38
3 Geometrically Nonlinear Theories
can further be simplified to 1-D beam element using Euler-Bernoulli, Timoshenko
or higher-order beam hypothesis.
The simplest hypothesis for plates and shells is the Kirchhoff-Love theory, which
is also known as classical lamination theory (CLT), see the Refs. [1–7]. The classical
lamination theory assumes that the straight lines normal to the mid-surface in the
undeformed configuration remain straight and normal after deformation. The EulerBernoulli beam theory has a same assumption as that in classical theory. Due to
the assumption imposed by classical theory, there is no transverse shear strain in
the mathematical model, which may result in an inadequate prediction of the elastic
behavior of layered composite and smart structures.
In order to introduce transverse shear strain, Reissner-Mindlin plate/shell theory [8] known as FOSD hypothesis was proposed. The FOSD hypothesis assumes
that the straight lines normal to the mid-surface in the undeformed configuration
remain straight after deformation, but not necessarily normal. By the assumption,
there exists an additional angle between the FOSD line and CLT line, defined as
transverse shear strain. Analogously, the FOSD plate/shell theory can be treated as
an extension of the Timoshenko beam theory. It can seen that both the classical theory
and Reissner-Mindlin theory assume a linear variation of displacement through the
shell thickness.
For moderately thick structures, classical theory and the FOSD theory may not
be accurate enough. In order to deal with thick structures, SOSD (Second-order
Shear Deformation), TOSD or other HOSD hypotheses were proposed. Due to different higher-order hypotheses, the through-thickness displacements can be distributed
quadratically, cubicly or other higher-order functions, see Fig. 2.1. With these higherorder shear deformation hypotheses, one can satisfy zero transverse shear strains at
outer surfaces while the transverse shear strains vary nonlinearly inside the structure,
which is a more practical way that structures usually occur.
Concerning laminated shell structures made of different materials, all the above
mentioned hypotheses exist inter-layer transverse shear stress discontinuity. Zigzag
theory assumes independent transverse shear strain for each layer. This satisfies the
inter-layer shear stress continuity. The first-order zigzag theory describes the throughthickness displacement as a fold line. For more accurately, second- or third-order
zigzag hypothesis can be employed.
3.2 Mathematical Preliminaries
3.2.1 Introduction of Coordinates
Two coordinate systems are introduced for the mathematical modeling, as shown
in Fig. 3.1. One is the Cartesian coordinate system, represented by X
1 , X
2 and X
3 ,
acting as global coordinate system. The other one is the curvilinear coordinate system
represented by Θ
1 , Θ
2 and Θ
3 , acting as convective coordinate system. The global
3 Geometrically Nonlinear Theories
can further be simplified to 1-D beam element using Euler-Bernoulli, Timoshenko
or higher-order beam hypothesis.
The simplest hypothesis for plates and shells is the Kirchhoff-Love theory, which
is also known as classical lamination theory (CLT), see the Refs. [1–7]. The classical
lamination theory assumes that the straight lines normal to the mid-surface in the
undeformed configuration remain straight and normal after deformation. The EulerBernoulli beam theory has a same assumption as that in classical theory. Due to
the assumption imposed by classical theory, there is no transverse shear strain in
the mathematical model, which may result in an inadequate prediction of the elastic
behavior of layered composite and smart structures.
In order to introduce transverse shear strain, Reissner-Mindlin plate/shell theory [8] known as FOSD hypothesis was proposed. The FOSD hypothesis assumes
that the straight lines normal to the mid-surface in the undeformed configuration
remain straight after deformation, but not necessarily normal. By the assumption,
there exists an additional angle between the FOSD line and CLT line, defined as
transverse shear strain. Analogously, the FOSD plate/shell theory can be treated as
an extension of the Timoshenko beam theory. It can seen that both the classical theory
and Reissner-Mindlin theory assume a linear variation of displacement through the
shell thickness.
For moderately thick structures, classical theory and the FOSD theory may not
be accurate enough. In order to deal with thick structures, SOSD (Second-order
Shear Deformation), TOSD or other HOSD hypotheses were proposed. Due to different higher-order hypotheses, the through-thickness displacements can be distributed
quadratically, cubicly or other higher-order functions, see Fig. 2.1. With these higherorder shear deformation hypotheses, one can satisfy zero transverse shear strains at
outer surfaces while the transverse shear strains vary nonlinearly inside the structure,
which is a more practical way that structures usually occur.
Concerning laminated shell structures made of different materials, all the above
mentioned hypotheses exist inter-layer transverse shear stress discontinuity. Zigzag
theory assumes independent transverse shear strain for each layer. This satisfies the
inter-layer shear stress continuity. The first-order zigzag theory describes the throughthickness displacement as a fold line. For more accurately, second- or third-order
zigzag hypothesis can be employed.
3.2 Mathematical Preliminaries
3.2.1 Introduction of Coordinates
Two coordinate systems are introduced for the mathematical modeling, as shown
in Fig. 3.1. One is the Cartesian coordinate system, represented by X
1 , X
2 and X
3 ,
acting as global coordinate system. The other one is the curvilinear coordinate system
represented by Θ
1 , Θ
2 and Θ
3 , acting as convective coordinate system. The global