式(23)中F(β)是第一类椭圆积分；E(β)是第二类椭圆积分；η'是是根据坐标变换公式
Figure PCTCN2020120393-appb-000051
Figure PCTCN2020120393-appb-000052
产生的新变量；η' 0是当弧坐标S＝0，即弧坐标S处的转角γ＝β 0时的η'值。
在综合上述A、B、C、D四种情况，根据大变形下的欧拉应力可得：
Figure PCTCN2020120393-appb-000053
其中，σ' x和σ' y表示X轴方向上承受荷载时的情况，σ″ x和σ″ y表示Y轴方向上承受荷载时的情况。大变形下的阿尔曼希应变可以表示为：
Figure PCTCN2020120393-appb-000054
其中：
Figure PCTCN2020120393-appb-000055
为胞元竖板DB和竖板AC的变形之和(受拉为正，受压为负)，A s＝tb为蜂窝壁板的横截面积，P是施加在代表胞元上的等效荷载，h是蜂窝胞元竖板的长度，E S为胞元材料的杨氏模量，ζ为无量纲荷载，t是蜂窝结构的壁板厚度，l是蜂窝结构斜边的长度，A s＝tb是蜂窝壁板的横截面积；同时公式(2)可以进一步表示为：
Figure PCTCN2020120393-appb-000056
S04：根据所述修正因子建立所述内凹蜂窝结构的非线性本构关系；
参照图11，为本发明具体实施方式中阿尔曼希应变与无量纲欧拉应力在理论模型和有限元模型中的结果示意图。在理论模型中的阿尔曼希应变曲线和有限元模型中的阿尔曼希应变曲线大体一致，说明本实施例中的内凹蜂窝结构与相对精细的有限元内凹蜂窝结构吻合良好。其中结构及材料参数为：φ＝60°，l＝10mm，h＝10mm，b＝1mm，t＝1mm，E s＝12MPa，σ s＝10MPa。
参照图12、图13、图14和图15，为本发明实施例中内凹蜂窝结构参数φ对蜂窝结构参数的影响示意图，正应变表示的是蜂窝结构承受的拉荷载，负应变表示蜂窝结构承受的压荷载。其中结构及材料参数为：φ＝30°、45°、60°、75°，l＝10mm，h＝10mm，b＝1mm，t＝1mm，E s＝12MPa，σ s＝10MPa。从结果可以看出：
1、当蜂窝结构承受拉荷载时，胞元角度φ越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越大，X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而变大，Y轴方向上杨氏模量修正因子k 2随着胞元变形的变大而变小；胞元角度φ越大，X轴方向上泊松比修正因子m 1越小，Y轴方向上泊松比修正因子m 2越大，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变小，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变大；
2、当蜂窝结构承受压荷载时，胞元角度φ越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越大，X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而变小，Y轴方向上杨氏模量修正因子k 2随着胞元变形的变 大而变大；胞元角度φ越大，X轴方向上泊松比修正因子m 1越小，Y轴方向上泊松比修正因子m 2越大，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变大，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变小；
3、X轴方向上杨氏模量修正因子和泊松比修正因子变化效果明显强于Y轴方向上的杨氏模量修正因子和泊松比修正因子的变化效果。
参照图16、17、18和图19，本发明实施例中内凹蜂窝结构细长比t/l对结构参数的影响示意图，正应变表示的是蜂窝结构承受的拉荷载，负应变表示蜂窝结构承受的压荷载。其中结构及材料参数为：φ＝60°，t/l＝0.05、0.1、0.2、0.4、0.6，l＝10mm，h＝10mm，b＝1mm，E s＝12MPa，σ s＝10MPa；从结果可以看出：
1、当蜂窝结构承受拉荷载时，细长比t/l越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越小，X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而先变小后变大，Y轴方向上杨氏模量修正因子k 2随着胞元变形的变大而变小；细长比t/l越大，X轴方向上泊松比修正因子m 1越小，Y轴方向上泊松比修正因子m 2无明显变化，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变小，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变大；
2、当蜂窝结构承受压荷载时，细长比t/l越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越小，X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而变小，Y轴方向上杨氏模量修正因子k 2随着胞元变形的变大而变大；细长比t/l越大，X轴方向上泊松比修正因子m 1越大，Y轴方向上泊松比修正因子m 2无明显变化，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变大，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变小；
参照图20、图21、图22和图23，为本发明实施例中蜂窝材料屈服强度σ s对结构参数的影响示意图，正应变表示的是蜂窝结构承受的拉荷载，负应变表示蜂窝结构承受的压荷载。其中结构及材料参数为：φ＝60°，l＝10mm，h＝10mm，b＝1mm，E s＝12MPa，σ s＝3MPa、5MPa、8MPa、10MPa、15MPa；从结果可以看出：
1、在弹性阶段，蜂窝材料屈服强度σ s对结构参数没有影响；
2、蜂窝材料屈服强度σ s越大，结构越晚出现塑性变形；
3、当蜂窝结构承受拉荷载时，蜂窝材料屈服强度σ s越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越大；X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而先变小后变大，Y轴方向上杨氏模量修正因子k 2随着胞元变形的变大而变小；蜂窝材料屈服强度σ s越大，X轴方向上泊松比修正因子m 1越大，Y轴方向上泊松比修正因子m 2无明显变化，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变小，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变大；
4、当蜂窝结构承受压荷载时，蜂窝材料屈服强度σ s越大，X轴方向上和Y轴方向上杨氏模量修正因子k 1和k 2越大；X轴方向上杨氏模量修正因子k 1随着胞元变形的变大而变小，Y轴方向上杨氏模量修正因子k 2随着胞元 变形的变大而变大；蜂窝材料屈服强度σ s越大，X轴方向上泊松比修正因子m 1越小，Y轴方向上泊松比修正因子m 2越大，X轴方向上泊松比修正因子m 1随着胞元变形的变大而变大，Y轴方向上泊松比修正因子m 2随着胞元变形的变大而变小。
S05：根据所述非线性本构关系设计和生产对应结构；
相较于现有技术，本负泊松比结构的非线性本构关系的分析方法的实施例具有以下优点：
1)本发明实施例通过变形分析并结合结构几何特征(角度、相对密度)、材料杨氏模量得到能够反映同一类内凹蜂窝结构非线性本构关系的修正因子，并建立反映同一类内凹蜂窝结构的非线性本构关系，并在分析过程中充分考虑了材料的弹塑性变形，相比于数值方法和试验方法，它具有更好的计算精度和适用范围；
2)本发明实施例在对材料或结构有殊结需求的场景业，可以方便快捷地设计出合适的蜂窝结构，使得设计、生产及制作过程更加便捷，进一步缩短产品的设计周期。
其次，本发明实施例提出一种负泊松比结构的非线性本构关系的分析系统。
所述系统具体包括：
目标获取单元，用于获取内凹蜂窝结构的胞元；
变形分析单元，用于对所述胞元进行变形分析；
核心处理单元，用于根据变形分析的结果得到修正因子以及根据所述修正因子建立所述负泊松比结构的非线性本构关系；
输出单元，用于根据所述非线性本构关系对结构进行仿真以及输出。
可见，上述方法实施例中的内容均适用于本系统实施例中，本系统实施例所具体实现的功能与上述方法实施例相同，并且达到的有益效果与上述方法实施例所达到的有益效果也相同。
同时，本发明实施例提供了一种负泊松比结构的非线性本构关系的分析装置，包括：
至少一个处理器；
至少一个存储器，用于存储至少一个程序；
当所述至少一个程序被所述至少一个处理器执行时，使得所述至少一个处理器实现所述的负泊松比结构的非线性本构关系的分析方法。
同理，上述方法实施例中的内容均适用于本装置实施例中，本装置实施例所具体实现的功能与上述方法实施例相同，并且达到的有益效果与上述方法实施例所达到的有益效果也相同。
在一些可选择的实施例中，在方框图中提到的功能/操作可以不按照操作示图提到的顺序发生。例如，取决于所涉及的功能/操作，连续示出的两个方框实际上可以被大体上同时地执行或所述方框有时能以相反顺序被执行。此外，在本发明的流程图中所呈现和描述的实施例以示例的方式被提供，目的在于提供对技术更全面的理解。 所公开的方法不限于本文所呈现的操作和逻辑流程。可选择的实施例是可预期的，其中各种操作的顺序被改变以及其中被描述为较大操作的一部分的子操作被独立地执行。
此外，虽然在功能性模块的背景下描述了本发明，但应当理解的是，除非另有相反说明，所述的功能和/或特征中的一个或多个可以被集成在单个物理装置和/或软件模块中，或者一个或多个功能和/或特征可以在单独的物理装置或软件模块中被实现。还可以理解的是，有关每个模块的实际实现的详细讨论对于理解本发明是不必要的。更确切地说，考虑到在本文中公开的装置中各种功能模块的属性、功能和内部关系的情况下，在工程师的常规技术内将会了解该模块的实际实现。因此，本领域技术人员运用普通技术就能够在无需过度试验的情况下实现在权利要求书中所阐明的本发明。还可以理解的是，所公开的特定概念仅仅是说明性的，并不意在限制本发明的范围，本发明的范围由所附权利要求书及其等同方案的全部范围来决定。
所述功能如果以软件功能单元的形式实现并作为独立的产品销售或使用时，可以存储在一个计算机可读取存储介质中。基于这样的理解，本发明的技术方案本质上或者说对现有技术做出贡献的部分或者该技术方案的部分可以以软件产品的形式体现出来，该计算机软件产品存储在一个存储介质中，包括若干指令用以使得一台计算机设备(可以是个人计算机，服务器，或者网络设备等)执行本发明各个实施例所述方法的全部或部分步骤。而前述的存储介质包括：U盘、移动硬盘、只读存储器(ROM，Read-Only Memory)、随机存取存储器(RAM，Random Access Memory)、磁碟或者光盘等各种可以存储程序代码的介质。
在流程图中表示或在此以其他方式描述的逻辑和/或步骤，例如，可以被认为是用于实现逻辑功能的可执行指令的定序列表，可以具体实现在任何计算机可读介质中，以供指令执行系统、装置或设备(如基于计算机的系统、包括处理器的系统或其他可以从指令执行系统、装置或设备取指令并执行指令的系统)使用，或结合这些指令执行系统、装置或设备而使用。
应当理解，本发明的各部分可以用硬件、软件、固件或它们的组合来实现。在上述实施方式中，多个步骤或方法可以用存储在存储器中且由合适的指令执行系统执行的软件或固件来实现。例如，如果用硬件来实现，和在另一实施方式中一样，可用本领域公知的下列技术中的任一项或他们的组合来实现：具有用于对数据信号实现逻辑功能的逻辑门电路的离散逻辑电路，具有合适的组合逻辑门电路的专用集成电路，可编程门阵列(PGA)，现场可编程门阵列(FPGA)等。
在本说明书的上述描述中，参考术语“一个实施方式/实施例”、“另一实施方式/实施例”或“某些实施方式/实施例”等的描述意指结合实施方式或示例描述的具体特征、结构、材料或者特点包含于本发明的至少一个实施方式或示例中。在本说明书中，对上述术语的示意性表述不一定指的是相同的实施方式或示例。而且，描述的具体特征、结构、材料或者特点可以在任何的一个或多个实施方式或示例中以合适的方式结合。
尽管已经示出和描述了本发明的实施方式，本领域的普通技术人员可以理解：在不脱离本发明的原理和宗旨的情况下可以对这些实施方式进行多种变化、修改、替换和变型，本发明的范围由权利要求及其等同物限定。
以上是对本发明的较佳实施进行了具体说明，但本发明并不限于所述实施例，熟悉本领域的技术人员在不违背本发明精神的前提下还可做作出种种的等同变形或替换，这些等同的变形或替换均包含在本申请权利要求所限定的范围内。

Claims (10)

  1. 负泊松比结构的非线性本构关系的分析方法，其特征在于，包括以下步骤：
    获取内凹蜂窝结构的胞元；
    对所述胞元进行变形分析；
    根据变形分析的结果得到修正因子；
    根据所述修正因子建立所述内凹蜂窝结构的非线性本构关系；
    根据所述非线性本构关系设计和生产对应结构；
    所述修正因子包括：杨氏模量修正因子和泊松比修正因子。
  2. 根据权利要求1所述的负泊松比结构非线性本构关系的分析方法，其特征在于，所述对所述胞元进行变形分析这一步骤，其具体包括：
    获取所述胞元的斜壁板端点处的弯矩以及极限弯矩；
    对比所述弯矩与所述极限弯矩，确定所述胞元所处的变形阶段；
    所述变形阶段包括弹性变形阶段和塑性变形阶段。
  3. 根据权利要求1所述的负泊松比结构非线性本构关系的分析方法，其特征在于，所述杨氏模量修正因子包括水平坐标方向上杨氏模量修正因子以及竖直坐标方向上的杨氏模量修正因子；所述泊松比修正因子包括水平坐标方向上泊松比修正因子以及竖直坐标方向上的泊松比修正因子。
  4. 根据权利要求1所述的负泊松比结构非线性本构关系的分析方法，其特征在于：所述根据变形分析的结果得到修正因子这一步骤，其具体包括：
    建立所述胞元的半斜壁板的控制方程；
    将所述控制方程转化为无量纲方程，并对所述无量纲方程进行简化；
    根据简化后的无量纲方程得到无量纲荷载以及斜壁板的投影率；
    根据所述无量纲载荷以及斜壁板的投影率得到修正因子。
  5. 根据权利要求4所述的负泊松比结构非线性本构关系的分析方法，其特征在于：所述根据所述无量纲载荷以及斜壁板的投影率得到修正因子这一步骤，具体包括下步骤：
    获取所述胞元的几何特征以及所述内凹蜂窝结构材料杨氏模量；
    结合所述几何特征、所述材料杨氏模量、所述无量纲载荷以及斜壁板的投影率得到修正因子。
  6. 根据权利要求4-5任一项所述的负泊松比结构非线性本构关系的分析方法，其特征在于：所述投影率包括水平坐标方向上的投影率和竖直坐标方向上的投影率。
  7. 根据权利要求4所述的负泊松比结构非线性本构关系的分析方法，其特征在于：所述建立所述胞元的半斜壁板的控制方程这一步骤，其包括以下步骤至少一项：
    当荷载为拉荷载，针对半斜壁板的弹性变形，建立第一控制方程；
    当荷载为拉荷载，针对半斜壁板的塑性变形，建立第二控制方程；
    当荷载为压荷载，针对半斜壁板的弹性变形，建立第三控制方程；
    当荷载为压荷载，针对半斜壁板的塑性变形，建立第四控制方程。
  8. 根据权利要求7所述的负泊松比结构非线性本构关系的分析方法，其特征在于：所述荷载包括所述胞元在水平坐标方向上承受的荷载以及在所述胞元竖直坐标方向上承受的荷载。
  9. 负泊松比结构非线性本构关系的分析系统，其特征在于，包括：
    目标获取单元，用于获取内凹蜂窝结构的胞元；
    变形分析单元，用于对所述胞元进行变形分析；
    核心处理单元，用于根据变形分析的结果得到修正因子以及根据所述修正因子建立所述负泊松比结构的非线性本构关系；
    输出单元，用于根据所述非线性本构关系对结构进行仿真以及输出。
  10. 负泊松比结构非线性本构关系的分析装置，其特征在于，包括：
    至少一个处理器；
    至少一个存储器，用于存储至少一个程序；
    当所述至少一个程序被所述至少一个处理器执行，使得所述至少一个处理器实现如权利要求1-8中任一项所述的负泊松比结构非线性本构关系的分析方法。
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