# Assumed Strain Rate

With,

$\left\{\dot{\epsilon}\right\}={\langle \begin{array}{cccccc}{\dot{\epsilon}}_{xx}& {\dot{\epsilon}}_{yy}& {\dot{\epsilon}}_{zz}& 2{\dot{\epsilon}}_{xy}& 2{\dot{\epsilon}}_{yz}& 2{\dot{\epsilon}}_{xz}\end{array}\rangle}^{t}$

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

^{1}of the shape functions written by:

Where,

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

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.

with:

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

^{2}ASQBI assumed strain is used:

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

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 ${\dot{\epsilon}}_{xy}$ 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: `I`_{cpre} =1,2,3

`I`

_{cpre}has been introduced for new solid elements.

`I`_{cpre}- =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.