[12 pts] Consider a microalloyed steel whose structure consists of a ferrite (BCC iron) matrix containing 0.2% (volume percent) niobium carbide (NbC) particles. The carbide particles are spherical and have an average radius ($r$) is 10 nm. The NbC particles serve two functions. First, the NbC particles can restrict the grain growth of ferrite during processing, thus leading to grain boundary strengthening. The grain diameter $d$ of the microalloyed steel varies with $r$ and the volume fraction of the carbide, $f$, as $d = r/f$. Second, carbide particles present within grains also result in particle strengthening. NbC is structurally incoherent with the ferrite matrix. The effective spacing between the carbide particles within the grain can be approximated as $L' = r (\pi/f)^{1/2}$. Data for steel: $k_y = 0.60 \, MN/m^{3/2}$; $G = 81 \, GPa$; $b = 0.248 \, nm$. The tensile yield strength of ferrite having a grain diameter of 50 µm is 260 MN/m². The Taylor factor for BCC metals is approximately 2.75.
(a) Calculate the increase in tensile yield strength resulting from grain refinement induced by the carbides.
(b) Calculate the tensile yield strength increment due to the NbC particles situated within the grains?
(c) What is the tensile yield strength of this microalloyed steel if we assume dislocation strengthening and solid solution strengthening are negligible.
