Coupled Traction Law

The Traction Law

(This documentation is implementation of this law OSParticulas that is coming soon to NairnMPM. The current implementation in NairnMPM is different and should not be used until updated to this new implementation).

This traction law uses either the triangular cohesive law or the exponential cohesive law but couples the normal and tangential displacements by using an effective displacement. This implementation follows methods derived in a paper by Högberg. The Högberg effective displacement is defined by

      [math]\displaystyle{ \lambda = \sqrt{\left(\frac{u_n}{\delta_c^{(n)}}\right)^2 +\left(\frac{u_t}{\delta_c^{(t)}}\right)^2} }[/math]

where u is displacement, δ_c is displacement at failure (see figure), and "n" and "t" refer to normal an tangential directions. In terms of λ, the normal an tangential tractions are:

      [math]\displaystyle{ \left(T^{(n)},T^{(t)}\right) = \frac{H(\lambda_\omega)}{\lambda_\omega} \left(\frac{\sigma_{c}^{(n)} u_n}{\delta_c^{(n)}},\frac{\sigma_{c}^{(t)} u_t}{\delta_c^{(t)}}\right) }[/math]

where λ_ω is a damage parameter equal to the maximum value of λ during a simulation. These traction apply after initiation and correctly account for elastic unloading and reloading. Prior to initiation, deformations are elastic. Initiation using an elliptical failure criterion occurs when λ reaches λ_p(θ) defined by

      [math]\displaystyle{ \frac{1}{ \lambda_p(\theta)} = \sqrt{\left(\frac{\delta_c^{(n)}\sin\theta}{\delta_e^{(n)}}\right)^2 + \left(\frac{\delta_c^{(t)}\cos\theta}{\delta_e^{(t)}}\right)^2} \quad {\rm with} \quad \theta = \frac{\delta_c^{(t)}u_n}{\delta_c^{(n)}u_t} }[/math]

and δ_e is displacement at end of the elastic regime. Finally, Högberg assumed H(λ) is defined by linear decay from 1 when λ=λ_p(θ) to zero when λ=1:

      [math]\displaystyle{ H(\lambda) = \frac{1-\lambda}{1-\lambda_p(\theta)} }[/math]

This implementation has extended the Höberg approach to also allow an exponential decay using:

      [math]\displaystyle{ H(\lambda) = \frac{e^{-\alpha\frac{\lambda-\lambda_p(\theta)}{1-\lambda_p(\theta)}}-e^{-\alpha}}{1-e^{-\alpha}} }[/math]

See figure above for effect of α on the decay rate (note that α approaching zero is identical to linear decay).

Failure

Under pure mode I or pure mode II, a coupled traction law and its failure are identical to a triangular cohesive law or an exponential cohesive law. Under mixed-mode conditions, however, these laws differ. Triangular and exponential cohesive law have to impose a mixed-mode failure criterion. This model does not need such a criterion. It fails naturally when λ reaches one. Despite this advantage of failing "naturally", the Högberg model has limitations explained in the next section.

Property Limitations

The Höberg model seemingly depends on independent normal and traction law properties that could be entered as for triangular or the exponential cohesive laws. In fact Ref. claims independent properties are allowed. A new analysis of cohesive zone modeling, however, has revealed that the Högberg model is always invalid for mixed-mode loading unless properties are chosen such that

      [math]\displaystyle{ \frac{\delta_e^{(n)}}{\delta_c^{(n)}} = \frac{\delta_e^{(t)}}{\delta_c^{(t)}} }[/math]

In other words, the ratio of displacement at peak stress to displacment and failure must be the same for both normal and tangential directions. When using the exponential extension of the Högberg model the α coefficient must also be the same for normal and tangential directions.

Note that this implementation lets you choose properties that make the modeling invalid. That option is allowed for comparison of the published model to other models. If you do select invalid properties, the output file will include a warning. The preferred model for simulations of mixed-mode failure, which remains valid for independent normal and tangential laws, is the Mixed Mode Traction Law.

Alternate Effective Displacement Method

Other mixed-mode cohesive laws have used unscaled effective displacement models or simply define it is magnitude of the crack opening displacement:

      [math]\displaystyle{ \lambda = \sqrt{u_n^2+u_t^2} }[/math]

Examples are found in Comanho and Dàvila and Comacho and Ortiz (these models are implement in Abaqus cohesive elements. These models are note implemented in NairnMPM (or in OSParticulas) because they are superfluous. Then can be used simply by choosing

      [math]\displaystyle{ \delta_c^{(n)} = \delta_c^{(t)} }[/math]

Combining this restriction with the Högerg limitations, means this alternate effective displacement model is invalid unless:

      [math]\displaystyle{ \frac{\delta_e^{(n)}}{\delta_e^{(t)}} = \frac{\delta_c^{(n)}}{\delta_c^{(t)}} = 1 }[/math]

or critical displacements in the normal and tangential cohesive laws are identical (their magnitudes, or normal and tangential strengths, can differ). Implementations of this approach, such as in Abaqus, unfortunately let users select properties that invalidate the modeling.

Traction Law Properties

The following properties are used to create a coupled traction law:

Property	Description	Units	Default
(Sawtooth Laws)	Enter properties to define normal and tangential cohesive laws assuming triangular traction laws	varies	none
model	Choose whether to model using linear decay (12) or exponential decay (34). The numerical values are material IDs for linear and exponential traction laws	none	12
alpha	When using exponential decay (model=34), this parameter selects the α coefficient for the decay rate. Note than when using exponential decay, this implementation first assumes law properties assuming triangluar traction laws and then adjusts δc such that toughness Gc (or area under the traction law) of the exponential law matches the input properties for triangular laws.	none	none

As described above, this implementation lets tyou select properties that invalidate the models when used to mode mixed-mode failure. If you choose invalid properties, a warning is printed in the outpout files.