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. 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 the critical opening displacements are 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. 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.. 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) and used in Bardenhagen et al. (2011) and Matsumoto and Nairn (2012). The first reference 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 (2020) derivesa new approach to using cohesive zone when modeling mixed-mode failure. It fixes errors common on 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 Dnt = Dt = 0 and Dnc defining compression stiffness).

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
LinearTraction	13	A linear elastic traction law (no failure)
CubicTraction	14	A cubic traction law
TrilinearTraction	20	A trilinear traction law
CoupledTraction	23	A coupled triangular traction law
PressureTraction	26	A constant normal stress traction law (no failure)
MixedModeTraction	33	An improved, coupled, mixed-mode traction law

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