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Heat management in thermoelectric and power devices as well as in random access memories poses a grand challenge and can make the difference between a working and an abandoned device design. Despite the prevalence of dislocations in all these technologies, the modeling of their thermal resistance is based on 50-year-old analytical approximations, whose simplicity was driven by practical limitations rather than physical insight. We introduce an efficient ab initio approach based on Green's functions computed by two-dimensional reciprocal space integration. By combining elasticity theory and density functional theory, we calculate the scattering strength of a 90^0 misfit edge dislocation in Si. Because of the breakdown of the Born approximation, earlier literature models fail, even qualitatively. We find that a dislocation density larger than 10^9cm^(−2) is necessary to substantially influence thermal conductivity at room temperature and above. We quantify how much of the reduction of thermal conductivity measured in nanograined samples can be explained by realistic dislocation concentrations.