Isotropic Phase Field Softening Material

Constitutive Law

This material implements phase field fracture model using the viscous regularization method described in Miehe and extends it in a few areas.

Phase Field Methods

In phase field fracture model of small-strain elastic materials, the total strain energy is partitioned into damaging, [math]\displaystyle{ \Psi^{(+)} }[/math], and nondamaging, , [math]\displaystyle{ \Psi^{(+)} }[/math], terms by

     [math]\displaystyle{ \Psi = g(d)\Psi^{(+)} + \Psi^{(-)} }[/math]

where [math]\displaystyle{ g(d) }[/math] is a softening law that depends on damage modeled by a phase value tracked on each material point that varied from 0 for undamaged to 1 for complete damage.

Using viscous regularization (see Miehe ), the phase field evolves by the equation

     [math]\displaystyle{ \eta \frac{d\phi}{dt} = G_c \ell \nabla^2\phi - \frac{G_c}{\ell}\phi - g'(\phi)\mathcal{H} }[/math]

where [math]\displaystyle{ G_c }[/math] is toughness, [math]\displaystyle{ \ell }[/math] is a length scale (describing width of diffuse cracks), and [math]\displaystyle{ \mathcal{H} }[/math] is history variable given by [math]\displaystyle{ \max(\Psi^{(+)}) }[/math]. This equation is a diffusion equation that describes evolution of damage driven by damaging energy. Simulations using this material must couple solution of this diffusion equation of the phase field material mechanics calculations. For that to work, all simulations must include a Diffusion command and choose (style) of "fracture. If needed, you can set phase value on the grid. Although you can set phase flux on particle surfaces, such conditions like have no physical meaning.

Phase Field Softening Law

The vast majority of phase field fracture papers set [math]\displaystyle{ g(d) = (1-d)^2 }[/math] under the misunderstood concept that [math]\displaystyle{ g'(1) }[/math] needs to be zero to stopp dissipating energy. In dynamic codes, any [math]\displaystyle{ g(d) }[/math] can be used provided [math]\displaystyle{ d }[/math] is prevented from exceeding 1.

Material Properties

The isotropic variational mechanics model using a single energy release rate that scales evolution of damage. The critical energy release rate is enter using the base material JIc property. The other needed material properties are as follows:

Property	Description	Units	Default
(Isotropic Properties)	Enter all properties needed to define the underlying isotropic material response	varies	varies
ell	Length scale parameter used in variational fracture mechanics	length units	none
viscosity	Viscosity to use when solving coupled phase field evolution in a diffusion tasks	viscosity units	none
gd	Softening law with options 0 = quadratic, 1 = exponential, 2 = linear softening	none	0
garg	An optional argument for use within the softening law. If not provided, default values depend on gd and are 1, 3, and 4, for gd = 0, 1, or 2, respectively	none	varies
stability	A stability factor thought to stabilize post-failure analysis	none	0
partition	Chose the method used to partition energy into energy that causes fracture and energy that does not cause fracture. The options are 0 = using eigenstrain analysis and 1 = divide into pressure and deviatoric strains	none	1
(other)	Properties common to all materials	varies	varies

The results in Miehe correspond to gd = 1, garg = 1, and partition = 0. These choices give poor results in some problems. This material has extension that can explore different phase field options.

History Variables

This material stores several history variables that track the extent of the damage and evolution of the phase field:

1. Maximum energy history term that provides source terms for phase field evolution
2. Damage state equation to 0 if not failed and 1 if failure (i.e., phase value has reached 1)
3. Current phase field value
4. Change in phase field since the last time step. It is used in constitutive law modeled and is scaled by 0.5 when using USAVG method.