CN113191053A - Dot matrix cell differentiated configuration design method for high bearing performance - Google Patents

Dot matrix cell differentiated configuration design method for high bearing performance Download PDF

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CN113191053A
CN113191053A CN202110482592.5A CN202110482592A CN113191053A CN 113191053 A CN113191053 A CN 113191053A CN 202110482592 A CN202110482592 A CN 202110482592A CN 113191053 A CN113191053 A CN 113191053A
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李宝童
尹鹏
陈豪
刘宏磊
洪军
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Xian Jiaotong University
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Abstract

A dot matrix cell element differentiation configuration design method for high bearing performance is characterized in that firstly RUC cell elements with positive and negative Poisson ratio extreme values are searched on an mesoscopic layer, then the two cell elements are combined in a proportion and arrangement mode to form a structure with higher equivalent Young modulus and positive and negative Poisson ratio values, the structure with higher positive and negative Poisson ratios is taken as a forming cell element for cyclic design, and finally, the jump of the bearing performance of a macroscopic dot matrix structure under the condition of small strain is realized in a mode of multiple cyclic arrangement of the cell element configuration on the mesoscopic layer; the invention realizes the obvious improvement of the bearing performance of the macroscopic lattice structure through the simple arrangement of mesoscopic cell elements, and has good design reference and application values.

Description

Dot matrix cell differentiated configuration design method for high bearing performance
Technical Field
The invention belongs to the technical field of dot matrix structures, and particularly relates to a high-bearing-performance dot matrix cell differentiated configuration design method.
Background
With the continuous development of high-end equipment manufacturing industry, the requirement for fully excavating the structural performance potential under the condition of limited material consumption is continuously improved. As a new functional structure independent of constituent materials, the lattice structure has the advantages of high specific stiffness, high specific strength, large design space and the like, has important application value in the fields of vehicles, ships, aerospace and the like, and is paid more and more attention by engineering designers. The lattice structure is formed by periodically arranging Repeating Unit Cells (RUCs) on an mesoscopic layer, and as the bearing performance of the lattice structure is sensitive to the topology and arrangement mode of the RUC cells, the corresponding equivalent bearing performance of a macroscopic structure can be improved by designing and arranging the mesoscopic RUC cells, so that the bearing performance advantages brought by different RUC Cell combinations can be fully exerted under the constraint of a certain volume, and the urgent requirements of the structure on excellent bearing performance and limited material consumption in practical application can be met.
Research shows that the equivalent bearing performance of the macroscopic lattice structure can be influenced to a certain extent through the change of the RUC cell configuration, and the optimization design of the lattice structure performance at present mainly focuses on changing the RUC cell configuration and the two-dimensional plane stretching thickness and the like. The design of the RUC cell element mainly carries out parametric modeling on the configuration, and the macroscopic lattice structure is more excellent in the aspects of equivalent Young modulus, Poisson ratio, shear modulus and the like by optimizing the model parameters, such as: the limit configuration with a negative Poisson ratio lattice structure and a Poisson ratio theoretical value reaching +/-1 can be constructed through parameter optimization of the cell configuration, and the bearing characteristic of the structure is further enhanced; and the tensile thickness of the two-dimensional plane is increased, so that the performance requirements of the macroscopic lattice structure such as torsion resistance, bearing property and the like can be met under the condition of sacrificing the material consumption. The existing method for optimally designing the bearing performance of the lattice structure focuses on the change of the configuration of the cell, and ignores the influence of the interaction between different RUC cells on the mesoscopic layer on the bearing performance of the macroscopic lattice structure. Through the specific proportion and arrangement among the RUC cells with different bearing attributes, a combination rule among mesoscopic cells needs to be established to greatly strengthen the equivalent bearing performance parameters of the macroscopic lattice structure so as to meet the severe requirements of various application fields on the structural performance.
Disclosure of Invention
In order to overcome the drawbacks of the prior art, an object of the present invention is to provide a design method for differential configuration of a lattice cell with high load-carrying capability, so as to greatly improve the load-carrying capability of the structure.
In order to achieve the purpose, the invention adopts the technical scheme that:
a differential configuration design method for a high-bearing-performance-oriented dot matrix cell comprises the following steps:
the method comprises the following steps: based on a structural Poisson ratio calculation formula of a classical beam theory, obtaining mesoscopic cell configurations with positive and negative Poisson ratio values close to +/-1 in theory, and naming the mesoscopic cell configurations as a cell 1 and a cell 2;
step two: carrying out finite element analysis on the obtained mesoscopic cell element configuration to obtain a finite element solution of the positive and negative Poisson ratio values of the mesoscopic cell element configuration;
step three: the method comprises the following steps of (1) carrying out composition in proportion and arrangement mode on a cell element 1 and a cell element 2 with extreme positive and negative Poisson ratios to obtain a mesoscopic configuration with higher equivalent Young modulus and bearing performance parameters of the positive and negative Poisson ratios, and verifying the equivalent bearing performance of a macroscopic lattice structure after the obtained mesoscopic configuration is periodically arranged by a finite element analysis method;
step four: judging whether the combination times reach the preset combination times or not, if not, selecting the configuration with the maximum positive and negative Poisson ratio values obtained in the previous step according to the result of finite element analysis, and recombining the configurations in the same proportion and arrangement mode;
step five: and if the number of times of the preset composition is reached, carrying out finite element analysis on the periodically arranged macroscopic lattice structure of the mesoscopic structure to obtain reliable structure equivalent bearing performance, and outputting the final lattice structure design.
The mathematical expression of the structure Poisson ratio calculation method in the step two finite element analysis is as follows:
Figure BDA0003049003020000031
wherein mu is the Poisson's ratio of the structure, epsilon12Is strain in direction 2,. epsilon11Is the strain in direction 1, direction 1 refers to the load bearing direction of the structure, and direction 2 refers to the direction perpendicular to direction 1; since the transverse and longitudinal lengths of the structure are equal, the above equation is also equal to the displacement of the structure in the loaded direction 2, Δ u2Displacement Δ u from direction 11The ratio of (a) to (b).
The mathematical expression of the structural equivalent Young modulus calculation method in the finite element analysis is as follows:
E＝σ/ε (2)
wherein E is the Young modulus of the structure, sigma is the structural stress, epsilon is the structural strain, and epsilon is 0.005;
Figure BDA0003049003020000032
equivalent Young's modulus of structure
Figure BDA0003049003020000033
Equal to the Young's modulus E of the structure and the Young's modulus E of the constituent materialSThe specific mathematical expression is shown as formula (3);
σ＝∑F/A (4)
and (3) calculating the structural stress by adopting the ratio A of the sum sigma F of the supporting and reacting forces of the loading surface to the acting area, wherein the calculation formula is shown as the formula (4).
The lattice structure with the periodic arrangement of the cell component configuration needs the periodic boundary conditions applied to the configuration during the finite element analysis:
the periodic boundary condition is acted on boundary nodes of the RUC cell configuration to equivalently fit the macroscopic lattice structure, and the displacement field of the lattice structure which is periodically distributed on the configuration is expressed as:
Figure BDA0003049003020000041
wherein u isiIn order to be able to displace the field for the structure,
Figure BDA0003049003020000042
the overall average strain tensor for the periodically arranged structure,
Figure BDA0003049003020000043
is a linearly distributed displacement field and is,
Figure BDA0003049003020000044
the periodic fluctuating displacement field is a periodic fluctuating displacement field, and the periodic fluctuating displacement field is required to correct the linear displacement field;
Figure BDA0003049003020000045
Figure BDA0003049003020000046
where "j +" and "j-" refer to the corresponding j-th pair of nodes on the parallel boundary of a RUC cell, for a relatively parallel cell boundary,
Figure BDA0003049003020000047
is a constant;
Figure BDA0003049003020000048
wherein, for a given ε in the finite element analysis, the rightmost side in equation (8)
Figure BDA0003049003020000049
Is a constant term, and is directly applied to the node of the RUC cell as a constraint.
The proportion and arrangement combination mode of mesoscopic cells with different Poisson ratios in the third step are as follows: the RUC cells with positive and negative Poisson ratio values close to +/-1 are combined in three specific combining modes: (1) arranging the cells 1 and 2 at intervals according to the ratio of 1:1 to form 4 x 4 configuration 1, wherein the configuration 1 is the 4 x 4 configuration with the largest equivalent Young modulus; (2) directly connecting cell 1 and cell 2 in a ratio of 3:1 to form 4 x 4 configuration 2, wherein the configuration 2 is the 4 x 4 configuration with the maximum positive Poisson ratio; (3) if cell 1 and cell 2 are directly connected to form 4 x 4 configuration 3 in a ratio of 1:3, configuration 3 is the 4 x 4 configuration with the largest negative poisson's ratio.
In the fourth step, the configuration with the improved equivalent bearing performance parameter is subjected to cyclic permutation and combination design, and the method specifically comprises the following steps: and (3) equivalent to 4 × 4 configuration 2 and configuration 3 with the maximum positive and negative Poisson ratios are combined according to the same proportion and arrangement mode, so that 16 × 16 different configurations which are further improved on equivalent bearing performance parameters are formed, and the circular arrangement is continued.
The invention has the beneficial effects that:
(1) the invention adopts the cell elements with positive and negative values close to +/-1 to combine according to the proportion and the arrangement mode, and different combination modes can form an RUC structure with better bearing performance;
(2) the invention circularly combines the structures after the configurations are rearranged, and the configurations after the equivalent bearing performance parameters are improved are equivalent to the cell elements to be continuously combined, thereby obtaining a lattice structure which has more excellent performance on certain bearing performance, realizing the full excavation of the structural performance under the limited material consumption and further meeting the requirements on the structural performance which is more severe.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 shows the cell configuration with a theoretical Poisson's ratio close to. + -. 1.
Fig. 3 is a schematic diagram of finite element deformation of cell 1 and cell 2.
Fig. 4 is a schematic diagram of configuration 1 (maximum equivalent young's modulus), configuration 2 (maximum positive poisson's ratio) and configuration 3 (maximum negative poisson's ratio) formed by combining cell 1 and cell 2 to 4 × 4.
Figure 5 is three 16 x 16RUC configurations consisting of 4 x 4 configuration 2 and configuration 3.
Detailed Description
To better explain the technical solutions, objects and advantages of the present invention, the present invention is further described below with reference to the accompanying drawings and embodiments. In addition, the embodiments described herein are only for explaining the present invention and are not intended to limit the present invention.
As shown in fig. 1, a design method for differential configuration of a high-load-carrying performance dot matrix cell includes the following steps:
the method comprises the following steps: based on a structural Poisson ratio calculation formula of a classical beam theory, mesoscopic cell configurations with positive and negative Poisson ratio values close to +/-1 are obtained theoretically, and the two mesoscopic cell configurations are named as cell 1 and cell 2 respectively;
as shown in fig. 2, wherein cell 1 is mesoscopic cell configuration with theoretical poisson's ratio close to-1, and cell 2 is mesoscopic cell configuration with theoretical poisson's ratio close to 1;
step two: carrying out finite element analysis on the obtained mesoscopic cell element configuration to obtain a finite element solution of the positive and negative Poisson ratio values of the mesoscopic cell element configuration;
the mathematical expression of the structure Poisson ratio calculation method in finite element analysis is as follows:
Figure BDA0003049003020000061
wherein mu is the Poisson's ratio of the structure, epsilon12Is strain in direction 2,. epsilon11Is the strain in direction 1, direction 1 refers to the load bearing direction of the structure, and direction 2 refers to the direction perpendicular to direction 1; since the transverse and longitudinal lengths of the structure are equal, the above equation is also equal to the displacement of the structure in the loaded direction 2, Δ u2Displacement Δ u from direction 11The ratio of (A) to (B);
the mathematical expression of the structural equivalent Young modulus calculation method in the finite element analysis is as follows:
E＝σ/ε (2)
wherein E is the Young modulus of the structure, sigma is the structural stress, epsilon is the structural strain, and epsilon is 0.005 because the structure is under the action of small strain;
Figure BDA0003049003020000071
equivalent Young's modulus of structure
Figure BDA0003049003020000072
Equal to the Young's modulus E of the structure and the Young's modulus E of the constituent materialSThe specific mathematical expression is shown as formula (3);
σ＝∑F/A (4)
the structural stress can not be directly measured in finite element analysis, so the structural stress is calculated by adopting the ratio A of the sum sigma F of the supporting and reacting forces of the loading surface and the action area, and the calculation formula is shown as the formula (4);
the lattice structure with the periodic arrangement of the cell component configuration needs the periodic boundary conditions applied to the configuration during the finite element analysis: the periodic boundary condition is a macro lattice structure which is equivalently fitted by acting on boundary nodes of the RUC cell configuration, and is a common means for analyzing equivalent bearing performance of the lattice structure, especially in the case of complex cell configuration, the displacement field of the lattice structure which is periodically distributed on the configuration is expressed as:
Figure BDA0003049003020000073
wherein u isiIn order to be able to displace the field for the structure,
Figure BDA0003049003020000074
the overall average strain tensor for the periodically arranged structure,
Figure BDA0003049003020000075
is a linearly distributed displacement field and is,
Figure BDA0003049003020000076
the periodic fluctuation displacement field is a periodic fluctuation displacement field, and the configuration of the cells in the periodic arrangement structure is deformed and influenced by the boundary cells, so that the periodic fluctuation displacement field is required to correct the linear displacement field;
Figure BDA0003049003020000077
Figure BDA0003049003020000078
where "j +" and "j-" refer to the corresponding j-th pair of nodes on the parallel boundary of a RUC cell, for a relatively parallel cell boundary,
Figure BDA0003049003020000081
is a constant;
Figure BDA0003049003020000082
wherein, for a given ε in the finite element analysis, the rightmost side in equation (8)
Figure BDA0003049003020000083
The constant term can be directly applied to the node of the RUC cell as a constraint condition;
step three: the mesoscopic cell elements 1 and 2 with the extreme positive and negative Poisson ratios are formed in a proportion and arrangement mode, mesoscopic configurations with higher equivalent Young modulus, positive and negative Poisson ratios and other bearing performance parameters are obtained, and the equivalent bearing performance of the macroscopic lattice structure after the obtained mesoscopic configurations are periodically arranged is verified through a finite element analysis method;
here, the combination of the RUC cells with positive and negative poisson ratio values close to ± 1, there are three specific ratios and arrangement combinations of mesoscopic cells with different poisson ratio values, here, 4 × 4 three combinations are first given, as shown in fig. 3, wherein configuration 1 is a RUC structure in which cell 1 and cell 2 are arranged at intervals in a ratio of 1:1 to form 4 × 4, and configuration 1 is a 4 × 4 configuration with the largest equivalent young modulus, as shown in fig. 3 (a); configuration 2 directly connects cell 1 and cell 2 in a ratio of 3:1 to form a 4 x 4 RUC structure, and configuration 2 is the 4 x 4 configuration with the largest positive Poisson's ratio, as shown in FIG. 3 (b); configuration 3 is a configuration 4 x 4 formed by directly connecting cell 1 and cell 2 in a ratio of 1:3, and configuration 3 is a configuration 4 x 4 with the largest negative poisson's ratio, as shown in fig. 3 (c);
step four: judging whether the combination times reach the preset combination times or not, if not, selecting the configuration with the maximum positive and negative Poisson ratio values obtained in the previous step according to the result of finite element analysis, and recombining the configurations in the same proportion and arrangement mode;
and (3) carrying out cyclic permutation and combination design on the configuration with the improved equivalent bearing performance: combining the equivalent cells of the 4 x 4 configuration 2 and the configuration 3 with the maximum positive and negative Poisson ratios according to the same proportion and arrangement mode, thereby forming 16 x 16 different configurations which are further improved in equivalent bearing performance, and continuously carrying out cyclic arrangement according to the configuration;
finite element analysis of equivalent load bearing performance was performed on the lattice structure in this example within 3 cycles, as shown in fig. 4, where a 16 × 16RUC structure formed by cyclic arrangement is given; wherein, similarly to the 4 x 4 structure given in fig. 3, three configurations are also given here which reach the maximum values in young's modulus, positive poisson's ratio and negative poisson's ratio;
step five: and if the number of times of the preset composition is reached, carrying out finite element analysis on the periodically arranged macroscopic lattice structure of the mesoscopic structure to obtain reliable structure equivalent bearing performance, and outputting the final lattice structure design.
In order to fully understand the effect of the present invention, the equivalent load-bearing performance of the structure was analyzed and verified by the finite element method, three configurations of the RUC with the analysis objects of cell 1, cell 2, 4 × 4 and 16 × 16 were selected, and the material parameters were: young modulus is E ═ 71GPa, Poisson ratio is mu ═ 0.33; the unit division is performed by using a quadrilateral mesh, a fixed constraint is applied to the left end of the structure, a displacement load with epsilon being 0.005 is applied to the right end of the structure, periodic boundary conditions are applied to the RUC structures of 4 x 4 and 16 x 16, deformation graphs of the RUC cell 1 and the cell 2 are shown in FIG. 5, and the obtained structure equivalent Young modulus, positive and negative Poisson ratios are shown in Table 1.
TABLE 1
Figure BDA0003049003020000091
Figure BDA0003049003020000101
It can be seen from table 1 that the young's modulus and the positive poisson ratio of the structure are continuously increased along with the progress of the cyclic alignment, and the negative poisson ratio is also improved compared with the cell configuration, thereby further proving the effectiveness of the invention.
The above examples do not set any limit to the scope of the present invention; any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A differential configuration design method for a high-bearing-performance-oriented dot matrix cell is characterized by comprising the following steps:
the method comprises the following steps: based on a structural Poisson ratio calculation formula of a classical beam theory, obtaining mesoscopic cell configurations with positive and negative Poisson ratio values close to +/-1 in theory, and naming the mesoscopic cell configurations as a cell 1 and a cell 2;
step two: carrying out finite element analysis on the obtained mesoscopic cell element configuration to obtain a finite element solution of the positive and negative Poisson ratio values of the mesoscopic cell element configuration;
step three: the method comprises the following steps of (1) carrying out composition in proportion and arrangement mode on a cell element 1 and a cell element 2 with extreme positive and negative Poisson ratios to obtain a mesoscopic configuration with higher equivalent Young modulus and bearing performance parameters of the positive and negative Poisson ratios, and verifying the equivalent bearing performance of a macroscopic lattice structure after the obtained mesoscopic configuration is periodically arranged by a finite element analysis method;
step four: judging whether the combination times reach the preset combination times or not, if not, selecting the configuration with the maximum positive and negative Poisson ratio values obtained in the previous step according to the result of finite element analysis, and recombining the configurations in the same proportion and arrangement mode;
step five: and if the number of times of the preset composition is reached, carrying out finite element analysis on the periodically arranged macroscopic lattice structure of the mesoscopic structure to obtain reliable structure equivalent bearing performance, and outputting the final lattice structure design.
2. The method according to claim 1, wherein the design method for differential configuration of lattice cells with high loading performance is as follows: the mathematical expression of the structure Poisson ratio calculation method in the step two finite element analysis is as follows:
Figure FDA0003049003010000011
wherein mu is the Poisson's ratio of the structure, epsilon12Is strain in direction 2,. epsilon11Is the strain in direction 1, direction 1 refers to the load bearing direction of the structure, and direction 2 refers to the direction perpendicular to direction 1; since the transverse and longitudinal lengths of the structure are equal, the above equation is also equal to the displacement of the structure in the loaded direction 2, Δ u2Displacement Δ u from direction 11The ratio of (a) to (b).
3. The method according to claim 2, wherein the design method for differential configuration of the lattice cell with high loading performance is as follows: the mathematical expression of the structural equivalent Young modulus calculation method in the finite element analysis is as follows:
E＝σ/ε (2)
wherein E is the Young modulus of the structure, sigma is the structural stress, epsilon is the structural strain, and epsilon is 0.005;
Figure FDA0003049003010000021
equivalent Young's of structureModulus of elasticity
Figure FDA0003049003010000022
Equal to the Young's modulus E of the structure and the Young's modulus E of the constituent materialSThe specific mathematical expression is shown as formula (3);
σ＝∑F/A (4)
and (3) calculating the structural stress by adopting the ratio A of the sum sigma F of the supporting and reacting forces of the loading surface to the acting area, wherein the calculation formula is shown as the formula (4).
4. The method according to claim 3, wherein the design method for differential configuration of the lattice cell with high loading performance is as follows: the lattice structure with the periodic arrangement of the cell component configuration needs the periodic boundary conditions applied to the configuration during the finite element analysis:
the periodic boundary condition is acted on boundary nodes of the RUC cell configuration to equivalently fit the macroscopic lattice structure, and the displacement field of the lattice structure which is periodically distributed on the configuration is expressed as:
Figure FDA0003049003010000023
wherein u isiIn order to be able to displace the field for the structure,
Figure FDA0003049003010000031
the overall average strain tensor for the periodically arranged structure,
Figure FDA0003049003010000032
is a linearly distributed displacement field and is,
Figure FDA0003049003010000033
the periodic fluctuating displacement field is a periodic fluctuating displacement field, and the periodic fluctuating displacement field is required to correct the linear displacement field;
Figure FDA0003049003010000034
Figure FDA0003049003010000035
where "j +" and "j-" refer to the corresponding j-th pair of nodes on the parallel boundary of a RUC cell, for a relatively parallel cell boundary,
Figure FDA0003049003010000036
is a constant;
Figure FDA0003049003010000037
wherein, for a given ε in the finite element analysis, the rightmost side in equation (8)
Figure FDA0003049003010000038
Is a constant term, and is directly applied to the node of the RUC cell as a constraint.
5. The method according to claim 1, wherein the design method for differential configuration of lattice cells with high loading performance is as follows: the proportion and arrangement combination mode of mesoscopic cells with different Poisson ratios in the third step are as follows: the RUC cells with positive and negative Poisson ratio values close to +/-1 are combined in three specific combining modes: (1) arranging the cells 1 and 2 at intervals according to the ratio of 1:1 to form 4 x 4 configuration 1, wherein the configuration 1 is the 4 x 4 configuration with the largest equivalent Young modulus; (2) directly connecting cell 1 and cell 2 in a ratio of 3:1 to form 4 x 4 configuration 2, wherein the configuration 2 is the 4 x 4 configuration with the maximum positive Poisson ratio; (3) if cell 1 and cell 2 are directly connected to form 4 x 4 configuration 3 in a ratio of 1:3, configuration 3 is the 4 x 4 configuration with the largest negative poisson's ratio.
6. The method according to claim 5, wherein the design method for differential configuration of lattice cells with high loading performance is as follows: in the fourth step, the configuration with the improved equivalent bearing performance parameter is subjected to cyclic permutation and combination design, and the method specifically comprises the following steps: and (3) equivalent to 4 × 4 configuration 2 and configuration 3 with the maximum positive and negative Poisson ratios are combined according to the same proportion and arrangement mode, so that 16 × 16 different configurations which are further improved on equivalent bearing performance parameters are formed, and the circular arrangement is continued.
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