Linear buckling
===================

Linear buckling is a particular case of the geometric nonlinear problem. It is based on the assumption of a linear dependence of the fields of displacements, deformations and stresses on the load level, before the critical load is reached.

In a total Lagrangian formulation, we recall that linearized equilibrium can be written in the variational form:

:math:`\Delta \delta {\pi }_{\text{int}}\mathrm{-}\Delta \delta {\pi }_{\text{ext}}\mathrm{=}\delta {\pi }_{\text{ext}}\mathrm{-}\delta {\pi }_{\text{int}}`

or in matrix form after discretization:

:math:`\delta u\mathrm{\cdot }{K}_{T}\Delta u\mathrm{=}\delta u\mathrm{\cdot }(\lambda f\mathrm{-}r)`

where the dependence of the tangent stiffness matrix :math:`{K}_{T}` is nonlinear with respect to the generalized nodal displacement vector :math:`p\mathrm{=}\underset{\text{e=}1\text{,Nel}}{U}{p}^{e}`.

If we assume the linear dependence of the displacement on the load level, we can write:

:math:`u\mathrm{=}\lambda {u}_{0}`

where :math:`{u}_{0}` is the solution obtained following a linear analysis for :math:`\lambda \mathrm{=}1` by:

:math:`{K}_{0}{u}_{0}\mathrm{=}f`

where :math:`{K}_{0}` is the initial stiffness tangent matrix. It is then possible to develop the tangent stiffness matrix in a linear manner with respect to the load level:

:math:`\underset{e\mathrm{=}\mathrm{1,}\text{Nel}}{U}{K}_{T}^{e}\mathrm{=}{K}_{0}^{e}+\lambda ({K}_{u}^{e}+{K}_{\sigma }^{e})+\text{.}\text{.}\text{.}\text{.}`

where :math:`{K}_{u}^{e}` is the matrix of initial displacements dependent on :math:`{p}_{0}^{e}`, traditionally overlooked in *Code_Aster,* and :math:`{K}_{\sigma }^{e}` is the matrix of initial stresses depending on the global tensor of Piola-Kirchhoff stresses of the second kind :math:`\left[{S}_{0}\right]` and on the local vector :math:`{\tilde{S}}_{0}`. These constraints are deliberately confused with the Cauchy constraints. They are obtained by post-processing the linear analysis.

For the rotation part of :math:`{p}^{e}`, the assumption of linearity of the deformations as a function of the load level results in the equality of:

:math:`{n}_{I}^{\varphi }\mathrm{=}{n}_{I}`

which leads us to confuse the initial normals :math:`{n}_{I}` with their :math:`{n}_{I}^{\varphi }` transforms.

The initial stress matrix :math:`{K}_{\sigma }^{e}` represents the constant part in :math:`\lambda` of the geometric part of the tangent stiffness matrix. It is evaluated at points :math:`{p}_{0}^{e}` and :math:`\lambda \mathrm{=}1` with a normal transform replaced by the initial normal:

:math:`{K}_{\sigma }^{e}\mathrm{=}{K}_{\sigma }^{{e}_{\text{classique}}}+{K}_{\sigma }^{{e}_{\text{non classique}}}`

with the classical part of the geometric part (see [:ref:`bib4 <bib4>`] volume 1 p. 141):

:math:`{K}_{\sigma }^{{e}_{\text{classique}}}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\sigma }\right]}_{2}\right]}^{T}\stackrel{ˉ}{S}\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\right]\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`

where the second matrix of derivatives of form functions becomes:

:math:`\begin{array}{c}\mathrm{\cdots }\left[\left[\begin{array}{ccc}{N}_{I,{\xi }_{1}}^{(1)}& 0& 0\\ {N}_{I,{\xi }_{2}}^{(1)}& 0& 0\\ 0& 0& 0\\ 0& {N}_{I,{\xi }_{1}}^{(1)}& 0\\ 0& {N}_{I,{\xi }_{2}}^{(1)}& 0\\ 0& 0& 0\\ 0& 0& {N}_{I,{\xi }_{1}}^{(1)}\\ 0& 0& {N}_{I,{\xi }_{2}}^{(1)}\\ 0& 0& 0\end{array}\right]\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}\\ 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}\\ 0& {N}_{I}^{(2)}{n}_{z}& \mathrm{-}{N}_{I}^{(2)}{n}_{y}\\ \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}\\ \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}\\ \mathrm{-}{N}_{I}^{(2)}{n}_{z}& 0& {N}_{I}^{(2)}{n}_{x}\\ {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}& 0\\ {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}& 0\\ {N}_{I}^{(2)}{n}_{y}& \mathrm{-}{N}_{I}^{(2)}{n}_{x}& 0\end{array}\right]\right]\mathrm{\cdots }I\mathrm{=}\mathrm{1,}\text{NB}1\\ \mathrm{\mid }\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}\\ 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}\\ 0& {N}_{\text{NB}2}^{(2)}{n}_{z}& \mathrm{-}{N}_{1}^{(2)}{n}_{y}\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}\\ \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{z}& 0& {N}_{\text{NB}2}^{(2)}{n}_{x}\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}& 0\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}& 0\\ {N}_{\text{NB}2}^{(2)}{n}_{y}& \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{x}& 0\end{array}\right]\\ {\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\mathrm{=}\begin{array}{c}\left[\right]\left[\right]\end{array}\mathrm{[}\mathrm{]}\end{array}`

and the generalized tensor of global constraints:

:math:`\stackrel{9\mathrm{\times }9}{\stackrel{ˉ}{S}}\mathrm{=}\left[\begin{array}{ccc}\mathrm{[}\stackrel{3\mathrm{\times }3}{S}\mathrm{]}& 0& 0\\ 0& \left[S\right]& 0\\ 0& 0& \left[S\right]\end{array}\right]`

The non-classical part of the geometric part represents the decoupled, non-symmetric terms of rotation. Since the current algorithm for solving the problem with eigenvalues [:external:ref:`R5.06.01 <R5.06.01>`] :math:`\left[{K}_{0}+\lambda ({K}_{u}+{K}_{\sigma })\right]F\mathrm{=}0` (:math:`\lambda` being the critical load level) only considers symmetric matrices, we make the matrix symmetric, by dividing by two the sum with its transpose, the matrix:

:math:`{K}_{\sigma }^{{e}_{\text{non classique}}}(I,I)\mathrm{=}\frac{1}{2}(\left[{z}_{I}\mathrm{\times }\right]\left[{n}_{I}\mathrm{\times }\right])`

where :math:`{n}_{I}` is the normal to node :math:`I` and :math:`{z}_{I}` is a vector :math:`3\mathrm{\times }3` to node :math:`I\mathrm{=}\mathrm{1,}\text{NB}2` of the node vector :math:`3\mathrm{\times }(3\mathrm{\times }\text{NB}2){Z}_{I}`:

:math:`{Z}_{I}\mathrm{=}(\begin{array}{c}\text{.}\\ \text{.}\\ \text{.}\\ {z}_{I}\\ \text{.}\\ \text{.}\\ \text{.}\\ I\mathrm{=}\mathrm{1,}\text{NB}2\end{array})`

Nodal vector :math:`{Z}_{I}` is similar to an internal force vector and its expression is:

:math:`{Z}_{I}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{z}}{\tilde{B}}_{3}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`

with the operator for the iterative variation of virtual deformations (third operator of deformations):

:math:`{\tilde{B}}_{3}\mathrm{=}\text{HQ}{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}`

which is evidenced by the relationship:

:math:`{\mathrm{\int }}_{{\Omega }_{z}}{\tilde{B}}_{3}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}\mathrm{=}{\mathrm{\int }}_{\Omega }\Delta \delta {\tilde{E}}_{\text{non classique}}\mathrm{\cdot }\tilde{S}d\Omega`

**Note:**

For the numerical integration into the thickness of the various stiffness terms, we use a two-point Gauss diagram just as in elasticity for geometric linear shells [:ref:`R3.07.04 <R3.07.04>`].