The strain rates exhibit very high frequency vibrations which are not physical. The
strain rate filtering option will enable to damp those oscillations and; therefore obtain
more physical strain rate values.

If there is no strain rate filtering, the equivalent strain rate is the maximum value
of the strain rate components:(7)
$${\dot{\epsilon}}_{eq}=\mathrm{max}\left({\dot{\epsilon}}_{{x}^{\prime}}{\dot{\epsilon}}_{{y}^{\prime}}{\dot{\epsilon}}_{{z}^{\prime}}\text{\hspace{0.17em}}2{\dot{\epsilon}}_{x{y}^{\prime}}\text{\hspace{0.17em}}2{\dot{\epsilon}}_{y{z}^{\prime}}\text{\hspace{0.17em}}2{\dot{\epsilon}}_{XZ}\right)$$

For thin-walled structures, the equivalent strain is computed by the following
approach. If ε is the main component of strain tensor, the kinematic assumptions of
thin-walled structures allows to decompose the in-plane strain into membrane and
flexural deformations:(8)
$\epsilon =KZ={\epsilon}_{m}$

Then, the expression of internal energy can by written as:(9)
$${E}_{i}={\displaystyle \underset{-\frac{t}{2}}{\overset{\frac{t}{2}}{\int}}\sigma \epsilon \text{\hspace{0.05em}}dz=}{\displaystyle \underset{-\frac{t}{2}}{\overset{\frac{t}{2}}{\int}}E{\epsilon}^{2}\text{\hspace{0.05em}}dz=}{\displaystyle \underset{-\frac{t}{2}}{\overset{\frac{t}{2}}{\int}}E{\left(\kappa z+{\epsilon}_{m}\right)}^{2}dz}$$

Therefore:(10)
${E}_{i}={\displaystyle \underset{-\frac{t}{2}}{\overset{\frac{t}{2}}{\int}}E({\kappa}^{2}{z}^{2}+{\epsilon}_{m}^{2}}+2\kappa {\epsilon}_{m}z)\text{\hspace{0.17em}}dz=E{\left(\frac{1}{3}{\kappa}^{2}{z}^{3}+{\epsilon}_{m}^{2}z+\kappa {\epsilon}_{m}^{}{z}^{2}\right)}_{-\frac{t}{2}}^{\frac{t}{2}}$

The expression can be simplified to:(11)
$${E}_{i}=E\left[\frac{1}{12}{\kappa}^{2}{t}^{3}+{\epsilon}_{m}^{2}t\right]=E{\epsilon}_{eq}^{2}\text{}\text{\hspace{0.05em}}t$$
(12)
$${\epsilon}_{eq}=\sqrt{\frac{1}{12}{\kappa}^{2}{t}^{2}+{\epsilon}_{m}^{2}}$$

The expression of the strain rate is derived from

Equation 8:

(13)
$\dot{\epsilon}=\dot{K}Z+{\dot{\epsilon}}_{m}$
Admitting the assumption that the strain rate is proportional to the strain, that
is:(14)
${\dot{\epsilon}}_{m}=\alpha {\epsilon}_{m}$
(15)
$\dot{K}=\alpha K$

Therefore:(16)
$\dot{\epsilon}=\alpha \epsilon $

Referring to

Equation 12, it can be seen that
an equivalent strain rate can be defined using a similar expression to the
equivalent strain:

(17)
${\dot{\epsilon}}_{eq}=\alpha {\epsilon}_{eq}$
(18)
$${\dot{\epsilon}}_{eq}=\sqrt{\frac{1}{12}{\kappa}^{2}{t}^{2}+{\dot{\epsilon}}_{m}^{2}}$$
For solid elements, the strain rate is computed using the maximum element
stretch:(19)
$${\dot{\epsilon}}_{eq}={\dot{\lambda}}_{\mathrm{max}}$$

The strain rate at integration point, $$i$$ in /ANIM/TENS/EPSDOT/i $$\left(1<i<n\right)$$ is calculated by:(20)
$${\dot{\epsilon}}_{i}={\dot{\epsilon}}_{\mathrm{m}}-\frac{1}{2}\left(\frac{\left(2i-1\right)}{n}-1\right)t{\dot{\epsilon}}_{b}$$

Where,

- $${\dot{\epsilon}}_{m}$$
- Membrane strain rate /ANIM/TENS/EPSDOT/MEMB
- $${\dot{\epsilon}}_{b}$$
- Bending strain rate /ANIM/TENS/EPSDOT/BEND.

The strain rate in upper and lower layers is computed by:(21)
${\dot{\epsilon}}_{u}={\epsilon}_{m}+\frac{1}{2}t{\dot{\epsilon}}_{b}$

/ANIM/TENS/EPSDOT/UPPER(22)
${\dot{\epsilon}}_{1}={\dot{\epsilon}}_{m}-\frac{1}{2}t{\dot{\epsilon}}_{b}$

/ANIM/TENS/EPSDOT/LOWER

The strain rate is filtered by using:(23)
$${\dot{\epsilon}}_{f}\left(t\right)=\text{a}\dot{\epsilon}\left(t\right)+\left(1-\text{a}\right){\dot{\epsilon}}_{f}\left(t-1\right)$$

Where,

- $$\text{a=2}\pi {\text{dtF}}_{cut}$$
- $$\text{dt}$$
- Time interval
`F`_{cut}
- Cutting frequency
- ${\dot{\epsilon}}_{f}$
- Filtered strain rate