Establishment method of local laminated micro-beam flexoelectric response size dependence effect theoretical model

Abstract

The invention discloses a method for establishing a local laminated micro-beam flexoelectric response size dependence effect theoretical model. The method comprises the following steps: establishing a local laminated micro-beam flexoelectric response size effect model considering a strain gradient force effect, a surface effect, a flexoelectric effect and a polarization gradient electric effect; and on the basis of the established flexoelectric response size effect model of the local laminated micro-beam, obtaining an overall control equation and boundary conditions of the bending characteristics of the local laminated micro-beam under the action of external force. According to the method, the influence of the strain gradient force effect, the surface effect, the flexoelectric effect and the polarization gradient electric effect is considered and incorporated into the model for optimization, so that the problem that the flexoelectric response of the local laminated micro-beam cannot be accurately predicted by an existing local laminated micro-beam theoretical model is solved; the change rule of the mechanical and mechanical-electric coupling characteristics of the local laminated beam micro-component is accurately described, and a theoretical basis is provided for application and optimization of the local laminated beam micro-component in a micro electro mechanical system.

Description

Technical Field

[0001] The invention relates to the technical field of micro-electromechanical systems, and in particular to a method for establishing a theoretical model of size-dependent effect of flexural electrical response of a local laminated micro-beam. Background Art

[0002] At present, the research on modeling of laminated microbeams is mostly focused on the flexoelectric response characteristics of single-layer or laminated microbeams, but less on the size dependence of the flexoelectric response of local laminated microbeams.

[0003] The theoretical models of the flexoelectric effect of local laminated microbeams established by current scholars do not include strain gradients and polarization gradients, and cannot accurately predict the flexoelectric response of local laminated microbeams.

[0004] A method for theoretical modeling of the size effect of the flexoelectric response of local laminated microbeams including strain gradient and polarization gradient is needed in order to accurately predict the size dependence of the flexoelectric response of microbeams. Summary of the invention

[0005] The invention provides a method for establishing a theoretical model of size dependence effect of flexural electrical response of a local laminated microbeam, so as to realize a reasonable prediction of size dependence of flexural electrical response of a local laminated microbeam.

[0006] A method for predicting the size dependence of the flexoelectric response of a localized laminated microbeam comprises:

[0007] A size effect model of the flexoelectric response of local laminated microbeams is established, which takes into account the strain gradient force effect, surface effect, flexoelectric effect and polarization gradient electric effect.

[0008] Based on the established size effect model of the flexural electrical response of local laminated microbeams, the overall control equations and boundary conditions of the bending characteristics of local laminated microbeams under external forces are obtained.

[0009] As a preferred solution of the theoretical model of size-dependent effect of flexural electrical response of laminated microbeams described in the present invention, the establishment of a size-effect model of flexural electrical response of local laminated microbeams considering full strain gradient and polarization gradient comprises:

[0010] Step 1: Obtain the structural parameters of the laminated microbeam, where the structural parameters of the laminated microbeam include the bottom beam thickness h1, the upper beam thickness h2, the elastic layer width b1, the flexoelectric layer width b2, the total thickness of the microbeam h, the total length of the microbeam L, the external load q(x) and the voltage V; and according to formula (1), based on the generalized flexoelectric theory, obtain the theoretical internal energy density function w0 of the centrosymmetric material:

[0011] According to the generalized flexoelectric theory, the theoretical internal energy density function w0 of the centrosymmetric material is obtained:

[0012]

[0013] Where λ and μ are the Lamé constants. ε ij is the strain tensor, ε ij =0.5(u i,j +u j,i ), η ijk is the strain gradient tensor, η ijk =0.5(u i,kj +u j,ik );u i is the displacement vector, the comma indicates the differential of the displacement; β i (i=1,2,3) is the electrical scale parameter related to the polarization gradient; l i (i=0,1,2) is the material scale parameter related to the strain gradient; α is the parameter related to the dielectric polarizability; P i and Q ij =P i,j ) are the polarization vector and polarization gradient tensor respectively; f i (i=1,2) is the flexoelectric coefficient;

[0014] Step 2: Assuming that the distance between the neutral axis of the laminated area and the centroid axis of the bottom beam is d, according to the Euler Bernoulli beam theory, the displacement mode equation of the non-laminated area of ​​the local laminated microbeam is obtained as formula (2):

[0015]

[0016] The neutral axis of the lamination zone is moved along the z-axis by a distance d to the bottom beam center axis, and the displacement mode equation of the lamination zone of the local laminated microbeam is obtained as formula (3):

[0017]

[0018] Step 3: Substitute equations (2) and (3) into equation (1) to obtain the strain energy density function w1 of the actual laminated area of ​​the local laminated microbeam as equation (4):

[0019]

[0020] Among them, w (i) represents the i-th order derivative of w with respect to x; the polarization gradient electric scale parameter β is expressed as

[0021]

[0022] e n (n=1,2,..,5) is expressed as

[0023]

[0024] Among them, E (1) and E(2) represent the elastic moduli of the lower beam and the upper beam; μ (1) and μ (2) represent the shear moduli of the lower beam and the upper beam; l i(1) and l i(2) (i = 0, 1, 2) represent the material scale parameters of the lower beam and the upper beam.

[0025] Step 4: According to the Euler - Bernoulli beam theory, calculate the strain energy functions U1 and U2 of the two non - laminated regions of the local laminated micro - beam as shown in Equation (5) and Equation (6) and the strain energy function U3 of the laminated region as shown in Equation (7):

[0026] Strain energy function U1 of the non - laminated region (0 < x < L1):

[0027]

[0028] Strain energy function U2 of the non - laminated region (L2 < x < L):

[0029]

[0030] Strain energy function U3 of the laminated region (L1 < x < L2):

[0031]

[0032] where L1 is the length of the first non - laminated region of the local laminated micro - beam; L2 is the sum of the length of the first non - laminated region and the laminated region of the local laminated micro - beam.

[0033] Step 5: Use Equation (4), (5), (6), (7) to obtain the strain energy function U of the local laminated beam of the local laminated micro - beam as shown in Equation (8):

[0034]

[0035] where u0 represents the axial displacement;

[0036]

[0037] A1, A2, A3, A4, A5, A6 are expressed as

[0038]

[0039] where H0 is expressed as H0 = H(x - L1)-H(x - L2), H(x) is the unit step function; α 1(1) 、α 1(2) 、α 2(1) 、α 2(2) 、α 3(1) 、α 3(3) 、α 4(1) 、α4(2) , α 5(1) , α 5(2) , α 6(1) , α 6(2) Expressed as

[0040]

[0041] Among them, I (1) and I (2) Respectively represent the section inertia moment of the lower beam and the upper beam; A (1) and A (2) Respectively represent the cross-sectional area of ​​the lower beam and the upper beam; S (1) and S (2) represent the static moments of the lower beam and the upper beam respectively.

[0042] Step 6: According to the surface elasticity theory, the constitutive equation of the surface layer of the local laminated beam is calculated as formula (9):

[0043] (9)

[0044] where τ nα represents the out-of-plane surface stress, τ αβ represents the in-plane surface elastic stress τ s represents the residual surface stress under unconstrained conditions. μ s and λ s represents the surface Lamé constant;

[0045] The non-zero surface stress is expressed as formula (10):

[0046] (10)

[0047] Step 7: According to the surface elasticity theory, the surface energy Us of the local laminated beam is expressed as formula (11):

[0048] (11)

[0049] Where c is the circumference of the beam section; Substituting equation (10) into equation (11), the surface energy Us of the local laminated beam can be obtained as equation (12):

[0050] (12)

[0051] where s s , k s Expressed as

[0052]

[0053] Where b is the width of the beam, I p Represents the second bending moment about the perimeter of the cross section.

[0054]

[0055] Step 8: Calculate the electrical enthalpy function H of the local laminated beam total Formula (13):

[0056]

[0057] Here, φ is the electric potential.

[0058] Step 7: Derive the work done by the external force W as formula (14):

[0059]

[0060] Where q(x) is the external load, V is the shear force on the boundary, M is the bending moment, and M h is the higher order bending moment.

[0061] Step 8: Using equations (13) and (14), and applying the variational principle, the electrical control equation of the local laminated microbeam is obtained as equation (15) and the force control equation (16):

[0062]

[0063]

[0064] When z=-0.5h1 and -(0.5h1+h2), the electrical boundary condition of the local laminated microbeam is obtained as equation (17);

[0065]

[0066] When x=0 and x=L, the local laminated micro-beam force boundary condition is obtained as formula (18):

[0067] BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1 The figure is a flow chart of an embodiment of the present invention.

[0069] Figure 2 A laminated microbeam model according to an embodiment of the present invention DETAILED DESCRIPTION

[0070] The technical solutions in the embodiments of the present invention are fully and accurately described below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, other embodiments obtained by those skilled in the art without creative work are within the protection scope of the present invention.

[0071] To facilitate a comprehensive understanding of the present invention, many specific details are listed in the following description, but the present invention may also be implemented in other ways different from those described herein, and those skilled in the art may make similar generalizations without violating the connotation of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0072] Reference Figure 1 , is an embodiment of the present invention, comprising:

[0073] Step 1: Calculate the local laminated microbeam strain energy U b , the steps are as follows:

[0074] The structural parameters of the laminated microbeam are obtained, wherein the structural parameters of the laminated microbeam include the bottom beam thickness h1, the upper beam thickness h2, the elastic layer width b1, the flexural electrical layer width b2, the total thickness of the microbeam h, the total length of the microbeam L, the external load q(x) and the voltage V.

[0075] According to the generalized flexoelectric theory, the theoretical internal energy density function w0 of the centrosymmetric material is obtained:

[0076]

[0077] Where λ and μ are the Lamé constants. ε ij is the strain tensor, ε ij =0.5(u i,j +u j,i ), η ijk is the strain gradient tensor, η ijk =0.5(u i,kj +u j,ik );u i is the displacement vector, the comma indicates the differential of the displacement; β i (i=1,2,3) is the electrical scale parameter related to the polarization gradient; l i (i=0,1,2) is the material scale parameter related to the strain gradient; α is the parameter related to the dielectric polarizability; P i and Q ij =P i,j ) are the polarization vector and polarization gradient tensor respectively; f i (i=1,2) is the flexoelectric coefficient;

[0078] Assuming that the distance between the neutral axis of the laminated area and the centroid axis of the bottom beam is d, according to the Euler Bernoulli beam theory, the displacement mode equation of the non-laminated area of ​​the local laminated microbeam is obtained as formula (2);

[0079]

[0080] Move the neutral axis of the laminated region along the z-axis by a distance d to the centroid axis of the bottom beam, and the displacement mode equation of the laminated region of the local laminated microbeam is Equation (3):

[0081]

[0082] Substitute Equations (2) and (3) into Equation (1) to obtain the strain energy density function W of the actual laminated region of the local laminated microbeam as Equation (4)

[0083]

[0084] where, w (i) represents the i-th derivative of w with respect to x; the polarization gradient electrical scale parameter β is expressed as

[0085]

[0086] e n (n = 1, 2,.., 5) is expressed as

[0087]

[0088] where, E (1) and E (2) represent the elastic moduli of the bottom beam and the top beam; μ (1) and μ (2) represent the shear moduli of the bottom beam and the top beam; l i(1) and l i(2) (i = 0, 1, 2) represent the material scale parameters of the bottom beam and the top beam.

[0089] According to the Euler-Bernoulli beam theory, calculate the strain energy functions U1 and U2 of the two non-laminated regions of the local laminated microbeam as Equation (5) and Equation (6) and the strain energy function U3 of one laminated region as Equation (7):

[0090] Strain energy function U1 of the non-laminated region (0 < x < L1):

[0091]

[0092] Strain energy function U2 of the non-laminated region (L2 < x < L):

[0093]

[0094] Strain energy function U3 of the laminated region (L1 < x < L2):

[0095]

[0096] where, L1 is the length of the first non-laminated region of the local laminated microbeam; L2 is the sum of the lengths of the first non-laminated region and the laminated region of the local laminated microbeam.

[0097] Step 5: Using equations (4), (5), (6), and (7), the strain energy function U of the local laminated microbeam is obtained as equation (8):

[0098]

[0099] Where, u0 represents the axial displacement;

[0100]

[0101] A1, A2, A3, A4, A5, A6 are represented by

[0102]

[0103] Among them, H0 is expressed as H0=H(x-L1)-H(x-L2), H(x) is the unit transition function; α 1(1) , α 1(2) , α 2(1) , α 2(2) , α 3(1) , α 3(3) , α 4(1) , α 4(2) , α 5(1) , α 5(2) , α 6(1) , α 6(2) Expressed as

[0104]

[0105] Among them, I (1) and I (2) Respectively represent the section inertia moment of the lower beam and the upper beam; A (1) and A (2) Respectively represent the cross-sectional area of ​​the lower beam and the upper beam; S (1) and S (2) represent the static moments of the lower beam and the upper beam respectively.

[0106] Step 2: According to the surface elasticity theory, calculate the surface energy U of the local laminated microbeam s .

[0107] Furthermore, the constitutive equation of the surface layer of the local laminated microbeam is calculated as formula (9):

[0108] (9)

[0109] where τ nα represents the out-of-plane surface stress, τ αβ represents the in-plane surface elastic stress τ s represents the residual surface stress under unconstrained conditions. μ s and λs represents the surface Lamé constant;

[0110] The non-zero surface stress is expressed as formula (10):

[0111] (10)

[0112] According to the surface elasticity theory, the surface energy Us of the local laminated beam is expressed as formula (11):

[0113] (11)

[0114] Where c is the circumference of the beam section; Substituting equation (10) into equation (11), the surface energy Us of the local laminated beam can be obtained as equation (12):

[0115] (12)

[0116] where s s , k s Expressed as

[0117]

[0118] Where b is the width of the beam, I p Represents the second bending moment about the perimeter of the cross section.

[0119]

[0120] Step 3: Calculate the electrical enthalpy function H of the local laminated microbeam total Formula (13):

[0121]

[0122] Here, φ is the electric potential.

[0123] Step 4: Derive the work done by the external force under uniform load as formula (14):

[0124]

[0125] Where q(x) is the external load, V is the shear force on the boundary, M is the bending moment, and M h is the higher order bending moment.

[0126] Step 5: Using equations (13) and (14), and applying the variational principle, the electrical control equation of the local laminated microbeam is obtained as equation (15) and the electrical boundary condition (16):

[0127]

[0128] When z = -0.5h1 and -(0.5h1+h2), the electrical boundary condition of the local laminated microbeam is obtained as formula (17)2

[0129]

[0130] Step 6: The polarization P3 calculated according to the electrical control equation and electrical boundary conditions is formula (17) and the potential is formula (18)

[0131] (17)

[0132] (18)

[0133] Among them A s1 , A s2 , A s3 , A s4 , A s5 , A s6 Expressed as

[0134]

[0135] Step 7: Calculate the local laminated microbeam force control equation as equation (19) and the force boundary condition as (20):

[0136]

[0137]

[0138] Furthermore, the polarization P3 and the potential φ are substituted into equation (19) to calculate the local laminated microbeam bending control equation as equation (21):

[0139]

[0140] When x=0 and x=L, the shear boundary condition of the local laminated microbeam is formula (22):

[0141]

[0142] The local laminated microbeam bending moment boundary condition is formula (23):

[0143]

[0144] The boundary condition of the high-order bending moment of the local laminated microbeam is formula (24):

[0145]

[0146] The boundary condition of the axial force of the local laminated microbeam is formula (25):

[0147]

[0148] The boundary condition of the high-order axial force of the local laminated microbeam is formula (26):

[0149]

[0150] Among them, T1, T2, T3, T4 are expressed as

[0151]

[0152] Step 8: For the clamped partially laminated beam, the boundary condition is expressed as (27)

[0153]

[0154] According to formula (28), the deflection and axial displacement are expressed as

[0155]

[0156] Among them, A n and B n is the unknown coefficient, P n is the basis function, P n Expressed as

[0157]

[0158] The larger the number of terms n, the higher the accuracy of formula (28). Furthermore, the uniformly distributed load q is expressed as formula (29) using trigonometric series:

[0159]

[0160] Step 9: Substitute equation (28) into the control equation (21) and the boundary condition equation (27) and solve them to obtain the local laminated microbeam deflection w and axial displacement u0; since the approximate function cannot completely satisfy the control equation and boundary conditions, there will be margins R1(x) and R2(x) expressed as

[0161]

[0162] According to formula (30), select the unknown coefficient A n and B n The weighted integral value of the residuals R1(x) and R2(x) is made zero.

[0163]

[0164] Where W n and Y n The weight function is expressed as

[0165]

[0166] Step 10: Obtain the unknown coefficient A n and B n After that, the deflection w and axial displacement u0 of the local laminated microbeam can be determined; substituting the obtained deflection into equations (17) and (18), the polarization and potential generated by the positive flexoelectric effect and the charge density ρ(x) generated on the surface of the flexoelectric layer can be obtained as equation (31):

[0167]

[0168] Where C is the capacitance per unit area expressed as

[0169]

[0170] where ε r is the relative dielectric constant.

[0171] Step 11: According to equation (32), the total charge Q(x) collected on the upper surface of the flexoelectric layer is

[0172]

Claims

1. A method for establishing a theoretical model of the size-dependent effect of the flexural electrical response of a partially laminated microbeam, characterized in that: include: A size effect model of the flexoelectric response of local laminated microbeams is established, which takes into account the strain gradient force effect, surface effect, flexoelectric effect and polarization gradient electric effect. Based on the established size effect model of the flexural electrical response of local laminated microbeams, the overall control equations and boundary conditions of the bending characteristics of local laminated microbeams under external forces are obtained.

2. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 1, characterized in that: The method of establishing a size effect model of the flexural electrical response of a local laminated microbeam considering the full strain gradient and the polarization gradient comprises the following steps: Step 1: Obtain the structural parameters of the laminated microbeam, where the structural parameters of the laminated microbeam include the bottom beam thickness h1, the upper beam thickness h2, the elastic layer width b1, the flexoelectric layer width b2, the total thickness of the microbeam h, the total length of the microbeam L, the external load q(x) and the voltage V; and according to formula (1), based on the generalized flexoelectric theory, obtain the theoretical internal energy density function w0 of the centrosymmetric material: ; Where λ and μ are the Lamé constants. ε ij is the strain tensor, ε ij =0.5(u i,j +u j,i ), η ijk is the strain gradient tensor, η ijk =0.5(u i,kj +u j,ik );u i is the displacement vector, the comma indicates the differential of the displacement; β i (i=1,2,3) is the electrical scale parameter related to the polarization gradient; l i (i=0,1,2) is the material scale parameter related to the strain gradient; α is the parameter related to the dielectric polarizability; P i and Q ij =P i,j ) are the polarization vector and polarization gradient tensor respectively; f i (i=1,2) is the flexoelectric coefficient; Step 2: Assuming that the distance between the neutral axis of the laminated area and the centroid axis of the bottom beam is d, according to the Euler Bernoulli beam theory, the displacement mode equation of the non-laminated area of ​​the local laminated microbeam is obtained as formula (2); the neutral axis of the laminated area is moved along the z-axis by a distance d to the centroid of the bottom beam, and the displacement mode equation of the laminated area of ​​the local laminated microbeam is obtained as formula (3): ; Step 3: Substitute equations (2) and (3) into equation (1) to obtain the strain energy density function w1 of the actual laminated area of ​​the local laminated microbeam as equation (4): ; Among them, w (i) represents the i-th order derivative of w with respect to x; the polarization gradient electric scale parameter β is expressed as ;e n (n=1,2,..,5) is expressed as ; Among them, E (1) and E (2) represents the elastic modulus of the lower beam and the upper beam; μ (1) and μ (2) represents the shear modulus of the lower beam and the upper beam; l i(1) and l i(2) (i=0,1,2) represents the material scale parameters of the lower beam and the upper beam.

3. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 2, characterized in that: The method further comprises: Step 4: According to the Euler Bernoulli beam theory, the strain energy functions U1 and U2 of the two non-laminated areas of the local laminated microbeam are calculated as equations (5) and (6) and the strain energy function U3 of one laminated area is calculated as equation (7): Strain energy function U1 in the non-laminated region (0 < x < L1): ; Strain energy function U2 of the non-laminated region (L2 < x < L): ; Strain energy function U3 of the laminated region (L1 < x < L2): ; Among them, L1 is the length of the first non-laminated area of ​​the local laminated microbeam; L2 is the sum of the lengths of the first non-laminated area and the laminated area of ​​the local laminated microbeam.

4. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 2 or 3, characterized in that: The method further comprises: Step 5: Use equations (4), (5), (6), and (7) to obtain the strain energy function U of the local laminated microbeam: b Formula (8): ; Where u0 represents the axial displacement, ; A1, A2, A3, A4, A5, A6 are represented by ; Among them, H0 is expressed as H0=H(x-L1)-H(x-L2), H(x) is the unit transition function; α 1(1) , α 1(2) , α 2(1) , α 2(2) , α 3(1) , α 3(3) , α 4(1) , α 4(2) , α 5(1) , α 5(2) , α 6(1) , α 6(2) Expressed as ; Among them, I (1) and I (2) Respectively represent the section inertia moment of the lower beam and the upper beam; A (1) and A (2) Respectively represent the cross-sectional area of ​​the lower beam and the upper beam; S (1) and S (2) represent the static moments of the lower beam and the upper beam respectively.

5. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 4, characterized in that: The method further comprises: Step 6: According to the surface elasticity theory, the constitutive equation of the surface layer of the local laminated beam is calculated as equation (9) and equation (10): (9) where τ nα represents the out-of-plane surface stress, τ αβ represents the in-plane surface elastic stress τ s represents the residual surface stress under unconstrained conditions. μ s and λ s represents the surface Lamé constant; Non-zero surface stress is expressed as: (10) Among them, E s is the surface elastic modulus.

6. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 5, characterized in that: The method further comprises: Step 7: According to the surface elasticity theory, the surface energy U of the local laminated beam is s It is expressed as formula (11): (11) Where c is the circumference of the beam section; Substituting equation (10) into equation (11) yields the surface energy U of the local laminated beam: s Formula (12): (12) where s s , k s Expressed as Among them, I p Second bending moment representing the perimeter of the cross section , b is the width of the beam.

7. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 5 or 6, characterized in that: The method further comprises: Calculation of the electrical enthalpy function H of a partially laminated beam total Formula (13): ; Here, φ is the electric potential.

8. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 1, characterized in that: The control equations and boundary conditions of the local laminated microbeam under the action of uniformly distributed load or voltage include: The work done by the external force W is derived as formula (10): ; Where V is the shear force on the boundary, M is the bending moment, and M h is the higher order bending moment.

9. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 7 or 8, characterized in that: The method further comprises: Using equations (13) and (14) and applying the variational principle, the electrical control equation of the local laminated microbeam is obtained as equation (15) and the force control equation (16): ; 10. The method for establishing a theoretical model of size dependence of flexural electrical response of a partially laminated microbeam according to claim 7, characterized in that: The method further comprises: When z=-0.5h1 and -(0.5h1+h2), the electrical boundary condition of the local laminated microbeam is obtained as formula (17); when x=0 and x=L, the force boundary condition of the local laminated microbeam is obtained as formula (18):

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