A program of research spanning several years ending in November 2005 was dedicated primarily to formulation and analysis of canonical boundary- value problems in mathematical modeling of dynamic fracture in brittle materials. The sub-topics within the broad topic of dynamic fracture in brittle materials that were studied, and the accomplishments in each sub-topic, are summarized as follows:

*Unsteady mode-I (opening-mode) crack growth in elastic material.*A Dirichlet-to-Neumann map needed to relate crack-opening displacement to driving tractions for a semi-infinite, Mode-I crack in an infinite, linearly elastic body was constructed. At the time of reporting the information for this article, the map was in use in a study of a crack-tip cohesive zone and of the use of a critical-crack-opening-displacement fracture criterion for prediction of crack-tip motion.

*Dynamic crack growth in functionally graded elastic bodies.*One study addressed the problem of a semi-infinite, tearing-mode crack propagating dynamically along an interface between two half-planes of linear elastic material, the shear modulus of which varies spatially in the direction transverse to the fracture plane. Because this problem is analytically intractable, it was approximated by performing an asymptotic analysis based upon the assumption that the properties of the material throughout its bulk are given by an asymptotic series, the zero-order term of which corresponds to a homogeneous elastic material. The asymptotic analysis effectively converted the problem to a simpler problem involving a quasi-static case for which one can construct analytic solutions for a few special classes of nonhomogeneous material models. The stress-intensity factor (SIF) values calculated in the asymptotic analysis were found to be in reasonable agreement with true SIF values.

*Dynamic, steady-state, generalized plane-strain crack growth in anisotropic linear viscoelastic material.*This study produced a general solution method based upon transform and complex-variable techniques that produces a full field solution to a semi-infinite crack problem under general crack-face loadings leading to generalized plane-strain deformations. A key step in the analysis is the construction of the Dirichlet-to-Neumann map on the bounding plane for an anisotropic viscoelastic half-space under dynamic steady-state conditions. Two classes of crack-face tractions were simulated: a point load placed a prescribed distance behind the crack tip and a uniform traction applied over a finite interval placed a prescribed distance behind the crack tip. Each simulation exhibited the surprising feature of vanishing of the SIF at a crack speed strictly less than the glassy Rayleigh wave speed. It was proved that (1) if the loading interval begins at the crack tip, then the SIF cannot vanish until the glassy Rayleigh wave speed is reached, but (2) if the loading interval is moved back from the crack tip, then the SIF can vanish at a lower speed.

*Dynamic, transient, mode-III (tearing-mode) crack growth in linear viscoelastic material.*A previously developed method of solving problems of dynamic unsteady crack growth in an isotropic linear elastic body was generalized to a viscoelastic body. The generalized method was applied in simulations of a single semi-infinite tearing-mode shear crack in a viscoelastic body. This was asserted to be the first dynamic, unsteady crack analysis valid for a general linear viscoelastic body, and the results of the simulations were interpreted as demonstrating clearly the critical role played by viscoelasticity in controlling the speed of a propagating crack and the length to which it grows.

In addition to the studies summarized above, studies were initiated in new sub-topics, including various issues concerning finite deformations of nonlinear elastic bodies, a new approach to incorporation of molecular-scale effects into continuum models of brittle fracture without incurring the logical inconsistencies inherent in classical linear elastic fracture mechanics, and a novel approach to modeling residual stress accumulation in a continuously growing oxide layer on a metallic substrate.

*This work was done by Jay Walton of Texas A&M University for the Air Force Office of Scientific Research.*

* AFRL-0087*

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