Figure 2

Minimal model for the straight-line crack propagation. (a) Two-dimensional square-lattice model of a sheet with a line crack. Here, l is the lattice spacing. Each lattice point interacts with the nearest-neighbor points. We introduce a line crack by cutting bonds, i.e., we set the interactions of the corresponding bonds to zero. (b) Minimal model obtained by coarse-graining the lattice in (a). We decimate all the lattice points except for the points on the two horizontal lines on which the two surfaces of the line crack are positioned, where L is the height of the sheet under zero strain. (c) Mechanism of the crack propagation. When the spring at the crack tip (encircled by a blue ellipse) is stretched to the critical strain ε _c , the bond at the tip breaks, and, after a certain time, the next bond at the tip is stretched to ε _c . This cycle continues during the crack propagation. (d) Forces acting on a lattice point. On each point in (c) located at the front side of the crack tip, four forces act: (i) one from the top boundary, (ii) one from the point below, and (iii, iv) the remaining two reflecting shear and acting from the left and right nearest-neighbor points. For each point located at the rear side, one force from the point below is missing. (e) Zener element. This element is a parallel connection of two components: a spring (elastic modulus E ₀) and a Maxwell element, i.e., a serial connection of another spring (modulus E ₁) and a dashpot (viscosity η). (f) Kelvin-Voigt element obtained from a Zener element in the large E ₁ limit, in which λ ≡ E _∞/E ₀ → ∞.