/MAT/LAW38 (VISC_TAB)

Format

Definition

Comments

1. Engineering stress versus engineering strain can be input as functions for different strain rates. The stress and strain is positive in compression and negative in tension. By default (Itota=0), the tension behavior is linear elastic using Young’s modulus, E₀. If Itota=1, the engineering stress strain behavior should be input using the functions, fct_ID_iL, with the stress strain curve defined both in compression and tension
2. When stress strain functions are defined at different strain rates, the stresses are computed by interpolation from input functions: $\sigma =f(\epsilon ,\dot{\epsilon })$
   for given $\dot{\epsilon }$ , read two values of function at $\epsilon$ for the two immediately lower and higher strain rates.

   
   
   Figure 1. Two Strain Rate Curves (up to five may be input)

   with(1)

   $$\sigma ={\sigma }_2+\left({\sigma }_1-{\sigma }_2\right){\left[1-{\left(\frac{\dot{\epsilon }-{\dot{\epsilon }}_1}{{\dot{\epsilon }}_2-{\dot{\epsilon }}_1}\right)}^a\right]}^b$$

   The parameters $a$ and $b$ define the shape of the interpolation function within each interval. If $a$ = $b$ = 1, the interpolation is linear.

3. A "coupled" set of principal nominal stresses is computed with anisotropic Poisson's ratios:

   ${\nu }_{ij}={\nu }_c+\left({\nu }_t-{\nu }_c\right)\left(1-\exp \left(-R_v\left|{\epsilon }_{ij}\right|\right)\right)$ in tension ( ${\epsilon }_{ij}\ge 0$ )

   ${\nu }_{ij}={\nu }_c$ in compression.

   Where,

   ${\epsilon }_{ij}=\frac{({\epsilon }_i+{\epsilon }_j)}{2}$

   ${\epsilon }_{ij}\ge 0$

4. Analysis formulation type Iflag.

   Iflag=0: corresponds to the visco-elastic foam tabulated material (visco-elasticity is computed in each principal stress direction).

   Iflag=1: behavior will be linear in both tension and compression, following Hook's relations.

   For compression, Young's modulus $E_0$ and Poisson's ratio ${\nu }_c$ are used.

   In tension, the instantaneous Young's modulus ratio $E_t$ is used.

   The instantaneous Young’s modulus is updated using:(2)

   $$E_t=E_{final}+(E_0-E_{final})\left[1-e^{-\lambda (V_R-1+{\epsilon }_{final})}\right]$$
   with (3)

   $$E_0<E<E_f{}_{inal}$$
   Where,

   $E_0$

   Minimum tension modulus

   $E_{final}$

   Maximum tension modulus

   $V_R$

   Relative volume computed in Radioss

   ${\epsilon }_{final}$

   Absolute value of the strain corresponding to the maximum compression modulus.

   The instantaneous modulus is only used for tension.

5. For stability, $\dot{\epsilon }$ is filtered using:(4)
   $${\dot{\epsilon }}_{\mathit{filt}}^n={\dot{\epsilon }}_{\mathit{filt}}^{n-1}+R_D\left({\dot{\epsilon }}^n-{\dot{\epsilon }}_{\mathit{filt}}^{n-1}\right)$$
6. Hysteresis is applied in linear tension case.

   If K_R=1, Hysteresis is only applied in compression.

   If K_R=2, Hysteresis is applied both in compression and in tension.

7. For air pressure $P_{air}$ (when K_air=1)
   If fct_ID_p≠0:(5)

   $$P_{\mathit{air}}={\mathit{Fscale}}_p.\mathrm{f}\left(\frac{V}{V_0}\right)$$

   Where, $f$ refers to function number fct_ID_p.

   If fct_ID_p=0:(6)

   $$P_{\mathit{air}}=P_0\frac{\left(1-\frac{V}{V_0}\right)}{\left(\frac{V}{V_0}-\mathrm{\Phi }\right)}$$
   Relaxation is applied as:(7)

   $$P_{\mathit{air}}=\min \left(P_{\mathit{air}},P_{\max }\right)\exp \left(-R_{p}t\right)$$

   Where, $R_p$ is the relaxation rate of pressure and t is the time.

8. During unloading, without an unloading curve defined fct_ID_iul = fct_ID_ul=0, $\sigma$ is computed from the first loading curve, fct_ID_1L.

   
   
   Figure 2.

   If the unloading curve is defined, $\sigma$ is interpolated between the first loading curve fct_ID_1L and the defined unloading curve fct_ID_ul or fct_ID_iul. In this case, fct_ID_1L must correspond to a quasi-static state.

9. Unloading functions fct_ID_iul (Line 12) are used only if the unloading curve fct_ID_ul is not defined.
10. If Visc is input, interpolated stress will be limited by this value to have a larger timestep:(8)
    $$\sigma \le {\sigma }_1+\mathit{Visc}\left(\dot{\epsilon }-{\dot{\epsilon }}_1\right)$$
11. The behavior is strain rate independent when $\dot{\epsilon }\le {\dot{\epsilon }}_1$ .
12. /VISC/PRONY can be used with this material law to include viscous effects.