Assumed Strain Rate

With,

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

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

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:

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

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 ${\dot{\epsilon }}_{xy}$ 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.