7.5 Stability of the Size-Dependent Graded Curvilinear Timoshenko Beams
233
7.5.2 Theoretical Background
In the modified couple stress theory, for infinitely small deformations, the energy of
deformation U accumulated in a linear elastic body of the volume V is as follows:
U =
1
2
V
σ i j ε i j + m i j χ i j
dV .
(7.55)
In the isotropic case, we have
σ i j = λε mm δ i j + 2με i j ,
(7.56)
ε i j =
1
2
u i, j + u j, i + u m, i u m, j
,
(7.57)
m i j = βχ i j = 2μl
2
χ i j ,
(7.58)
χ i j =
1
2
θ i, j + θ j, i
.
(7.59)
The coordinate system, kinematic parameters and the load for the curvilinear
Timoshenko FG beam are shown in Fig. 7.11. It is assumed that the properties of the
curvilinear beam do not change along the axis O X. The kinematic relations regarding
Timoshenko beam follow [122]
u x = u (x, t) + zψ(x, t), u y = 0, u z = w (x, t) .
(7.60)
It is assumed that transverse beam cross sections remain flat after a deformation
process. Nevertheless, they may undergo a rigid displacement in the plane X O Z as
well as the rotation around the axis OY .
Fig. 7.11 Geometry of
Timoshenko beam and the
acting load [reprinted with
permission from the Journal
of Computational and
Nonlinear Dynamics
publishers]
q(x,t)
G(x,t)
z
x
L
h
z c
z
C(x,t)
~
~