Elastic fracture in random materials

We analyze a simple model of elastic failure in randomly inhomogeneous materials such as mineralsand ceramics. We study a two-dimensional triangular lattice with nearest-neighbor harmonic springs. The springs are present with probability p. The springs can only withstand a small strain before they fail completely and irreversibly. The applied breakdown stress in a large, but finite, sample tends to zero as the fraction of springs in the material approaches the rigidity percolation threshold. The average initial breakdown stress, σ_b, behaves as σ_b^U  ≈ [A(p)+B(p)ln(L)]^-1, where L is the linear dimension of the system and the exponent μ, is between 1 and 2. The coefficient B(p) diverges as p approaches the rigidity percolation threshold. The breakdown-stress distribution function F_L(σ) has the form F_L(σ) ≈ 1-exp[-cL²exp(-k/σ^μ)]. The parameters c and k are constants characteristic of the microscopic properties of the system. The parameter k tends to zero at the rigidity percolation threshold. These predictions are verified by computer simulations of random lattices. The breakdown process can continue untila macroscopic elastic failure occurs in the system. The failure occurs in two steps. First, a number of springs fail at approximately the strain which causes the initial failure. This results in a system which has zero elastic modulus. Finally, at a considerably larger strain a macroscopic crack forms across the entire sample.