# Assumed Strain Rate

Using Voigt convention, the strain rate of Strain Rate, Equation 5 can be written as:

$\left\{\stackrel{˙}{\epsilon }\right\}=\left[B\right]\left\{v\right\}=\sum _{I=1}^{8}\left[{B}_{I}\right]\left\{{v}_{I}\right\}$

With,

$\left\{\stackrel{˙}{\epsilon }\right\}={〈\begin{array}{cccccc}{\stackrel{˙}{\epsilon }}_{xx}& {\stackrel{˙}{\epsilon }}_{yy}& {\stackrel{˙}{\epsilon }}_{zz}& 2{\stackrel{˙}{\epsilon }}_{xy}& 2{\stackrel{˙}{\epsilon }}_{yz}& 2{\stackrel{˙}{\epsilon }}_{xz}\end{array}〉}^{t}$

$\left[{B}_{I}\right]={\left[\begin{array}{cccccc}\frac{\partial {\Phi }_{I}}{\partial x}& 0& 0& 0& \frac{\partial {\Phi }_{I}}{\partial x}& \frac{\partial {\Phi }_{I}}{\partial y}\\ 0& \frac{\partial {\Phi }_{I}}{\partial y}& 0& \frac{\partial {\Phi }_{I}}{\partial x}& 0& \frac{\partial {\Phi }_{I}}{\partial z}\\ 0& 0& \frac{\partial {\Phi }_{I}}{\partial z}& \frac{\partial {\Phi }_{I}}{\partial y}& \frac{\partial {\Phi }_{I}}{\partial z}& 0\end{array}\right]}^{t}$

It is useful to take the Belytschko-Bachrach's mix form 1 of the shape functions written by:

${\Phi }_{I}\left(x,y,z,\xi ,\eta ,\zeta \right)={\text{Δ}}_{I}+{b}_{xI}\cdot x+{b}_{yI}\cdot y+{b}_{zI}\cdot z+\sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }{\varphi }_{\alpha }$

Where,

$\begin{array}{l}{b}_{iI}=\frac{\partial {\Phi }_{I}}{\partial {x}_{i}}\left(\xi =\eta =\zeta =0\right);\\ {\gamma }_{I}^{\alpha }=\frac{1}{8}\left[{\Gamma }_{I}^{\alpha }-\left(\sum _{J=1}^{8}{\Gamma }_{J}^{\alpha }{x}_{J}\right){b}_{xI}-\left(\sum _{J=1}^{8}{\Gamma }_{J}^{\alpha }{y}_{J}\right){b}_{yI}-\left(\sum _{J=1}^{8}{\Gamma }_{J}^{\alpha }{z}_{J}\right){b}_{zI}\right];\\ 〈\varphi 〉=〈\begin{array}{cccc}\eta \zeta & \xi \zeta & \xi \eta & \xi \eta \zeta \end{array}〉\end{array}$

The derivation of the shape functions is given by:

$\frac{\partial {\Phi }_{I}}{\partial {x}_{i}}={b}_{iI}+\sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial {x}_{i}}$

It is decomposed by a constant part which is directly formulated with the Cartesian coordinates, and a non-constant part which is to be approached separately. For the strain rate, only the non-constant part is modified by the assumed strain. You can see in the following that the non-constant part or the high order part is just the hourglass terms.

You now have the decomposition of the strain rate:

$\left\{\stackrel{˙}{\epsilon }\right\}=\sum _{I=1}^{8}\left[{B}_{I}\right]\left\{{v}_{I}\right\}=\sum _{I=1}^{8}\left({\left[{B}_{I}\right]}^{0}+{\left[{B}_{I}\right]}^{H}\right)\left\{{v}_{I}\right\}={\left\{\stackrel{˙}{\epsilon }\right\}}^{0}+{\left\{\stackrel{˙}{\epsilon }\right\}}^{H}$

with:

${\left[{B}_{I}\right]}^{0}=\left[\begin{array}{ccc}{b}_{xI}& 0& 0\\ 0& {b}_{yxI}& 0\\ 0& 0& {b}_{zI}\\ {b}_{yxI}& {b}_{xI}& 0\\ {b}_{zI}& 0& {b}_{xI}\\ 0& {b}_{zI}& {b}_{yxI}\end{array}\right]$ ; ${\left[{B}_{I}\right]}^{H}={\left[\begin{array}{cccccc}\sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial x}& 0& 0& 0& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial x}& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial y}\\ 0& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial y}& 0& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial x}& 0& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial z}\\ 0& 0& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial z}& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial y}& \sum _{\alpha =1}^{4}{\gamma }_{I}^{\alpha }\frac{\partial {\varphi }_{\alpha }}{\partial z}& 0\end{array}\right]}^{t}$

Belvtschko and Bindeman 2 ASQBI assumed strain is used:

$\left\{\stackrel{˙}{\epsilon }\right\}=\sum _{I=1}^{8}\left({\left[{B}_{I}\right]}^{0}+{\left[{\overline{B}}_{I}\right]}^{H}\right)\left\{{v}_{I}\right\}$

with ${\left[{\overline{B}}_{I}\right]}^{H}=\left[\begin{array}{ccc}{X}_{I}^{1234}& -\overline{\nu }{Y}_{I}^{3}-\nu {Y}_{I}^{24}& -\overline{\nu }{Z}_{I}^{2}-\nu {Z}_{I}^{34}\\ -\overline{\nu }{X}_{I}^{3}-\nu {X}_{I}^{14}& {Y}_{I}^{1234}& -\overline{\nu }{Z}_{I}^{1}-\nu {Z}_{I}^{34}\\ -\overline{\nu }{X}_{I}^{2}-\nu {X}_{I}^{14}& -\overline{\nu }{Y}_{I}^{1}-\nu {Y}_{I}^{24}& {Z}_{I}^{1234}\\ {Y}_{I}^{12}& {X}_{I}^{12}& 0\\ {Z}_{I}^{13}& 0& {X}_{I}^{13}\\ 0& {Z}_{I}^{23}& {Y}_{I}^{23}\end{array}\right]$

Where, ${X}_{I}^{13}={\gamma }_{I}^{1}\frac{\partial {\phi }_{1}}{\partial x}+{\gamma }_{I}^{3}\frac{\partial {\phi }_{3}}{\partial x}$ ; ${Y}_{I}^{13}={\gamma }_{I}^{1}\frac{\partial {\phi }_{1}}{\partial y}+{\gamma }_{I}^{3}\frac{\partial {\phi }_{3}}{\partial y}$ ; and $\overline{\nu }=\frac{\nu }{1-\nu };$ .

To avoid shear locking, some hourglass modes are eliminated in the terms associated with shear so that no shear strain occurs during pure bending. That is, ${Y}_{I}^{3},{X}_{I}^{3}$ in ${\stackrel{˙}{\epsilon }}_{xy}$ terms and all fourth hourglass modes in shear terms are also removed since this mode is non-physical and is stabilized by other terms in ${\left[{\overline{B}}_{I}\right]}^{H}$ .

The terms with Poisson coefficient are added to obtain an isochoric assumed strain field when the nodal velocity is equivoluminal. This avoids volumetric locking as $\nu =0.5$ . In addition, these terms enable the element to capture transverse strains which occurs in a beam or plate in bending. The plane strain expressions are used since this prevents incompatibility of the velocity associated with the assumed strains.

## Incompressible or Quasi-incompressible Cases

Flag for new solid element: Icpre =1,2,3

For incompressible or quasi- incompressible materials, the new solid elements have no volume locking problem due to the assumed strain. Another way to deal with this problem is to decompose the stress field into the spherical part and the deviatory part and use reduced integration for spherical part so that the pressure is constant. This method has the advantage on the computation time, especially for the full integrated element. For some materials which the incompressibility can be changed during computation (for example: elastoplastic material, which becomes incompressible as the growth of plasticity), the treatment is more complicated. Since the elastoplastic material with large strain is the most frequently used, the constant pressure method has been chosen for Radioss usual solid elements. The flag Icpre has been introduced for new solid elements.
Icpre
=0
Assumed strain with $\nu$ terms is used.
=1
Assumed strain without $\nu$ terms and with a constant pressure method is used. The method is recommended for incompressible (initial) materials.
=2
Assumed strain with $\nu$ terms is used, where $\nu$ is variable in function of the plasticity state. The formulation is recommended for elastoplastic materials.
=3
Assumed strain with $\nu$ terms is used.
1 Belytschko T. and Bachrach W.E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Computer Methods in Applied Mechanics and Engineering, 54:279-301, 1986.
2 Belytschko Ted and Bindeman Lee P., Assumed strain stabilization of the eight node hexahedral element, Computer Methods in Applied Mechanics and Engineering, vol.105, 225-260, 1993.