7.7 Mathematical Model of Three-Layer Micro- and Nano-Beams
263
it has been reported that the torsion stiffness of a copper made wires increases simultaneously decreasing their diameter from 170 down to 12 microns (observe that the
size decrease should imply zero stiffness of a wire). In Ref. [16], the increase in the
bending stiffness of a nickel foil corresponding to its height yielded the decrease in
its thickness from 50 to 12.5 microns. The micro- and nano-oriented investigations
[44, 172] show that the stiffness of the pure and poly-crystal metallic materials can
be doubled while decreasing the thickness from 10 to 1 microns. In order to estimate
the material resistance and keep the optimal design of the mentioned products, the
key role is played by the reliable and highly accurate analysis of the stress-strain
states of the investigated micro- and nanomechanical objects.
Importance of the size-dependent behaviour being an inherent property of materials has been presented experimentally by McFarland and Colton [15] and Kong et
al. [17]. It has been demonstrated that in micron/submicron-scale regions, i.e. when
the characteristic diameter of thickness is close to the internal material length scale
parameter, the classical continuum mechanical theories cannot be employed to study
the beam static/dynamic behaviour.
In Refs. [173, 174], investigations regarding the explanation of the so far mentioned effects on a basis of the strain gradient theory for the problems of bending
and torsions have been carried out.
More recently, Srinivasa and Reddy [177] proposed a systematic treatment of
higher gradient theories in a nonlinear context. The size-dependent models of
Bernoulli-Euler beam have been studied in Ref. [43, 176], whereas Timoshenko
beam has been analysed in [178], and Reddy-Levinson beam has been studied in
Refs. [42, 60] using the modified couple stress theory proposed in Ref. [179] (the
latter mainly employs the only size-dependent material length).
The origin and development of the couple stress elasticity theory can be found in
the seminal works of Koiter [37], Toupin [35], Mindlin and Tiersten [36] and Mindlin
[180].
As it has been already mentioned, a modified couple stress theory has been developed by Yang et al. [44], where only one scale parameter of material length as well as
symmetric couple stress tensor have been employed. This concept has been expanded
in Ref. [178], where the variational formalism associated with this theory has been
introduced.
Park and Gao [43] utilized the concepts of the modified couple stress theory
(MCST) to develop a new model for the bending of a Bernoulli-Euler beam. It contains an internal material length scale parameter and in contrary to the classical
Bernoulli-Euler beam model, it captures the size scale effects. Considering a cantilever beam, it has been shown, that the bending rigidity estimated through their
model is larger than that yielded by the classical model.
The seminal work of Eringen [181] serves as a source to develop microstructuredependent nonlocal theories of beams being based on Hamilton’s principle and
the nonlocal constitutive relations including Bernoulli-Euler, Timoshenko, Reddy/
Levinson models [26, 44, 182–184]. Ma et al. [106] constructed a microstructuredependent Timoshenko beam model based on modified couple stress bending and
263
it has been reported that the torsion stiffness of a copper made wires increases simultaneously decreasing their diameter from 170 down to 12 microns (observe that the
size decrease should imply zero stiffness of a wire). In Ref. [16], the increase in the
bending stiffness of a nickel foil corresponding to its height yielded the decrease in
its thickness from 50 to 12.5 microns. The micro- and nano-oriented investigations
[44, 172] show that the stiffness of the pure and poly-crystal metallic materials can
be doubled while decreasing the thickness from 10 to 1 microns. In order to estimate
the material resistance and keep the optimal design of the mentioned products, the
key role is played by the reliable and highly accurate analysis of the stress-strain
states of the investigated micro- and nanomechanical objects.
Importance of the size-dependent behaviour being an inherent property of materials has been presented experimentally by McFarland and Colton [15] and Kong et
al. [17]. It has been demonstrated that in micron/submicron-scale regions, i.e. when
the characteristic diameter of thickness is close to the internal material length scale
parameter, the classical continuum mechanical theories cannot be employed to study
the beam static/dynamic behaviour.
In Refs. [173, 174], investigations regarding the explanation of the so far mentioned effects on a basis of the strain gradient theory for the problems of bending
and torsions have been carried out.
More recently, Srinivasa and Reddy [177] proposed a systematic treatment of
higher gradient theories in a nonlinear context. The size-dependent models of
Bernoulli-Euler beam have been studied in Ref. [43, 176], whereas Timoshenko
beam has been analysed in [178], and Reddy-Levinson beam has been studied in
Refs. [42, 60] using the modified couple stress theory proposed in Ref. [179] (the
latter mainly employs the only size-dependent material length).
The origin and development of the couple stress elasticity theory can be found in
the seminal works of Koiter [37], Toupin [35], Mindlin and Tiersten [36] and Mindlin
[180].
As it has been already mentioned, a modified couple stress theory has been developed by Yang et al. [44], where only one scale parameter of material length as well as
symmetric couple stress tensor have been employed. This concept has been expanded
in Ref. [178], where the variational formalism associated with this theory has been
introduced.
Park and Gao [43] utilized the concepts of the modified couple stress theory
(MCST) to develop a new model for the bending of a Bernoulli-Euler beam. It contains an internal material length scale parameter and in contrary to the classical
Bernoulli-Euler beam model, it captures the size scale effects. Considering a cantilever beam, it has been shown, that the bending rigidity estimated through their
model is larger than that yielded by the classical model.
The seminal work of Eringen [181] serves as a source to develop microstructuredependent nonlocal theories of beams being based on Hamilton’s principle and
the nonlocal constitutive relations including Bernoulli-Euler, Timoshenko, Reddy/
Levinson models [26, 44, 182–184]. Ma et al. [106] constructed a microstructuredependent Timoshenko beam model based on modified couple stress bending and