Nonlocal elasticity theory for vibration and buckling of nanoplates
The nonlocal continuum mechanics has been introduced by Eringen in order to account for
the small-scale effect. Recently different beam and plate models are developed based on
nonlocal elasticity theory. These models include classical plate theory CLPT, first order shear
deformation theory (FSDT), higher order shear deformation theory (HSDT), but as rule these
models are based on linear strain-displacement relation.
According to the nonlocal elasticity theory it is assumed that the stress at a point is a
function of strains at all points in the continuum. Nonlocal theory consider long-range interatomic interaction and yields results dependent on the size of a body. In the following the
simplified form of the Eringen’s nonlocal constitutive equation is employed (originally it is
given in integral form and is too complex for constitutive modelling)
1. Stress-strain relation in the case of local elasticity theory (Hookean solid)
Hooke’s law is valid: linear elasticity is considered
1D :
  E
l
(1)
3D:
   C  
l
(2)
For orthotropic material:









 C
 
 
 
 
 
 
 
 
l
11
22
33
23
31
12
1111
C
C
1122
2222
C
C
C
1133
2233
3333
Sym
0
0
0
C
2323
0
0
0
0
C
3131
 
 

 

 
0  

C  
0
0
0
0
1212
11
22
33
23
31
12









(3)
2. Stress-strain relation in the case of nonlocal elasticity theory
According to Eringen the nonlocal constitutive behavior of a Hookean solid is represented by the
following differential constitutive relation [1-4]
(4)
where
is the Laplacian operator,
is nonlocal parameter,
a - internal characteristic length
e0 - constant.
It follows from (4) and Hooke’s law that
(1  2 ) nl   C  
(5)
In as assume that thickness of the plate ia << than length and width of the plate and
consider plane stress state:  zz   xz   yz  0 .
In the case of isotropic material , the stiffness matrix C reads
 E /(1   2 ) E /(1   2 ) 0 
C   E /(1   2 ) E /(1   2 ) 0 

0
0
G 
(6)
and the relation (5) takes the following form
 x 
  x   E /(1   2 ) E /(1   2 ) 0   x 
 
 
 
2
2
2
 y     y   E /(1   ) E /(1   ) 0   y 
 
  
0
0
G  xy 
 xy 
 xy  
(7)
In comparison it can be noted that in the case of local elasticity we have
  x   E /(1   2 ) E /(1   2 ) 0   x 
  
 
2
2
 y   E /(1   ) E /(1   ) 0   y 
  
0
0
G  xy 
 xy  
Thus, main differnce between nonlocal and local elasticity is stress strain relation.
(8)
3. Strain - displacement relation
Let us consider CPT theory.
First, the displacements at any point can be written in terms of middle surface
displacement as
(9)
where u, v and w are thedisplacementfunctionsofthemiddle surface oftheplateand t is
time. The formulas (9) are explained by the following figure
Next, the strain components are defined as
x 
U
,
x
 xy   yx 
y 
V
W
, z 
,
z
y
U V
V W

 yz   zy 

y x ,
z
y ,
(10)
 zx   xz 
W U

x
z
Finally, inserting (9) in (10) yields stain-displacement relations as
(11)
4.
G
overning equations
Principal of virtual work is independent of constitutive relations. So this can be
applied to derive the equilibrium equations of the nonlocal plates. Using the principle
of virtual work, the following governing equations can be obtained for a thin
nanoplate.
(12)
Here q(x, t) is the transverse force per unit area, Nx and Ny are the in-plane
compression or tension forces and Nxy is the in-plane shear. Mass inertias are defined
by:
(13)
 - material density.
(14)
Multiplying the relation (7) by z, and integrating yields
 x 
 x 
 E /(1   2 ) E /(1   2 ) 0  z / 2   x 
z/2
 
 

  
2
2
2
 y  zdz     y zdz  E /(1   ) E /(1   ) 0     y  zdz

z / 2
z / 2
z / 2
 
 

0
0
G   xy 
 xy 
 xy 
(15)
z/2
Force and moment resultants used in (12) can be written in terms of stresses as:
(16)
Considering the second equation of (16) (in left hand side of (15)) and strain
displacement relation (on right hand side of (15)) one obtains from (15)
,
(17)
where D 
Eh3
, also it is considered that the shear modulus G can be written in
12(1   2 )
terms of elatic mudulus and Poisson ration as G 
E
2(1   ) .
Similarly, integrating (7)
 x 
 x 
 E /(1   2 ) E /(1   2 ) 0  z / 2   x 
z/2
 
 

  
2
2
2
 y dz     y dz  E /(1   ) E /(1   ) 0     y dz

z / 2
z / 2
z / 2
 
 

0
0
G   xy 
 xy 
 xy 
z/2
(18)
considering the first equation of (16) (in left hand side of (18)) and strain displacement
relation (on right hand side of (18)) one obtains from (18)
(19)
Applying the differential operator (1  2 ) to governing equation (12), considering
(18) and some simplifications (m2=0) yields

2w
2w
2w
2w 
 D w  (1   ) q   2  N x 2  N y 2  2 N xy
t
x
y
xy 

4
2
(19)
Obviously, the force resultants N x , N y and N xy can be evaluate dby use of formulas (19),
but this substitution is omitted due to consiseness sake.
5.
Buc
kling and vibration of nanoplates
Buckling andvibrationproblemofnanoplatescanbeobtained by setting q =0 in (19)
The boundary conditions of plates are given as
Simply Supported: w=0, Mx=0
Clamped : w=0, w’=0.
Navier type solution method-series expansion
,
(20)
Where m and n are the half wave numbers in the x and y directions, respectively,  is
the unknown circular frequency in the case of vibration problem and Wmn is
amplitude of motion.
Levy type solution
Levy-type solutions are sekked for case where the edges x=0, and x=Lx are assumed
to be simply supported, while the remaining two edges y=0 and y=Ly can have any
arbitrary conditions.
(21)
Inserting solution (21) in goverment equation (19) leads to differential equation with
respect to y and time
(22)
General solution of the latter equation is given by
(23)
where
.
(24)
The constants A, B, C and D must be determined from the four conditions on the edge
y=0 and y=b. In determining the constants we obtain four homogeneous equations.The
nondimensional frequency parameter is determined by setting the determinant of this
system to zero. For SSSS boundary conditions one obtains:
(25)
For SCSC boundary conditions:
(26)
In order to avoid trivial solution(A=B=C=D=0), the determinant of the coefficient
matrix in Eqs.(22) and (23) must vanish. This condition gives nondimensional
frequency parameters and critical buckling loads.
Results: