Fig. 2: Displacement field solution in the one dimensional bar for a fully developed phase-field (predefined, φ(x = 0) := 1) and different kg (0.0, 10^-5, 10^-3)

Predefined phase-field function

\phi = exp(-sqrt(x^2)/l)

representing fully broken state in the process of a developing crack were applied to a one-dimensional bar with L =[−1, 1]. The regularising internal length was chosen to be l = 2. The mechanical part of the phase-field formulation was solved via FE-simulation for the displacement field u(x) across the crack, with the applied boundary conditions u(x =−1) =−1 and u(x = 1) = 1. 10,000 elements were used over the bar to ensure a sufficiently accurate approximation of the different slopes using linear ansatz functions. The resulting displacement functions are depicted.

The artificial stiffness k_g is chosen differently each time.

Columns
1: x-coordinates of the 1D bar 
2: displacement result with k_g = 0.0
3: displacement result with k_g = 0.00001
4: displacement result with k_g = 0.001
