26
2 Size-Dependent Theories of Beams, Plates and Shells
The mentioned size effects can be explained with a use of the modelling of molecular dynamics (MD) or mechanics of continuum of a higher order. Though the MD
method can yield accurate and reliable results, it is expensive from a point of view
of computations. Therefore, the methods of higher orders of mechanics of a solid
medium are widely employed while modelling of nanoscale structures. Development
of theory of continuum of a higher order begun with the work of Dell’Isola in nineteenth century [9, 10] as well as the pioneering works of Cosserat and Cosserat [11]
published in 1909. However, the ideas proposed by the Cosserat brothers attracted
researchers at the beginning of 1960 yielding a large number of theories of continuum
of higher order.
In general, the developed theories can be divided into three parts: (i) theory of
gradient deformations, (ii) theory of micro-continuum and (iii) nonlocal theories of
elasticity.
The class of theory of gradient deformations consists of the modified couple
stress theory and the modified theory of deformations gradient. In the class of the
deformation gradients, the deformation energy consists of the stress and deformation
gradients, and hence the size effect can be accounted with the help of size parameters
characterizing the material length. In the stress couple theory worked out by Toupin
[12], Mindlin and Tiersten [13], and Koiter [14], in the deformation energy, only a
vector of rotation gradient is employed, and hence only two size-dependent length
parameters are used. Yang et al. [15], based on the change of the couple stress
theory, proposed a modified couple stress theory. In the latter case, the number
of the size-dependent parameters is decreased from two to one. The first theory of
deformation gradient has been proposed by Mindlin [16] who considered only the first
gradient of deformations. One year later, Mindlin [17] developed the second theory
of deformation gradient, where the first and the second gradients of deformations
have been taken into account. Lam et al. [18] proposed the modified couple stress
theory based on three size-dependent length parameters. The latter approach was
based on the modification of the Mindlin theory with the help of approach analogous
to that of Yang et al. [15].
Theory of micro-continuum has been worked out by Eringen [19–21], and it
includes micropolar, microcracks and micromorphic (3M) theories. The micropolar
theory, practically proposed by Cosserat brothers [11], belongs to the most simple
among 3M theories, whereas the micromorphic theory stands for the most general
among 3M theories. In the 3M theories, each particle can rotate and undergoes
deformation independently on the centroid of motion of a particle (see [22–28]).
Nonlocal theory used of elasticity has been initially proposed by Kroner [29] and
then improved by Eringen [30, 31] and Eringen and Edelen [32]. In the latter theory,
stress in a control point of a continuum depends on deformation of all solid body
points, and therefore the size-dependent effect is accounted with the help of state
equations using a nonlocal parameter. The nonlocal theory of elasticity has been first
formulated in the integral form, and later on [33] reformulated into its counterpart
differential form with the help of a defined kernel function. Owing to simplicity of
the differential models, the differential theory has been widely used in modelling of
nanostructures. Besides, more recently one more class of theories of higher order
2 Size-Dependent Theories of Beams, Plates and Shells
The mentioned size effects can be explained with a use of the modelling of molecular dynamics (MD) or mechanics of continuum of a higher order. Though the MD
method can yield accurate and reliable results, it is expensive from a point of view
of computations. Therefore, the methods of higher orders of mechanics of a solid
medium are widely employed while modelling of nanoscale structures. Development
of theory of continuum of a higher order begun with the work of Dell’Isola in nineteenth century [9, 10] as well as the pioneering works of Cosserat and Cosserat [11]
published in 1909. However, the ideas proposed by the Cosserat brothers attracted
researchers at the beginning of 1960 yielding a large number of theories of continuum
of higher order.
In general, the developed theories can be divided into three parts: (i) theory of
gradient deformations, (ii) theory of micro-continuum and (iii) nonlocal theories of
elasticity.
The class of theory of gradient deformations consists of the modified couple
stress theory and the modified theory of deformations gradient. In the class of the
deformation gradients, the deformation energy consists of the stress and deformation
gradients, and hence the size effect can be accounted with the help of size parameters
characterizing the material length. In the stress couple theory worked out by Toupin
[12], Mindlin and Tiersten [13], and Koiter [14], in the deformation energy, only a
vector of rotation gradient is employed, and hence only two size-dependent length
parameters are used. Yang et al. [15], based on the change of the couple stress
theory, proposed a modified couple stress theory. In the latter case, the number
of the size-dependent parameters is decreased from two to one. The first theory of
deformation gradient has been proposed by Mindlin [16] who considered only the first
gradient of deformations. One year later, Mindlin [17] developed the second theory
of deformation gradient, where the first and the second gradients of deformations
have been taken into account. Lam et al. [18] proposed the modified couple stress
theory based on three size-dependent length parameters. The latter approach was
based on the modification of the Mindlin theory with the help of approach analogous
to that of Yang et al. [15].
Theory of micro-continuum has been worked out by Eringen [19–21], and it
includes micropolar, microcracks and micromorphic (3M) theories. The micropolar
theory, practically proposed by Cosserat brothers [11], belongs to the most simple
among 3M theories, whereas the micromorphic theory stands for the most general
among 3M theories. In the 3M theories, each particle can rotate and undergoes
deformation independently on the centroid of motion of a particle (see [22–28]).
Nonlocal theory used of elasticity has been initially proposed by Kroner [29] and
then improved by Eringen [30, 31] and Eringen and Edelen [32]. In the latter theory,
stress in a control point of a continuum depends on deformation of all solid body
points, and therefore the size-dependent effect is accounted with the help of state
equations using a nonlocal parameter. The nonlocal theory of elasticity has been first
formulated in the integral form, and later on [33] reformulated into its counterpart
differential form with the help of a defined kernel function. Owing to simplicity of
the differential models, the differential theory has been widely used in modelling of
nanostructures. Besides, more recently one more class of theories of higher order