Bistable Isotropic Material
Constitutive Law
This MPM material is two small-strains materials (each with same constitutive law as an isotropic material). The two material states are linked by a transition rule and the transition between the two states can be reversible or irreversible.
The transition between the two states is determined either by a "dilation" rule, a "distortion" rule, or a "Von Mises" stress rule (as determined by the transition property). With a dilation rule, the transition occurs when the volumetric strain
[math]\displaystyle{ {\Delta V\over V} = \varepsilon_{xx}+\varepsilon_{yy}+\varepsilon_{zz} }[/math]
reaches the entered critical value. WIth a distortion rule, the transition occurs when the second strain invariant:
[math]\displaystyle{ I_2 = \sqrt{{1\over 2}\sum_{i,j}\varepsilon_{ij}'\varepsilon_{ij}'} }[/math]
where [math]\displaystyle{ \varepsilon_{ij}' }[/math] is deviatoric strain tensor, reaches a critical value. By a Von Mises stress rule, the transition occurs when the Von Mises stress
[math]\displaystyle{ \sigma_{VM} = \sqrt{\sum_{i,j}\sigma_{ij}'\sigma_{ij}'} }[/math]
where [math]\displaystyle{ \sigma_{ij}' }[/math] is is deviatoric stress, reaches a critical value. When using a dilation rule, the new stress-strain relation can include a changed offset in volumetric strain corresponding to stress-free conditions at a non-zero dilation relative to the initial state (see DeltaVOffset property). This change normally leads to an instantaneous change in stress upon transition. When using a distortion or Von Mises stress rule, the offset is ignored and the change is only a change in slope of mechanical properties.
Material Properties
The material properties for each state and the transition rules are set using:
| Property | Description | Units | Default |
|---|---|---|---|
| K0 | Bulk modulus for initial state | MPa | none |
| G0 | Shear modulus for initial state | MPa | none |
| alpha0 | Thermal expansion coefficient for initial state | ppm/K | 40 |
| kCond0 | Thermal conductivity for initial state | W/(m-K) | 0 |
| beta0 | Solvent expansion coefficien for initial statet | 1/(wt fraction) | 0 |
| D0 | Solvent diffusion constant for initial state | mm2/sec | 0 |
| Kd | Bulk modulus for transformed state | MPa | none |
| Gd | Shear modulus for transformed state | MPa | none |
| alphad | Thermal expansion coefficient for transformed state | ppm/K | 40 |
| kCondd | Thermal conductivity for transformed state | W/(m-K) | 0 |
| betad | Solvent expansion coefficien for transformed statet | 1/(wt fraction) | 0 |
| Dd | Solvent diffusion constant for transformed state | mm2/sec | 0 |
| transition | Set to "dilation" (or 1), "distortion" (or 2), or "vonmises" (or 3) to select the transition model | none | dilation |
| critical | The critical volumetric strain to induce a dilation transition (in precent strain), critical strain invariant to induce a distortion transition (in percent strain), or critical Von Mises stress to induce a vonmises transition (in MPa). | varies | none |
| DeltaVOffset | An offset volumetric strain in the transformed state. This property only applies for dilation rule. | % | 0 |
| reversible | Make the transition reversible (if "yes" or 1) or irreversible (if "no" or 0). | none | yes |
| (other) | Properties common to all materials | varies | varies |
History Variables
History variable 1 will be 0 for the initial state and 1 for the deformed state after a transition.