Rigidity prediction method of carbon nanotube reinforced composite material laminated plate

Abstract

The invention discloses a rigidity prediction method of a carbon nano tube reinforced composite material laminated plate, a geometric accurate two-dimensional beam section modeling method is adopted to perform structural modeling on a carbon nano tube composite material beam, and the method can perform structural modeling on a thin beam under any boundary condition and any deformation, including large deformation displacement and section warping. And obtaining a 6 * 6 section stiffness matrix of the composite material beam, wherein the matrix elements comprise stretching, shearing, torsion and corresponding coupling stiffness coefficients. The method is suitable for predicting the rigidity of the carbon nanotube reinforced composite material laminated plate under the conditions of small deformation, medium deformation and large deformation (namely any deformation).

Description

Technical Field

[0001] The present invention relates to the field of helicopter composite materials, and in particular to a method for predicting the stiffness of a carbon nanotube reinforced composite laminate. Background Art

[0002] Composite laminates are composite structures formed by stacking multiple single-layer plates in a specific pattern. Their unique multilayer construction represents a richer set of mechanical properties than single-layer plates. Therefore, this multilayered structure offers significant flexibility and controllability in structural design. Composite beams, a specific form of composite plate structure, are characterized by a geometric dimension significantly greater than their width and height. Depending on the load type to which the laminate is subjected, composite beams can be subdivided into laminated beams, plate beams, and thin-walled beams. In helicopter engineering, composite beams are widely used as load-bearing structures. Thin-walled beams, due to their unique structural advantages, are generally preferred as the primary component for bearing bending loads. Carbon nanotubes (CNTs) possess excellent mechanical properties, a large aspect ratio, and a high specific surface area, making them attractive polymer-matrix fillers. The large specific surface area of ​​CNTs enhances interaction with the resin matrix, resulting in higher stress transfer and energy dissipation, significantly improving the stiffness, strength, and toughness of the composite. CNT-reinforced composite laminates exhibit superior mechanical properties, such as increased stiffness and strength, compared to conventional composite laminates. Existing methods for predicting the stiffness of carbon nanotube-reinforced composite laminates are primarily based on classical beam theory. However, these methods are limited to predicting stiffness under small and moderate deformations and cannot accurately predict stiffness under large deformations. Therefore, there is a need for a stiffness prediction method for carbon nanotube-reinforced composite laminates that can be applied to small, moderate, and large (i.e., arbitrary) deformations. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to address the defect that the existing methods involved in the background technology cannot accurately predict the stiffness under large deformation conditions, and provide a stiffness prediction method for carbon nanotube reinforced composite laminates. The provided method can be applied to the stiffness prediction of carbon nanotube reinforced composite laminates under small deformation, medium deformation and large deformation (i.e., arbitrary deformation).

[0004] The present invention adopts the following technical solutions to solve the above technical problems:

[0005] A method for predicting the stiffness of a carbon nanotube reinforced composite laminate comprises the following steps:

[0006] Step 1), establishing a beam deformation coordinate system to describe the deformation of the beam of the carbon nanotube reinforced composite laminate;

[0007] Let x1 be the arc length coordinate along the axis r of the beam before deformation, the normal of the beam section at any x1 is along the tangent direction of the axis r, and the coordinates of any point in the beam section are represented by x2 and x3; introduce a set of orthogonal undeformed reference coordinate vectors b1, b2, and b3 at the point where the axis r before deformation of the beam intersects the beam section at x1, where b1 is along the tangent direction of the undeformed beam axis, b1=b2×b3, b2 and b3 are along the coordinate axes x2 and x3 respectively, and the position of the material point represented by each point on the beam passes through the position vector of any spatial point P Determine; introduce a set of orthogonal deformed reference coordinate systems T1, T2, and T3 at the point where the deformed axis R of the beam intersects the beam section. T1 is tangent to the deformed axis of the beam, and T2 and T3 are determined by T1.

[0008] Step 2) Set a boundary for the warping displacement phenomenon that occurs when the beam deforms, and calculate the strain at any point in the beam in coordinate system b;

[0009] Step 3) Considering the warping deformation, the finite element method is used to discretize the warping deformation and calculate the cross-sectional strain energy at this time;

[0010] Step 4), considering shear deformation, derive the one-dimensional generalized shear strain, substitute it into the cross-sectional strain energy formula of the beam, and obtain the generalized Timoshenko strain energy;

[0011] Step 5), for the condition of static analysis without external distributed forces and moments, the composite beam has initial torsion and bending, and a one-dimensional nonlinear equation system of the beam is obtained;

[0012] Step 6) Solve the derivative and transformation relationship of Timoshenko one-dimensional generalized strain with respect to x1, combine the standard equation of generalized Timoshenko strain energy and its corresponding constitutive equation, solve the nonlinear equation system expression related to the stiffness coefficient matrix, and then recombine the expression to obtain the cross-sectional equation system expression corresponding to the generalized Timoshenko strain energy and Timoshenko one-dimensional generalized strain, and then obtain the cross-sectional stiffness matrix; finally, the stiffness of the carbon nanotube reinforced composite laminate is obtained according to the cross-sectional stiffness matrix.

[0013] As a further optimization scheme of the stiffness prediction method of a carbon nanotube reinforced composite laminate of the present invention, the classical one-dimensional strain vector of any point in the beam in tension, torsion and bending in two directions in the coordinate system b in step 2) is The expression is:

[0014]

[0015] Where, Γ={Γ 11 ,2Γ 12 ,2Γ 13 ,Γ 22 ,2Γ23 ,Γ 33} T , w is the warping deformation, and the respective meanings of the matrices are:

[0016]

[0017] Where k = {k1, k2, k3} T , Δ represents the identity matrix, O3 represents the 3×3 zero matrix,

[0018] As a further optimization scheme of the stiffness prediction method of a carbon nanotube reinforced composite laminate of the present invention, the specific steps of calculating the strain energy during discrete warping deformation in step 3) are as follows:

[0019] Considering the warping deformation, the expression of the cross-sectional strain energy is as follows:

[0020]

[0021] Where D represents the stiffness matrix,

[0022] Use the finite element method to discretize the warping deformation:

[0023] w(x1,x2,x3)=N(x2,x3)V(x1);

[0024] Where N(x2,x3) represents the element shape function matrix, V(x1) is the warping displacement of the discrete node;

[0025] Substituting into the cross-sectional strain energy expression, the strain Γ is simplified to:

[0026]

[0027] The strain energy is further expressed as:

[0028]

[0029] Where,

[0030]

[0031] As a further optimization scheme of the stiffness prediction method of the carbon nanotube reinforced composite laminate of the present invention, the expression of the Timoshenko one-dimensional generalized strain in step 4) is as follows:

[0032] γ= C bB R′-r′=γ 11 b1+2γ 12 b2+2γ 13b3;

[0033] κ= C bB Kk = κ1b1 + κ2b2 + κ3b3;

[0034] Where K = K i B i represents the curvature of the beam in shear deformation, C bB is the rotation tensor; γ 11 , 2γ 12 , 2γ 13 , k1, k2, k3 represent the Timoshenko one-dimensional generalized strains corresponding to tension, transverse shear in two directions, torsion, and bending in two directions after the beam is deformed, respectively. Let ε = {γ 11 ,κ1,κ2,κ3} T , γ s ={2γ 12 ,2γ 13} T ,γ={γ 11 ,2γ 12 ,2γ 13} T , κ={κ1,κ2,κ3} T ;

[0035] The following transformation equation exists between coordinate system T and coordinate system B:

[0036]

[0037] Through the conversion relationship of the above formula, the Timoshenko one-dimensional generalized strain expression is obtained as follows:

[0038]

[0039] Where,

[0040] Substituting Timoshenko's one-dimensional generalized strain expression into the expression of the cross-sectional strain energy of the beam, we obtain the further optimized quadratic asymptotically accurate strain energy. The relevant expression is as follows:

[0041]

[0042] The standard equation for the generalized Timoshenko strain energy is:

[0043]

[0044] The corresponding constitutive equation is:

[0045]

[0046] Where, F i 、M i are the resultant force and moment of the deformed beam; and in the above formula, the matrices X, Y, and G are unknown, X is a 4×4 matrix, Y is a 4×2 matrix, and G is a 2×2 matrix.

[0047] As a further optimization scheme of the stiffness prediction method of the carbon nanotube reinforced composite laminate of the present invention, the one-dimensional nonlinear equation group of the beam in step 5) is:

[0048]

[0049] Where F = {F1, F2, F3} T ,M={M1,M2,M3} T ,e1={1,0,0} T ;

[0050] In order to establish a two-dimensional cross-section analysis model, according to the order of magnitude used in this paper, some high-order terms are ignored, so the above formula is simplified to the following expression:

[0051] {F2′F3} T +D1{F2 F3} T +D2{F1 M1 M2 M3} T =0;

[0052] {F1′ M1′ M′2 M3} T +D3{F1 M1 M2 M3} T +D4{F2 F3} T =0;

[0053] Where,

[0054] As a further optimization scheme of the stiffness prediction method of a carbon nanotube reinforced composite laminate of the present invention, the specific steps of step 6) are as follows:

[0055] Substituting the one-dimensional nonlinear equations of the beam into the two-dimensional cross-sectional analysis model yields the following expression:

[0056] Y T ε′+Gγ′ s +D1(Y T ε+Gγ s )+D2(Xε+Yγ s )=0;

[0057] Xε′+Yγ′ s +D3(Xε+Yγ s )+D4(Y T ε+Gγs )=0;

[0058] The derivative and transformation relationship of Timoshenko one-dimensional generalized strain with respect to x1 are solved from the above equations. Then, combined with the standard equation of generalized Timoshenko strain energy and its corresponding constitutive equation, the nonlinear equations related to matrices X, Y, and G are solved. The solved nonlinear equations of X, Y, and G are recombined to obtain the cross-sectional equations corresponding to the generalized Timoshenko strain energy and Timoshenko one-dimensional generalized strain:

[0059]

[0060] The 6×6 matrix in the above formula is the section stiffness matrix calculated by Timoshenko beam theory; the subscript 1 represents tension, 2 and 3 represent shear in two directions, 4 represents torsion, and 5 and 6 represent bending in two directions; the main diagonal elements represent the main stiffness coefficients; the off-diagonal elements represent the coupling stiffness coefficients.

[0061] Compared with the prior art, the present invention adopts the above technical solution and has the following technical effects:

[0062] This paper uses a geometrically precise two-dimensional beam cross-section modeling method to structurally model carbon nanotube composite beams. This method can model thin beams with arbitrary boundary conditions and deformations, including large deformation displacements and cross-sectional warping. A 6×6 cross-sectional stiffness matrix for the composite beam is obtained, with matrix elements encompassing tensile, shear, torsion, and the corresponding coupled stiffness coefficients. This method is simple, practical, and highly accurate, capable of predicting the stiffness of thin carbon nanotube composite beams. This method provides essential reference values ​​for helicopter rotor blade design, and possesses significant academic and engineering value. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 Schematic diagram of the deformation coordinate system of the composite beam;

[0064] Figure 2 Effect of ply angle on axial tensile stiffness of carbon nanotube reinforced composite laminates;

[0065] Figure 3 Carbon fiber layup is [45° / 0° / 45°] 2s Schematic cross-sectional view of a carbon nanotube reinforced composite laminate;

[0066] Figure 4 The deflection of the end of a composite laminate reinforced with carbon nanotubes containing different mass fractions under the action of vertical force at the end;

[0067] Figure 5 Torsion angle at the end of a composite laminate reinforced with carbon nanotubes containing different mass fractions under end torque. DETAILED DESCRIPTION

[0068] The technical solution of the present invention is further described in detail below with reference to the accompanying drawings:

[0069] The present invention can be implemented in many different forms and should not be considered to be limited to the embodiments described herein. On the contrary, these embodiments are provided to make this disclosure thorough and complete and will fully convey the scope of the invention to those skilled in the art. In the accompanying drawings, components are enlarged for clarity.

[0070] This application provides a method for predicting the stiffness of a carbon nanotube reinforced composite laminate, which is described using the application of the method to a terminal as an example, and includes the following steps:

[0071] S1, such as Figure 1 As shown, the coordinate system of the beam deformation is established; the reference line r is specified to represent the axis of the undeformed beam, x1 represents the arc length coordinate on this axis, and x2 and x3 represent the cross-sectional coordinates along the axis of the beam. A set of orthogonal undeformed reference coordinate vectors b1, b2, and b3 are introduced here. b1 is along the tangential direction of the undeformed beam axis, b1 = b2 × b3, and b2 and b3 are along the coordinate axes x2 and x3 respectively. The position of the material point represented by each point on the beam is represented by the position vector Determine. P is an arbitrary point in space, s represents the arc length coordinate along the beam axis R after deformation, and a set of orthogonal deformed reference coordinate systems T1, T2, and T3 are introduced along the deformed beam axis. T1 is tangent to the deformed reference line of the beam, and T2 and T3 are determined by T1. These two coordinate systems are used to describe the deformation of the entire beam.

[0072] S2. Set boundaries for the warping displacement phenomenon that occurs when the beam deforms, and calculate the strain at any point in the beam in coordinate system b. The strain expression is:

[0073]

[0074] Where, Γ={Γ 11 ,2Γ 12 ,2Γ 13 ,Γ 22 ,2Γ 23 ,Γ 33} T , the respective meanings of the matrices are:

[0075]

[0076] Where k = {k1, k2, k3} T , Δ represents the identity matrix, O3 represents the 3×3 zero matrix,

[0077] S3. Considering the warping deformation, the finite element method is used to discretize the warping deformation and calculate the strain energy at this time. The expression is:

[0078]

[0079] Where D represents the stiffness matrix,

[0080] Use the finite element method to discretize the warping deformation:

[0081] w(x1,x2,x3)=N(x2,x3)V(x1)

[0082] Where N(x2,x3) represents the element shape function matrix, and V(x1) is the warping displacement of the discrete node.

[0083] Substituting into the cross-sectional strain energy expression, the strain Γ is simplified to:

[0084]

[0085] The strain energy can be further expressed as:

[0086]

[0087] Where,

[0088] E=<<[Γ a N] T D[Γ a N]>> D aε =<<[Γ a N] T D[Γ ε ]>>

[0089] D aR =<<[Γ a N] T D[Γ R N]>> D al =<<[Γ d N] T L[Γ l N]>>

[0090] D εε =<<[Γ ε ] T D[Γ ε ]>> D RR =<<[Γ R N] T D[Γ R N]>>

[0091] D ll =<<[Γl N] T D[Γ l N]>> D Rε =<<[Γ R N] T D[Γ ε ]>>

[0092] D lε =<<[Γ l N] T D[Γ ε ]>>D Rl =<<[Γ R N] T D[Γ ε ]>>

[0093] S4. Considering shear deformation, we can derive the one-dimensional generalized shear strain and substitute it into the cross-sectional strain energy formula of the beam to obtain the generalized Timoshenko strain energy. The one-dimensional generalized Timoshenko strain can be expressed as follows:

[0094] γ= C bB R′-r′=γ 11 b1+2γ 12 b2+2γ 13 b3

[0095] κ= C bB Kk=κ1b1+k2b2+κ3b3

[0096] Where K = K i B i represents the curvature of the beam in shear deformation, C bB is the rotation tensor. γ 11 , 2γ 12 , 2γ 13 , κ1, κ2, k3 represent the Timoshenko one-dimensional generalized strains corresponding to tension, transverse shear in two directions, torsion, and bending in two directions after the beam is deformed, respectively. Let ε = {γ 11 ,k1,k2,k3} T , γ s ={2γ 12 ,2γ 13} T ,γ={γ 11 ,2γ 12 ,2γ 13} T , κ={κ1,κ2,κ3} T .

[0097] The following transformation equation exists between coordinate system T and coordinate system B:

[0098]

[0099] Through the conversion relationship of the above formula, the Timoshenko one-dimensional generalized strain expression can be obtained as:

[0100]

[0101] Where,

[0102] Substituting Timoshenko's one-dimensional generalized strain expression into the expression of the cross-sectional strain energy of the beam, we can obtain a further optimized quadratic asymptotically accurate strain energy. The relevant expression is as follows:

[0103] 2U=(ε+Qγ′ s +Pγ s ) T A(ε+Qγ′ s +Pγ s )+2(ε+Qγ′ s +Pγ s ) T

[0104] B(ε+Qγ′ s +Pγ s )′+(ε+Qγ′ s +Pγ s ) ′T C(ε+Qγ′ s +Pγ s )′

[0105] +2(ε+Qγ′ s +Pγ s ) T D(ε+Qγ′ s +Pγ s )″

[0106] The standard equation for the generalized Timoshenko strain energy is:

[0107]

[0108] The corresponding constitutive equation is:

[0109]

[0110] Where, F i 、M i are the cross-sectional force and moment of the deformed beam. In the above formula, the matrices X, Y, and G are unknown, X is a 4×4 matrix, Y is a 4×2 matrix, and G is a 2×2 matrix.

[0111] S5. For static analysis without external distributed forces and moments, the composite beam has initial torsion and bending, and the one-dimensional nonlinear equations of the beam are obtained. The nonlinear equations of the one-dimensional beam are:

[0112]

[0113] Where F = {F1, F2, F3} T ,M={M1,M2,M3} T ,e1={1,0,0} T .

[0114] In order to establish a two-dimensional cross-sectional analysis model, according to the order of magnitude adopted in the present invention, some high-order terms can be ignored. Therefore, the nonlinear equations of the one-dimensional beam can be simplified to the following expression:

[0115] {F2′ F3} T +D1{F2 F3} T +D2{F1 M1 M2 M3} T =0

[0116] {F1′ M1′ M′2 M′3} T +D3{F1 M1 M2 M3} T +D4{F2 F3} T =0

[0117] Where,

[0118] S6. Solve the derivative and transformation relationship of Timoshenko's one-dimensional generalized strain with respect to x1. Combined with the standard equation for the generalized Timoshenko strain energy and its corresponding constitutive equation, we can solve the nonlinear equation system related to the matrices X, Y, and G. Substituting the nonlinear equation system of the one-dimensional beam into the two-dimensional cross-section analysis model yields the following expression:

[0119] Y T ε′+Gγ′ s +D1(Y T ε+Gγ s )+D2(Xε+Yγ s )=0

[0120] Xε′+Yγ′ s +D3(Xε+Yγ s )+D4(Y T ε+Gγ s )=0

[0121] The above equations can be used to solve the derivative and transformation relationship of Timoshenko's one-dimensional generalized strain with respect to x1. Combined with the standard equation for the generalized Timoshenko strain energy and its corresponding constitutive equation, the nonlinear equations related to the matrices X, Y, and G can be solved. Recombining the solved nonlinear equations for X, Y, and G, we can obtain the cross-sectional equations corresponding to the generalized Timoshenko strain energy and Timoshenko's one-dimensional generalized strain:

[0122]

[0123] The 6x6 matrix in the above formula is the section stiffness matrix calculated by Timoshenko beam theory. The subscript 1 represents tension, 2 and 3 represent shear in two directions, 4 represents torsion, and 5 and 6 represent bending in two directions. The main diagonal elements represent the main stiffness coefficients; the off-diagonal elements represent the coupled stiffness coefficients, for example, S 12 represents the tensile-shear coupling stiffness coefficient; to obtain other coupling stiffness coefficients, simply follow the corresponding subscripts of the calculated section stiffness matrix. This stiffness matrix can be used to calculate the stiffness of carbon nanotube-reinforced composite beams.

[0124] In order to demonstrate the effectiveness of the method of the present invention, the stiffness calculation analysis of carbon nanotube reinforced composite laminates was carried out. The thickness of the single layer was 0.175 mm and the laying method was (±θ) s , the section stiffness S is calculated and analyzed when the ply angle θ changes in the range of -90° to 90°. ij (i,j=1,2,3,4,5,6), calculated section tensile stiffness coefficient S 11 As the ply angle θ changes, Figure 2 shown.

[0125] To further demonstrate the effectiveness of the method of the present invention, the deformation calculation of a carbon nanotube reinforced carbon fiber composite laminate under load was performed. The carbon nanotube reinforced carbon fiber composite laminate is 0.23m long, 0.038m wide, 0.0021m thick, with a layer thickness of 0.000175, a total of 12 layers, and a layup angle of [45° / 0° / 45°]. 2s , the cross-sectional diagram is as follows Figure 3 As shown in the figure. During the calculation, it is assumed that one end of the laminate is fixed and the other end is free. A vertical load and a torque are applied at the center of the end, and the deflection and torsion angle of the end of the laminate are calculated respectively. The ability of the composite beam to resist bending and torsional deformation after adding different contents of carbon nanotubes to the carbon fiber composite laminate is calculated and analyzed. The calculation results are shown in the figure. Figure 4 and Figure 5As shown, the calculation results show that after adding carbon nanotubes with mass fractions of 2%, 3%, 4% and 5%, the deflection and torsion angle of the laminate are significantly reduced. The more carbon nanotubes are added, the smaller the deflection and torsion angle are, indicating that the greater the improvement in the bending stiffness and torsional stiffness of the laminate, the greater the improvement in the bending and torsional resistance.

[0126] It will be understood by those skilled in the art that, unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by those skilled in the art in the art to which the present invention belongs. It should also be understood that terms such as those defined in common dictionaries should be understood to have meanings consistent with their meanings in the context of the prior art and, unless defined as such, will not be interpreted in an idealized or overly formal sense.

[0127] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for predicting the stiffness of a carbon nanotube reinforced composite laminate, characterized in that: The steps are as follows: Step 1), establishing a beam deformation coordinate system to describe the deformation of the beam of the carbon nanotube reinforced composite laminate; Let x1 be the arc length coordinate along the axis r of the beam before deformation, the normal of the beam section at any x1 is along the tangent direction of the axis r, and the coordinates of any point in the beam section are represented by x2 and x3; introduce a set of orthogonal undeformed reference coordinate vectors b1, b2, and b3 at the point where the axis r before deformation of the beam intersects the beam section at x1, where b1 is along the tangent direction of the undeformed beam axis, b1=b2×b3, b2 and b3 are along the coordinate axes x2 and x3 respectively, and the position of the material point represented by each point on the beam passes through the position vector of any spatial point P Determine; introduce a set of orthogonal deformed reference coordinate systems T1, T2, and T3 at the point where the deformed axis R of the beam intersects the beam section. T1 is tangent to the deformed axis of the beam, and T2 and T3 are determined by T1. Step 2) Set a boundary for the warping displacement phenomenon that occurs when the beam deforms, and calculate the strain at any point in the beam in coordinate system b; Step 3) Considering the warping deformation, the finite element method is used to discretize the warping deformation and calculate the cross-sectional strain energy at this time; Step 4), considering shear deformation, derive the one-dimensional generalized shear strain, substitute it into the cross-sectional strain energy formula of the beam, and obtain the generalized Timoshenko strain energy; Step 5), for the condition of static analysis without external distributed forces and moments, the composite beam has initial torsion and bending, and a one-dimensional nonlinear equation system of the beam is obtained; Step 6) Solve the derivative and transformation relationship of Timoshenko one-dimensional generalized strain with respect to x1, combine the standard equation of generalized Timoshenko strain energy and its corresponding constitutive equation, solve the nonlinear equation system expression related to the stiffness coefficient matrix, and then recombine the expression to obtain the cross-sectional equation system expression corresponding to the generalized Timoshenko strain energy and Timoshenko one-dimensional generalized strain, and then obtain the cross-sectional stiffness matrix; finally, the stiffness of the carbon nanotube reinforced composite laminate is obtained according to the cross-sectional stiffness matrix.

2. The method for predicting stiffness of a carbon nanotube reinforced composite laminate according to claim 1, characterized in that: In step 2), the classical one-dimensional strain vector of any point in the beam in the coordinate system b in tension, torsion and bending in two directions is The expression is: Where, Γ={Γ 11 ,2Γ 12 ,2Γ 13 ,Γ 22 ,2Γ 23 ,Γ 33 } T , w is the warping deformation, and the respective meanings of the matrices are: Where k = {k1, k2, k3} T , Δ represents the identity matrix, O3 represents the 3×3 zero matrix, 3. The method for predicting stiffness of a carbon nanotube reinforced composite laminate according to claim 2, wherein: The specific steps for calculating the strain energy during discrete warping deformation in step 3) are as follows: Considering the warping deformation, the expression of the cross-sectional strain energy is as follows: Where D represents the stiffness matrix, Use the finite element method to discretize the warping deformation: w(x1,x2,x3)=N(x2,x3)V(x1); Where N(x2,x3) represents the element shape function matrix, V(x1) is the warping displacement of the discrete node; Substituting into the cross-sectional strain energy expression, the strain Γ is simplified to: The strain energy is further expressed as: Where, 4. The method for predicting stiffness of a carbon nanotube reinforced composite laminate according to claim 3, wherein: The expression of Timoshenko one-dimensional generalized strain in step 4) is as follows: c= C bB ·R′-r′=γ 11 b1+2c 12 b2+2c 13 b3; κ= C bB ·Kk=k1b1+k2b2+k3b3; Where K = K i B i represents the curvature of the beam in shear deformation, C bB is the rotation tensor; γ 11 , 2γ 12 , 2γ 13 , κ1, κ2, κ3 represent the Timoshenko one-dimensional generalized strains corresponding to tension, transverse shear in two directions, torsion, and bending in two directions after the beam is deformed, respectively. Let ε = {γ 11 ,κ1,κ2,κ3} T , γ s ={2γ 12 ,2γ 13 } T ,γ={γ 11 ,2γ 12 ,2γ 13 } T , κ={κ1,κ2,κ3} T ; The following transformation equation exists between coordinate system T and coordinate system B: Through the conversion relationship of the above formula, the Timoshenko one-dimensional generalized strain expression is obtained as follows: Where, Substituting Timoshenko's one-dimensional generalized strain expression into the expression of the cross-sectional strain energy of the beam, we obtain the further optimized quadratic asymptotically accurate strain energy. The relevant expression is as follows: The standard equation for the generalized Timoshenko strain energy is: The corresponding constitutive equation is: Where, F i 、M i are the resultant force and moment of the deformed beam; and in the above formula, the matrices X, Y, and G are unknown, X is a 4×4 matrix, Y is a 4×2 matrix, and G is a 2×2 matrix.

5. The method for predicting stiffness of a carbon nanotube reinforced composite laminate according to claim 4, wherein: The one-dimensional nonlinear equations of the beam in step 5) are: Where F = {F1, F2, F3} T ,M={M1,M2,M3} T ,e1={1,0,0} T ; In order to establish a two-dimensional cross-section analysis model, according to the order of magnitude used in this paper, some high-order terms are ignored, so the above formula is simplified to the following expression: {F′2 F′3} T +D1{F2 F3} T +D2{F1 M1 M2 M3} T =0; {F′1 M′1 M′2 M′3} T +D3{F1 M1 M2 M3} T +D4{F2 F3} T =0; Where, 6. The method for predicting stiffness of a carbon nanotube reinforced composite laminate according to claim 5, characterized in that: The specific steps of step 6) are as follows: Substituting the one-dimensional nonlinear equations of the beam into the two-dimensional cross-sectional analysis model yields the following expression: Y T ε′+Gγ′ s +D1(Y T e+Gγ s )+D2(Xε+Yγ s )=0; Xε′+Yγ′ s +D3(Xε+Yγ s )+D4(Y T e+Gγ s )=0; The derivative and transformation relationship of Timoshenko one-dimensional generalized strain with respect to x1 are solved from the above equations. Then, combined with the standard equation of generalized Timoshenko strain energy and its corresponding constitutive equation, the nonlinear equations related to matrices X, Y, and G are solved. The solved nonlinear equations of X, Y, and G are recombined to obtain the cross-sectional equations corresponding to the generalized Timoshenko strain energy and Timoshenko one-dimensional generalized strain: The 6×6 matrix in the above formula is the section stiffness matrix calculated by Timoshenko beam theory; the subscript 1 represents tension, 2 and 3 represent shear in two directions, 4 represents torsion, and 5 and 6 represent bending in two directions; the main diagonal elements represent the main stiffness coefficients; the off-diagonal elements represent the coupling stiffness coefficients.

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