13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- Here, A denotes the area of the cross section, is the shear coefficient, 0C and 1C are the surface elastic modulus and the Steigman-Ogden constant, respectively, S denotes the circumference of the cross section. The potential energy of the lateral and axial load is given by: 2 0 0 1 ( ) ( ) ( ) ( ) 2 L L f w U w f x w x dx N dx x , (4) where s f f x q x q x q x . (5) Here, ( ) sq x stands for the equivalent lateral force due to the residual surface stress and ( ) f q x represents the distributed force arising from the substrate medium. To minimize the total potential energy, we apply the variational theory, ( ) ( ) 0 Total f U w U w . (6) Finally, we get: 2 2 2 2 0 1 ( ) ( ) 0 ( ) (( * ) ) 0 w w GA f x N x x x w GA EI C I CS x x x , (7) where 2 A I y dA , 2 * S I y dS , 2 * y S S n dS . Take partial derivative of the second equation with respect tox，and substitute the first equation into the second equation, we have: 4 2 4 2 ( )* ( )* ( )* ( )* ( ) ( ) 0 p w p w EI EI w EI w EI H N C N C H C C w x GA GA x GA x . (8) The general solution of Eq. (8) can be written as: () mx w x Ae . (9) Substituting Eq. (9) into Eq. (8), we can solvem. The boundary conditions of a simply supported NW are given as follows: (0) 0 w , '' (0) 0 w , (10a) ( ) 0 w L , '' ( ) 0 w L . (10b) Substitute Eqs.(10a,b) into Eq. (9), we can get the characteristic equation to solve the critical buckling force. The corresponding buckling mode is given by Eq. (9). 3. Results and Discussions
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