12
2 Literature Review
Bufler [74], Wriggers and Gruttmann [75], Sansour and Bednarczyk [76], Brank et
al. [77], Bischoff and Ramm [78], Kreja and Schmidt [79], Lentzen [80] and others.
Kuznetsov and Levyakov [81] developed a fully geometrically nonlinear model with
large rotations based on the Kirchhoff-Love theory. Moreover, large rotation nonlinear models were developed for beam or arch structures by Saravia et al. [82], Miller
and Palazotto [83].
For relatively thick plates and shells, the TOSD hypothesis was implemented into
the large rotation theory by Basar et al. [84, 85] for composite structures, which
assumes inextensible shell director yielding seven parameters. Similar TOSD nonlinear models were developed by Bischoff and Ramm [78], Gummadi and Palazotto [86, 87]. Later, Arciniega and Reddy [88] implemented the second-order shear
deformation (SOSD) hypothesis into large rotation theory. The SOSD hypothesis
assumes a quadratic displacement distribution along the thickness direction. In the
model, 3-dimensional constitutive equations was applied, which indicates that the
shell director is considered as extensible.
Concerning with soft materials, Basar and Ding [89] developed a nonlinear model
considering large strains by taking into account the the transverse normal strain based
on SOSD hypothesis. To avoid shear locking phenomenon, large rotation models
with four-node assumed strain elements were developed by Dvorkin and Bathe [90],
Stander et al. [91], and a nonlinear model with four-node mixed interpolation elements was proposed by Sze et al. [92]. In addition, fully geometrically nonlinear
models with using solid elements were developed by Koˇ zar and Ibrahimbegovi´ c [93],
Masud et al. [94], Lopez and Sala [95] for static analysis of shell structures.
Large or finite rotation theories presented in some publications were not permitting arbitrarily large rotations of the shell director, even though fully geometrically
nonlinear strain-displacement relations were considered. Large or finite rotation theories are those which not only consider fully geometrically nonlinear phenomena
but also take into account unrestricted rotations. There are two typical approach
for large rotation representation, namely Euler angles formulation and Rodrigues
rotation formulation, see [96] for the detailed classification. In the FOSD hypothesis, large rotation theory usually includes six independent kinematic parameters.
Neglecting the drilling rotation in plates and shells, two rotational variables are proposed to represent last three kinematic parameters. The first approach, Euler angle
formulation, was implemented to represent large rotations by Gruttmann et al. [71],
Bruechter and Ramm [97], Basar et al. [73], Wriggers and Gruttmann [75], Brank et
al. [77], Kreja and Schmidt [79] and others. Additionally, the Rodrigues rotation
formulation was proposed by Simo et al. [98, 99]. Later, it was implemented and
applied by Sansour and Bufler [74], Betsch et al. [96, 100], Basar et al. [101], Wang
and Thierauf [102], Lentzen [80].
2 Literature Review
Bufler [74], Wriggers and Gruttmann [75], Sansour and Bednarczyk [76], Brank et
al. [77], Bischoff and Ramm [78], Kreja and Schmidt [79], Lentzen [80] and others.
Kuznetsov and Levyakov [81] developed a fully geometrically nonlinear model with
large rotations based on the Kirchhoff-Love theory. Moreover, large rotation nonlinear models were developed for beam or arch structures by Saravia et al. [82], Miller
and Palazotto [83].
For relatively thick plates and shells, the TOSD hypothesis was implemented into
the large rotation theory by Basar et al. [84, 85] for composite structures, which
assumes inextensible shell director yielding seven parameters. Similar TOSD nonlinear models were developed by Bischoff and Ramm [78], Gummadi and Palazotto [86, 87]. Later, Arciniega and Reddy [88] implemented the second-order shear
deformation (SOSD) hypothesis into large rotation theory. The SOSD hypothesis
assumes a quadratic displacement distribution along the thickness direction. In the
model, 3-dimensional constitutive equations was applied, which indicates that the
shell director is considered as extensible.
Concerning with soft materials, Basar and Ding [89] developed a nonlinear model
considering large strains by taking into account the the transverse normal strain based
on SOSD hypothesis. To avoid shear locking phenomenon, large rotation models
with four-node assumed strain elements were developed by Dvorkin and Bathe [90],
Stander et al. [91], and a nonlinear model with four-node mixed interpolation elements was proposed by Sze et al. [92]. In addition, fully geometrically nonlinear
models with using solid elements were developed by Koˇ zar and Ibrahimbegovi´ c [93],
Masud et al. [94], Lopez and Sala [95] for static analysis of shell structures.
Large or finite rotation theories presented in some publications were not permitting arbitrarily large rotations of the shell director, even though fully geometrically
nonlinear strain-displacement relations were considered. Large or finite rotation theories are those which not only consider fully geometrically nonlinear phenomena
but also take into account unrestricted rotations. There are two typical approach
for large rotation representation, namely Euler angles formulation and Rodrigues
rotation formulation, see [96] for the detailed classification. In the FOSD hypothesis, large rotation theory usually includes six independent kinematic parameters.
Neglecting the drilling rotation in plates and shells, two rotational variables are proposed to represent last three kinematic parameters. The first approach, Euler angle
formulation, was implemented to represent large rotations by Gruttmann et al. [71],
Bruechter and Ramm [97], Basar et al. [73], Wriggers and Gruttmann [75], Brank et
al. [77], Kreja and Schmidt [79] and others. Additionally, the Rodrigues rotation
formulation was proposed by Simo et al. [98, 99]. Later, it was implemented and
applied by Sansour and Bufler [74], Betsch et al. [96, 100], Basar et al. [101], Wang
and Thierauf [102], Lentzen [80].