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The topological derivative describes the variation of a response functional with respect to infinitesimal changes in topology, such as the introduction of an infinitesimal crack or hole. In this three-dimensional fracture mechanics work, we propose an approximation of the energy release rate field associated with a small surface crack of any boundary location, direction, and orientation combination using the topological derivative. This work builds on the work of Silva et al. (“Energy Release Rate Approximation for Small Surface-Breaking Cracks Using the Topological Derivative,” J. Mech. Phys. Solids 59(5), pp. 925–939), in which the authors proposed an approximation of the energy release rate field which was limited to two-dimensional domains. The proposed method is computationally advantageous because it only requires a single analysis. By contrast, current boundary element and finite element-based methods require an analysis for each crack length-location-direction-orientation combination. Furthermore, the proposed method is evaluated on the non-cracked domain, obviating the need for refined meshes in the crack tip region.

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We derive a continuum-level plasticity model for polycrystalline materials in the high energy density regime, based on a single dislocation density and single mobility mechanism, with an evolution model for the dislocation density. The model is formulated explicitly in terms of quantities connected closely with equation of state (EOS) theory, in particular the shear modulus and Einstein temperature, which reduces the number of unconstrained parameters while increasing the range of applicability. The least constrained component is the Peierls barrier EP , which is however accessible by atomistic simulations. We demonstrate an efficient method to estimate the variation of EP with compression, constrained to fit a single flow stress datum. The formulation for dislocation mobility accounts for some or possibly all of the stiffening at high strain rates usually attributed to phonon drag. The configurational energy of the dislocations is accounted for explicitly, giving a self-consistent calculation of the conversion of plastic work to heat. The configurational energy is predicted to contribute to the mean pressure, and may reach several percent in the terapascal range, which may be significant when inferring scalar EOS data from dynamic loading experiments. The bulk elastic strain energy also contributes to the pressure, but appears to be much smaller. Although inherently describing the plastic relaxation of elastic strain, the model can be manipulated to estimate the flow stress as a function of mass density, temperature, and strain rate, which is convenient to compare with other models and inferences from experiment. The deduced flow stress reproduces systematic trends observed in elastic waves and instability growth experiments, and makes testable predictions of trends versus material and crystal type over a wide range of pressure and strain rate.