CN112784460B - Stability analysis method for mechanical metamaterial compression bar - Google Patents

Stability analysis method for mechanical metamaterial compression bar Download PDF

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CN112784460B
CN112784460B CN202110118958.0A CN202110118958A CN112784460B CN 112784460 B CN112784460 B CN 112784460B CN 202110118958 A CN202110118958 A CN 202110118958A CN 112784460 B CN112784460 B CN 112784460B
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孙伟福
林高建
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Beijing Institute of Technology BIT
Chongqing Innovation Center of Beijing University of Technology
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Abstract

The invention provides a method for analyzing stability of a compression bar of a mechanical metamaterial, which comprises the following steps: acquiring a sample period unit of a sample metamaterial compression bar, and constructing a finite element analysis model according to the sample period unit; acquiring a target periodic unit of a target metamaterial compression bar, and carrying out finite element analysis on the target periodic unit according to the finite element analysis model to acquire a plurality of characteristic stiffness parameters; according to the deformation coordination condition of the target metamaterial compression bar, combining the characteristic stiffness parameters to obtain an overall stiffness matrix of the target metamaterial compression bar; establishing a buckling control equation, bringing the set compression bar boundary condition and the integral stiffness matrix into the buckling control equation, and solving and obtaining a critical instability load; and comparing the design load with the critical instability load, and judging the stability of the target metamaterial compression bar. The invention shortens the time required for judging the structural stability of the metamaterial compression bar and greatly improves the structural design efficiency.

Description

Stability analysis method for mechanical metamaterial compression bar
Technical Field
The invention relates to the technical field of metamaterial structure performance analysis, in particular to a method for analyzing stability of a mechanical metamaterial compression bar.
Background
The metamaterial refers to a composite material which has a structure designed manually and presents supernormal physical properties which are not possessed by natural materials, and breaks through the limitation of certain apparent natural laws, so that supernormal material functions are obtained. The metamaterial has wide application prospect and comprises the fields of electronic engineering, condensed state physics, microwaves, optoelectronics, classical optics, material science, semiconductor science, nano-technology and the like. The compression bar structure made of the metamaterial can be applied to manufacturing industries such as aircraft automobile manufacturing, engineering machinery manufacturing and the like.
In the prior art, for mechanical stability analysis of a compression bar structure constructed by adopting a metamaterial, a 3D printing prototype experiment verification or a finite element analysis method based on a full-size model is generally adopted. However, both of these methods require a long time and are inefficient.
Disclosure of Invention
Based on the above, it is necessary to provide a method for analyzing the stability of a mechanical metamaterial compression bar.
A stability analysis method of a mechanical metamaterial compression bar comprises the following steps: acquiring a sample period unit of a sample metamaterial compression bar, and constructing a finite element analysis model according to the sample period unit; acquiring a target periodic unit of a target metamaterial compression bar, and carrying out finite element analysis on the target periodic unit according to the finite element analysis model to acquire a plurality of characteristic stiffness parameters; according to the deformation coordination condition of the target metamaterial compression bar, combining the characteristic stiffness parameters to obtain an overall stiffness matrix of the target metamaterial compression bar; establishing a buckling control equation, bringing the set compression bar boundary condition and the integral stiffness matrix into the buckling control equation, and solving and obtaining a critical instability load; and comparing the design load with the critical instability load, and judging the stability of the target metamaterial compression bar.
In one embodiment, the acquiring the target period unit of the target metamaterial compression bar specifically includes: the target metamaterial compression bar is formed by periodically arranging basic units, and the basic units forming the target metamaterial compression bar are taken as the target periodic units.
In one embodiment, the finite element analysis is performed on the target periodic unit according to the finite element analysis model to obtain a plurality of characteristic stiffness parameters, which specifically includes: and carrying out four finite element analyses on the target periodic unit, loading a displacement load in a specific form in the four finite element analyses, wherein the displacement load comprises strain and curvature, strain energy and reaction force are obtained in a finite element analysis model according to the displacement load, and the characteristic rigidity parameter is obtained according to the strain energy and the reaction force.
In one embodiment, the four times finite element analysis comprises;
in the first finite element analysis, the first strain and the first curvature are set to ε= {0, ε, respectively 3 The first strain energy U is obtained by the sum of the two components of the sequence [ kappa ] = {0, 0} 1 And a first reaction torque according to the first strain energy and formula U 1 ＝C 3 ε 3 2 Obtaining a first characteristic rigidity parameter C 3s Obtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strain s If the metamaterial is achiral, H s Is 0;
in the second finite element analysis, the second strain and the second curvature are set to ε= {0, 0} and κ= {0, τ } respectively, and the second strain energy U is obtained 2 According to the second strain energy and the formula U 2 ＝D 3 τ 2 Obtaining third characteristic rigidity parameter D 3s ；
In the third finite element analysis, the third strain and the third curvature are set to ε= { ε, respectively 1 0,0 and κ= {0, 0} to obtain a third strain energy U 3 According to the third strain energy and the formula U 3 ＝C 1s (1-H s 2 /C 3s D 3s )ε 1 2 Obtaining fourth characteristic rigidity parameter C 1s And obtain a fifth characteristic rigidity parameter C according to symmetry 2s ；
In the fourth finite element analysis, the fourth strain and the fourth curvature are set to ε= {0, 0} and κ= { κ, respectively 1 0, obtain a fourth strain energy U 4 According to the fourth strain energy and equation U 4 ＝D 1s (1-H s 2 /C 3s D 3s )κ 1 2 Obtaining a sixth characteristic rigidity parameter D 1s And obtain a seventh characteristic rigidity parameter D according to symmetry 2s 。
In one embodiment, the obtaining the overall stiffness matrix of the metamaterial compression rod according to the deformation coordination condition of the target metamaterial compression rod and combining the plurality of characteristic stiffness parameters specifically includes: if the target metamaterial compression bar is formed by repeatedly arranging m×n periodic units, and the size of each target periodic unit is a, constructing a mechanical model by combining the characteristic stiffness parameters, and obtaining the overall stiffness matrixes C, B and D of the target metamaterial compression bar according to the mechanical model, wherein the overall stiffness matrixes C, B and D are as follows:
Figure BDA0002921309050000031
Figure BDA0002921309050000032
Figure BDA0002921309050000033
wherein h= mnH s ，C 1 ＝C 2 ＝mnC 1s ，C 3 ＝mnC 3s ，
Figure BDA0002921309050000034
Figure BDA0002921309050000035
Figure BDA0002921309050000036
In one embodiment, the buckling control equation is:
Figure BDA0002921309050000037
Figure BDA0002921309050000038
simultaneous equations (1) and (2), the general solution is obtained as:
Figure BDA0002921309050000039
Figure BDA00029213090500000310
wherein, let the
Figure BDA0002921309050000041
Figure BDA0002921309050000042
Then there are:
Figure BDA0002921309050000043
Figure BDA0002921309050000044
Figure BDA0002921309050000045
in one embodiment, the boundary condition includes:
for the cantilever beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ 1 (0)＝θ 2 (0)＝θ′ 1 (L)-ηPθ 2 (L)＝θ′ 2 (L)+ηPθ 1 (L)＝0；
for the simple beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ 1 ′(0)-ηPθ 2 (0)＝θ′ 2 (0)+ηPθ 1 (0)＝θ′ 1 (L)-ηPθ 2 (L)＝θ′ 2 (L)+ηPθ 1 (L)＝0。
in one embodiment, the comparing the design load with the critical destabilizing load to determine the stability of the target metamaterial compression rod specifically includes: if the design load is greater than the critical instability load, the target metamaterial compression bar is unstable and buckling can occur to cause instability; and if the design load is smaller than or equal to the critical instability load, the target metamaterial is stable and buckling does not occur.
Compared with the prior art, the invention has the advantages that:
1. according to the invention, the critical instability load of the metamaterial compression bar can be rapidly calculated or verified through the repeatable cycle unit, so that the design time required for judging the structural stability of the metamaterial compression bar is shortened, and the structural design efficiency is greatly improved.
2. According to the invention, the rigidity matrix of the metamaterial is obtained through the characterization of a plurality of characteristic rigidity parameters, so that the method can be suitable for chiral metamaterial or achiral metamaterial, and the application range is enlarged.
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FIG. 1 is a flow chart of a method for analyzing stability of a mechanical metamaterial compression bar in an embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail by the following detailed description with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
In one embodiment, as shown in fig. 1, a method for analyzing stability of a mechanical metamaterial compression rod is provided, which comprises the following steps:
step S101, a sample period unit of a sample metamaterial compression bar is obtained, and a finite element analysis model is constructed according to the sample period unit.
Specifically, since the mechanical metamaterial compression bar structure is generally formed by periodic arrangement of basic units, the basic units constituting the metamaterial compression bar can be taken as periodic units, and thus, the periodic units can be obtained by analyzing the basic units of the metamaterial of the compression bar structure. The method comprises the steps of acquiring sample period units of a plurality of sample metamaterial compression bars, and constructing a finite element analysis model according to the sample period units.
Step S102, a target periodic unit of a target metamaterial compression bar is obtained, finite element analysis is carried out on the target periodic unit according to a finite element analysis model, and a plurality of characteristic rigidity parameters are obtained.
Specifically, a target periodic unit of a target metamaterial compression bar is obtained, and multiple times of finite element analysis is carried out on the target periodic unit according to a constructed finite element analysis model, so that a plurality of characteristic rigidity parameters are obtained.
In this embodiment, four finite element analyses are required to be performed on the target periodic unit according to the finite element analysis model, so as to obtain seven characteristic stiffness parameters, and therefore, stability of the target metamaterial compression bar structure is determined according to the seven characteristic stiffness parameters. In the four finite element analysis process, specific forms of displacement loads including strain and curvature are required to be loaded respectively, so that strain energy and reaction force are obtained, and characteristic rigidity parameters are obtained according to the strain energy and the reaction force.
And step S103, according to the deformation coordination condition of the target metamaterial compression bar, combining a plurality of characteristic stiffness parameters to obtain the overall stiffness matrix of the target metamaterial compression bar.
The deformation coordination condition is a condition for ensuring that the integrity and the continuity of the target metamaterial compression bar can be maintained after the continuous solid deformation. The deformation coordination condition of the target metamaterial compression bar can be obtained according to parameters when the metamaterial compression bar is designed.
Specifically, a mechanical model can be constructed according to deformation coordination conditions of a single target period unit in the whole deformation of the target metamaterial compression bar, and the mechanical model is obtained on the condition that strain energy is equal. For example, when the compression rod is axially compressed by a certain strain, each cyclic unit also needs to be compressed by the corresponding strain, so the strain energy of the compression rod as a whole can be expressed as a multiple of the strain energy of a single target cyclic unit. Wherein the principle of compression of the compression bar is similar to the principle of bending and torsion of the compression bar. And the overall stiffness matrix of the target metamaterial compression bar is obtained by combining a plurality of characteristic stiffness parameters according to the deformation coordination conditions of a single target period unit.
And S104, constructing a buckling control equation, bringing the set compression bar boundary condition and the set integral stiffness matrix into the buckling control equation, and solving and obtaining the critical instability load.
Specifically, according to the boundary condition of the target metamaterial compression bar and the overall stiffness matrix, a constructed buckling control equation is brought in, and the equation is solved to obtain the critical instability load of the target metamaterial compression bar.
Step S105, comparing the design load with the critical instability load, and judging the stability of the target metamaterial compression bar.
Specifically, since the design load is set during the design of the compression bar, the design load is compared with the critical instability load, so that the stability of the target metamaterial compression bar is judged. If the design load is greater than the critical instability load, the target metamaterial compression bar is unstable and buckling can occur to cause instability; otherwise, the target metamaterial compression bar is stable, and buckling is not sent to cause instability.
In the embodiment, firstly, a sample period unit of a sample metamaterial compression bar is obtained, and a finite element analysis model is constructed according to the sample period unit; according to the finite element analysis model, finite element analysis is carried out on a target periodic unit of a target metamaterial compression rod, a plurality of characteristic rigidity parameters are obtained, the deformation coordination condition of the compression rod structure is combined according to the plurality of characteristic rigidity parameters, the overall rigidity matrix of the target metamaterial compression rod is obtained, the overall rigidity matrix and compression rod boundary conditions are brought into a buckling control equation, the critical instability load is solved and obtained, the temporary instability load is compared with the design load, so that the stability of the target metamaterial is judged, the repeatable periodic unit represents the overall compression rod structure, the critical instability load of the target metamaterial compression rod structure is calculated, the stability of the target metamaterial compression rod structure is judged, the time required for calculating the critical instability load is shortened, the structural design efficiency is improved, the rigidity matrix is suitable for chiral or achiral materials, and the application range is enlarged.
The step S102 specifically includes: the target metamaterial compression bar is formed by periodically arranging basic units, and the basic units forming the target metamaterial compression bar are taken as the target periodic units.
Wherein, step S102 further comprises: and carrying out four times of finite element analysis on the target periodic unit, loading a displacement load in a specific form in the four times of finite element analysis, wherein the displacement load comprises strain and curvature, obtaining strain energy and reaction force in a finite element analysis model according to the displacement load, and obtaining the characteristic rigidity parameter according to the strain energy and the reaction force.
Wherein the four times of finite element analysis includes:
in the first finite element analysis, the first strain and the first curvature are set to ε= {0, ε, respectively 3 The first strain energy U is obtained by the sum of the two components of the sequence [ kappa ] = {0, 0} 1 And a first reaction torque according to the first strain energy and formula U 1 ＝C 3 ε 3 2 Obtaining a first characteristic rigidity parameter C 3s Obtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strain s If the metamaterial is achiral, H s Is 0;
in the second finite element analysis, the second strain and the second curvature are set to ε= {0, 0} and κ= {0, τ } respectively, and the second strain energy U is obtained 2 According to the second strain energy and the formula U 2 ＝D 3 τ 2 Obtaining third characteristic rigidity parameter D 3s ；
In the third finite element analysis, the third strain and the third curvature are set to ε= { ε, respectively 1 0,0 and κ= {0, 0} to obtain a third strain energy U 3 According to the third strain energy and the formula U 3 ＝C 1s (1-H s 2 /C 3s D 3s )ε 1 2 Obtaining fourth characteristic rigidity parameterC 1s And obtain a fifth characteristic rigidity parameter C according to symmetry 2s ；
In the fourth finite element analysis, the fourth strain and the fourth curvature are set to ε= {0, 0} and κ= { κ, respectively 1 0, obtain a fourth strain energy U 4 According to the fourth strain energy and formula U 4 ＝D 1s (1-H s 2 /C 3s D 3s )κ 1 2 Obtaining a sixth characteristic rigidity parameter D 1s And obtain a seventh characteristic rigidity parameter D according to symmetry 2s 。
Specifically, fourth characteristic stiffness parameter C 1s And a fifth characteristic stiffness parameter C 2s With symmetry between them, then there is C 1s ＝C 2s . Similarly, a sixth characteristic stiffness parameter D 1s And a seventh characteristic stiffness parameter D 2s Between them, there is D 1s ＝D 2s 。
The step S103 specifically includes: if the target metamaterial compression bar is formed by repeatedly arranging m×n periodic units, and the size of each periodic unit is a, a mechanical model is built by combining a plurality of characteristic stiffness parameters, and the overall stiffness matrixes C, B and D of the metamaterial compression bar are obtained according to the mechanical model, wherein the overall stiffness matrixes C, B and D are as follows:
Figure BDA0002921309050000081
Figure BDA0002921309050000082
Figure BDA0002921309050000083
wherein h= mnH s ，C 1 ＝C 2 ＝mnC 1s ，C 3 ＝mnC 3s ，
Figure BDA0002921309050000084
Figure BDA0002921309050000085
Figure BDA0002921309050000086
Wherein, the buckling control equation in step S104 is:
Figure BDA0002921309050000087
Figure BDA0002921309050000088
simultaneous equations (1) and (2), the general solution is obtained as:
Figure BDA0002921309050000089
Figure BDA00029213090500000810
wherein, let the
Figure BDA0002921309050000091
Figure BDA0002921309050000092
Then there are:
Figure BDA0002921309050000093
Figure BDA0002921309050000094
Figure BDA0002921309050000095
the boundary conditions in step S104 include:
for the cantilever beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ 1 (0)＝θ 2 (0)＝θ′ 1 (L)-ηPθ 2 (L)＝θ′ 2 (L)+ηPθ 1 (L)＝0；
for the simple beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ′ 1 (0)-ηPθ 2 (0)＝θ′ 2 (0)+ηPθ 1 (0)＝θ′ 1 (L)-ηPθ 2 (L)＝θ′ 2 (L)+ηPθ 1 (L)＝0。
the step S105 specifically includes: if the design load is greater than the critical instability load, the target metamaterial compression bar is unstable and buckling can occur to cause instability; if the design load is less than or equal to the critical instability load, the target metamaterial compression bar is stable and buckling cannot occur.
The foregoing is a further detailed description of the invention in connection with specific embodiments, and is not intended to limit the practice of the invention to such descriptions. It will be apparent to those skilled in the art that several simple deductions or substitutions may be made without departing from the spirit of the invention, and these should be considered to be within the scope of the invention.

Claims (6)

1. The method for analyzing the stability of the mechanical metamaterial compression bar is characterized by comprising the following steps of:
acquiring a sample period unit of a sample metamaterial compression bar, and constructing a finite element analysis model according to the sample period unit;
obtaining a target periodic unit of a target metamaterial compression bar, performing finite element analysis on the target periodic unit according to the finite element analysis model, and obtaining a plurality of characteristic rigidity parameters, wherein the method specifically comprises the following steps of: carrying out four finite element analyses on the target periodic unit, loading a displacement load in a specific form in the four finite element analyses, wherein the displacement load comprises strain and curvature, strain energy and reaction force are obtained in a finite element analysis model according to the displacement load, and the characteristic rigidity parameter is obtained according to the strain energy and the reaction force;
wherein in the first finite element analysis, the first strain and the first curvature are set to ε= {0, ε, respectively 3 The first strain energy U is obtained by the sum of the two components of the sequence [ kappa ] = {0, 0} 1 And a first reaction torque according to the first strain energy and formula U 1 ＝C 3 ε 3 2 Obtaining a first characteristic rigidity parameter C 3s Obtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strain s If the metamaterial is achiral, H s Is 0;
in the second finite element analysis, the second strain and the second curvature are set to ε= {0, 0} and κ= {0, τ } respectively, and the second strain energy U is obtained 2 According to the second strain energy and the formula U 2 ＝D 3 τ 2 Obtaining third characteristic rigidity parameter D 3s ；
In the third finite element analysis, the third strain and the third curvature are set to ε= { ε, respectively 1 0,0 and κ= {0, 0} to obtain a third strain energy U 3 According to the third strain energy and the formula U 3 ＝C 1s (1-H s 2 /C 3s D 3s )ε 1 2 Obtaining fourth characteristic rigidity parameter C 1s And obtain a fifth characteristic rigidity parameter C according to symmetry 2s ；
In the fourth finite element analysis, the fourth strain and the fourth curvature are set to ε= {0, 0} and κ= { κ, respectively 1 0, obtain a fourth strain energy U 4 According to the fourth strain energy and equation U 4 ＝D 1s (1-H s 2 /C 3s D 3s )κ 1 2 Obtaining a sixth characteristic rigidity parameter D 1s And obtain a seventh characteristic rigidity parameter D according to symmetry 2s ；
According to the deformation coordination condition of the target metamaterial compression bar, combining the characteristic stiffness parameters to obtain an overall stiffness matrix of the target metamaterial compression bar;
establishing a buckling control equation, bringing the set compression bar boundary condition and the integral stiffness matrix into the buckling control equation, and solving and obtaining a critical instability load;
and comparing the design load with the critical instability load, and judging the stability of the target metamaterial compression bar.
2. The method for analyzing stability of a mechanical metamaterial compression rod according to claim 1, wherein the step of obtaining the target periodic unit of the target metamaterial compression rod specifically comprises the steps of:
the target metamaterial compression bar is formed by periodically arranging basic units, and the basic units forming the target metamaterial compression bar are taken as the target periodic units.
3. The method for analyzing stability of a mechanical metamaterial compression rod according to claim 1, wherein the method for obtaining the overall stiffness matrix of the metamaterial compression rod by combining the characteristic stiffness parameters according to the deformation coordination condition of the target metamaterial compression rod specifically comprises the following steps:
if the target metamaterial compression bar is formed by repeatedly arranging m×n periodic units, and the size of each target periodic unit is a, constructing a mechanical model by combining the characteristic stiffness parameters, and obtaining the overall stiffness matrixes C, B and D of the target metamaterial compression bar according to the mechanical model, wherein the overall stiffness matrixes C, B and D are as follows:
Figure FDA0004069636440000021
Figure FDA0004069636440000022
Figure FDA0004069636440000023
wherein h= mnH s ，C 1 ＝C 2 ＝mnC 1s ，C 3 ＝mnC 3s ，
Figure FDA0004069636440000024
Figure FDA0004069636440000025
Figure FDA0004069636440000026
4. The method for analyzing stability of a mechanical metamaterial compression rod according to claim 3, wherein the buckling control equation is as follows:
Figure FDA0004069636440000031
Figure FDA0004069636440000032
simultaneous equations (1) and (2), the general solution is obtained as:
Figure FDA0004069636440000033
Figure FDA0004069636440000034
/>
wherein, let the
Figure FDA0004069636440000035
Figure FDA0004069636440000036
Then there are:
Figure FDA0004069636440000037
Figure FDA0004069636440000038
Figure FDA0004069636440000039
5. the method for analyzing stability of a mechanical metamaterial compression rod according to claim 4, wherein the boundary conditions include:
for the cantilever beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ 1 (0)＝θ 2 (0)＝θ 1 ′(L)-ηPθ 2 (L)＝θ 2 ′(L)+ηPθ 1 (L)＝0；
for the simple beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ 1 ′(0)-ηPθ 2 (0)＝θ 2 ′(0)+ηPθ 1 (0)＝θ 1 ′(L)-ηPθ 2 (L)＝θ 2 ′(L)+ηPθ 1 (L)＝0。
6. the method for analyzing the stability of the mechanical metamaterial compression rod according to claim 1, wherein comparing the design load with the critical instability load to determine the stability of the target metamaterial compression rod specifically comprises:
if the design load is greater than the critical instability load, the target metamaterial compression bar is unstable and buckling can occur to cause instability;
and if the design load is smaller than or equal to the critical instability load, the target metamaterial is stable and buckling does not occur.
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