# Traction Laws

Traction laws can be placed on crack surfaces to model cohesive zones.

## Introduction

MPM implements explicit cracks by defining a series or massless particles that define the crack path. The method is called the CRAMP algorithm.^{[1]} The CRAMP algorithm takes care of the crack geometry and can handle crack-surface contact or imperfect interface contact. In addition, MPM can implement traction laws on the crack surfaces by assigning a traction laws to one or more crack particles along the crack. The traction laws can be assigned when creating the crack or during crack propagation (*i.e*., new crack surfaces can be dynamically create traction laws).

Traction laws have several uses. The most common is to implement cohesive zones where the traction laws will naturally debond if a critical opening displacement or some other condition is reached. When they are modeling cohesive zones, the visualization tools can plot total crack length or debonded crack length (which is length with no traction laws). Their difference is the length of crack surface with traction law materials still bonded. The tools can also plot the opening and shear displacements at the actual crack tip or the transition from debonded crack into the traction zone, or total amount of mode I and mode II dissipated energy, and various traction history variables.

The term "Traction Law" is the common term used in cohesive zone modeling, but it is a misleading term. When cohesive zones unload, the traction decreases back to the origin. As a result, the zone traction is not defined by the "Traction law." A better term is to call these function the zone's "Cohesive Law." Even better, is to recognize a better interpretation of these laws is as a "Strength Model" or as providing the zone's evolve strength as a function of its current damage state.^{[2]} This documentation intermixes these terms.

Another use for traction laws is to model an imperfect interface for simulations where imperfect interface contact law cannot be used. Two common situations are when the interfacial failure displacement is larger then a cell size or then the problem needs to also model dynamic contact using the `ContactPosition` command.

This section explains the possible traction laws. See the crack creation and crack propagation commands for how to use traction laws on cracks.

The use of traction laws on MPM cracks is described in Nairn (2009)^{[3]} and used in Bardenhagen *et al.* (2011)^{[4]}, Matsumoto and Nairn (2012).^{[5]}, and Nairn (2015)^{[6]} The first reference^{[3]} showed how MPM can model fracture using a cohesive zone or a combination of fracture mechanics and adhesive zone resulting in a simulated R curve; this R curve can be predicted from the shape of the traction law. Nairn and Aimene (2021)^{[2]} derives a new approach to using cohesive zone when modeling mixed-mode failure. It fixes errors common in commercial software, such as Abaqus.

## Define a Traction

You create traction law materials using a `Material` command block. Within that block all traction properties are set using property commands. Refer to each traction law type to learn about its possible properties.

Note that normal traction is added only when the crack is opened while tangential traction is added under all conditions. The handling of crack contact in tandem with crack tractions is done in the CRAMP algorithm by assigning a crack-surface contact law to the crack. To avoid conflict between tangential traction law forces and tangential forces that occur during crack contact, the crack contact law should always be frictionless (such that contact applies no tangential forces). In other words, whenever a crack has traction laws, the crack contact law is normally a frictionless Coulomb friction law. One alternative is to use an imperfect interface contact law on the crack, but if used, the interface must have zero stiffness in the tangential direction and in the normal direction when opened (to avoid conflict with traction law forces). The interface law, however, may choose to define a finite stiffness in compression (*e.g.*, use a Linear Imperfect Interface with D_{nt} = D_{t} = 0 and D_{nc} defining compression stiffness).

## Cohesive Zone Debonding

When a traction law on a crack reaches critical conditions (and those conditions depend on the traction law being used), the cohesive zone it is modeling debonds. Each debonding event is reported in the output file's "ARCHIVED ANALYSIS RESULTS" section in a line such as

# Debond: t=8.23338 (x,y) = (148.28,-0.612942) GI(%) = 51.1585 G = 228.465

where

`t`is the time of the debonding event`(x,y)`is the location of the cohesive zone that debonded`G`is total dissipated energy per unit area of the debonding cohesive zone`GI(%)`is the percent of the reported`G`that occurred in mode I (and`100-GI(%)`is the percent that occurred in mode II).

### Fracture Mechanics Energy Release Rate

In fracture mechanics, the energy release rate is the amount of dissipated energy dissipated per unit crack area. When modeling crack growth with cohesive zones, the energy release rate depends on your definition of crack area and, in general, differs from the reported `G` at each debonding event.^{[2]} A general method to find fracture mechanics energy release rate is:^{[2]}

- Model crack propagation using traction laws on a crack
- Set the simulation to archive czmdisp or the mode I and mode II energy dissipated per unit area by each cohesive zone
- Determine
*effective*crack length as a function of time (see below) - Find total dissipated energy for all cohesive zones on one crack by summing czmdisp results for each zone times that zone's area. The net result is a force (in NairnFEAMPM or NairnFEAMPMViz, this summation is done by plotting "CZM Mode I Force" and "CZM Mode II Force").
- Cross plot mode forces as a function of
*effective*crack length - The slopes of the cross plots are mode I and mode II energy release rates as functions of crack length. Their sum is total energy release rate.

When using cohesive zones to model experiments, the key challenge is to determine *effective* crack length by methods that correspond to the experimentally-measured crack length. Some options are:

- If the cohesive zone is modeling microcracks or microvoids that are not visible during experiments, the experiments likely correspond to the "Debonded Crack Length" (
*i.e.*, the crack length not counting any with a still-bonded cohesive zone and a plotting option in NairnFEAMPM or NairnFEAMPMViz).

- If the cohesive zone is modeling a visible process, such as fiber bridging in composite or wood crack growth, the experiment might be closer to sum of "Debonded Crack Length" and "Cohesive Damage Length" (
*i.e.*, the crack length including a process zone defined as any cohesive zones beyond their elastic limit and the sum or two plotting options in NairnFEAMPM or NairnFEAMPMViz). Note that the Cubic Traction Law has no elastic regime, which means its cohesive damage length will include all cohesive zones after any finite deformation occurs. It is therefore better to avoid the Cubic Traction Law when modeling needs to determine the cohesive damage length.

### Fracture Mechanics J Integral

The total J Integral for a crack with a cohesive zone is equal to the reported `G` for each debonding event.^{[7]} Unfortunately, this J integral is only equal to energy release rate when the crack growth is "self-similar crack growth."^{[7]} When modeling crack growth with cohesive zones, "self-similar crack growth" requires the cohesive damage length to remain constant. Thus J integral is equal to energy release rate if, and only if, the cohesive damage length remains constant. Whenever it is not constant, J integral still exists, but the method in the previous section has to be used instead to find energy release.^{[2]} Furthermore, mode mixity during non-self-similar crack growth must use the mode I and mode II energy release found by the general methods and not rely on the report `GI(%)`.

## Traction Law Materials

This table lists the available traction law materials. Click each one for more details and information on their properties.

Name | ID | Description |
---|---|---|

TriangularTraction | 12 | A triangular traction law |

ExponentialTraction | 34 | An exponential traction law |

CubicTraction | 14 | A cubic traction law |

TrilinearTraction | 20 | A trilinear traction law |

MixedModeTraction | 33 | An improved, coupled, mixed-mode traction law |

CoupledTraction | 23 | A coupled law using effective displacements |

PressureTraction | 26 | A constant normal stress traction law (no failure) |

LinearTraction | 13 | A linear elastic traction law (no failure) |

It is relatively easy to write code for new traction laws, if needed.

## References

- ↑ J. A. Nairn, "Material Point Method Calculations with Explicit Cracks,"
*Computer Modeling in Engineering & Sciences*,**4**, 649-664 (2003). (See PDF) - ↑
^{2.0}^{2.1}^{2.2}^{2.3}^{2.4}J. A. Nairn and Y. E. Aimene "A re-evaluation of mixed-mode cohesive zone modeling based on strength concepts instead of traction laws" submitted (2021). - ↑
^{3.0}^{3.1}J. A. Nairn, "Analytical and Numerical Modeling of R Curves for Cracks with Bridging Zones"*Int. J. Fracture*,**155**, 167-181 (2009). (See PDF) - ↑ S. G. Bardenhagen, J.A. Nairn, and H. Lu, "Simulation of dynamic fracture with the Material Point Method using a mixed J-integral and cohesive law approach,"
*Int. J. Fracture*,**170**, 49-66. - ↑ J.A. Nairn, "Fracture Toughness of Wood and Wood Composites During Crack Propagation,"
*Wood and Fiber Science*,**44**, 121-133 (2012). - ↑ J. A. Nairn. Numerical simulation of orthogonal cutting using the material point method. Engineering Fracture Mechanics, 149:262–275, 2015.
- ↑
^{7.0}^{7.1}J. R. Rice. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Applied Mech., June:379–386, 1968.