Cubic Traction Law

The Traction Law

This traction law assumes use a cubic function for traction as a function of crack opending displacement (COD). This cubic relation is choosen such that the slope is zero at δ_c and the peak always occurs at δ_c. This traction law was first proposed by Needleman. It has a convenient smooth shape for numerical calculations and the zero slope at δ_c may be desirable. There is no physical basis to claim it is more realistic than other traction laws. The resulting function is

      [math]\displaystyle{ \sigma = {27\over 4}\sigma_{max} {\delta\over\delta_c}\left(1-{\delta\over\delta_c}\right)^2 }[/math]

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 = {9\over 16} \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 = {16J_c\over 9000\sigma}, \qquad \sigma = {16J_c\over 9000\delta_c}, \qquad {\rm or} \qquad J_c = {9000\over 16}\sigma\delta_c }[/math]

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

Besides specifying area and toughness properties, the cubic traction law has no other adjustable parameters. The initial slope and peak location of give once the area is specfied. Thus unlike most other traction laws, this law has fewer parameters, which can be helpful when studying role of traction law on failure simulations.

Traction Law Properties

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

Property	Description	Units	Default
(other)	Properties common to all traction laws	varies	varies