Question 3 of 7: Question III: Lath-Martensite Yield-Strength Equation, and Alloying Roles in a T1 High-Speed Tool Steel

Question III: Lath-Martensite Yield-Strength Equation, and Alloying Roles in a T1 High-Speed Tool Steel (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

3.1 — (i) The four strengthening terms in the lath-martensite yield-strength equation

The equation is a superposition of four INDEPENDENT strengthening mechanisms, each contributing additively to the total flow stress of as-quenched lath martensite (Norstrom, 1976; Hertzberg, Ch. 4):

1. $\sigma_i$ — intrinsic lattice friction stress. This is the Peierls–Nabarro stress: the stress needed to move a dislocation through the "perfect" body-centred (tetragonal) iron lattice with no solute atoms, boundaries or other dislocations present. It is a baseline strength floor set purely by the lattice's own crystallography and bonding, independent of composition or microstructure.
2. $k(C)^{1/2}$ — interstitial solid-solution strengthening by carbon. Carbon atoms trapped interstitially in the martensite (the same carbon responsible for the tetragonal distortion discussed in Question IV(iii)) create local elastic strain fields around each solute site; these strain fields interact with, and impede, the stress fields of moving dislocations. The square-root dependence on carbon content is the classical statistical scaling for randomly distributed point obstacles (the same $\sqrt{c}$ form as substitutional Fleischer-type solid-solution hardening), so this term grows with carbon content but sub-linearly.
3. $k_y(d)^{-1/2}$ — Hall–Petch strengthening from the martensite lath/packet substructure. $d$ is the characteristic size of the martensite substructure (packet or block size, sometimes lath width). Packet/block/lath boundaries block dislocation glide and force pile-ups, exactly as high-angle grain boundaries do in an ordinary polycrystal; a finer substructure packs more boundary area into a given volume, so the barrier spacing shrinks and the strengthening contribution rises as $d^{-1/2}$ — the identical functional form (and the identical dislocation pile-up physics) as classical Hall–Petch grain-size strengthening.
4. $\alpha Gb(\rho)^{1/2}$ — forest-dislocation (Taylor) hardening from the as-quenched dislocation density. The diffusionless martensitic shear transformation itself generates an extremely high dislocation density (typically $\sim10^{11}$–$10^{12}$ cm$^{-2}$, comparable to a heavily cold-worked metal) as a lattice-invariant shear needed to accommodate the transformation strain. These dislocations tangle and interact with one another ("forest" hardening); by the classical Taylor relation, the resulting flow-stress contribution scales with $G$ (shear modulus), $b$ (Burgers vector) and the square root of the dislocation density $\rho$, with $\alpha$ a geometric constant of order $0.2$–$0.4$.

3.2 — (ii) Roles of C, Cr and W in T1 high-speed tool steel

1. Carbon (0.65–0.80 wt%). Carbon plays a double role: it is the interstitial solute that hardens the quenched martensitic matrix itself (the same $k(C)^{1/2}$ term from part (i)), and it combines with W, Cr and V to form the hard alloy carbides (predominantly M₆C and MC-type) that give the steel its wear resistance and, on tempering, its secondary-hardening carbide dispersion.
2. Chromium (∼4.00 wt%). Chromium's primary role is to raise hardenability, not to harden the matrix directly: as a substitutional, carbide-forming solute it slows the diffusional decomposition of austenite to pearlite/bainite (solute drag plus the extra diffusional step of partitioning Cr between phases or into alloy carbides), pushing the TTT/CCT "nose" to much longer times. This lets a large, complex tool section be through-hardened on a comparatively mild air or oil quench instead of the severe water quench that would otherwise be needed — and that would crack or distort the tool. A secondary role is forming M₂₃C₆/M₇C₃ carbides that add modestly to hardness and to oxidation/temper resistance.
3. Tungsten (∼18 wt%). Tungsten is the defining "high-speed" alloying addition. It forms extremely stable, high-melting tungsten-rich carbides (M₆C-type) that contribute wear resistance directly, but far more importantly these carbides dissolve into the austenite at the very high austenitizing temperature this class of steel requires (of order 1250 °C) and then reprecipitate as an extremely fine, thermally stable dispersion during tempering. This secondary-hardening reaction is what gives high-speed steels their signature "hot hardness" (red hardness) — retained cutting-edge hardness at the elevated temperatures ($400$–$600$ °C) generated during high-speed machining, long after a plain-carbon tool steel would already have over-tempered and softened.