Fig. 2: The crack driving force as a function of its propagation velocity in 2D and 3D, under different levels of disorder, and the emergence of a limiting crack velocity.

(Main) The crack driving force G/Γ₀ vs. steady-state crack velocity v/c_s for a 2D homogeneous material (green circle) and two 3D heterogeneous materials with σ > 0 (yellow diamonds for σ = 0.25 and brown diamonds for σ = 0.50), obtained in phase-field simulations. The thick-boundary diamonds correspond to the 5 snapshots shown in Fig. 3. Steady-state solutions in 2D homogeneous materials do not exist for G > G_B (horizontal dashed-dotted line), upon which a linear branching instability sets in (see Fig. 1b). In 3D materials, the average steady-state crack velocity is bounded by a σ-dependent limiting velocity \({v}_{\lim }(\sigma )\) (vertical dashed lines), where G increases in a critical-like manner. (Inset) G (normalized by the first data point G₁) vs. steady-state crack velocity v/c_R measured in experiments on a glassy polymer (Polymethyl Methacrylate, PMMA), data extracted from Fig. 3 of ref. ³⁰ (the different symbols correspond to different values of L_z, see ref. ³⁰). The vertical dashed line, highlighting the similarity to the simulational data in the main panel, is added as a guide to the eye. Similar observations for additional materials are provided in Fig. 17 of ref. ². Source data are provided as a Source Data file.