Bilinear mixed mode traction separation law

This class implements the bilinear mixed mode traction separation law described in Camanho and Dávila (2002).

Softening onset prediction

The initiation of the softening process is predicted using the quadratic failure criterion given below,

The total mixed-mode relative displacement is defined as where represents the norm of the vector defining the tangential relative displacements of the element.

Using the same penalty stiffness in Modes I, II and III, the tractions before softening onset are:

Assuming , the single mode relative displacements at softening onset are:

For an opening displacement greater than zero, the mode mixity ratio is defined as:

The mixed-mode relative displacement corresponding to the onset of softening is given as

Delamination propagation prediction

Power law criterion

The power law criterion is given as

The mixed-mode displacements corresponding to total decohesion is given as:

B-K criterion

The mixed-mode criterion proposed by Benzeggagh and Kenane is given as (B-K criterion):

The mixed-mode displacements corresponding to total decohesion is given as:

Constitutive equation for mixed-mode loading

The constitutive equation for mixed-mode loading is given as

(1)

(2)

Solver options

Viscous regularization

Cohesive zone models exhibiting softening behavior and stiffness degradation often lead to convergence difficulties in an implicit solver. The traction-separation laws can be regularized using viscosity. The viscous damage variable is defined by where is the viscosity parameter representing the relaxation time of the viscous system. An analytical expression of can be obtained by using the backward Euler method. With viscous regularization, the will be replaced by in Eq. (1) to compute traction.

Lag separation state

It is typically useful to improve convergence by lagging the separation state. When lag_separation_state = true, the , , and will be replaced by their old values from previous time step.

Use Regularized Heaviside Function

The step (Heaviside) function in Eq. (1) usually makes convergence bad. In the code, we replaced it with the regularized Heaviside function which provides a C0 continuity. The regularization parameter can be set by alpha parameter.

Examples

[Materials]
  [czm]
    type = BiLinearMixedModeTraction
    boundary = 'interface'
    penalty_stiffness = 1e6
    GI_c = 1e3
    GII_c = 1e2
    normal_strength = 1e4
    shear_strength = 1e3
    displacements = 'disp_x disp_y'
    eta = 2.2
    viscosity = 1e-3
  []
[]
(moose/modules/solid_mechanics/test/tests/cohesive_zone_model/bilinear_mixed.i)

References

  1. Pedro P. Camanho and Carlos G. Dávila. Mixed-mode decohesion finite elements for the simulation of delamination in composite materials. Technical Report NASA/TM-2002-211737, National Aeronautics and Space Administration, 2002.[BibTeX]