/MAT/LAW79 (JOHN_HOLM)

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Definition

Input Example

Comments

1. The equation describing the normalized equivalent stress is:(1)
   $${\sigma }^{*}=\left(1-D\right){\sigma }_i^{*}+D{\sigma }_f^{*}$$
   with the equivalent stress of the intact material:(2)

   $${\sigma }_i^{*}=a{\left(P^{*}+T^{*}\right)}^n\left(1+c\ln \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)$$
   and the equivalent stress of the failed material:(3)

   $${\sigma }_f^{*}=b{\left(P^{*}\right)}^m\left(1+c\ln \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)<{\sigma }_{f\max }^{*}$$
   The stresses are normalized to the stress at the Hugoniot elastic limit:(4)

   $${\sigma }_{\mathit{HEL}}=\frac{3}{2}\left(\mathit{HEL}-P_{\mathit{HEL}}\right)$$

   ${\sigma }^{*}=\frac{\sigma }{{\sigma }_{\mathit{HEL}}}$ and pressure are normalized to P_HEL:

   $P^{*}=\frac{P}{P_{HEL}}$ and $T^{*}=\frac{T}{P_{HEL}}$

2. If damage parameters are not specified ( $D_1=D_2=0$ ), plastic strain evolution is not computed and an instant failure is obtained when the element behavior reaches the elastic limit. Otherwise, if damage parameters are mentioned, the evolution of plastic strain is computed and the accumulated damage is:(5)
   $$D=\frac{{\displaystyle \sum \mathrm{\text{Δ}}{\epsilon }_f^p}}{{\epsilon }_f^p}$$
   where, the plastic strain to failure is:(6)

   $${\epsilon }_f^p=D_1{\left(P^{*}+T^{*}\right)}^{D_2}$$
   The maximum pressure tensile strength is decreased during damage as follows:(7)

   $$P^{*}=-(1-D)T^{*}$$

   This equation is not used when IDEL = 1.

3. The equation of state is: (8)
   $$\left\{\begin{array}{l}P=K_1\mu \\ P=K_1\mu +K_2{\mu }^2+K_3{\mu }^3\end{array}\right.\begin{array}{cc}\text{ in}& \text{tension}\\ \text{ in}& \text{compression}\end{array}$$
   Where, (9)

   $$\mu =\frac{\rho }{{\rho }_0}-1$$
   When damage starts, a bulking pressure increment $\text{Δ}P$ is computed as a function of the elastic energy loss $\text{Δ}U$ converted into potential hydrostatic energy:(10)

   $$\mathrm{\text{Δ}}{P}_{t+\mathrm{\text{Δ}}t}=-K_1\mu +\sqrt{{\left(K_1\mu +\mathrm{\text{Δ}}{P}_t\right)}^2+2\beta K_1\mathrm{\text{Δ}}U}$$

   Where, $\text{Δ}U=U\left(D\right)-U\left(D_{n+1}\right)$ with $U\left(D\right)=\frac{\sigma }{6G}$ .

   This increment is then added to the equation of state:(11)

   $$\text{Δ}{P}_{t+\text{Δ}t}=P_t+\text{Δ}{P}_{t+\text{Δ}t}$$
4. Time history and animation output is available using these USRi variables:
   • USR1: Bulking Pressure $\text{Δ}P$
   • USR2: Old Yield Stress
5. Strain rate filtering can be used and activated when a cutoff frequency F_cut for filtering is defined.