Chapter 2
Size-Dependent Theories of Beams,
Plates and Shells
2.1 Introduction
This chapter concerns the latest literature review devoted to micro/nano-sizedependent mathematical models of beams, plates and shells. The review includes
nonlocal theory of elasticity, surface theory of elasticity, modified couple stress theory and modified theory of deformation gradient taking into account higher order
shear deformation theory. Particular emphasis is placed on nonlocal models of the
Euler-Bernoulli and Timoshenko beams, Kirchhoff plates and Kirchhoff-Love shells.
Both the review and the real-world vibrational behaviour of the size-dependent structural member implies the need to consider physical, geometric, material and design
nonlinearity. It is expected that the novel approach based on employment of the earlier developed concepts of nonlinear dynamics like the Fourier and wavelet spectra,
phase portraits, Poincaré maps, the Lyapunov exponents computation (at least the
largest one), the autocorrelation functions, etc. will improve mathematical models of
partial differential equations (PDEs) to be in a good fit with experimentally observed
nonlinear phenomena exhibited by the size-dependent structural members.
2.2 Literature Review
Micro- and nanosize-dependent beams, plates and shells are widely employed in
micro- and nanoelectromechanical systems (MEMS and NEMS) serving as vibration
sensors [1], micro-drives [2] and micro-switchers [3]. The mentioned objects exhibited the size-dependent effects with regard to other mechanical properties [4–8]. The
classical solid mechanics cannot give a proper interpretation of the size-dependent
behaviour occurring in structures exhibiting scale effects. In the last years, many
novel theories have been proposed allowing for modelling of the scale effects in
continuum.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_2
25
Size-Dependent Theories of Beams,
Plates and Shells
2.1 Introduction
This chapter concerns the latest literature review devoted to micro/nano-sizedependent mathematical models of beams, plates and shells. The review includes
nonlocal theory of elasticity, surface theory of elasticity, modified couple stress theory and modified theory of deformation gradient taking into account higher order
shear deformation theory. Particular emphasis is placed on nonlocal models of the
Euler-Bernoulli and Timoshenko beams, Kirchhoff plates and Kirchhoff-Love shells.
Both the review and the real-world vibrational behaviour of the size-dependent structural member implies the need to consider physical, geometric, material and design
nonlinearity. It is expected that the novel approach based on employment of the earlier developed concepts of nonlinear dynamics like the Fourier and wavelet spectra,
phase portraits, Poincaré maps, the Lyapunov exponents computation (at least the
largest one), the autocorrelation functions, etc. will improve mathematical models of
partial differential equations (PDEs) to be in a good fit with experimentally observed
nonlinear phenomena exhibited by the size-dependent structural members.
2.2 Literature Review
Micro- and nanosize-dependent beams, plates and shells are widely employed in
micro- and nanoelectromechanical systems (MEMS and NEMS) serving as vibration
sensors [1], micro-drives [2] and micro-switchers [3]. The mentioned objects exhibited the size-dependent effects with regard to other mechanical properties [4–8]. The
classical solid mechanics cannot give a proper interpretation of the size-dependent
behaviour occurring in structures exhibiting scale effects. In the last years, many
novel theories have been proposed allowing for modelling of the scale effects in
continuum.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_2
25