7.7 Mathematical Model of Three-Layer Micro- and Nano-Beams
267
7.7.2 Theory of Bending Including Shear Effects and the
Modified Couple Stress Theory
In the process of the beam deformation the transversal cross sections of the middle
layer being perpendicular to the beam axis, in contrary to Bernoulli-Euler hypothesis,
rotate as a rigid body on amount of angle ψ. Here, on contrary to the hypothesis of
the plane cross sections, we do not require cross sections to be perpendicular to the
bended beam axis while deformation process, but in general, this behaviour is not
excluded. This stands for a general hypothesis, which is not contradicting BernoulliEuler one. Namely, it tends to the latter one if the middle layer stiffness against shear
is of infinite magnitude. The external layers material is assumed to be linearly elastic,
and they obey Hooke’s law. The homogenous internal layer also obeys Hooke’s law.
The beam is analysed using the rectangular coordinates Oxz (see Fig. 7.25). The
axis x overlaps with the middle beam layer line, whereas the axis z goes in the
direction opposite to the Earth gravity. The carrying out load layer located in the
positive z axis direction is called the first layer and the middle layer is called the
third layer.
Let h k (k = 1, 2, 3) denote the layer thickness (h 3 = 2c); h = h 1 + h 2 + h 3
stands for the thickness and b is the width of a beam wall, respectively; E k is the
elasticity material modulus, whereas ν k denotes Poisson’s coefficient of a layer with
number k; G 3 is the modulus regarding the transversal shear of the middle beam, l k
stands for the internal material length scale parameter. In order to keep a compact
formulation of the problem, we introduce the following averaged elasticity modulus:
E = (E 1 h 1 + E 2 h 2 + E 3 h 3 ) / h,
(7.100)
as well as the non-dimensional stiffness characteristics γ k and the non-dimensional
thickness of the layer t k as follows:
Fig. 7.25 The longitudinal cross-section of the three layers beams [reprinted with permission from
the International Journal of Solids and Structures publishers]
267
7.7.2 Theory of Bending Including Shear Effects and the
Modified Couple Stress Theory
In the process of the beam deformation the transversal cross sections of the middle
layer being perpendicular to the beam axis, in contrary to Bernoulli-Euler hypothesis,
rotate as a rigid body on amount of angle ψ. Here, on contrary to the hypothesis of
the plane cross sections, we do not require cross sections to be perpendicular to the
bended beam axis while deformation process, but in general, this behaviour is not
excluded. This stands for a general hypothesis, which is not contradicting BernoulliEuler one. Namely, it tends to the latter one if the middle layer stiffness against shear
is of infinite magnitude. The external layers material is assumed to be linearly elastic,
and they obey Hooke’s law. The homogenous internal layer also obeys Hooke’s law.
The beam is analysed using the rectangular coordinates Oxz (see Fig. 7.25). The
axis x overlaps with the middle beam layer line, whereas the axis z goes in the
direction opposite to the Earth gravity. The carrying out load layer located in the
positive z axis direction is called the first layer and the middle layer is called the
third layer.
Let h k (k = 1, 2, 3) denote the layer thickness (h 3 = 2c); h = h 1 + h 2 + h 3
stands for the thickness and b is the width of a beam wall, respectively; E k is the
elasticity material modulus, whereas ν k denotes Poisson’s coefficient of a layer with
number k; G 3 is the modulus regarding the transversal shear of the middle beam, l k
stands for the internal material length scale parameter. In order to keep a compact
formulation of the problem, we introduce the following averaged elasticity modulus:
E = (E 1 h 1 + E 2 h 2 + E 3 h 3 ) / h,
(7.100)
as well as the non-dimensional stiffness characteristics γ k and the non-dimensional
thickness of the layer t k as follows:
Fig. 7.25 The longitudinal cross-section of the three layers beams [reprinted with permission from
the International Journal of Solids and Structures publishers]