Supplementary Information for “An analytic model for accurate spring constant calibration of rectangular atomic force microscope cantilevers”
Rui Li[1],2,Hongfei Ye1,Weisheng Zhang1,Guojun Ma1Yewang Su3,4
In this file, we elaborate on the solution of governing equation for static problem of a rectangular AFM cantilever, by which we find theanalytic scaling law [equation (5) of the main text]for spring constant calibration.
S.1. Governing equation within the framework of Hamiltonian system
The Hellinger-Reissner variational principleS1 for bending of a thin plateis
,(S1)
where the generalized potential energy functional with two kinds of variables is
,(S2)
herein the surface integral is taken over the domain and the line integrals with respect to boundary conditions are not taken into account. is the distributed transverse load, is the Poisson’s ratio, is the flexural rigidity, is the transverse deflection of the plate mid-plane, and are the bending moments, and is the torsional moment. Assuming the independence of , , and and the arbitrariness of their variations, equation(S1)gives the natural conditions of stationary ,two of which are and .Substituting theminto equation(S2),we have a new functional :
.(S3)
Define
(S4)
as well as the Lagrange multiplier , is transformed into :
.(S5)
Noting the arbitrariness of , , and ,
(S6)
yieldsequation (3) of the main text, i.e.,
,(S7)
where , , , ,,and . is inferred from equation (S7), where is the equivalent shear force. It is noted that is a Hamiltonian operator matrixS1andequation(S7) is the governing equation for thin plate bending within the framework of Hamiltonian system.
S.2. Analytic solution of rectangular cantilever under point load
The problem (see Fig. 1 of the main text) is solved by superposing three sub-problems: (A) the plate is simply supported at the edge andslidingly clamped at the other three edges, with the point load Papplied at ; (B) the plate is slidingly clamped at and ,with the slope and moment applied at the slidingly clamped edge and simply supportededge , respectively; (C) the plate is slidingly clamped at and simply supportedat , with the slopesand applied at the slidingly clamped edges and , respectively. Here , and are the Fourier expansion coefficients.
In the Hamiltonian system, the homogeneous equation of equation (S7)
(S8)
could be handled by separation of variablesS2, i.e., imposing
(S9)
such that
,(S10a,b)
where is the eigenvalue and is the corresponding eigenvector.
We begin with the plate with slidingly clamped and simply supported. The eigenvalue problem (S10b) and the boundary conditions
(S11)
determine the eigenvalue equation, where is anx-component of equivalent shear force. This gives the eigenvalues and for . The corresponding eigenvectors are
,(S12)
where anddenote the zeroth-order and first-order eigenvectors of whileand denote those of . The symplectic conjugacy and orthogonalityS2are satisfied between any two eigenvectors, i.e., and while any other combinations of two eigenvectors are orthogonal. The solution of equation (S7) is
,(S13)
where
.(S14)
Substituting equation (S13) into equation (S7) yields
,(S15)
where , , , is the column matrix of expansion coefficients of .For the case of point loading at , we obtain
,(S16)
where is the Diracdeltafunction. Substituting equations (S16) into equation (S15) gives
,(S17)
where is the Heaviside theta function, are to be determined by imposing the remaining boundary conditions at and . For the plate with and slidingly clamped, we have
.(S18)
Substituting equations (S12) and (S17) into equations (S14) then equation (S13) and imposing equation (S18), are obtained. Thus we obtain the analytic solution of the plate with simply supported and the other three edges slidingly clamped under the point load P at . By conversion of coordinates, we obtain the analytic solution of the plate with simply supported and the other three edges slidingly clamped under the point load P at . For convenience, we present the normalized load-point deflection solution as
,(S19)
where and.
By way of the similar procedure as stated above, we obtain the analytic solution of the plate with and slidingly clamped and the slope and moment applied at the slidingly clamped edge and simply supportededge , respectively. The normalized deflection at is
.(S20)
Here and henceforth the normalized constants , , and , where the subscript “s”could be 0, m, n, or i.
For the plate slidingly clamped at and simply supportedat , with the slopes and applied at the slidingly clamped edges and , respectively, we obtain
.(S21)
Noting, the sum of equations (S19)-(S21) gives the explicit expression ofthe function of the main text, i.e. equation (4):
.(4)
The requirements of at , at and at yield
, (S22)
, (S23)
(S24)
and
(S25)
for , and
(S26)
for .Equations (S22)-(S26) determine the normalized constants, and in equation (4). Accordingly, the scaling law is found, as shown in equation (5) of the main text.
References
S1.Yao, W., Zhong W. Lim, C. W. Symplectic elasticity (World Scientific, 2009).
S2.Li, R., Zhong, Y. Li, M.Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method. Proc. R. Soc. A-Math. Phys. Eng. Sci.469, 20120681 (2013).
1
[1]State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China. 2State Key Laboratory of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan 430074, China. 3State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China. 4Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA. Correspondence and requests for materials should be addressedto R.L. (email: ) and Y.S. (email: )