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22.2 Incompatible Modes Elements

Another assumed strain approach is to approximate the strain field using not only the compatible displacements but also an additional incompatible strain field which is defined by interpolations for strain or displacement variables. An example of an element which has additional displacement variables is the incompatible modes element, introduced by Wilson et al. [21]. More recently Simo & Rifai [17] proposed to use strain variables. The derivation of elements with displacement variables is analogous.

Consider the Hu-Washizu functional

F = $\displaystyle \int_{{V}}^{}$$\displaystyle \left(\vphantom{ \tfrac{1}{2}\boldsymbol{\varepsilon}^{\mathrm{\s...
...tscriptstyle{T}}}\mathbf{N}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{f} }\right.$$\displaystyle {\tfrac{{1}}{{2}}}$$\displaystyle \boldsymbol\varepsilon$TD$\displaystyle \boldsymbol\varepsilon$ - $\displaystyle \boldsymbol\sigma$TBu - uTNTf$\displaystyle \left.\vphantom{ \tfrac{1}{2}\boldsymbol{\varepsilon}^{\mathrm{\s...
...tscriptstyle{T}}}\mathbf{N}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{f} }\right)$ dV (22.8)

in which $ \boldsymbol\varepsilon$ is the strain vector, D is the stress-strain relation, $ \boldsymbol\sigma$ is the stress vector, u is the displacement vector, B is the standard discrete strain operator matrix and N are the interpolation functions of an element conveniently collected in a matrix. The strain is now interpolated as

$\displaystyle \boldsymbol\varepsilon$ = Bu + M$\displaystyle \boldsymbol\alpha$ (22.9)

with M containing the interpolation function of the additional strains and $ \boldsymbol\alpha$ collecting the additional strain variables. Substitution of (22.9) in (22.8) yields

F = $\displaystyle \int_{{V}}^{}$$\displaystyle \left(\vphantom{ \tfrac{1}{2}\left( \mathbf{B} \mathbf{u} + \math...
...tscriptstyle{T}}}\mathbf{N}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{f} }\right.$$\displaystyle {\tfrac{{1}}{{2}}}$$\displaystyle \left(\vphantom{ \mathbf{B} \mathbf{u} + \mathbf{M} \boldsymbol{\alpha} }\right.$Bu + M$\displaystyle \boldsymbol\alpha$$\displaystyle \left.\vphantom{ \mathbf{B} \mathbf{u} + \mathbf{M} \boldsymbol{\alpha} }\right)^{{\mathrm{\scriptscriptstyle{T}}}}_{}$D$\displaystyle \left(\vphantom{ \mathbf{B} \mathbf{u} + \mathbf{M} \boldsymbol{\alpha} }\right.$Bu + M$\displaystyle \boldsymbol\alpha$$\displaystyle \left.\vphantom{ \mathbf{B} \mathbf{u} + \mathbf{M} \boldsymbol{\alpha} }\right)$ - $\displaystyle \boldsymbol\sigma$TM$\displaystyle \boldsymbol\alpha$ - uTNTf$\displaystyle \left.\vphantom{ \tfrac{1}{2}\left( \mathbf{B} \mathbf{u} + \math...
...tscriptstyle{T}}}\mathbf{N}^{\mathrm{\scriptscriptstyle{T}}}\mathbf{f} }\right)$ dV (22.10)

We wish to eliminate the statically admissible stress field from the functional, therefore the operator M$ \boldsymbol\alpha$ has to be constructed such that

$\displaystyle \int_{{V}}^{}$$\displaystyle \boldsymbol\sigma$TM$\displaystyle \boldsymbol\alpha$ dV = 0 (22.11)

Taking the variation of the functional with respect to u and $ \boldsymbol\alpha$ yields the following system of equations.

$\displaystyle \left[\vphantom{ \negthickspace \begin{array}{cc} \mathbf{K} & \m...
...scriptstyle{T}}}\\  \mathbf{L} & \mathbf{Q} \end{array} \negthickspace }\right.$ $\displaystyle \begin{array}{cc} \mathbf{K} & \mathbf{L}^{\mathrm{\scriptscriptstyle{T}}}\\  \mathbf{L} & \mathbf{Q} \end{array}$ $\displaystyle \left.\vphantom{ \negthickspace \begin{array}{cc} \mathbf{K} & \m...
...scriptstyle{T}}}\\  \mathbf{L} & \mathbf{Q} \end{array} \negthickspace }\right]$$\displaystyle \left\{\vphantom{ \negthickspace \begin{array}{c} \mathbf{u} \\  \boldsymbol{\alpha} \end{array} \negthickspace }\right.$ $\displaystyle \begin{array}{c} \mathbf{u} \\  \boldsymbol{\alpha} \end{array}$ $\displaystyle \left.\vphantom{ \negthickspace \begin{array}{c} \mathbf{u} \\  \boldsymbol{\alpha} \end{array} \negthickspace }\right\}$ = $\displaystyle \left\{\vphantom{ \negthickspace \begin{array}{c} \mathbf{p} \\  \mathbf{0} \end{array} \negthickspace }\right.$ $\displaystyle \begin{array}{c} \mathbf{p} \\  \mathbf{0} \end{array}$ $\displaystyle \left.\vphantom{ \negthickspace \begin{array}{c} \mathbf{p} \\  \mathbf{0} \end{array} \negthickspace }\right\}$ (22.12)

in which

\begin{displaymath}\begin{split}&\mathbf{K} = \int_{V} \mathbf{B}^{\mathrm{\scri...
...hrm{\scriptscriptstyle{T}}}\mathbf{f} \,\mathrm{d}V \end{split}\end{displaymath} (22.13)

Since the degrees of freedom $ \boldsymbol\alpha$ are not continuous across inter-element boundaries, they can be eliminated from system (22.12) on element level.



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DIANA-9.4.4 User's Manual - Element Library
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