M-K Damage

The model introduces thickness imperfections, in the form of grooves, in the sheet material to simulate defects. Damage occurs when the ratio of deformation in the grooves relative to deformation in the original sheet thickness exceeds a critical value. By default, four grooves are evaluated at equally spaced angles of 0°, 45°, 90°, and 135° with respect to the local 1-direction of the material at each time increment, and the worst result is used to determine damage initiation. The M-K criterion can be used in conjunction with the von Mises and Johnson-Cook plasticity models, including kinematic hardening.

You can specify an anisotropic distribution of thickness imperfections as a function of angle with respect to the local material orientation, $f_{0} (θ)$ . The stress-strain field in the nominal area is solved first, ignoring the presence of imperfections; then the effect of each groove alone is computed by enforcing the strain compatibility condition

ε_{t t}^{b} = ε_{t t}^{a}

and the force equilibrium equations

F_{n n}^{b} = F_{n n}^{a} a n d F_{n t}^{b} = F_{n t}^{a} .

The subscripts $n$ and $t$ refer to the directions normal and tangential to the groove. In the above equilibrium equations $F_{n n}$ and $F_{n t}$ are forces per unit width in the $t$ -direction.

The onset of necking instability is assumed to occur when the ratio of the rate of deformation inside a groove relative to the rate of deformation if no groove were present is greater than a critical value. In addition, it may not be possible to find a solution that satisfies equilibrium and compatibility conditions once localization initiates at a particular groove; consequently, failure to find a converged solution is also an indicator of the onset of localized necking. For the evaluation of the damage initiation criterion the following measures of deformation severity are used:

F_{e q} = \frac{Δ {\bar{ε}}_{b}^{p l}}{Δ {\bar{ε}}_{a}^{p l}}, F_{n n} = \frac{Δ ε_{n n}^{b}}{Δ ε_{n n}^{a}}, F_{n t} = \frac{Δ ε_{n t}^{b}}{Δ ε_{n t}^{a}} .

These deformation severity factors are evaluated on each of the specified groove directions and compared with the critical values. (The evaluation is performed only if the incremental deformation is primarily plastic; the M-K criterion will not predict damage initiation if the deformation increment is elastic.) The most unfavorable groove direction is used for the evaluation of the damage initiation criterion, which is given as

ω_{M K} = \max (\frac{f_{e q}}{f_{e q}^{c r i t}}, \frac{f_{n n}}{f_{n n}^{c r i t}}, \frac{f_{n t}}{f_{n t}^{c r i t}}),

where $f_{e q}^{c r i t}$ , $f_{n n}^{c r i t}$ , and $f_{n t}^{c r i t}$ are the critical values of the deformation severity indices. Damage initiation occurs when $ω_{M K} = 1$ or when a converged solution to the equilibrium and compatibility equations cannot be found. By default, it is assumed that $f_{e q}^{c r i t} = f_{n n}^{c r i t} = f_{n t}^{c r i t} = 10$ ; you can specify different values. If one of these parameters is set equal to zero, its corresponding deformation severity factor is not included in the evaluation of the damage initiation criterion. If all of these parameters are set equal to zero, the M-K criterion is based solely on nonconvergence of the equilibrium and compatibility equations.

There can be a substantial increase in the overall computational cost when the M-K criterion is used. The computational cost per element with the M-K criterion increases by approximately a factor of

1 + 0.25 \frac{n_{i m p}}{N_{i n c r}},

where $n_{i m p}$ is the number of imperfections specified for the evaluation of the M-K criterion and $N_{i n c r}$ is the frequency, in number of increments, at which the M-K computations are performed.


Input Data	Description
Feq	Critical deformation severity factor, $F_{e q}$ .
Fnn	Critical deformation severity factor, $F_{n n}$ .
Fnt	Critical deformation severity factor, $F_{n t}$ .
Number of imperfections	The number of angular groove positions to evaluate. The groove positions are equally spaced, starting at 0° and ending at $180 \frac{(n - 1)}{n}$ with respect to the local 1-direction of the material.
Frequency	The number of increments between calculating the M-K criterion. Using the default frequency, 1, can be expensive because each groove is evaluated at every increment.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specify material parameters that depend on field variables. Field columns appear in the data table for each field variable you add. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Groove Size	The ratio of the thickness at the groove to the nominal material thickness.
Angle	The starting angle (in degrees) with respect to the 1-direction of the local material orientation.
Use damage evolution	Damage evolution defines how the material degrades after one or more damage initiation criteria are met. Select the check box, and specify settings in the Damage Evolution group box; these settings are described in Damage Evolution.