Bammann1990Modeling

The following equations are those employed in the Bammann (1990) paper that are implemented in the Bammann1990Modeling type, constructor, and that method of the kernel function, update associate with this type. As such, the internal equations use the same nomenclature for plastic strain rate, $\mathbf{D}^{p} \equiv \dot{\epsilon}^{(p)}$; the second-rank, deviatoric tensors for Cauchy stress, $\underset{\sim}{\sigma}'$ and kinematic hardening, $\underset{\sim}{\alpha}'$ which is an ISV; the scalar isotropic hardening, $\kappa$, and the constants associated with the dynamic and static recovery temperature equations. Of major importance is that, although not explicitly listed in the publication, the equations for $h$ and $H$ are included in this implementation (c. f. DYNA3D User Manual (1993)).

\[\begin{aligned} % plastic strain rate \mathbf{D}^{p} &= f(\theta)\sinh\left[ \frac{ |\underset{\sim}{\xi}| - \kappa - Y(\theta) }{ V(\theta) } \right]\frac{\underset{\sim}{\xi}'}{|\underset{\sim}{\xi}'|}\text{, let }\underset{\sim}{\xi}' = \underset{\sim}{\sigma}' - \underset{\sim}{\alpha}' \\ % kinematic hardening \dot{\underset{\sim}{\alpha}} &= h\mu(\theta)\mathbf{D}^{p} - [r_{d}(\theta)|\mathbf{D}^{p}| + r_{s}(\theta)]|\underset{\sim}{\alpha}|\underset{\sim}{\alpha} \\ % isotropic hardening \dot{\kappa} &= H\mu(\theta)\mathbf{D}^{p} - [R_{d}(\theta)|\mathbf{D}^{p}| + R_{s}(\theta)]\kappa^{2} \\ % flow rule F &= |\underset{\sim}{\sigma}' - \underset{\sim}{\alpha}'| - \kappa - \beta(|\mathbf{D}^{p}|, \theta) \\ % initial yield stress beta \beta(\mathbf{D}^{p}, \theta) &= Y(\theta) + V(\theta)\sinh^{-1}\left(\frac{|\mathbf{D}^{p}|}{f(\theta)}\right) \\ V(\theta) &= C_{ 1} \exp\left( -\frac{ C_{ 2} }{ \theta } \right) \\ Y(\theta) &= C_{ 3} \exp\left( \frac{ C_{ 4} }{ \theta } \right) \\ f(\theta) &= C_{ 5} \exp\left( -\frac{ C_{ 6} }{ \theta } \right) \\ r_{d}(\theta) &= C_{ 7} \exp\left( -\frac{ C_{ 8} }{ \theta } \right) \\ r_{s}(\theta) &= C_{ 9} \exp\left( -\frac{ C_{10} }{ \theta } \right) \\ R_{d}(\theta) &= C_{11} \exp\left( -\frac{ C_{12} }{ \theta } \right) \\ R_{s}(\theta) &= C_{13} \exp\left( -\frac{ C_{14} }{ \theta } \right) \\ h(\theta) &= C_{15} \exp\left( \frac{ C_{16} }{ \theta } \right) \\ H(\theta) &= C_{17} \exp\left( \frac{ C_{18} }{ \theta } \right) \\ \end{aligned}\]

Types

Bammann1990Modeling(Ω, μ)

Functions

References

Index