GATE Metallurgical Engineering Mechanical Metallurgy Syllabus

Write the force per unit length between two parallel edge dislocations (glide component) and state the sign convention.

Glide force: $F_x=\frac{Gb^{2}}{2\pi(1-\nu)}\frac{x(x^{2}-y^{2})}{(x^{2}+y^{2})^{2}}$. Like-sign edges repel when on the same side; they form a stable vertical wall (low-angle tilt boundary) when stacked one above another at $90^{\circ}$.

What is work (strain) hardening, and define the strain-hardening exponent in the Hollomon equation.

Work hardening is the increase in flow stress with plastic strain due to dislocation multiplication and interaction. Hollomon equation: $\sigma=K\varepsilon^{n}$, where $n$ is the strain-hardening exponent ($0\leq n\leq1$) and $K$ is the strength coefficient.

In a true stress-strain curve obeying $\sigma=K\varepsilon^{n}$, at what true strain does necking (the onset of plastic instability) begin?

Necking begins at the maximum-load (Considère) condition $\frac{d\sigma}{d\varepsilon}=\sigma$, which gives the true uniform strain $\varepsilon_u=n$ (the strain-hardening exponent).

What are the three stages of work hardening in single crystals?

Stage I: easy glide (low hardening, single slip system active). Stage II: linear hardening (high rate, multiple slip and dislocation forest interactions). Stage III: parabolic/dynamic recovery (rate decreases as cross-slip relieves stresses).

State the Hall-Petch relation for grain-boundary strengthening and identify each term.

$\sigma_y=\sigma_0+k_y\,d^{-1/2}$, where $\sigma_y$ is the yield strength, $\sigma_0$ the friction stress (lattice resistance), $k_y$ the Hall-Petch (locking) coefficient, and $d$ the average grain diameter.

Physically, why do grain boundaries strengthen a polycrystal (basis of Hall-Petch)?

Grain boundaries act as barriers to dislocation motion because slip planes are discontinuous across them and misorientation requires nucleating slip in the neighboring grain. Dislocations pile up at boundaries; finer grains mean shorter pile-ups, higher stress concentration needed to propagate slip, hence higher yield strength.

What is the inverse Hall-Petch effect?

At very small grain sizes (typically below about $10$–$20$ nm), the strength decreases with further grain refinement instead of increasing, because deformation shifts from dislocation-mediated slip to grain-boundary sliding/diffusion (Coble-type) accommodation.

How does solid-solution strengthening scale with solute concentration, and what factors control its magnitude?

Strengthening typically scales as $\Delta\tau\propto G\,\varepsilon^{3/2}c^{1/2}$ (often written $\Delta\sigma\propto c^{1/2}$ or $c^{2/3}$). It depends on the size misfit $\varepsilon_b=\frac{1}{b}\frac{db}{dc}$ and the modulus misfit $\varepsilon_G=\frac{1}{G}\frac{dG}{dc}$.

Compare substitutional and interstitial solid-solution strengthening and their lattice strain fields.

Substitutional solutes produce mainly symmetric (spherical) distortions and interact chiefly with edge dislocations. Interstitial solutes (e.g., C, N in BCC Fe) produce asymmetric (tetragonal) distortions that interact with both edge and screw dislocations, making them far more potent strengtheners per atom.

Distinguish the two mechanisms by which dislocations overcome precipitates: shearing vs. Orowan looping.

Shearing (cutting) occurs for small, coherent, soft precipitates; strength increases with particle size. Orowan looping (bypassing) occurs for large, hard, or incoherent precipitates; the dislocation bows between and leaves loops, and strength decreases with increasing particle spacing. Peak strength occurs at the crossover (critical) radius.

Write the Orowan equation for the increase in shear stress due to bypassing of impenetrable particles spaced $L$ apart.

$\Delta\tau=\frac{Gb}{L}$ (more completely $\Delta\tau\approx\frac{Gb}{L-2r}$, where $L$ is the inter-particle spacing and $r$ the particle radius). Strength is inversely proportional to particle spacing.

How does precipitation strengthening differ from dispersion strengthening in terms of particle origin and stability?

Precipitation (age) hardening uses coherent/semi-coherent particles formed by solid-state precipitation from a supersaturated solution; they are thermodynamically metastable and can over-age/coarsen and dissolve at high temperature. Dispersion strengthening uses incoherent, thermally stable, insoluble particles (e.g., oxides like $\ce{Y2O3}$, $\ce{ThO2}$) added externally, retaining strength at high temperature.

Sketch the sequence of GP zones and precipitates in age-hardening of Al-Cu, and where peak hardness occurs.

Sequence: supersaturated solid solution $\rightarrow$ GP zones $\rightarrow$ $\theta''$ (coherent) $\rightarrow$ $\theta'$ (semi-coherent) $\rightarrow$ $\theta$ ($\ce{CuAl2}$, incoherent equilibrium). Peak hardness corresponds to the $\theta''/\theta'$ stage; over-aging (coarse incoherent $\theta$) reduces hardness.

State Griffith's energy criterion for brittle fracture and the resulting critical fracture stress.

Crack propagates when the elastic strain energy released equals the surface energy created. Critical stress: $\sigma_f=\sqrt{\frac{2E\gamma_s}{\pi a}}$ (plane stress), where $\gamma_s$ is the surface energy and $2a$ the central crack length.

How did Orowan modify the Griffith criterion for crystalline (metallic) materials?

Orowan added the plastic work term $\gamma_p$ to the surface energy: $\sigma_f=\sqrt{\frac{2E(\gamma_s+\gamma_p)}{\pi a}}$. Since $\gamma_p\gg\gamma_s$ for metals, this accounts for the much higher fracture stress of metals than predicted by pure surface energy.

Define the stress intensity factor $K$ and write its general form for mode I.

$K$ characterizes the magnitude of the elastic crack-tip stress field. Mode I: $K_I=Y\sigma\sqrt{\pi a}$, where $Y$ is a dimensionless geometry factor, $\sigma$ the applied stress, and $a$ the crack length. The crack-tip stresses scale as $\sigma_{ij}=\frac{K_I}{\sqrt{2\pi r}}f_{ij}(\theta)$.

Name the three loading modes in fracture mechanics.

Mode I: opening (tensile, crack faces pulled apart normal to crack plane). Mode II: in-plane sliding (shear normal to crack front). Mode III: out-of-plane tearing (anti-plane shear parallel to crack front).

Relate the strain energy release rate $G$ to the stress intensity factor $K_I$.

$G=\frac{K_I^{2}}{E}$ (plane stress) or $G=\frac{K_I^{2}(1-\nu^{2})}{E}$ (plane strain). $G$ is the elastic energy released per unit area of crack advance; fracture occurs when $G\geq G_c$.

Define fracture toughness $K_{IC}$ and the condition for it to be a valid material property.

$K_{IC}$ is the critical mode-I stress intensity factor (plane-strain fracture toughness) at which a crack propagates unstably: $K_{IC}=Y\sigma_f\sqrt{\pi a}$. Validity (plane strain, small-scale yielding) requires thickness and crack length $\geq2.5\left(\frac{K_{IC}}{\sigma_y}\right)^{2}$.

Estimate the plastic zone size at a crack tip under plane stress and plane strain.

Plane stress: $r_p=\frac{1}{2\pi}\left(\frac{K_I}{\sigma_y}\right)^{2}$. Plane strain: $r_p=\frac{1}{6\pi}\left(\frac{K_I}{\sigma_y}\right)^{2}$ (about one-third the plane-stress value due to triaxial constraint).

How does specimen thickness affect fracture toughness and the fracture appearance?

Thin sections are in plane stress, giving higher (thickness-dependent) toughness with slant/shear (45 degree) fracture lips. As thickness increases, constraint rises toward plane strain, toughness falls to the minimum plateau value $K_{IC}$ and fracture becomes flat/square. $K_{IC}$ is the conservative, thickness-independent property.

In fractography, what surface features distinguish ductile, brittle cleavage, and fatigue fractures?

Ductile (microvoid coalescence): dimples (cup-and-cone macroscopically). Brittle transgranular cleavage: flat facets with river patterns and feathers; intergranular brittle fracture shows faceted grain boundaries. Fatigue: macroscopic beach marks and microscopic striations, each striation representing one load cycle.

What is the difference between beach marks and striations in fatigue fractography?

Beach marks (clamshell marks) are macroscopic features visible to the eye, formed by changes in loading or environment over many cycles (marking crack-front rest positions). Striations are microscopic ridges (seen under SEM/TEM), each typically corresponding to a single load cycle of crack advance.

What does the ductile-to-brittle transition temperature (DBTT) represent, and which crystal structures show it?

The DBTT is the temperature below which a material's fracture mode changes from ductile (high energy) to brittle (low energy), measured by a sharp drop in impact toughness (e.g., Charpy). BCC metals (e.g., ferritic steels) and HCP metals show a pronounced DBTT; FCC metals generally do not and remain ductile to very low temperatures.

List factors that raise the ductile-to-brittle transition temperature (make a steel more brittle).

Higher strain rate, triaxial stress state / notches (constraint), larger grain size, increased carbon content, interstitials (P, N), and irradiation embrittlement all raise the DBTT. Grain refinement, Mn, and Ni lower it (improve low-temperature toughness).

Why do BCC metals exhibit a DBTT while FCC metals usually do not?

In BCC metals the yield/flow stress is strongly temperature-dependent (high Peierls stress for screw dislocations); at low temperature yield stress exceeds the cleavage stress, so fracture occurs before yielding. FCC metals have low, temperature-insensitive Peierls stress and abundant slip systems, so they yield before cleaving at all temperatures.

Describe a hysteresis loop in cyclic stress-strain behaviour and what its area represents.

Under cyclic loading, plotting stress versus strain gives a closed hysteresis loop; its width is the plastic strain range $\Delta\varepsilon_p$ and its height the stress range $\Delta\sigma$. The enclosed area equals the plastic strain energy dissipated per cycle.

Distinguish cyclic hardening from cyclic softening and how to predict which occurs.

Cyclic hardening: stress amplitude rises with cycles (typical of initially soft/annealed metals). Cyclic softening: stress amplitude falls (typical of cold-worked/hard metals). Rule of thumb: if $\frac{\sigma_{UTS}}{\sigma_y}>1.4$ the material cyclically hardens; if $<1.2$ it cyclically softens.

State the Coffin-Manson and Basquin relations describing strain-life fatigue.

Coffin-Manson (plastic, low-cycle): $\frac{\Delta\varepsilon_p}{2}=\varepsilon_f'(2N_f)^{c}$. Basquin (elastic, high-cycle): $\frac{\Delta\varepsilon_e}{2}=\frac{\sigma_f'}{E}(2N_f)^{b}$. Total: $\frac{\Delta\varepsilon}{2}=\frac{\sigma_f'}{E}(2N_f)^{b}+\varepsilon_f'(2N_f)^{c}$.