Keywords: fracture, stiffness, compliance, cracks, elasticity, cracked solids, chaos, first-order iterative maps, chaotic crack size, logistic map.

The effect of chaotic crack size on the fracture stiffness of a distribution of coplanar flat elliptic cracks is examined. The crack lengths are computed as discrete elements of a chaotic sequence resulting from the solutions of different first order non-linear iterative maps. The iterative maps considered are generalized logistic maps in the sense that a succeeding crack length is given as the product of the preceding crack length raised to a positive power times the difference of the preceding crack length from the maximum crack length raised to a different in general positive power. The resulting crack length distribution is chaotic depending on the two positive powers and a positive parameter multiplying the aforementioned product. When the positive powers are equal to one the iterative map becomes the classical logistic map. The crack length involved in the generating map is non-dimensional having being divided by a characteristic length such that the maximum value of the ratio is equal to one. This yields a maximum value for the positive parameter of the generating map for which the resulting non-dimensional crack lengths are fully chaotic. From the computed crack lengths, the chaotic crack opening areas are obtained for mode I loading and non-interacting cracks. Subsequently, the fracture stiffness is computed from the inverse of the sum of the squared crack lengths. A non-dimensional fracture compliance is defined as the average squared crack length so that it is identified with the average power produced by the non-linear iteration map. Numerical results are presented for four different generating maps. The logistic map, a map with the power of the difference involved in the map equal to 1.1, a map with this power equal to 1.5, and the elliptic map defined by the two powers involved in the map equal to 0.5. For full chaos in each of these maps, the sequence of four hundred crack lengths is obtained. Then, the fracture compliance is computed not only for full chaos but for a range of the positive parameter of each generating map. The results show that the chaotic sequence of crack lengths is very sensitive to the powers involved in the non-linear iterative map. In turn, the fracture compliance is in general larger for smaller map powers, generally independent of the map multiplying parameter. Furthermore, the fracture compliance exhibits a much stronger dependence on this parameter near its maximum value that yields fully chaotic crack lengths.

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