Chapter 3
Geometrically Nonlinear Theories
Abstract This chapter starts with discussing various hypotheses, and the differences
between these hypotheses are outlined. Afterwards, the mathematical preliminaries,
including position vectors, covariant and contravariant base vectors, Christoffel symbols, shifter tensor, curvature tensor, etc., will be defined and discussed. Based on
the FOSD hypothesis, through-thickness displacement distribution is assumed, where
six parameters are introduced. Using these predefined quantities, Green-Lagrange
strain tensor with fully geometrically nonlinear strain-displacement relations is developed in terms of six parameters for geometrically nonlinear theory with unrestricted
finite rotations (LRT56). Imposing different assumptions, various simplified nonlinear strain-displacement relations are developed for the theories of von Kármán
type nonlinear (RVK5), moderate rotation nonlinear (MRT5), fully geometrically
nonlinear with moderate rotations (LRT5).
3.1 Shear Deformation Hypotheses
The FE method with 3-D solid elements is one of the possible solutions for modeling
of thin-walled composite and smart structures. Even though the thickness of plates
and shells are very small compared to the in-plane dimensions, the elements through
the thickness direction must reach a certain number to ensure the computation accuracy. Therefore, using 3-D solid element for modeling of thin-walled smart structures
certainly results in large model size and high computation time. Because of small
thickness in thin-walled plate and shell structures, FE methods with 2-D surface elements based on various hypotheses (shown in Fig. 2.1) are more frequently used in
numerical analysis. The main advantage of 2-D FE models is that less computation
time is needed due to small size of the models compared to 3-D ones, but they are
still retaining a relatively high accuracy. For beam structures, 2-D surface element
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_3
37
Geometrically Nonlinear Theories
Abstract This chapter starts with discussing various hypotheses, and the differences
between these hypotheses are outlined. Afterwards, the mathematical preliminaries,
including position vectors, covariant and contravariant base vectors, Christoffel symbols, shifter tensor, curvature tensor, etc., will be defined and discussed. Based on
the FOSD hypothesis, through-thickness displacement distribution is assumed, where
six parameters are introduced. Using these predefined quantities, Green-Lagrange
strain tensor with fully geometrically nonlinear strain-displacement relations is developed in terms of six parameters for geometrically nonlinear theory with unrestricted
finite rotations (LRT56). Imposing different assumptions, various simplified nonlinear strain-displacement relations are developed for the theories of von Kármán
type nonlinear (RVK5), moderate rotation nonlinear (MRT5), fully geometrically
nonlinear with moderate rotations (LRT5).
3.1 Shear Deformation Hypotheses
The FE method with 3-D solid elements is one of the possible solutions for modeling
of thin-walled composite and smart structures. Even though the thickness of plates
and shells are very small compared to the in-plane dimensions, the elements through
the thickness direction must reach a certain number to ensure the computation accuracy. Therefore, using 3-D solid element for modeling of thin-walled smart structures
certainly results in large model size and high computation time. Because of small
thickness in thin-walled plate and shell structures, FE methods with 2-D surface elements based on various hypotheses (shown in Fig. 2.1) are more frequently used in
numerical analysis. The main advantage of 2-D FE models is that less computation
time is needed due to small size of the models compared to 3-D ones, but they are
still retaining a relatively high accuracy. For beam structures, 2-D surface element
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_3
37