Exponential Traction Law

The Traction Law

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 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 are 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 or \sigma_c = k \delta_{peak}\delta_c }[/math]

Finally J_c is found from the expression above.

      [math]\displaystyle{ \delta_c = {2J_c\over \sigma_c}, \qquad \sigma_c = {2J_c\over \delta_c}, \qquad {\rm or} \qquad J_c = {1\over 2}\sigma_c\delta_c }[/math]

The initial linear-elastic rise can be defined by slope k or by the relative crack opening displacement at the peak stress, δ_peak (which is entered relative to the critical opening displacement or the elastic COD to reach the peak is δ_e = δ_peakδ_c). You can only enter one of these two properties (i.e., one of k_Ie and δ_I,peak and one of k_IIe and δ_II,peak). The other will be calculated from: This traction law assumes the triangular shape for traction as a function of crack opending displacement (COD) during uniaxial, monotonic loading. There are separate and uncoupled 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 three 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 = {2J_c\over \sigma}, \qquad \sigma = {2J_c\over \delta_c}, \qquad {\rm or} \qquad J_c = {1\over 2}\sigma\delta_c }[/math]