CN112784460A - Method for analyzing stability of mechanical metamaterial compression bar - Google Patents

Method for analyzing stability of mechanical metamaterial compression bar Download PDF

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CN112784460A
CN112784460A CN202110118958.0A CN202110118958A CN112784460A CN 112784460 A CN112784460 A CN 112784460A CN 202110118958 A CN202110118958 A CN 202110118958A CN 112784460 A CN112784460 A CN 112784460A
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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 the stability of a mechanical metamaterial compression bar, which comprises the following steps: obtaining 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 period unit of a target metamaterial compression bar, and performing finite element analysis on the target period unit according to the finite element analysis model to acquire a plurality of characteristic stiffness parameters; obtaining an integral rigidity matrix of the target metamaterial compression bar by combining the characteristic rigidity parameters according to the deformation coordination condition of the target metamaterial compression bar; establishing a buckling control equation, substituting the set boundary condition of the compression bar and the integral rigidity matrix into the buckling control equation, and solving to obtain a critical buckling load; and comparing the design load with the critical instability load, and judging the stability of the target metamaterial compression bar. The method shortens the time required by judging the structural stability of the metamaterial compression bar, and greatly improves the structural design efficiency.

Description

Method for analyzing stability of 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 the stability of a mechanical metamaterial compression bar.
Background
The metamaterial refers to a composite material which has an artificially designed structure and shows extraordinary physical properties which natural materials do not have, and the metamaterial breaks through the limitation of certain apparent natural laws so as to obtain extraordinary material functions. The metamaterial has wide application prospect and comprises the fields of electronic engineering, condensed state physics, microwave, optoelectronics, classical optics, material science, semiconductor science, nanotechnology and the like. The compression bar structure made of the metamaterial can be applied to the manufacturing industries of aircraft automobile manufacturing, engineering machinery manufacturing and the like.
In the prior art, for mechanical stability analysis of a compression bar structure constructed by using metamaterials, a 3D printing prototype experimental verification or a finite element analysis method based on a full-scale model is generally adopted. However, these two methods require a long time and have low working efficiency.
Disclosure of Invention
In view of the above, it is necessary to provide a method for analyzing the stability of a mechanical metamaterial compression bar.
A method for analyzing the stability of a mechanical metamaterial compression bar comprises the following steps: obtaining 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 period unit of a target metamaterial compression bar, and performing finite element analysis on the target period unit according to the finite element analysis model to acquire a plurality of characteristic stiffness parameters; obtaining an integral rigidity matrix of the target metamaterial compression bar by combining the characteristic rigidity parameters according to the deformation coordination condition of the target metamaterial compression bar; establishing a buckling control equation, substituting the set boundary condition of the compression bar and the integral rigidity matrix into the buckling control equation, and solving to obtain a critical buckling 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 target cycle unit for acquiring 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 used as the target periodic units.
In one embodiment, the performing finite element analysis on the target cycle unit according to the finite element analysis model to obtain a plurality of characteristic stiffness parameters specifically includes: and performing four finite element analyses on the target periodic unit, loading displacement loads in a specific form in the four finite element analyses, wherein the displacement loads comprise strain and curvature, acquiring strain energy and reaction force in a finite element analysis model according to the displacement loads, and acquiring the characteristic stiffness parameters according to the strain energy and the reaction force.
In one embodiment, the quartic finite element analysis comprises;
in the first finite element analysis, the first strain and the first curvature are set to ∈ {0,0, ∈ respectively3Acquiring first strain energy U, wherein k is {0,0,0}1And a first reaction torque according to the first strain energy and the formula U1=C3ε3 2Acquiring a first characteristic stiffness parameter C3sObtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strainsIf the metamaterial is achiral, then HsIs 0;
in the second finite element analysis, the second strain and the second curvature are set to ∈ {0,0,0} and κ {0,0, τ } respectively, and the second strain energy U is acquired2According to the second strain energy and formula U2=D3τ2Acquiring a third characteristic stiffness parameter D3s
In the third finite element analysis, the third strain and the third curvature are respectively set to be ∈ ═ epsilon { (epsilon)10,0 and k ═ 0,0,0, and a third strain energy U is obtained3According to the third strain energy and formula U3=C1s(1-Hs 2/C3sD3s1 2Acquiring a fourth characteristic stiffness parameter C1sAnd acquiring a fifth characteristic stiffness parameter C according to the symmetry2s
In the fourth finite element analysis, the fourth strain and the fourth curvature are set to e ═ 0,0,0, and k ═ k { κ { (0, 0, 0) } respectively10,0}, and obtaining fourth strain energy U4According to the fourth strain energy and equation U4=D1s(1-Hs 2/C3sD3s1 2Acquiring a sixth characteristic stiffness parameter D1sAnd obtaining a seventh characteristic stiffness parameter D according to the symmetry2s
In one embodiment, the obtaining an overall stiffness matrix of the metamaterial compression bar according to the deformation coordination condition of the target metamaterial compression bar by 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 integral stiffness matrixes C, B and D of the target metamaterial compression bar according to the mechanical model, wherein the integral stiffness matrixes C, B and D are as follows:
Figure BDA0002921309050000031
Figure BDA0002921309050000032
Figure BDA0002921309050000033
wherein, H is mnHs,C1=C2=mnC1s,C3=mnC3s
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, it is made
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:
θ1(0)=θ2(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0;
for the simply supported beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ1′(0)-ηPθ2(0)=θ′2(0)+ηPθ1(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0。
in one embodiment, the comparing the design load with the critical buckling load to determine the stability of the target metamaterial compression bar specifically includes: if the design load is larger than the critical buckling load, the target metamaterial compression bar is not stable, and buckling can occur to cause buckling; and if the design load is less than or equal to the critical buckling load, the target metamaterial is stable, and buckling cannot occur.
Compared with the prior art, the invention has the advantages and beneficial effects that:
1. according to the invention, the critical instability load of the metamaterial compression bar can be rapidly calculated or verified through a repeatable periodic unit, the design time required by judging the structural stability of the metamaterial compression bar is shortened, and the structural design efficiency is greatly improved.
2. The stiffness matrix of the metamaterial is obtained through characterization of a plurality of characteristic stiffness parameters, and the metamaterial can be suitable for chiral metamaterials or non-chiral metamaterials, so that the application range is expanded.
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FIG. 1 is a schematic flow chart of a method for analyzing the stability of a mechanical metamaterial compression bar according to an embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings by way of specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
In one embodiment, as shown in fig. 1, a method for analyzing the stability of a mechanical metamaterial compression bar is provided, which comprises the following steps:
and S101, obtaining a sample period unit of the sample metamaterial compression bar, and constructing a finite element analysis model according to the sample period unit.
Specifically, since the mechanical metamaterial compression bar structure is generally periodically arranged by the basic units, the basic units constituting the metamaterial compression bar can be used as the periodic units, and therefore, the periodic units can be obtained by analyzing the basic units of the metamaterial of the compression bar structure. Sample period units of a plurality of sample metamaterial compression bars can be obtained, and a finite element analysis model is constructed according to the sample period units.
And S102, acquiring a target period unit of the target metamaterial compression bar, and performing finite element analysis on the target period unit according to a finite element analysis model to acquire a plurality of characteristic stiffness parameters.
Specifically, a target period unit of the target metamaterial compression bar is obtained, and finite element analysis is carried out on the target period unit for multiple times according to the constructed finite element analysis model, so that a plurality of characteristic rigidity parameters are obtained.
In this embodiment, four times of finite element analysis needs to be performed on the target periodic unit according to the finite element analysis model to obtain seven characteristic stiffness parameters, so as to determine the stability of the target metamaterial compression bar structure according to the seven characteristic stiffness parameters. In the process of four times of finite element analysis, a specific form of displacement load is required to be loaded respectively, wherein the displacement load comprises strain and curvature, 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 S103, obtaining an overall stiffness matrix of the target metamaterial compression bar by combining a plurality of characteristic stiffness parameters according to the deformation coordination condition of the target metamaterial compression bar.
The deformation coordination condition is a condition which can still keep the integrity and continuity of the target metamaterial compression bar after the continuous solid is deformed. The deformation coordination condition of the target metamaterial compression bar can be obtained according to the 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 overall deformation of the target metamaterial compression bar, and the mechanical model is obtained under the condition that the strain energy is equal. For example, when the strut is axially compressed to a certain strain, each periodic unit also needs to compress the corresponding strain, so the strain energy of the strut as a whole can be expressed as a multiple of the strain energy of a single target periodic unit. Wherein the bending and twisting principle of the pressure rod is similar to the compression principle of the pressure rod. Therefore, the integral rigidity matrix of the target metamaterial compression bar is obtained by combining a plurality of characteristic rigidity parameters according to the deformation coordination condition of a single target period unit.
And step S104, constructing a buckling control equation, substituting the set boundary condition of the compression bar and the integral rigidity matrix into the buckling control equation, and solving to obtain the critical instability load.
Specifically, a constructed buckling control equation is substituted according to boundary conditions of the target metamaterial compression bar and an integral rigidity matrix, the equation is solved, and the critical instability load of the target metamaterial compression bar is obtained.
And 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 as to judge the stability of the target metamaterial compression bar. If the design load is larger than the critical instability load, the target metamaterial compression bar is not stable, and buckling can occur to cause instability; otherwise, the target metamaterial compression bar is stable, and buckling cannot be sent to cause instability.
In the embodiment, a sample period unit of the sample metamaterial compression bar is obtained, and a finite element analysis model is constructed according to the sample period unit; carrying out finite element analysis on a target periodic unit of the target metamaterial compression bar according to the finite element analysis model to obtain a plurality of characteristic rigidity parameters, obtaining an integral rigidity matrix of the target metamaterial compression bar according to a plurality of characteristic rigidity parameters and the deformation coordination condition of the compression bar structure, substituting the integral rigidity matrix and the boundary condition of the compression bar into a buckling control equation, solving and obtaining a critical destabilizing load, comparing the temporary destabilizing load with a design load, thereby judging the stability of the target metamaterial, representing the whole compression bar structure through repeatable periodic units, the critical destabilization load of the target metamaterial compression bar structure is calculated, so that the stability of the target metamaterial compression bar structure is judged, the time for calculating the critical destabilization load is shortened, the structural design efficiency is improved, the rigidity matrix is suitable for chiral or achiral materials, and the application range is expanded.
Wherein, 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 used as the target periodic units.
Wherein, step S102 further includes: and carrying out quartic finite element analysis on the target periodic unit, loading a displacement load in a specific form in the quartic finite element analysis, wherein the displacement load comprises strain and curvature, acquiring strain energy and reaction force in a finite element analysis model according to the displacement load, and acquiring the characteristic stiffness parameter according to the strain energy and the reaction force.
Wherein the fourth finite element analysis comprises:
in the first finite element analysis, the first strain and the first curvature are set to ∈ {0,0, ∈ respectively3Acquiring first strain energy U, wherein k is {0,0,0}1And a first reaction torque according to the first strain energy and the formula U1=C3ε3 2Acquiring a first characteristic stiffness parameter C3sObtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strainsIf the metamaterial is achiral, then HsIs 0;
in the second finite element analysis, the second strain and the second curvature are set to ∈ {0,0,0} and κ {0,0, τ } respectively, and the second strain energy U is acquired2According to the second strain energy and formula U2=D3τ2Acquiring a third characteristic stiffness parameter D3s
In the third finite element analysis, the third strain and the third curvature are respectively set to be ∈ ═ epsilon { (epsilon)10,0 and k ═ 0,0,0, and a third strain energy U is obtained3According to the third strain energy and formula U3=C1s(1-Hs 2/C3sD3s1 2Acquiring a fourth characteristic stiffness parameter C1sAnd acquiring a fifth characteristic stiffness parameter C according to the symmetry2s
In the fourth finite element analysis, the fourth strain and the fourth curvature are set to e ═ 0,0,0, and k ═ k { κ { (0, 0, 0) } respectively10,0}, and obtaining fourth strain energy U4According to the fourth strain energy and formula U4=D1s(1-Hs 2/C3sD3s1 2Acquiring a sixth characteristic stiffness parameter D1sAnd obtaining a seventh characteristic stiffness parameter D according to the symmetry2s
In particular, the fourth characteristic stiffness parameter C1sAnd a fifth characteristic stiffness parameter C2sHas symmetry therebetween, then has C1s=C2s. Similarly, the sixth characteristic stiffness parameter D1sAnd a seventh characteristic stiffness parameter D2sBetween, there is D1s=D2s
Wherein, step S103 specifically includes: if the target metamaterial compression bar is formed by repeatedly arranging m × n periodic units, the size of each periodic unit is a, a mechanical model is constructed by combining a plurality of characteristic stiffness parameters, and the overall stiffness matrixes C, B and D of the metamaterial compression bar obtained according to the mechanical model are as follows:
Figure BDA0002921309050000081
Figure BDA0002921309050000082
Figure BDA0002921309050000083
wherein, H is mnHs,C1=C2=mnC1s,C3=mnC3s
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, it is made
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:
θ1(0)=θ2(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0;
for the simply supported beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ′1(0)-ηPθ2(0)=θ′2(0)+ηPθ1(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0。
wherein, step S105 specifically includes: if the design load is larger 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 buckling load, the target metamaterial compression bar is stable, and buckling cannot occur.
The foregoing is a more detailed description of the present invention that is presented in conjunction with specific embodiments, and the practice of the invention is not to be considered limited to those descriptions. For those skilled in the art to which the invention pertains, several simple deductions or substitutions can be made without departing from the spirit of the invention, and all shall be considered as belonging to the protection scope of the invention.

Claims (8)

1. A method for analyzing the stability of a mechanical metamaterial compression bar is characterized by comprising the following steps:
obtaining 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 period unit of a target metamaterial compression bar, and performing finite element analysis on the target period unit according to the finite element analysis model to acquire a plurality of characteristic stiffness parameters;
obtaining an integral rigidity matrix of the target metamaterial compression bar by combining the characteristic rigidity parameters according to the deformation coordination condition of the target metamaterial compression bar;
establishing a buckling control equation, substituting the set boundary condition of the compression bar and the integral rigidity matrix into the buckling control equation, and solving to obtain a critical buckling 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 the stability of the mechanical metamaterial compression bar according to claim 1, wherein the obtaining of the target period unit of the target metamaterial compression bar specifically comprises:
the target metamaterial compression bar is formed by periodically arranging basic units, and the basic units forming the target metamaterial compression bar are used as the target periodic units.
3. The method for analyzing the stability of the mechanical metamaterial compression bar according to claim 1, wherein the step of performing finite element analysis on the target periodic unit according to the finite element analysis model to obtain a plurality of characteristic stiffness parameters specifically comprises:
and performing four finite element analyses on the target periodic unit, loading displacement loads in a specific form in the four finite element analyses, wherein the displacement loads comprise strain and curvature, acquiring strain energy and reaction force in a finite element analysis model according to the displacement loads, and acquiring the characteristic stiffness parameters according to the strain energy and the reaction force.
4. The method for analyzing the stability of the compression bar of the mechanical metamaterial according to claim 3, wherein the quartic finite element analysis includes;
in the first finite element analysis, the first strain and the first curvature are set to ∈ {0,0, ∈ respectively3Acquiring first strain energy U, wherein k is {0,0,0}1And a first reaction torque according to the first strain energy and the formula U1=C3ε3 2Acquiring a first characteristic stiffness parameter C3sObtaining a second characteristic stiffness parameter H according to the ratio of the first reaction torque to the first strainsIf the metamaterial is achiral, then HsIs 0;
in the second finite element analysis, the second strain and the second curvature are set to ∈ {0,0,0} and κ {0,0, τ } respectively, and the second strain energy U is acquired2According to the second strain energy and formula U2=D3τ2Acquiring a third characteristic stiffness parameter D3s
In the third finite element analysis, the third strain and the third curvature are respectively set to be ∈ ═ epsilon { (epsilon)10,0 and k ═ 0,0,0, and a third strain energy U is obtained3According to the third strain energy and formula U3=C1s(1-Hs 2/C3sD3s1 2Acquiring a fourth characteristic stiffness parameter C1sAnd acquiring a fifth characteristic stiffness parameter C according to the symmetry2s
In a fourth finite element analysis, a fourth strain and a fourth curvature are dividedRespectively set as epsilon ═ {0,0,0} and kappa ═ { kappa { (kappa) }10,0}, and obtaining fourth strain energy U4According to the fourth strain energy and equation U4=D1s(1-Hs 2/C3sD3s1 2Acquiring a sixth characteristic stiffness parameter D1sAnd obtaining a seventh characteristic stiffness parameter D according to the symmetry2s
5. The method for analyzing the stability of the mechanical metamaterial compression bar according to claim 4, wherein the obtaining of the overall stiffness matrix of the metamaterial compression bar according to the deformation coordination condition of the target metamaterial compression bar by combining the plurality of characteristic stiffness parameters specifically comprises:
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 integral stiffness matrixes C, B and D of the target metamaterial compression bar according to the mechanical model, wherein the integral stiffness matrixes C, B and D are as follows:
Figure FDA0002921309040000021
Figure FDA0002921309040000022
Figure FDA0002921309040000023
wherein, H is mnHs,C1=C2=mnC1s,C3=mnC3s
Figure FDA0002921309040000024
Figure FDA0002921309040000031
Figure FDA0002921309040000032
6. The method for analyzing the stability of the mechanical metamaterial compression bar according to claim 5, wherein the buckling control equation is as follows:
Figure FDA0002921309040000033
Figure FDA0002921309040000034
simultaneous equations (1) and (2), the general solution is obtained as:
Figure FDA0002921309040000035
Figure FDA0002921309040000036
wherein, it is made
Figure FDA0002921309040000037
Figure FDA0002921309040000038
Then there are:
Figure FDA0002921309040000039
Figure FDA00029213090400000310
Figure FDA0002921309040000041
7. the method for analyzing the stability of the mechanical metamaterial compression bar according to claim 6, wherein the boundary conditions include:
for the cantilever beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ1(0)=θ2(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0;
for the simply supported beam of the target metamaterial compression bar, the boundary conditions are as follows:
θ′1(0)-ηPθ2(0)=θ′2(0)+ηPθ1(0)=θ′1(L)-ηPθ2(L)=θ′2(L)+ηPθ1(L)=0。
8. the method for analyzing the stability of the mechanical metamaterial compression bar according to claim 1, wherein the comparing the design load with the critical destabilizing load to judge the stability of the target metamaterial compression bar specifically comprises:
if the design load is larger than the critical buckling load, the target metamaterial compression bar is not stable, and buckling can occur to cause buckling;
and if the design load is less than or equal to the critical buckling load, the target metamaterial is stable, and buckling cannot occur.
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