GATE Metallurgical Engineering Mechanical Metallurgy Syllabus

Mechanical Metallurgy syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mechanical Metallurgy in GATE Metallurgical Engineering, not a summary of it.

1. 1

   Strain Tensor and Stress Tensor

   2 topics

   • Representation by Mohr’s Circle
   • Elasticity, Stiffness, and Compliance Tensor
2. 2

   Dislocation Theory

   5 topics

   • Edge, Screw, and Mixed Dislocations
   • Source and Multiplication of Dislocations
   • Stress Fields Around Dislocations
   • Partial Dislocations
   • Dislocation Interactions and Reactions
3. 3

   Strengthening Mechanisms

   5 topics

   • Work/Strain Hardening
   • Strengthening due to Grain Boundaries
   • Strengthening due to Solid Solution
   • Strengthening due to Precipitation
   • Strengthening due to Dispersion
4. 4

   Fracture Behaviour

   5 topics

   • Griffith Theory
   • Linear Elastic Fracture Mechanics
   • Fracture Toughness
   • Fractography
   • Ductile to Brittle Transition
5. 5

   Fatigue

   2 topics

   • Cyclic Stress Strain Behaviour
     • Low Cycle Fatigue
     • High Cycle Fatigue
   • Crack Growth
6. 6

   Mechanisms of High Temperature Deformation and Failure

   2 topics

   • Creep and Stress Rupture
   • Stress Exponent and Activation Energy

Mechanical Metallurgy flashcards for GATE Metallurgical Engineering

23 of 49 cards from the Mechanical Metallurgy deck — real questions with worked answers.

1. What does a Mohr's circle graphically represent, and what are the coordinates of its center and radius for a 2D stress state with normal stresses $\sigma_x,\sigma_y$ and shear stress $\tau_{xy}$?

   Mohr's circle is a graphical representation of the transformation of stress (or strain) on differently oriented planes. Center: $\left(\frac{\sigma_x+\sigma_y}{2},\,0\right)$; Radius: $R=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}}$.

2. Using Mohr's circle, give the formulas for the principal stresses and the maximum in-plane shear stress.

   Principal stresses: $\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}}$. Maximum in-plane shear stress: $\tau_{max}=R=\frac{\sigma_1-\sigma_2}{2}$.

3. On a Mohr's circle, how does the angle between physical planes relate to the angle subtended at the circle's center?

   Angles on Mohr's circle are doubled: a rotation of $\theta$ between physical planes corresponds to an arc of $2\theta$ on the circle. Principal planes (where $\tau=0$) are $90^{\circ}$ apart physically but $180^{\circ}$ apart on the circle.

4. State Hooke's law in generalized tensor form and define the stiffness and compliance tensors.

   $\sigma_{ij}=C_{ijkl}\,\varepsilon_{kl}$ and $\varepsilon_{ij}=S_{ijkl}\,\sigma_{kl}$. $C_{ijkl}$ is the fourth-rank elastic stiffness tensor and $S_{ijkl}$ is the compliance tensor; they are inverses of one another ($C=S^{-1}$).

5. How many independent elastic constants does the stiffness tensor have in the general anisotropic case, and how many for a cubic and an isotropic material?

   General anisotropic (triclinic): 21 independent constants. Cubic: 3 ($C_{11},C_{12},C_{44}$). Isotropic: 2 (e.g., $E$ and $\nu$, or the two Lamé constants $\lambda$ and $\mu$).

6. For an isotropic material, relate the shear modulus $G$, Young's modulus $E$, bulk modulus $K$ and Poisson's ratio $\nu$.

   $G=\frac{E}{2(1+\nu)}$ and $K=\frac{E}{3(1-2\nu)}$. The Zener anisotropy ratio for cubic crystals is $A=\frac{2C_{44}}{C_{11}-C_{12}}$, equal to 1 for an isotropic material.

7. Define the Burgers vector orientation relative to the dislocation line for edge and screw dislocations.

   Edge dislocation: Burgers vector $\vec{b}$ is perpendicular to the dislocation line ($\vec{b}\perp\vec{\xi}$). Screw dislocation: $\vec{b}$ is parallel to the line ($\vec{b}\parallel\vec{\xi}$). A mixed dislocation has $\vec{b}$ at an arbitrary angle and contains both components.

8. How do edge and screw dislocations differ in their freedom to glide and cross-slip?

   An edge dislocation has a uniquely defined slip plane (containing both $\vec{b}$ and $\vec{\xi}$) and can only climb out of it (non-conservative). A screw dislocation has no unique slip plane, so it can cross-slip onto any plane containing $\vec{b}$.

9. What is the magnitude of the Burgers vector for a perfect dislocation in an FCC crystal of lattice parameter $a$?

   Perfect FCC dislocation: $\vec{b}=\frac{a}{2}\langle110\rangle$, with magnitude $|\vec{b}|=\frac{a}{2}\sqrt{2}=\frac{a}{\sqrt{2}}$.

10. Describe how a Frank-Read source multiplies dislocations and give its critical activation stress.

    A dislocation segment pinned at two points of length $L$ bows out under shear stress; when it reaches a semicircle it becomes unstable, expands, and pinches off a loop while regenerating the segment, repeating to multiply dislocations. Critical stress: $\tau_c=\frac{Gb}{L}$.

11. Define the dislocation density and give the Taylor (forest) hardening relation between flow stress and dislocation density.

    Dislocation density $\rho$ = total dislocation line length per unit volume (units $\text{m}^{-2}$). Taylor relation: $\tau=\tau_0+\alpha G b\sqrt{\rho}$, where $\alpha\approx0.3$–$0.5$.

12. Give the stress field of a screw dislocation along the $z$-axis.

    A screw dislocation has only shear components: $\tau_{\theta z}=\frac{Gb}{2\pi r}$ (in cylindrical coordinates), or $\tau_{xz}=-\frac{Gb}{2\pi}\frac{y}{x^{2}+y^{2}}$, $\tau_{yz}=\frac{Gb}{2\pi}\frac{x}{x^{2}+y^{2}}$. The stress decays as $1/r$ and is axisymmetric.

13. Write the elastic strain energy per unit length stored in a dislocation and explain the cut-off radii.

    $E\approx\frac{Gb^{2}}{4\pi}\ln\!\left(\frac{R}{r_0}\right)$ for a screw (and $\frac{Gb^{2}}{4\pi(1-\nu)}\ln\frac{R}{r_0}$ for an edge). Often simplified to $E\approx\tfrac{1}{2}Gb^{2}$. $r_0$ is the core radius (continuum elasticity invalid inside) and $R$ is the outer cut-off (crystal size or spacing).

14. State the line-tension approximation for a dislocation and the stress to bend it to radius $R$.

    Line tension $T\approx\frac{1}{2}Gb^{2}$ (energy per unit length). The shear stress needed to curve a dislocation to radius of curvature $R$ is $\tau=\frac{T}{bR}=\frac{Gb}{2R}$.

15. What is a partial dislocation, and write the dissociation reaction of a perfect FCC dislocation into Shockley partials.

    A partial dislocation has a Burgers vector that is not a full lattice translation, bounding a stacking fault. FCC dissociation: $\frac{a}{2}[10\bar{1}]\rightarrow\frac{a}{6}[11\bar{2}]+\frac{a}{6}[2\bar{1}\bar{1}]$.

16. Why does a perfect dislocation dissociate into partials, and what sets the equilibrium separation distance?

    It dissociates because the energy criterion $b_1^{2}>b_2^{2}+b_3^{2}$ (Frank's rule: $\frac{a^{2}}{2}>\frac{a^{2}}{6}+\frac{a^{2}}{6}$) is satisfied, lowering energy. Equilibrium separation $d$ is set by the balance of the partials' elastic repulsion against the stacking-fault energy $\gamma$: $d\approx\frac{G b_p^{2}}{2\pi\gamma}$.

17. How does stacking fault energy (SFE) affect the separation of partials and the ease of cross-slip?

    Low SFE gives wide separation of partials (wide stacking-fault ribbons), making cross-slip and climb difficult and favoring planar slip and twinning. High SFE gives narrow/constricted partials, easy cross-slip, and wavy slip.

18. State Frank's rule for whether a dislocation reaction is energetically favorable.

    A reaction $\vec{b}_1\rightarrow\vec{b}_2+\vec{b}_3$ is favorable if it lowers the total energy, i.e. $|\vec{b}_1|^{2}>|\vec{b}_2|^{2}+|\vec{b}_3|^{2}$ (since energy $\propto b^{2}$).

19. What is a Lomer-Cottrell lock and why does it impede slip?

    A Lomer-Cottrell lock forms when two dislocations on intersecting {111} planes react to give a sessile (stair-rod) partial $\frac{a}{6}\langle110\rangle$ lying along the line of intersection. Because its Burgers vector lies in neither slip plane, it cannot glide, acting as a barrier that contributes to work hardening.

20. Give the Peach-Koehler equation for the force per unit length on a dislocation.

    $\vec{F}=(\boldsymbol{\sigma}\cdot\vec{b})\times\vec{\xi}$, where $\boldsymbol{\sigma}$ is the stress tensor, $\vec{b}$ the Burgers vector, and $\vec{\xi}$ the unit line vector. For a simple glide force: $F=\tau b$ per unit length.

21. 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 $90^{\circ}$.

22. 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.

23. 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).

See more Mechanical Metallurgy flashcards →

Planning Mechanical Metallurgy for GATE Metallurgical Engineering

Mechanical Metallurgy (GATE Metallurgical Engineering) FAQ