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). In XML files, use a <reversible/> or an <irreversible/> element to set the type of transition.	none	no
(other)	Properties common to all materials	varies	varies

The state properties can only be entered using bulk and shear moduli. See these relations to covert other properties (such as modulus and Poisson's ratio) to bulk and shear moduli.

History Variables

History variable 1 will be 0 for the initial state and 1 for the deformed state after a transition.