Compute Linear Elastic Phase Field Fracture Stress
Description
This material implements the unified phase-field model for mechanics of damage and quasi-brittle failure from Jian-Ying Wu Wu (2017). The pressure on the fracture surface can be optionally applied as described in Chukwudozie et al. (2019) and Mikelić et al. (2019).
Crack Surface Energy
The regularized functional is given as
The crack surface density function is expressed in terms of the crack phase-field and its spatial gradient as where the geometric function characterizes homogeneous evolution of the crack phase-field. is an internal length scale regularizing the sharp crack. is a scaling parameter such that the regularized functional represents the crack surface.
The crack geometric function generally satisfies the following properties,
In the classical phase-field models the modeling crack geometric function have been widely adopted for brittle fracture_energy
Elastic Energy
The elastic energy is defined as
The monotonically decreasing energetic function describes degradation of the initial strain energy as the crack phase-field evolves, satisfying the following properties Miehe et al. (2015)
The variation of the elastic energy gives constitutive relations where the thermodynamic force drives evolution of the crack phase-field with the reference energy related to the strain field .
Energetic Degradation Function
A genetic expression for degradation function is given as for and continuous function Q(d) > 0. Jian-Ying Wu considers following polynomials where the coefficients are calibrated from standard material properties.
The energetic function recovers some particular examples used in the literature, such as when and .
Decomposition Approaches
The elastic energy is usually decomposed additively to distinguish between tensile and compressive contributions. Three decomposition approaches are implemented.
Strain Spectral Decomposition
The total strain energy density is defined as where is the strain energy due to tensile stress, is the strain energy due to compressive stress. where is the th eigenvalue of the strain tensor and is an operator that provides the positive or negative part.
To be thermodynamically consistent, the stress is related to the deformation energy density according to Thus, where is the th eigenvector. The stress becomes
Strain Volumetric and Deviatoric Decomposition
The approach is based on the orthogonal decomposition of the linearized strain tensor in its spherical and deviatoric components: werhe deontes the n-dimensional identity tensor.
and is defined as
The stress is defined as and
Stress Spectral Decomposition
and is defined as
The compressive and tensile parts of the stress are computed from postive and negative projection tensors (computed from the spectral decomposition) according to
To be thermodynamically consistent, the stress is related to the deformation energy density according to Since then,
The Jacobian matrix for the stress is where is the elasticity tensor.
Note that stress spectral decomposition approach can be used for anisotropic elasticity tensor.
Evolution Equation (Allen-Cahn)
To avoid crack healing, a history variable is defined that is the maximum energy density over the time interval , where is the current time step, i.e.
Now, the total free energy is redefined as: with and
Its derivatives are
To further avoid crack phase-field going to negative, should overcome a barrier energy. The barrier energy is determined by and the is modified as
The evolution equation for the damage parameter follows the Allen-Cahn equation where and .The is scaled by the which is consistent with the definition given by Miehe at.al Miehe et al. (2015).
This equation follows the standard Allen-Cahn and thus can be implemented in MOOSE using the standard Allen-Cahn kernels, TimeDerivative, AllenCahn, and ACInterface. There is now an action that automatically generates these kernels: NonconservedAction. See the PhaseField module documentation for more information.
Pressure on the fracture surface
As suggested by Chukwudozie et al. (2019), the work of pressure forces acting along each side of the cracks that is added to the total free energy can be approximated by
Integration by parts yields
The boundary integral term can be neglected as in most applications on . Some authors Mikelić et al. (2019) have proposed to replace the indicator function with in the first term in order to make the functional convex. The indicator function is implemented as a generic material object that can be easily provided and modified in an input file. The stress equilibrium and damage evolution equations are also modified to account for the pressure contribution.
PETSc SNES variational inequalities solver option
Alternatively, the damage irreversibility condition can be enforced by using PETSc's SNES variational inequalities (VI) solver. In order to use PETSc's VI solver, upper and lower bounds for damage variable should be provided. Specifically, ConstantBoundsAux can be used to set the upper bound to be 1. VariableOldValueBoundsAux can be used to set the lower bound to be the old value. Note that in order for these bounds to have an effect, the user has to specify the PETSc options -snes_type vinewtonssls or -snes_type vinewtonrsls.
Example Input File
[./damage_stress]
type = ComputeLinearElasticPFFractureStress
c = c
E_name = 'elastic_energy'
D_name = 'degradation'
F_name = 'local_fracture_energy'
decomposition_type = strain_spectral
[../]
Example Input File with pressure
[./damage_stress]
type = ComputeLinearElasticPFFractureStress
c = c
E_name = 'elastic_energy'
D_name = 'degradation'
I_name = 'indicator_function'
F_name = 'local_fracture_energy'
decomposition_type = strain_spectral
[../]
[./pfbulkmat]
type = GenericConstantMaterial
prop_names = 'gc_prop l visco fracture_pressure'
prop_values = '1e-3 0.04 1e-4 1e-3'
[../]
References
- Chukwudi Chukwudozie, Blaise Bourdin, and Keita Yoshioka.
A variational phase-field model for hydraulic fracturing in porous media.
Computer Methods in Applied Mechanics and Engineering, 347:957 – 982, 2019.[BibTeX]
- Christian Miehe, Lisa-Marie Schänzel, and Heike Ulmer.
Phase field modeling of fracture in multi-physics problems. part i. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids.
Computer Methods in Applied Mechanics and Engineering, 294:449 – 485, 2015.
URL: http://www.sciencedirect.com/science/article/pii/S0045782514004423, doi:https://doi.org/10.1016/j.cma.2014.11.016.[BibTeX]
- A. Mikelić, M. F. Wheeler, and T. Wick.
Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium.
GEM - International Journal on Geomathematics, 10(1):2, 2019.[BibTeX]
- Jian-Ying Wu.
A unified phase-field theory for the mechanics of damage and quasi-brittle failure.
Journal of the Mechanics and Physics of Solids, 103:72 – 99, 2017.
URL: http://www.sciencedirect.com/science/article/pii/S0022509616308341, doi:https://doi.org/10.1016/j.jmps.2017.03.015.[BibTeX]