Triangular Traction Law

The Traction Law

This traction law assumes the triangular shape for traction as a function of crack opending displacement (COD). There are separate traction laws for opening displacement (mode I) and sliding displacement (mode II).

The toughness of this traction law is the area under the curve or:

      [math]\displaystyle{ J_c = {1\over 2} \sigma\delta_c }[/math]

When creating this traction law, you have to enter exactly two of these failure properties for both mode I and mode II (i.e., two of J_Ic, σ_I, and δ_Ic and two of J_IIc, σ_II, and δ_IIc). Whichever property is not specified will be calculated from the two provided properties using one of the following relations:

      [math]\displaystyle{ \delta_c = {J_c\over 500\sigma}, \qquad \sigma = {J_c\over 500\delta_c}, \qquad {\rm or} \qquad J_c = 500\sigma\delta_c }[/math]

These relations assume J_Ic in J/m², σ in MPa, and δ_c in mm.

To get the initial rise and location of the peak, you can provide, k, and the relative crack opening displacement at the peak stress, δ_peak (which is entered is relative to the critical opening displacement or the COD at the peak is δ_peakδ_c). You must enter zero or one of these terms above for both mode I and mode II (i.e., zero or one of k_Ie and δ_I,peak and zero or one of k_IIe and δ_II,peak). If one is provided, the other will be calculated from:

      [math]\displaystyle{ k = {\sigma\over \delta_{peak}\delta_c} \qquad {\rm or} \qquad \delta_{peak} = {\sigma\over k\delta_c} }[/math]

If neither k nor δ_peak are entered, the peak will be located at δ_peak = 0.225926299. This location provides the closest match to the cubic traction law for area under the curve as a function of COD. The initial slope will be set to

      [math]\displaystyle{ k = {\sigma\over 0.225926299 \delta_c} }[/math]

Failure

The entered traction laws give mode I or mode II traction law for pure mode I or pure mode II loading. Under pure mode lode, the traction law will fail (and release the crack surfaces) when the COD reaches the critical COD ()δ_c for that mode.

For mixed-mode loading, the normal and and shear tractions still depend only on normal and shear displacements, respectively, but the conditions for failure change. For mixed-mode loading, failure occurs when:

      [math]\displaystyle{ \left({G_I\over J_{Ic}}\right)^{n} + \left({G_{II}\over J_{IIc}}\right)^{n} = 1 }[/math]

where G_I and G_II are areas under mode I and mode II traction laws up to current COD and n is mixed-mode failure criterion parameter. This failure criterion will mean that failure may occur when normal or shear tractions are nonzero. In other words, each traction law will have COD less then their critical COD, which means their traction will be greater than zero. Because failure released the crack surface, mixed-mode failure by this criterion may result in a sudden drop of finite normal and shear traction to zero traction. This approach is identical to the one adopted by Thouless et. al to handle mixed-mode loading and appears to work well.

There has been work to modify the traction laws so they always approach zero at time of failure (e.g., work of Hogberg (2006) or Tvergaard and Hutchinson (1994)). These approaches were tried, gave poor results for certain problems, led to non-physical coupling between mode I and mode II, and therefore are not recommended. The coupled traction law available in NairnMPM lets you try a law that couples to COD such that failure occurs when tractions drop to zero.

Traction Law Properties

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

Property	Description	Units	Default
kIe	Initial elastic slope, k in mode I	MPa/mm	none
delpkI	COD for peak stress, δpeak, relative to critical COD in mode I (0 to 1)	dimensionless	none
kIIe	Initial elastic slope, k in mode I	MPa/mm	none
delpkII	COD for peak stress, δpeak, relative to critical COD in mode II (0 to 1)	dimensionless	none
(other)	Properties common to all traction laws	varies	varies

To get pure linear softening (i.e, peak at zero COD with linear decrease to the critical COD), you can enter zero for delpkI or delpkII of either mode. Although a pure linear softening law is allowed, experience suggest that having an initial elastic region before the peak with finite slope (kIe or kIIe less then ∞) works better. Thus if you experience problems with a pure linear softening law, the solution might be to increase the peak position to make the initial stiffness finite.