2.2 Geometrically Nonlinear Modeling in Composites
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2.2.3 Shear Locking Phenomena
Due to the inconsistencies between element representation and transverse shear
energy or membrane energy, plate and shell elements may exhibit over stiffening,
especially if the thickness tends to be zero. Locking problems are usually referred
to shear locking and membrane locking. Shear locking is caused by the Kirchhoff constraints or shear constraints of vanishing transverse shear strains, while
membrane locking results from hidden constraints in shell models. The details of
introduction of locking phenomena can be found in e.g. [88, 103, 104] among
others. To avoid locking problems many numerical methods were proposed and
developed e.g. assumed natural strain (ANS) [90, 105–107], enhanced assumed
strain (EAS) [108–111], selectively reduced integration (SRI) [112] and uniformly
reduced integration (URI) [113–115]. Alternatively, locking effects can be reduced
by increasing the number of elements for structures or the number of nodes in an
element. Increasing the number of nodes in an element will directly result in higherorder polynomial functions. The method is also known as h- p finite element method,
which was proposed and developed earlier by Pitkäranta et al. [103, 116], Leino and
Pitkäranta [104] and later by Ref. [88, 117, 118].
2.3 Geometrically Nonlinear Modeling for Smart
Structures
Linear models are only valid for smart structures undergoing small displacements
and under weak electric fields. When large displacements and rotations occur, geometrically nonlinear theories should be considered in FE models. With consideration
of different nonlinear effects and permission of different levels of rotations, various
geometrically nonlinear theories were proposed and developed, e.g. von Kármán
type nonlinear theory, moderate rotation nonlinear shell theory, fully geometrically
nonlinear theory with moderate rotations, and large rotation nonlinear theory. The
number of papers dealt with geometrically nonlinear analysis are much less than
those with linear analysis.
2.3.1 Von Kármán Type Nonlinear Theory
The von Kármán type nonlinear theory is the simplest nonlinear theory, which is used
very frequently in nonlinear analysis of smart structures. The theory contains only
the squares and products of derivatives of the transverse deflection in the in-plane
longitudinal and shear strain components. The theory is only valid for structures
undergoing moderate displacements and small rotations. Im and Atluri [119] first
applied von Kármán type nonlinear theory into analysis of piezoelectric integrated
13
2.2.3 Shear Locking Phenomena
Due to the inconsistencies between element representation and transverse shear
energy or membrane energy, plate and shell elements may exhibit over stiffening,
especially if the thickness tends to be zero. Locking problems are usually referred
to shear locking and membrane locking. Shear locking is caused by the Kirchhoff constraints or shear constraints of vanishing transverse shear strains, while
membrane locking results from hidden constraints in shell models. The details of
introduction of locking phenomena can be found in e.g. [88, 103, 104] among
others. To avoid locking problems many numerical methods were proposed and
developed e.g. assumed natural strain (ANS) [90, 105–107], enhanced assumed
strain (EAS) [108–111], selectively reduced integration (SRI) [112] and uniformly
reduced integration (URI) [113–115]. Alternatively, locking effects can be reduced
by increasing the number of elements for structures or the number of nodes in an
element. Increasing the number of nodes in an element will directly result in higherorder polynomial functions. The method is also known as h- p finite element method,
which was proposed and developed earlier by Pitkäranta et al. [103, 116], Leino and
Pitkäranta [104] and later by Ref. [88, 117, 118].
2.3 Geometrically Nonlinear Modeling for Smart
Structures
Linear models are only valid for smart structures undergoing small displacements
and under weak electric fields. When large displacements and rotations occur, geometrically nonlinear theories should be considered in FE models. With consideration
of different nonlinear effects and permission of different levels of rotations, various
geometrically nonlinear theories were proposed and developed, e.g. von Kármán
type nonlinear theory, moderate rotation nonlinear shell theory, fully geometrically
nonlinear theory with moderate rotations, and large rotation nonlinear theory. The
number of papers dealt with geometrically nonlinear analysis are much less than
those with linear analysis.
2.3.1 Von Kármán Type Nonlinear Theory
The von Kármán type nonlinear theory is the simplest nonlinear theory, which is used
very frequently in nonlinear analysis of smart structures. The theory contains only
the squares and products of derivatives of the transverse deflection in the in-plane
longitudinal and shear strain components. The theory is only valid for structures
undergoing moderate displacements and small rotations. Im and Atluri [119] first
applied von Kármán type nonlinear theory into analysis of piezoelectric integrated