Exercises: Equilibrium and iron-carbon diagrams

Explain what an equilibrium diagram (or phase diagram) is: what variables are represented on its axes (temperature and percentage of each component), what each region indicates, and why it is called "equilibrium" (cooling must be very slow so that atoms have time to rearrange). Use water as an analogy: to each temperature and pressure corresponds a stable state.

Match each constituent of the iron-carbon diagram with its description: (a) ferrite, (b) cementite, (c) pearlite, (d) austenite; with: solid solution of carbon in α-iron, soft and ductile; iron carbide Fe₃C, very hard and brittle; lamellar mixture of ferrite and cementite in alternating layers; solid solution of carbon in γ-iron, stable only at high temperature.

The figure shows the simplified iron-carbon diagram with a point marked with a question mark. a) State which region that point falls into and what phase or phases the steel exhibits at that temperature and composition. b) Explain what the two axes of the diagram represent. c) Reason in which direction the point would shift if the steel contained more carbon.

Classify these iron-carbon alloys as steel or cast iron and order them from lowest to highest expected hardness: (a) $0{,}10\,\%$ C, (b) $0{,}45\,\%$ C, (c) $0{,}90\,\%$ C, (d) $3{,}5\,\%$ C. Remember that the boundary between steels and cast irons is around $2\,\%$ carbon ($2{,}11\,\%$ in the actual diagram).

State whether true or false, justifying your answer: "Pearlite is a chemical element added to iron to harden it." Explain what pearlite actually is, from which constituent it forms upon cooling, and what appearance it has under the microscope (alternating light and dark lamellae).

Explain what it means for two metals to be completely soluble in the solid state (such as copper and nickel) as opposed to having partial solubility (such as iron and carbon). What shape does the phase diagram of each pair have, and what practical consequence does this have for the alloys that can be manufactured?

State the lever rule in your own words: what it is used for within a two-phase region, which segments are measured along the tie line (isotherm), and how the percentage of each phase is obtained. Illustrate it with the analogy of a see-saw: the fulcrum is the alloy composition and the two arms "weigh" the phases at the ends.

Describe the eutectoid transformation of steel: at what temperature it occurs ($727\ ^\circ\text{C}$), with what composition ($0{,}77\,\%$ C), and what is produced upon cooling through it (austenite transforms into pearlite). Compare it with the eutectic point of the diagram ($4{,}3\,\%$ C, $1\,147\ ^\circ\text{C}$), where a liquid transforms into two solids.

An alloy with $45\,\%$ of component B is at a temperature where an α phase with $20\,\%$ of B and a liquid phase with $60\,\%$ of B coexist, as shown in the figure. a) Read the length of the two lever arms from the isotherm. b) Apply the lever rule to calculate the percentage of α phase and liquid phase. c) Check that both percentages add up to $100\,\%$.

An isomorphous alloy (complete solubility) contains $25\,\%$ of component B. At a certain temperature, the α phase present has $15\,\%$ of B and the liquid has $45\,\%$. a) Calculate using the lever rule the percentage of each phase present. b) Reason what happens to these percentages if the temperature drops towards the solidus line (which phase gains ground?).

A steel containing $0{,}4\,\%$ carbon is cooled very slowly to room temperature. a) State which constituents its microstructure exhibits. b) Applying the lever rule between ferrite ($0{,}02\,\%$ C) and pearlite ($0{,}77\,\%$ C), calculate the percentage of pearlite in the steel. c) Express what approximate fraction of the microstructure will be ferrite.

Calculate the total percentage of cementite contained in a steel with $1{,}0\,\%$ carbon at room temperature, applying the lever rule between practically pure iron ($0{,}02\,\%$ C) and cementite ($6{,}67\,\%$ C). Comment on whether this result explains why tool steels, with carbon around $1\,\%$, are much harder than structural steels.

Classify these steels as hypoeutectoid, eutectoid, or hypereutectoid: (a) $0{,}25\,\%$ C, (b) $0{,}77\,\%$ C, (c) $1{,}10\,\%$ C. For each, state which primary constituent appears alongside pearlite upon slow cooling (proeutectoid ferrite or proeutectoid cementite) and cite an application consistent with their increasing hardness.

The micrograph of a steel shows white regions and regions with thin alternating light and dark bands, the latter occupying approximately half the surface area. a) Identify both constituents. b) Estimate the carbon percentage of the steel from the observed fraction of pearlite. c) Reason whether this is a typical steel for car bodywork sheet (very low carbon) or for a machine axle (medium carbon).

A eutectoid steel ($0{,}77\,\%$ C) is cooled slowly from $800\ ^\circ\text{C}$ to room temperature. a) What fraction of pearlite will it exhibit? b) Calculate, within that pearlite, the percentage of ferrite and cementite by applying the lever rule between $0{,}02\,\%$ and $6{,}67\,\%$ carbon. c) Comment on why cementite, despite being the minority component, dominates the hardness of the steel.

Compare the copper-nickel diagram (complete solid solubility) with the lead-tin diagram (partial solubility with a eutectic point): differences in shape, what occurs exactly at the eutectic point, and why alloys close to it melt at a lower temperature than pure metals. Relate this fact to the tin-lead soft solder used in electronics.

In the expert report on the failure of a transmission shaft manufactured from carbon steel ($0{,}6\,\%$ C) slowly cooled under equilibrium conditions, a technician states the following: "Since pearlite is composed mostly of ferrite ($88\,\%$) and proeutectoid ferrite is also present, nearly $90\,\%$ of the final microstructure is pure alpha ferrite; therefore, the shaft should have behaved as a ductile and soft metal, making its brittle fracture inexplicable." a) Quantitatively determine the microstructure at room temperature by calculating the percentage of microconstituents (proeutectoid ferrite and pearlite) and the total percentage of phases (total alpha ferrite and total cementite), assuming a solubility of $0{,}02\,\%$ and a eutectoid at $0{,}77\,\%$ C. b) Rebut the technician's statement using metallurgical arguments: explain why steel with that proportion of total ferrite does not behave like pure ferrite at all and possesses much higher strength and hardness. c) If it is necessary to increase the toughness of the shaft to prevent its brittle fracture without reducing the carbon percentage, what heat treatment would you propose and what microstructural change does it aim for compared to this slow cooling?

A workshop receives a batch of steel bars slowly cooled under equilibrium conditions, but unlabeled. Quantitative metallographic analyses reveal that all specimens exhibit $80\,\%$ pearlite.

1. a) Using the attached graph and based on the lever rule, show that there are two very different carbon compositions that exhibit exactly that $80\,\%$ pearlite, and calculate the carbon percentage of each.
2. b) To identify which of the two steels the batch corresponds to without resorting to destructive chemical analysis, an operator suggests observing under an optical microscope the microconstituent accompanying the pearlite colonies or measuring the surface hardness. Predict which proeutectoid microconstituent will accompany the pearlite in each case and how they will differ markedly in their mechanical properties.
3. c) What would happen to the microstructure and the percentage of pearlite if the hypoeutectoid bar from the graph ($0{,}2\,\%$ C) were subjected to continuous cooling at a critical rate higher than the quenching rate from the austenite region? Justify the result.

An agricultural machinery company needs to manufacture two components with very different requirements:

1) The articulated arm of an implement, subjected to strong impacts and bending stresses.

2) The housing of a gear reducer, of complex geometry with thin walls, subjected to compression and surface wear, but without dynamic shock loads.

The purchasing department proposes using the same white cast iron ($3{,}5\,\%$ C) for both parts via casting followed by hot-forge forming to adjust the tolerances of the arm.

1. a) Rebut the proposal to forge white cast iron for the articulated arm: justify why it is technically unfeasible to form it plastically at hot temperatures by comparing its microstructure with that of a hypoeutectoid steel ($0{,}2\,\%$ C).
2. b) Explain, in terms of solidification range, melting temperature, and fluidity, why the $3{,}5\,\%$ C cast iron is optimal for casting the complex housing compared to a steel with $0{,}2\,\%$ C.
3. c) If forging were ruled out and you wanted to cast the articulated arm in cast iron to reduce costs, what type of cast iron would you propose to withstand impacts without brittle fracture, and what heat treatment or inoculant is required to alter the carbon morphology?

In a metallurgical laboratory, three identical cylindrical specimens of a eutectoid steel ($0{,}77\,\%$ C), previously homogenized in the austenitic phase at $850\ ^\circ\text{C}$, are subjected to three different cooling regimes:

• Specimen A: cooled inside the turned-off furnace ($0{,}05\ ^\circ\text{C}/\text{s}$).

• Specimen B: cooled in still air ($5\ ^\circ\text{C}/\text{s}$).

• Specimen C: rapidly quenched in agitated water ($250\ ^\circ\text{C}/\text{s}$).

1. a) Justify the diffusion-controlled nucleation and growth mechanism experienced by specimens A and B upon crossing $727\ ^\circ\text{C}$. Explain why the greater undercooling of specimen B results in a finer interlamellar spacing and predict how this will affect its yield strength and hardness compared to A.
2. b) Explain why the diffusive eutectoid transformation is completely inhibited in specimen C. Identify the resulting microstructure, the crystallographic lattice change that takes place without diffusion, and the causes of its extreme hardness and brittleness.
3. c) Design the heat treatment that must be applied to specimen C to turn it into a tough and wear-resistant cutting tool, describing the microstructural changes it will undergo during the process.

To manufacture rails for a railway line, a hypoeutectoid steel is required that exhibits an equilibrium microstructure containing $75\,\%$ pearlite and $25\,\%$ proeutectoid ferrite, with an allowable pearlite range between $70\,\%$ and $80\,\%$.

1. a) Deduce the required nominal carbon percentage and the allowable carbon tolerance range in the steelmaking process, assuming the eutectoid point at $0{,}77\,\%$ C and the maximum solubility of ferrite at room temperature to be $0{,}008\,\%$ C.
2. b) A quality control manager argues that, since the steel contains $75\,\%$ pearlite, the total percentage of the ferrite phase in the steel will be approximately $25\,\%$. Analytically demonstrate their error by calculating the actual percentage of total ferrite and total cementite at room temperature.
3. c) Technically justify why this steel with $75\,\%$ pearlite is selected for a rail instead of a mild steel with $20\,\%$ pearlite or a hypereutectoid steel with grain boundary cementita, assessing the trade-off between friction wear and resistance to brittle impact fracture.

In the manufacture of machine tool beds, a material capable of effectively damping mechanical vibrations and exhibiting excellent machinability for grinding its slideways is required. During a casting with $3{,}2\,\%$ C, due to a technical failure, the addition of silicon (typically $2\,\%$) is omitted and the component cools excessively fast in a metal mold.

1. a) After demolding, the bed turns out to be impossible to machine and fractures in a brittle manner under the slightest impact. Justify what occurred from a metallurgical standpoint by comparing the metastable phase diagram (iron-cementite) with the stable one (iron-graphite): what microstructure was intended, and which one was produced instead?
2. b) Analyze the thermodynamic and kinetic role played by silicon content (graphitizing element) and cooling rate in the decomposition of metastable cementite into ferrite and graphite.
3. c) Explain at a micromechanical level why the presence of graphite flakes in gray cast iron provides exceptional vibration damping capacity and facilitates chip breaking during machining, compared to a steel with the same pearlitic matrix.

In a materials receiving control laboratory, three carbon steel bars of unknown composition, slowly cooled under annealing conditions, are tested. The mechanical tests yielded the following data:

• Bar A: hardness $105\ \text{HB}$, yield strength $\sigma_y = 210\ \text{MPa}$, elongation at break $A = 38\,\%$.

• Bar B: hardness $220\ \text{HB}$, yield strength $\sigma_y = 460\ \text{MPa}$, elongation at break $A = 15\,\%$.

• Bar C: hardness $320\ \text{HB}$, yield strength $\sigma_y = 780\ \text{MPa}$, elongation at break $A = 3\,\%$, brittle fracture.

1. a) Reasonably estimate the approximate carbon percentage of each bar and describe its expected equilibrium microstructure (identifying present phases and constituents).
2. b) Materials must be selected for: 1) deep drawing of a vehicle door; 2) a drive shaft subjected to torsion and fatigue; 3) a drill bit or file for woodworking. Assign and justify which bar you would use in each application based on its mechanical properties.
3. c) Predict what serious metallurgical difficulties will arise when attempting to join Bar C by fusion welding compared to Bar A, explaining the phenomenon that occurs in the Heat-Affected Zone (HAZ).

A materials engineering team must design the steel specification and heat treatment route for three mechanical components subjected to critical and opposing service requirements:

1) Deep drawing sheet metal for a vehicle fender (requires maximum cold formability and high energy absorption capacity upon impact).

2) A helical spring for a train suspension system (requires a very high yield strength and fatigue resistance without permanent deformation).

3) The industrial blade of a shear for cutting metal sections (requires extraordinary surface hardness and abrasive wear resistance without chipping).

1. a) Design the steel selection for each component: specify the recommended carbon range (% C) and justify the required microstructure prior to its forming or machining.
2. b) Design and justify the final heat treatment route for each part: indicate whether it requires treatment or not, the indicative temperatures with respect to the critical lines of the equilibrium diagram ($A_1$, $A_3$, or $A_{\text{cm}}$), the cooling medium, and subsequent stress relief or tempering treatments to optimize the hardness-toughness trade-off.
3. c) Predict the technical consequences if an operator mistakenly swaps the materials and manufactures the suspension springs using the fender steel, or manufactures the fender using the blade steel.

Unit challenge. A local company needs to select the suitable steel for the cable of a workshop hoist (working load of $300\text{ kg}$). Your mission is to issue a reasoned technical report linking its microstructure to its mechanical properties.

1. Data collection: Access the MatWeb materials database and look up the data sheets for two common carbon steels: AISI 1020 ($0,2\%$ C) and AISI 1080 ($0,8\%$ C). Record their yield strength ($\sigma_y$), tensile strength ($\sigma_u$), and elongation at break.

2. Phase analysis: Use the $Fe-C$ diagram to deduce the constituents present at room temperature in each steel (ferrite, pearlite, or cementite). Justify how the proportion of these phases and their crystal lattice influence the observed hardness and toughness.

3. Mechanical calculation: Using LibreOffice Calc, size the minimum cable diameter to support the load applying a safety factor $n = 2,5$. Graphically compare the elastic response of both materials.

4. Reasoned proposal: Write a brief report recommending the safest material against unforeseen brittle fractures.

Evaluation criteria:

• Rigorous deduction of phases and microstructures from the $Fe-C$ diagram.

• Technical correctness in tensile strength calculations and use of LibreOffice Calc.

• Coherent justification of the choice connecting internal structure, ductility, and service safety.