Exponential Traction Law

The Traction Law

(Currently only in OSParticulas but common soon to NairnMPM).

This traction law assumes a linear elastic response up to σ_c followed by a scaled exponential decrease to reach zero at critical crack opening displacement (COD) of δ_c. The exponential decay in the post peak region (i.e., δ>δ_e) is given by

      [math]\displaystyle{ S(\delta) = \sigma_c\frac{e^{-\alpha\frac{\delta-\delta_e}{\delta_c-\delta_e}}-e^{-\alpha}}{1-e^{-\alpha}} }[/math]

This cohesive law defines traction as a function of crack opening displacement (COD) during uniaxial, monotonic loading. There are separate and uncoupled cohesive laws for opening displacement (mode I) and sliding displacement (mode II).

Notice how the shape changes with α. In the limit of α=0, this law is identical to the Triangular Traction Law (and that one should be used instead because it is more efficient). Increasing α causes traction to drop faster, which might simulate increasingly brittle response. The toughness of this cohesive law is the area under the curve that can be cast as

      [math]\displaystyle{ J_c = {1\over 2} \sigma_c\delta_c\left(\delta_{peak} + \bigl(1-\delta_{peak}\bigr)f(\alpha)\right) \quad{\rm where}\quad f(\alpha) = 2\left(\frac{1}{\alpha}-\frac{e^{-\alpha}}{1-e^{-\alpha}}\right) }[/math]

The function f(α) is 1 when α=0 and decays to zero as α increases. The toughness is thus bracketed by

      [math]\displaystyle{ {1\over 2} \sigma_c\delta_e \le J_c \le {1\over 2} \sigma_c\delta_c }[/math]

The minimum corresponds to large α with an instant drop to zero traction at δ_e while the maximum corresponds to α=0 and is area under the corresponding Triangular Traction Law.

When creating this traction law, you have to enter α, exactly two of σ_c, δ_p, and k and either J_c or δ_c. These four required properties must be entered for both mode I and mode II. If you enter δ_c, then the missing one of σ_c, δ_p, and k is calculated from one of

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

Finally J_c is found from the expression above. If instead you enter J_c, the calculation of δ_c depends on which two of σ_c, δ_p, and k are provided:

      [math]\displaystyle{ {\rm from\ }\sigma_c\ {\rm and}\ \delta_{peak}:\quad \delta_c = \frac{2J_c}{\sigma_c\left(\delta_{peak} + \bigl(1-\delta_{peak}\bigr)f(\alpha)\right)} }[/math]

      [math]\displaystyle{ {\rm from\ }\sigma_c\ {\rm and}\ k:\quad \delta_c = \frac{2kJ_c-\bigl(1-f(\alpha)\bigr)\sigma_c^2}{k\sigma_cf(\alpha)} }[/math]

      [math]\displaystyle{ {\rm from\ }\delta_{peak}\ {\rm and}\ k:\quad \delta_c = \sqrt{\frac{2J_c}{k\delta_{peak}\left(\delta_{peak} + \bigl(1-\delta_{peak}\bigr)f(\alpha)\right)}} }[/math]

Once δ_c is found, the remaining property among σ_c, δ_p, and k is found using an expression above.

Comparison to Triangular Traction

While some a swayed that the use of exponential laws is some how much more realistic the linear laws, the details of cohesive law shape are of second-order importance. They have some minor effects (such as changes in predicted R curve shapes), but in general, any two laws with the same toughness (J_c) and peak (σ_c), will have very similar results.

The only different between this exponential law and the Triangular Traction Law is the α parameter. As shown in the figure above, this parameter changes the decay region of the cohesive law. If one does a series of simulations keeping J_c and σ_c constant, but varying α (as a consequence, δ_c will increase as α increases), the results will have only minor differences. Because Triangular Traction Law is more efficient, it is usually the preferred cohesive law for simulations. This exponential law is provided for comparison to others or if the minor differences become important.

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 exponential 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 triangular traction law:

Property	Description	Units	Default
alphaI	Mode I α coefficient	Dimensionless	none
alphaII	Mode II α coefficient	Dimensionless	none
(other)	All other traction law properties	varies	varies

The resulting laws are required to have alphaI and alphaII greater than zero.