Modeling Damage
Modeling problems with strain localization might turn into unstable behavior and convergence difficulties. The following techniques can be used to help in such situations:
•
Use a displacement-controlled loading scheme, since damage is often associated with a reduction in load carrying capacity, see the Cracking of a Notched Beam example in the Geomechanics Module or in the Nonlinear Structural Mechanics Module Application Libraries.
•
Better convergence is often obtained when the Nonlinear method is set to Constant (Newton) with a Damping factor equal to 1, and a Jacobian update on every iteration.
•
Reduce the smallest and/or largest allowable step size in an Auxiliary sweep, or restrict the time steps in a time-dependent study.
•
Introducing a soft spot where a crack is expected can make the localization of strains more stable, see the Brittle Damage in Uniaxial Tension example in the Geomechanics Module or in the Nonlinear Structural Mechanics Module Application Libraries. Alternatively, a random spatial distribution of the material parameters could be employed to obtain a more stable solution.
When using the crack band method or no regularization at all, the following steps are recommended:
•
The size of the biggest mesh element h should not exceed 2EGf/σts2, where E is the Young’s modulus, Gf is the fracture energy per unit area, and σts is the tensile strength. Larger values of h will cause a snap-back of the stress-strain curve at the material point level.
•
Use linear shape functions for the displacement field. When using higher-order shape functions, strains may localize in either rows of Gauss points, or entire elements, depending on the stress state and numerical rounding errors.
•
If cracks are located on a symmetry plane, the model parameters should be modified so that the amount of dissipated energy is reduced by one half in the elements adjacent to the symmetry plane. This can be achieved, for example, by reducing the fracture energy in that row of elements.
When using the implicit gradient method, the element size should be sufficiently small to resolve damaged zones. The same applies to the Phase field damage model, where it is recommended that size of the mesh elements in the expected crack path follows h < lint/2 for a linear displacement field, otherwise h < lint.
While it is possible to solve brittle fracture problems with the Phase field damage model by applying a fully coupled strategy, this can often exhibit poor or slow convergence. An alternative and often more stable approach is to use a segregated solution strategy, by splitting the evolution of the crack phase field and the displacement field in two groups. This type of algorithmic operator split can conceptually be summarized as follows for step n+1:
1
Initialization. The crack phase field, displacement field and state variables are known at step n.
2
Update state variables. Update internal state variables used by the phase field model with values from step n.
3
Solve for the Crack Phase Field. Compute the crack phase field variable in a Newton step, with the displacement field frozen at step n.
4
Solve for the Displacement field. Compute displacement field variables in a Newton step with the updated crack phase field.
This leads to a single-pass algorithm that is accurate only for sufficiently small parameter or time steps.
The default solver will suggest the above single-pass algorithm for the Phase field damage model when it is feasible to perform the operator split. Cases where this is not possible include when some multiphysics couplings are present in the model and when a segregated contact algorithm has to be used.
An improvement to the method is to add a multi-pass correction by iterating over steps 3 and 4 in each increment; either until convergence is achieved or for a predefined number of iterations. This type of strategy is demonstrated in the Brittle Fracture of a Holed Plate example in the Geomechanics Module or in the Nonlinear Structural Mechanics Module Application Libraries, where a multi-pass algorithm with a maximum of 3 outer iterations is used. Note that although the solution is accepted without requiring convergence of the outer problem, each sub group locally fulfills the defined convergence criterion. Hence the displacement field can be considered as a converged solution given the current crack phase field.