Chapter 6
Mathematical Models of Micro- and
Nano-cylindrical Panels in Temperature
Field
6.1 Introduction
Mathematical models of nonlinear micro- and nano-cylindrical panels in temperature
fields are introduced and studied. First, the application of the modified couple stress
theory of thermoelastic curvilinear panels based on the third-order hypotheses has
been described. Then, a technical theory of the Sheremetev–Pelekh, Timoshenko
and Bernoulli–Euler models is presented. The method of solving static problems
is outlined in Sect. 6.4. Chaotic dynamics of the size-dependent flexible Bernoulli–
Euler, Timoshenko and Sheremetev–Pelekh beams based on the modified couple
stress theory of elasticity is investigated in Sect. 6.5. The two last sections are devoted
to the construction of so-called charts of vibration character with regard to amplitude
and frequency of the external excitation and their study with the use of the first, second
and third kinematic hypotheses.
6.2 Literature Review
Occurrence of new polymer composite material (reinforced plastics, etc.) in engineering practice essentially increased motivation for the development and generalization
of the so far used classical theories of beams, plates and shells. Construction of
the theories focused on reliable computations of the structural members made from
the new materials with an account of specific features in their behaviour, grading
of material in one or two directions as well as the size-dependent behaviour under
dimensions of micro- and nano-meter orders requires novel challenging modelling
and mathematical/numerical approaches.
One of the ways to improve the accuracy of classical theory of shells is associated
with the employment of higher order models. Two Ukrainian scientists Sheremetev
© 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_6
131
Mathematical Models of Micro- and
Nano-cylindrical Panels in Temperature
Field
6.1 Introduction
Mathematical models of nonlinear micro- and nano-cylindrical panels in temperature
fields are introduced and studied. First, the application of the modified couple stress
theory of thermoelastic curvilinear panels based on the third-order hypotheses has
been described. Then, a technical theory of the Sheremetev–Pelekh, Timoshenko
and Bernoulli–Euler models is presented. The method of solving static problems
is outlined in Sect. 6.4. Chaotic dynamics of the size-dependent flexible Bernoulli–
Euler, Timoshenko and Sheremetev–Pelekh beams based on the modified couple
stress theory of elasticity is investigated in Sect. 6.5. The two last sections are devoted
to the construction of so-called charts of vibration character with regard to amplitude
and frequency of the external excitation and their study with the use of the first, second
and third kinematic hypotheses.
6.2 Literature Review
Occurrence of new polymer composite material (reinforced plastics, etc.) in engineering practice essentially increased motivation for the development and generalization
of the so far used classical theories of beams, plates and shells. Construction of
the theories focused on reliable computations of the structural members made from
the new materials with an account of specific features in their behaviour, grading
of material in one or two directions as well as the size-dependent behaviour under
dimensions of micro- and nano-meter orders requires novel challenging modelling
and mathematical/numerical approaches.
One of the ways to improve the accuracy of classical theory of shells is associated
with the employment of higher order models. Two Ukrainian scientists Sheremetev
© 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_6
131