Phase diagram of brittle fracture in the semi-grand-canonical ensemble

We present a simulation method to assess the quasistatic fracture resistance of materials. Set within a semi-grand-canonical Monte Carlo (SGCMC) simulation environment, an auxiliary field—the bond rupture potential—is introduced to generate a sufficiently large number of possible microstates in the semi-grand-canonical ensemble, and associated energy and bond fluctuations. The SGCMC approach permits identifying the full phase diagram of brittle fracture for harmonic and nonharmonic bond potentials, analogous to the gas-liquid phase diagram, with the equivalent of a liquidus line ending in a critical point. The phase diagram delineates a solid phase, a fractured phase, and a gas phase, and provides clear evidence of a first-order phase transition intrinsic to fracture. Moreover, energy and bond fluctuations generated with the SGCMC approach permit determination of the maximum energy dissipation associated with bond rupture, and hence of the fracture resistance of a widespread range of materials that can be described by bond potentials.

Figure 1

Work-conjugate pairs in the semi-grand-canonical ensemble. At fixed bond potential $\mathrm{\Delta }\mu$, successive MC simulations at different prescribed strains trace out a stress-strain curve for (a) harmonic and (b) Morse potentials. For a fixed volume strain, at different $\mathrm{\Delta }\mu$, bond number $N$ is measured generating bond isochores for (c) harmonic and (d) Morse potentials.

Figure 2

Phase diagrams of brittle fracture for harmonic (closed circles) and Morse (open squares) bond potential systems. The phase diagram is characterized by three domains corresponding to (I) $\mathrm{\Delta }\mu <\mathrm{\Delta }{\mu }_{\mathrm{gas}}$: the system is a collection of non-interacting particles and cannot undergo fracture. The line terminates in a critical point, CP (near-vertical line, $\mathrm{\Delta }\mu$ controlled). (II) $\mathrm{\Delta }{\mu }_{\mathrm{gas}}\le \mathrm{\Delta }\mu \le 0$: solid undergoes fracture when ${\epsilon }_V>{\epsilon }_{V_{\mathrm{crit}}}$ or $\mathrm{\Delta }\mu <\mathrm{\Delta }{\mu }_{\mathrm{crit}}$ (sloped line). (III) $\mathrm{\Delta }\mu >0$: fracture is controlled by constant ${\epsilon }_{V_{\mathrm{crit}}}$ (flat line). Insets: (a) Near zero strain the phase boundaries meet at a triple point. (b) Liquidus lines terminate at a CP.

Figure 3

Critical exponents fit from the simulation results. Lower and upper bounds of the fits (red dashed lines) for the stress critical exponent, $\alpha$ [(a) and (b), respectively], as well as for the bond number critical exponent, $\beta$ [(c) and (d), respectively] are in good agreement with the same exponents in the Ising model.

Figure 4

First-order phase transition of brittle fracture: Jump discontinuities in (a) stress, [(b),(d)] bond number, and (c) heat of bond rupture occur at the same point (red open circles) $q_{br}=0$ as predicted by the fluctuation-dissipation approach to fracture mechanics.

Figure 5

Radial distribution functions around fracture and in the gas phase. Starting from (a) the reference (unstrained) state, the system is (b) stretched, storing energy and shifting the RDF peaks to higher lattice distances. Comparing the RDFs before and after fracture, (c) peaks are noticeably shifted back to lower distances, indicative of a fracture-induced energy release in the system. (d) The gas RDF approaches ideal gas behavior as the system is a set of noninteracting particles. Insets: color-coded simulation snapshots of the potential energy (color bar on the far left).

Figure 6

Heat of bond rupture when crossing phase boundaries. There is no unique condition for fracture across all values of imposed bond rupture potential, across all domains. This is evidenced by the value of the heat of bond rupture when crossing a phase boundary in the different domains I, II, and III. Instead the value of $q_{br}$ when crossing a phase boundary serves to identify the different domains or phases of brittle fracture across all possible values of bond rupture potential.