CN110472320A - A kind of numerical method calculating the thermal buckling of graphene composite material laminate - Google Patents
A kind of numerical method calculating the thermal buckling of graphene composite material laminate Download PDFInfo
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- CN110472320A CN110472320A CN201910701259.1A CN201910701259A CN110472320A CN 110472320 A CN110472320 A CN 110472320A CN 201910701259 A CN201910701259 A CN 201910701259A CN 110472320 A CN110472320 A CN 110472320A
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Abstract
The invention discloses a kind of numerical methods for calculating the thermal buckling of graphene composite material laminate, based on first order shear deformation theory, the elastic parameter of composite material is calculated using the Halpin-Tsai model of extension, and Buckling Equation is obtained by Hamiton's principle, weak variation is finally carried out using non-mesh method and obtains discrete Buckling Equation, has finally calculated the buckling critical-temperature under laminate difference width-thickness ratio, boundary condition, laminated angle and initial temperature.Since graphene composite material has excellent mechanical property, the present invention can provide certain design for fields such as aerospace, automobiles and help.
Description
Technical field
The invention belongs to study thin plate stability problem, and in particular to a kind of calculating graphene composite material laminate heat is in the wrong
Bent numerical method.
Background technique
MATLAB language be mathematical problem computer solving, especially Control System Imitation and CAD development play it is huge
Big impetus.Its numerical operation function is extremely outstanding, and its tool box can efficiently, reliably solve it is various
Problem.
With the development of industrial technology, general material is no longer satisfied actual needs, and composite material is more taken seriously.
In view of the working environment of workpiece complexity, the thermal buckling effect for studying thin plate has comparable importance.
201810089235.0 disclose " numerical algorithm of the FG-GRC bucking of plate load factor based on gridless routing ", point
Analyse the buckling load factor of FG-GRC plate, but the influence without research temperature change to buckling.
Summary of the invention
The purpose of the present invention is to provide a kind of numerical methods for calculating the thermal buckling of graphene composite material laminate, are based on
First order shear deformation theory calculates the buckling critical-temperature of thin plate using mesh free kp-Ritz method.
The technical solution for realizing the aim of the invention is as follows: a kind of number calculating the thermal buckling of graphene composite material laminate
Value method, comprising the following steps:
The initial temperature of equally distributed graphene composite material laminate, sets simultaneously when step 1, determination are without temperature stress
Determine graphene content, base material parameter and laminated angle, by the Halpin-Tsai model of extension, it is compound to calculate graphene
The elastic constant of material laminate, is transferred to step 2；
Step 2 is based on first order shear deformation theory, in conjunction with Cauchy strain tensor and generalized Hooke law, calculates graphite
The strain energy of alkene composite laminated plate, is transferred to step 3；
Step 3, the thermal buckling equation that graphene composite material laminate is obtained by Hamiton's principle, are transferred to step 4；
Step 4, the buckling using the discrete graphene composite material laminate of mesh free kp-Ritz method, after acquisition is discrete
Equation carries out numerical value calculating by MATLAB, acquires the buckling temperature of graphene composite material laminate.
Compared with prior art, the present invention its remarkable advantage is:
(1) Modulus of Composites is calculated using newest extension Halpin-Tsai model, so that result is more quasi-
Really.
(2) non-mesh method is used, method is more novel.
Detailed description of the invention
Fig. 1 is the flow chart for the numerical method that the present invention calculates the thermal buckling of graphene composite material laminate.
Fig. 2 is that graphene of the present invention enhances composite panel schematic diagram.
Fig. 3 is that 300K graphene enhances composite panel [0] in the embodiment of the present inventions[0 pi/2,0 pi/2 0]sFace
Boundary's temperature line chart, wherein figure (a) is [0]sCritical-temperature line chart, figure (b) be [0 pi/2,0 pi/2 0]sCritical-temperature
Line chart.
Fig. 4 is that 400K graphene enhances composite panel [0] in the embodiment of the present inventions[0 pi/2,0 pi/2 0]sFace
Boundary's temperature line chart, wherein figure (a) is [0]sCritical-temperature line chart, figure (b) be [0 pi/2,0 pi/2 0]sCritical-temperature
Line chart.
Fig. 5 is 300K graphene composite material plate buckling shape ([0 pi/2,0 pi/2 at SSSS in the embodiment of the present invention
0]s) figure.
Fig. 6 is 300K graphene composite material plate buckling shape ([0 pi/2,0 pi/2 at CCCC in the embodiment of the present invention
0]s) figure.
Fig. 7 is 300K graphene composite material plate buckling shape ([0 pi/2,0 pi/2 at SCSC in the embodiment of the present invention
0]s) figure.
Fig. 8 is 300K graphene composite material plate buckling shape ([0 pi/2,0 pi/2 at CCFF in the embodiment of the present invention
0]s) figure.
Note: the above sheet thickness h is 2mm, and length-width ratio a/b is 1.
Specific embodiment
In conjunction with Fig. 1, a kind of numerical method calculating the thermal buckling of graphene composite material laminate of the present invention, step
It is as follows:
Initial temperature when step 1, determination are without temperature stress, setting graphene content, base material parameter and laminated angle
Degree calculates the elastic constant of graphene composite material laminate, is transferred to step 2；
Halpin-Tsai model according to extension calculates equally distributed graphene composite material elastic modulus E11、E22、
G12With Poisson's ratio ν12,
E in formulam、GmAnd VGIt is the volume ratio of the Young's modulus of substrate, modulus of shearing and graphene respectively,And vmIt is stone
The Poisson's ratio of black alkene and substrate, aG、bGAnd hGIt is the length and width and effective thickness geometric parameter of graphene film, VGIt is the volume of substrate
Than η1、η2And η3It is coefficient variation, variable γ11、22And γ33It is obtained by following relational expression
In formulaWithIt is the elasticity modulus of graphene, the temperature expansion coefficient α of composite material11And α22For
In formulaIt is expansion of graphene coefficient, αmIt is the coefficient of expansion of substrate.
Step 2 is based on first order shear deformation theory, in conjunction with Cauchy strain tensor and generalized Hooke law, calculates graphite
Alkene composite laminated plate strain energy, is transferred to step 3；
Graphene composite material laminate length is a, width b, with a thickness of h, is based on first order shear deformation theory, by
Cauchy strain tensor and generalized Hooke law calculate the strain energy of laminate.First order shear deformation theory is as follows:
U, v and w are graphene composite material laminates respectively in the displacement in the direction x, y and z, u in formula0、v0And w0It is graphite
Alkene composite laminated plate neutral surface is respectively in the displacement in the direction x, y and z, φxAnd φyFor neutral surface corner；Z is undeformed stone
The coordinate of point in a z-direction on black alkene composite laminated plate.
Geometric equation is obtained according to Cauchy strain tensor, the strain stress of laminate is
Wherein ε0For plane strain, εLIt is neutral surface in the strain of x/y plane, κ is torsional strain, and γ is shear strain.
Stress σ is obtained by generalized Hooke law to be expressed as follows
Δ T is temperature variation, Q in formula11、Q12、Q22、Q66、Q55And Q44There is following relationship:
Wherein G23And G13For modulus of shearing, G23=G13=G12, ν21For Poisson's ratio.
Graphene composite material laminate (thin plate) face internal force N, moment M, shearing QsWith temperature stress NtAnd moment MtRespectively
Are as follows:
Graphene composite material laminate layer close angle degree is θ, equivalent elastic constantIt is as follows with transition matrix T:
Wherein elastic constant matrix Q and equivalent elastic constant matrixIt is expressed as follows:
Enable variables Aij,Bij,DijIt respectively indicates as follows:
In formulaN is the number of plies of graphene composite material laminate, tk+1And tkIt is unchanged
A little respectively in the coordinate of z-axis on bottom surface a little corresponding on shape graphene composite material laminate kth layer top surface.
So Bulk stiffness matrix [A] and [B] and [D] and [As] be respectively
Equivalent temperature stressWith temperature moment of flexureFor
Strain energy can be written as
I.e.
Equivalent temperature stress in formulaWith equivalent temperature moment of flexureIt is expressed as follows
In formulaWithIt is equivalent temperature stress,For equivalent temperature moment of flexure.
Step 3 obtains laminate thermal buckling equation by Hamiton's principle, is transferred to step 4；
Variation by Hamiton's principle energy ∏ is zero, i.e.,
δ Π=δ (U-T)=0 (24)
Consider thermal buckling situation, kinetic energy T is zero, i.e. the variation of strain energy is zero, available following equilibrium equation:
Step 4 is carried out discrete, acquisition discrete equation by mesh free kp-Ritz method to graphene composite material laminate,
It is specific as follows:
Discrete node I=1,2 ..., NP, u in formulaIIt is the displacement of i-th node, ψI(x) be i-th node shape letter
Number, uI,vI,wIIt is the displacement in the direction x, y and z of i-th node, φxI,φyIIt is the bending angle of i-th node.Shape letter
Number is specific as follows:
ψI(x)=C (x；x-xI)fa(x-xI) (28)
F in formulaa(x-xI) it is kernel function, coefficient function C (x；x-xI) it is as follows:
X in formulaI,yIFor coordinate of the i-th node on x, the direction y, x, y are coordinate variable.H is secondary base vector, and B is
About the coefficient function of coordinate, obtained by following relationship:
Wherein M (x) is matrix of variables；
Kernel function fa(x-xI) it is expressed as follows:
A in formulaI=dmax·c！, dmaxIt is the scale parameter of control node domain of influence size, between 2.0 to 3.0, cIIt is
Node xITo the distance of nearest adjoint point,For cubic spline function；
Weak variation is carried out by (17), obtains Buckling Equation using non-mesh method
(K-λKg) u=0 (33)
K is the stiffness matrix of graphene composite material laminate, K in formulagIt is bent for the laminated plate temperature of graphene composite material
Stiffness matrix, temperature knots modification when thus λ is desired buckling are specific as follows:
K=Kb+Km+Ks (34)
Wherein, Kb,Km,KsRespectively bending resistance, tension and shear stiffness matrix, their I row J column are expressed as follows:
Wherein matrix of variablesIt is expressed as follows:
By matrix of variablesψ in expression formulaIIt is changed to ψJ, three matrixes are matrix of variablesTable
Up to formula.
The wherein laminated plate temperature flexion stiffness matrix K of graphene composite materialgI row J column be expressed as follows:
Wherein matrix of variablesWith equivalent temperature stress matrixIt is expressed as follows:
By matrix of variablesψ in expression formulaIIt is changed to ψJ, matrix is matrix of variablesExpression formula.
Embodiment
In conjunction with Fig. 1, a kind of numerical method calculating the thermal buckling of graphene composite material laminate of the present invention, step
It is as follows:
Initial temperature when step 1, determination are without temperature stress, setting graphene content, base material parameter and laminated angle
Degree calculates the elastic constant of graphene composite material laminate, elasticity modulus such as 1 institute of table when graphene 300K, 400K and 500K
Show, shown in springform scale 2 of the graphene composite material laminate in 300K, 400K and 500K, is transferred to step 2；
Elasticity modulus (the ν of 1 graphene of tableG=0.177)
Elastic constant (the V of 2 graphene composite material laminate of tableg=0.07)
Step 2, in conjunction with Fig. 2, be based on first order shear deformation theory, in conjunction with Cauchy strain tensor and generalized Hooke law,
The strain energy for obtaining graphene composite material laminate, is transferred to step 3；
Step 3, the thermal buckling equation that graphene composite material laminate is obtained by Hamiton's principle, are transferred to step 4；
Step 4, the buckling using the discrete graphene composite material laminate of mesh free kp-Ritz method, after acquisition is discrete
Equation carries out numerical value calculating by MATLAB, acquires the buckling temperature of graphene composite material laminate.The present invention calculates separately
Difference initial temperature (300K, 400K and 500K) when graphene content is 0.07, various boundary (simply supported on four sides SSSS,
Arbitrary loading CCCC, two opposite side freely-supporteds and two opposite clamped edges SCSC and two adjacent sides it is clamped with the free CCFF of two adjacent sides), it is different generous
Than with different laminar ways ([0]s[0 pi/2,0 pi/2 0]s) under critical-temperature Tcr[K], situation such as table 3, table 4 and table
Shown in 5.In conjunction with Fig. 5, Fig. 6, Fig. 7 and Fig. 8, above-mentioned boundary condition is all satisfied essential boundary condition.
Critical-temperature T when 3 300K of table when graphene content 0.07cr[K]
Critical-temperature T when 4 400K of table when graphene content 0.07cr[K]
Critical-temperature T when 5 500K of table when graphene content 0.07cr[K]
Fig. 3 is drawn by table 3, table 4 draws Fig. 4, and combines Fig. 3 and Fig. 4, and critical-temperature is obvious when clearly demonstrating that arbitrary loading
Higher than other boundary conditions, it is suitable for the biggish working environment of temperature change, with the increase of width-thickness ratio, critical-temperature is obvious
Reduce, simultaneously [0 pi/2,0 pi/2 0]sCritical-temperature ratio [0]sWant high.
Claims (5)
1. a kind of numerical method for calculating the thermal buckling of graphene composite material laminate, which comprises the following steps:
The initial temperature of equally distributed graphene composite material laminate, concurrently sets stone when step 1, determination are without temperature stress
Black alkene content, base material parameter and laminated angle calculate graphene composite material by the Halpin-Tsai model of extension
The elastic constant of laminate, is transferred to step 2；
Step 2 is based on first order shear deformation theory, and in conjunction with Cauchy strain tensor and generalized Hooke law, it is multiple to calculate graphene
The strain energy of condensation material laminate, is transferred to step 3；
Step 3, the thermal buckling equation that graphene composite material laminate is obtained by Hamiton's principle, are transferred to step 4；
Step 4, using the discrete graphene composite material laminate of mesh free kp-Ritz method, obtain it is discrete after Buckling Equation,
Numerical value calculating is carried out by MATLAB, acquires the buckling temperature of graphene composite material laminate.
2. the numerical method according to claim 1 for calculating the thermal buckling of graphene composite material laminate, it is characterised in that:
In step 1, the elasticity modulus of equally distributed graphene composite material laminate is calculated by the Halpin-Tsai model of extension
E11、E22、G12With Poisson's ratio ν12:
E in formulam、GmAnd VGIt is the volume ratio of the Young's modulus of substrate, modulus of shearing and graphene respectively,And νmIt is graphene
With the Poisson's ratio of substrate, aG、bGAnd hGIt is the length and width and effective thickness geometric parameter of graphene film, VGIt is the volume ratio of substrate,
η1、η2And η3It is coefficient variation, variable γ11、γ22And γ33It is obtained by following relational expression
In formulaWithIt is the elasticity modulus of graphene, the temperature expansion coefficient α of graphene composite material laminate11
And α22For
In formulaIt is expansion of graphene coefficient, αmIt is the coefficient of expansion of substrate.
3. the numerical method according to claim 1 for calculating the thermal buckling of graphene composite material laminate, it is characterised in that:
In step 2, it is based on first order shear deformation theory, by Cauchy strain tensor and generalized Hooke law, calculates graphene composite wood
The strain energy of bed of material plywood, specific as follows:
Graphene composite material laminate length is a, width b, and with a thickness of h, first order shear deformation theory is as follows:
U, v and w are graphene composite material laminates respectively in the displacement in the direction x, y and z, u in formula0、v0And w0It is that graphene is multiple
Condensation material laminate neutral surface is respectively in the displacement in the direction x, y and z, φxAnd φyFor neutral surface corner；Z is undeformed graphene
The coordinate of point in a z-direction on composite laminated plate；
Geometric equation is obtained according to Cauchy strain tensor, the strain stress of graphene composite material laminate is
Wherein ε0For plane strain, εLIt is neutral surface in the strain of x/y plane, κ is torsional strain, and γ is shear strain；
Stress σ is obtained by generalized Hooke law to be expressed as follows:
Δ T is temperature variation, elastic constant Q in formula11、Q12、Q22、Q66、Q55And Q44There is following relationship:
Wherein G23And G13For modulus of shearing, G23=G13=G12, ν21For Poisson's ratio；
Face internal force N, moment M, the shearing Q of graphene composite material laminatesWith temperature stress NtAnd moment MtIt is respectively as follows:
So, the strain energy U of graphene composite material laminate
Enable variables Aij,Bij,DijIt respectively indicates as follows:
In formulaN is the number of plies of graphene composite material laminate, tk+1And tkIt is undeformed stone
A little respectively in the coordinate of z-axis on bottom surface a little corresponding on black alkene composite laminated plate kth layer top surface, graphene is multiple
Condensation material laminate layer close angle degree is θ, equivalent elastic constant matrixIt is as follows with transition matrix T:
Q is elastic constant matrix in formula.
4. the numerical method according to claim 1 for calculating the thermal buckling of graphene composite material laminate, it is characterised in that:
In step 3, the thermal buckling equation of laminate is obtained by Hamiton's principle, specific as follows:
Hamiton's principle δ Π=δ (U-T)=0 is zero to the variation П of energy since kinetic energy T perseverance is zero, the side of being balanced
Journey:
In formulaWithIt is equivalent temperature stress, Nxx,Nxy,NyyFor face internal force, Mxx,Mxy,MyyFor moment of flexure, Qx,Qy
Shearing.
5. the numerical method according to claim 1 for calculating the thermal buckling of graphene composite material laminate, it is characterised in that:
In step 4, discrete, acquisition discrete equation is carried out to graphene composite material laminate by mesh free kp-Ritz method, specifically such as
Under:
Discrete node I=1,2 ..., NP, u in formulaIIt is the displacement of i-th node, ψI(x) be i-th node shape function,
uI,vI,wIIt is the displacement in the direction x, y and z of i-th node, φxI,φyIIt is the bending angle of i-th node, shape function tool
Body is as follows:
ψI(x)=C (x；x-xI)fa(x-xI) (28)
F in formulaa(x-xI) it is kernel function, coefficient function C (x；x-xI) it is as follows:
X in formulaI,yIFor coordinate of the i-th node on x, the direction y, x, y are coordinate variable.H is secondary base vector,
B is the coefficient function about coordinate, is obtained by following relationship:
Wherein M (x) is matrix of variables；
Kernel function fa(x-xI) it is expressed as follows:
A in formulaI=dmax·c！, dmaxIt is the scale parameter of control node domain of influence size, between 2.0 to 3.0, cIIt is node xI
To the distance of nearest adjoint point,For cubic spline function；
Weak variation is carried out by (17), obtains Buckling Equation using non-mesh method
(K-λKg) u=0 (33)
K is the stiffness matrix of graphene composite material laminate, K in formulagFor the laminated plate temperature flexion stiffness of graphene composite material
Matrix, temperature knots modification when thus λ is desired buckling are specific as follows:
K=Kb+Km+Ks (34)
Wherein, Kb,Km,KsRespectively bending resistance, tension and shear stiffness matrix.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN114512206A (en) * | 2022-04-21 | 2022-05-17 | 中国飞机强度研究所 | Airplane wallboard thermal buckling critical temperature determination method based on inflection point method |
| CN114512205A (en) * | 2022-04-21 | 2022-05-17 | 中国飞机强度研究所 | Thermal buckling critical temperature analysis method for aircraft wall panel |
-
2019
- 2019-07-31 CN CN201910701259.1A patent/CN110472320A/en not_active Withdrawn
Non-Patent Citations (1)
| Title |
|---|
| ZUXIANG LEI等: "Parametric studies on buckling behavior of functionally graded graphenereinforced composites laminated plates in thermal environment", 《COMPOSITE STRUCTURES》 * |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN114512206A (en) * | 2022-04-21 | 2022-05-17 | 中国飞机强度研究所 | Airplane wallboard thermal buckling critical temperature determination method based on inflection point method |
| CN114512205A (en) * | 2022-04-21 | 2022-05-17 | 中国飞机强度研究所 | Thermal buckling critical temperature analysis method for aircraft wall panel |
| CN114512205B (en) * | 2022-04-21 | 2022-07-19 | 中国飞机强度研究所 | Thermal buckling critical temperature analysis method for aircraft wall panel |
| CN114512206B (en) * | 2022-04-21 | 2022-07-19 | 中国飞机强度研究所 | Airplane wallboard thermal buckling critical temperature determination method based on inflection point method |
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