Trilinear Traction Law

The Traction Law

This traction law assumes a piece-wise linear relation with two arbitrarily-specifiable break points (i.e., three linear pieces). There are separate traction laws for opening displacement (mode I) and sliding displacement (mode II).

The toughness is

      [math]\displaystyle{ J_c = {1\over2}\bigl(\sigma_1\delta_2 + \sigma_2(\delta_c-\delta_1)\bigr) }[/math]

The traction law for each mode depends up seven properties - (δ₁,σ₁) and (δ₂,σ₂) breakpoints, critical COD (δ_c), total area or toughness (J_c), and the initial elastic slope (k). You must enter exactly 5 of these seven properties for each mode. Furthermore, if one of the 5 is the initial slope, then δ_c along with either σ₁ or δ₁ for that mode (but not both) must be among the specified properties. The two remaining unspecified parameters will be calculated from the five provided parameters. The final law must satisfy 0 < δ₁ ≤ δ₂ ≤ δ_c, δ_c > 0, σ₁ ≥ 0, σ₂ ≥ 0, and σ₁+σ₂ > 0. Note that these restriction prohibit infinite initial stiffness (i.e., δ=0) but allow zero initial stiffness (i.e., σ₁=0). It is not clear if zero initial stiffness has any realistic applications.

One useful interpretation of a trilinear law is that it is modeling two physically mechanisms. The total toughness is the total area under the curve and is given above. The first failure mechanism can be identified with the area under the first peak and bounded by the dotted red line in the above figure. This partial area is the toughness for the first mechanism and is equal to

      [math]\displaystyle{ J_1 = {1\over2}(\sigma_1\delta_2 - \sigma_2\delta_1) }[/math]

The remaining area is associated with the second mechanism. Its toughness is equal to

      [math]\displaystyle{ J_2 = {1\over2}\sigma_2\delta_c }[/math]

For example, the first peak might be associated with crack tip fracture toughness while the tail models a process zone such as fiber bridging. In fact, in the limit as σ₁ → ∞ and δ₂ → 0, a simulation with a pre-existing cohesive zone should approach a simulation propagating a crack tip using fracture mechanics along with a linear softening traction law having σ₂ as the cohesive stress. NairnMPM can do both these simulations and the results will be similar. The pure cohesive zone simulation is done by using a trilinear traction law on an existing crack. The combined fracture mechanics/cohesive zone simulation starts with a crack that propagates when energy release rate at the crack tip is equal to J₁. Furthermore, the crack propagation commands are set to leave a cohesive zone in the wake of the crack. This cohesive zone can be a triangular traction law with its toughness equal to J₂ and stress equal to σ₂. Both these types of simulations are described in Nairn (2009).

Failure

This laws uses uncoupled mode I and mode II cohesive laws. The failure is handled by the same methods used for the Triangular Traction Law. This trilinear cohesive law can also be used for mode I or mode II laws in coupled mixed-mode modeling by using the Mixed Mode Traction Law.

Traction Law Properties

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

Property	Description	Units	Default
sigmaI2	The stress at the second break point (σ2) in mode I or opening mode	pressure units	none
sigmaII2	The stress at the second break point in mode II or sliding mode	pressure units	none
delpkI2	The relative COD at the second break point in mode I or opening mode; it is equal to δ2/δc	dimensionless	none
delpkII2	The relative COD at the second break point in mode II or sliding mode; it is equal to δ2/δc	dimensionless	none
(first point)	Use the triangular traction law properties to enter the first break point COD and initial slope for each mode	varies	varies
(other)	Properties common to all traction laws, where sigmaI and sigmaII apply to the first break point stress	varies	varies

Traction History Variables

This material tracks two history variables:

1. Maximum normal opening displacement (or equal to normal δ_e prior to initiation).
2. Maximum shear opening displacement magnitude (or equal to tangential δ_e prior to initiation).

These history variables can be archived for later plotting. The cmdisp archiving option can archive total mode I and mode II cumulative dissipated energies.