UNIVERSITY OF TEHRAN • DEPARTMENT OF MECHANICAL ENGINEERING

SCIENCE OF MATERIALS

Comprehensive 400-Problem Midterm Examination Study & Mastery Suite

Prepared with meticulous analytical rigor by Teaching Assistant Sina Sharifi
Under the Academic Supervision of Prof. A. Sadeghi
Covering the Complete Undergraduate Curriculum from Chapter 1 through Chapter 8 (up to Phase Diagrams)

Explore All 8 Chapters (400 Problems)

Curricular Chapters

400

Sample Problems

Problems Per Chapter

100%

Step-by-Step Solutions

16+

Technical SVG Diagrams

How to Use This Problem Suite for Maximum Midterm Exam Success

1. Self-Testing Active Recall: Every problem has its full derivation and boxed answer hidden by default. Always attempt to solve the question independently on paper before clicking "Show Detailed Solution".

2. Categorized Learning: Each chapter is organized into distinct categories (Atomic Structure, Miller Indices, Fick's Laws, Schmid's Law, Hall-Petch, LEFM, Creep). Use the category filters to target specific weak areas.

3. Progress Tracking & Bookmarking: Click "Mark Mastered" on any problem you have conquered. Your browser will locally save your mastery tally in the header badge.

4. Formula Sheet Modal: Click the "Formula Sheet" button in the header at any time to consult the master compendium of governing equations, unit definitions, and physical constants.

Midterm Syllabus Directory

The Eight Curricular Chapters

Chapter 01 50 Problems • 100% Solutions

Introduction to Materials Science & Engineering

The Materials Science and Engineering paradigm, the central Processing-Structure-Properties-Performance tetrahedron, broad classification of materials (metals, ceramics, polymers, composites, advanced materials), and Ashby property charts for engineering design.

Materials ParadigmMSE TetrahedronMaterial ClassesAshby ChartsAdvanced Materials

Study Chapter 1 Problems

Chapter 02 50 Problems • 100% Solutions

Atomic Structure & Interatomic Bonding

Fundamental atomic physics, electronic configurations, quantum numbers, primary bonding (ionic, covalent, metallic), secondary bonding (van der Waals, hydrogen), and physical correlates of the net interatomic potential energy well.

Electronic ConfigurationPotential Energy WellIonic & Covalent BondingMetallic BondThermal Expansion

Study Chapter 2 Problems

Chapter 03 50 Problems • 100% Solutions

The Structure of Crystalline Solids

Lattice geometry, 7 crystal systems, 14 Bravais lattices, metallic crystal structures (FCC, BCC, HCP), APF and theoretical density, Miller indices for directions and planes, linear and planar densities, and X-ray diffraction (XRD / Bragg's Law).

FCC & BCC & HCPTheoretical DensityMiller IndicesPlanar & Linear DensityXRD & Bragg's Law

Study Chapter 3 Problems

Chapter 04 50 Problems • 100% Solutions

Imperfections in Solids

Point defects (vacancies, self-interstitials, impurities), Hume-Rothery solubility rules, composition conversions (wt% vs at%), ceramic defect chemistry (Schottky and Frenkel pairs), line defects (edge and screw dislocations, Burgers vectors), interfacial defects, and ASTM grain size.

Vacancies & ArrheniusHume-Rothery RulesComposition ConversionsDislocations & Burgers VectorASTM Grain Size

Study Chapter 4 Problems

Chapter 05 50 Problems • 100% Solutions

Diffusion in Solids

Atomic migration mechanisms (vacancy vs interstitial), steady-state diffusion and Fick's first law, non-steady-state diffusion and Fick's second law error function solutions, Arrhenius temperature dependence of diffusivity, carburization depth calculations, and structural factors.

Fick's 1st LawFick's 2nd LawError Function erf(z)Arrhenius DiffusivityIndustrial Carburization

Study Chapter 5 Problems

Chapter 06 50 Problems • 100% Solutions

Mechanical Properties of Metals

Engineering stress and strain, Hooke's law, elastic moduli and Poisson's ratio, yielding and the 0.2% offset method, tensile strength, ductility, resilience, toughness, true stress and true strain, Hollomon strain hardening law, Schmid's law for single crystals, and hardness testing.

Hooke's Law & Moduli0.2% Offset YieldingTrue Stress-StrainHollomon RelationSchmid's LawHardness

Study Chapter 6 Problems

Chapter 07 50 Problems • 100% Solutions

Dislocations & Strengthening Mechanisms

Principles of plastic deformation, critical shear stresses, the four classical strengthening mechanisms (Hall-Petch grain size refinement, solid solution strengthening, strain hardening/cold work, precipitation hardening/Orowan bowing), and the three stages of annealing (recovery, recrystallization, grain growth).

Hall-Petch EquationSolid Solution StrengtheningCold Work & Taylor LawOrowan PrecipitationAnnealing Kinetics

Study Chapter 7 Problems

Chapter 08 50 Problems • 100% Solutions

Failure: Fracture, Fatigue & Creep

Mechanisms of ductile vs brittle fracture, Griffith criterion, stress concentration factor, Mode I stress intensity and fracture toughness (K_Ic), Charpy impact testing and the ductile-to-brittle transition (DBTT), cyclic fatigue (S-N curves, fatigue limits, mean stress effects), and high-temperature creep power laws.

Fracture Toughness K_IcGriffith CriterionCharpy Impact & DBTTFatigue S-N & LimitsLarson-Miller Creep

Study Chapter 8 Problems

Domain / Law	Governing Mathematical Equation	Key Variables & Units
Theoretical Crystal Density	$$\rho = \frac{n \cdot A}{V_c \cdot N_A}$$	\(n\): atoms/cell; \(A\): g/mol; \(V_c\): cell volume (\(\text{cm}^3\)); \(\rho\): \(\text{g/cm}^3\)
Lattice Constants & APF	BCC: \(a = \frac{4R}{\sqrt{3}}\), \(\text{APF} = 0.68\), \(n=2\) FCC: \(a = 2\sqrt{2}R\), \(\text{APF} = 0.74\), \(n=4\)	\(R\): atomic radius; \(a\): lattice parameter; \(\text{APF} = \frac{n(4/3\pi R^3)}{V_c}\)
Cubic Interplanar Spacing	$$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$$	\((hkl)\): Miller indices; \(a\): unit cell lattice parameter
Bragg's Law (XRD)	$$n\lambda = 2 d_{hkl} \sin\theta$$	\(\lambda\): X-ray wavelength; \(\theta\): Bragg diffraction angle; \(n\): order (usually 1)
Equilibrium Vacancies	$$N_v = N \exp\left(-\frac{Q_v}{kT}\right)$$	\(N\): total atomic sites; \(Q_v\): vacancy activation energy (eV or J/mol); \(T\): Kelvin
ASTM Grain Size Number	$$N = 2^{n - 1}$$	\(N\): grains per square inch at 100\(\times\); \(n\): ASTM grain size number
Fick's First Law (Steady-State)	$$J = -D \frac{dC}{dx} = \frac{M}{A \cdot t}$$	\(J\): flux (\(\text{kg/m}^2\cdot\text{s}\)); \(D\): diffusivity (\(\text{m}^2/\text{s}\)); \(dC/dx\): gradient
Fick's Second Law (Semi-Infinite)	$$\frac{C_x - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$$	\(C_s\): surface conc; \(C_0\): initial base conc; \(C_x\): conc at depth \(x\) after time \(t\)
Arrhenius Temperature Law	$$D = D_0 \exp\left(-\frac{Q_d}{RT}\right)$$	\(D_0\): pre-exponential (\(\text{m}^2/\text{s}\)); \(Q_d\): activation energy (\(\text{J/mol}\)); \(R=8.314\)
Hooke's Law & Shear Modulus	$$\sigma = E\varepsilon, \quad \tau = G\gamma, \quad G = \frac{E}{2(1 + \nu)}$$	\(E\): Young's modulus; \(G\): Shear modulus; \(\nu\): Poisson's ratio (\(\sim 0.3\))
True Stress & True Strain	$$\sigma_T = \sigma(1+\varepsilon), \quad \varepsilon_T = \ln(1+\varepsilon)$$	Valid up to necking; \(\sigma, \varepsilon\): engineering stress/strain
Hollomon Power Law	$$\sigma_T = K \varepsilon_T^n$$	\(K\): strength coefficient; \(n\): strain hardening exponent; Necking: \(\varepsilon_T = n\)
Modulus of Resilience	$$U_r = \int_0^{\varepsilon_y} \sigma d\varepsilon \approx \frac{\sigma_y^2}{2E}$$	\(U_r\): elastic energy stored per unit volume (\(\text{J/m}^3\) or \(\text{Pa}\))
Schmid's Law (Single Crystals)	$$\tau_R = \sigma \cos\phi \cos\lambda \le \tau_{\text{CRSS}}$$	\(\phi\): angle to slip plane normal; \(\lambda\): angle to slip direction; \(m = \cos\phi\cos\lambda\)
Hall-Petch Strengthening	$$\sigma_y = \sigma_0 + k_y d^{-1/2}$$	\(\sigma_0\): friction stress; \(k_y\): locking coefficient (\(\text{MPa}\cdot\text{mm}^{1/2}\)); \(d\): grain size
Cold Work & Dislocation Density	$$\%\text{CW} = \frac{A_0 - A_d}{A_0} \times 100\%, \quad \tau = \tau_0 + \alpha G b \sqrt{\rho}$$	\(\rho\): dislocation density (\(\text{m}^{-2}\)); \(b\): Burgers vector; \(A_0, A_d\): cross-sections
Orowan Looping (Precipitation)	$$\Delta\tau = \frac{G b}{L - 2r} \approx \frac{G b}{L}$$	\(L\): inter-particle spacing; \(r\): particle radius; \(G\): matrix shear modulus
Stress Concentration (Crack Tip)	$$K_t = \frac{\sigma_m}{\sigma_0} = 2 \sqrt{\frac{a}{\rho_t}}$$	\(a\): crack length; \(\rho_t\): crack tip radius of curvature; \(\sigma_0\): nominal stress
Griffith Critical Fracture Stress	$$\sigma_c = \sqrt{\frac{2E\gamma_s}{\pi a}}$$	\(\gamma_s\): specific surface energy (\(\text{J/m}^2\)); \(E\): Young's modulus; \(a\): crack size
Mode I Fracture Toughness	$$K_I = Y \sigma \sqrt{\pi a} \le K_{Ic}$$	\(K_{Ic}\): plane strain fracture toughness (\(\text{MPa}\cdot\sqrt{\text{m}}\)); \(Y\): geometric factor
Fatigue Stress Parameters	$$\sigma_m = \frac{\sigma_{\max}+\sigma_{\min}}{2}, \quad \sigma_a = \frac{\sigma_{\max}-\sigma_{\min}}{2}, \quad R = \frac{\sigma_{\min}}{\sigma_{\max}}$$	\(\sigma_m\): mean stress; \(\sigma_a\): amplitude; \(R\): stress ratio
Steady-State Creep Rate	$$\dot{\varepsilon}_s = K_2 \sigma^n \exp\left(-\frac{Q_c}{RT}\right)$$	\(\dot{\varepsilon}_s\): steady creep rate (\(\text{s}^{-1}\)); \(n\): stress exponent; \(Q_c\): creep activation energy
Larson-Miller Parameter	$$P = T \left(20 + \log_{10} t_r\right)$$	\(T\): absolute temperature in Kelvin; \(t_r\): rupture life in hours

Domain / Law

Governing Mathematical Equation

Key Variables & Units

Theoretical Crystal Density

$$\rho = \frac{n \cdot A}{V_c \cdot N_A}$$

$n$: atoms/cell; $A$: g/mol; $V_c$: cell volume ($\text{cm}^3$); $\rho$: $\text{g/cm}^3$

Lattice Constants & APF

BCC: $a = \frac{4R}{\sqrt{3}}$, $\text{APF} = 0.68$, $n=2$
FCC: $a = 2\sqrt{2}R$, $\text{APF} = 0.74$, $n=4$

$R$: atomic radius; $a$: lattice parameter; $\text{APF} = \frac{n(4/3\pi R^3)}{V_c}$

Cubic Interplanar Spacing

$$d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$$

$(hkl)$: Miller indices; $a$: unit cell lattice parameter

Bragg's Law (XRD)

$$n\lambda = 2 d_{hkl} \sin\theta$$

$\lambda$: X-ray wavelength; $\theta$: Bragg diffraction angle; $n$: order (usually 1)

Equilibrium Vacancies

$$N_v = N \exp\left(-\frac{Q_v}{kT}\right)$$

$N$: total atomic sites; $Q_v$: vacancy activation energy (eV or J/mol); $T$: Kelvin

ASTM Grain Size Number

$$N = 2^{n - 1}$$

$N$: grains per square inch at 100$\times$; $n$: ASTM grain size number

Fick's First Law (Steady-State)

$$J = -D \frac{dC}{dx} = \frac{M}{A \cdot t}$$

$J$: flux ($\text{kg/m}^2\cdot\text{s}$); $D$: diffusivity ($\text{m}^2/\text{s}$); $dC/dx$: gradient

Fick's Second Law (Semi-Infinite)

$$\frac{C_x - C_0}{C_s - C_0} = 1 - \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$$

$C_s$: surface conc; $C_0$: initial base conc; $C_x$: conc at depth $x$ after time $t$

Arrhenius Temperature Law

$$D = D_0 \exp\left(-\frac{Q_d}{RT}\right)$$

$D_0$: pre-exponential ($\text{m}^2/\text{s}$); $Q_d$: activation energy ($\text{J/mol}$); $R=8.314$

Hooke's Law & Shear Modulus

$$\sigma = E\varepsilon, \quad \tau = G\gamma, \quad G = \frac{E}{2(1 + \nu)}$$

$E$: Young's modulus; $G$: Shear modulus; $\nu$: Poisson's ratio ($\sim 0.3$)

True Stress & True Strain

$$\sigma_T = \sigma(1+\varepsilon), \quad \varepsilon_T = \ln(1+\varepsilon)$$

Valid up to necking; $\sigma, \varepsilon$: engineering stress/strain

Hollomon Power Law

$$\sigma_T = K \varepsilon_T^n$$

$K$: strength coefficient; $n$: strain hardening exponent; Necking: $\varepsilon_T = n$

Modulus of Resilience

$$U_r = \int_0^{\varepsilon_y} \sigma d\varepsilon \approx \frac{\sigma_y^2}{2E}$$

$U_r$: elastic energy stored per unit volume ($\text{J/m}^3$ or $\text{Pa}$)

Schmid's Law (Single Crystals)

$$\tau_R = \sigma \cos\phi \cos\lambda \le \tau_{\text{CRSS}}$$

$\phi$: angle to slip plane normal; $\lambda$: angle to slip direction; $m = \cos\phi\cos\lambda$

Hall-Petch Strengthening

$$\sigma_y = \sigma_0 + k_y d^{-1/2}$$

$\sigma_0$: friction stress; $k_y$: locking coefficient ($\text{MPa}\cdot\text{mm}^{1/2}$); $d$: grain size

Cold Work & Dislocation Density

$$\%\text{CW} = \frac{A_0 - A_d}{A_0} \times 100\%, \quad \tau = \tau_0 + \alpha G b \sqrt{\rho}$$

$\rho$: dislocation density ($\text{m}^{-2}$); $b$: Burgers vector; $A_0, A_d$: cross-sections

Orowan Looping (Precipitation)

$$\Delta\tau = \frac{G b}{L - 2r} \approx \frac{G b}{L}$$

$L$: inter-particle spacing; $r$: particle radius; $G$: matrix shear modulus

Stress Concentration (Crack Tip)

$$K_t = \frac{\sigma_m}{\sigma_0} = 2 \sqrt{\frac{a}{\rho_t}}$$

$a$: crack length; $\rho_t$: crack tip radius of curvature; $\sigma_0$: nominal stress

Griffith Critical Fracture Stress

$$\sigma_c = \sqrt{\frac{2E\gamma_s}{\pi a}}$$

$\gamma_s$: specific surface energy ($\text{J/m}^2$); $E$: Young's modulus; $a$: crack size

Mode I Fracture Toughness

$$K_I = Y \sigma \sqrt{\pi a} \le K_{Ic}$$

$K_{Ic}$: plane strain fracture toughness ($\text{MPa}\cdot\sqrt{\text{m}}$); $Y$: geometric factor

Fatigue Stress Parameters

$$\sigma_m = \frac{\sigma_{\max}+\sigma_{\min}}{2}, \quad \sigma_a = \frac{\sigma_{\max}-\sigma_{\min}}{2}, \quad R = \frac{\sigma_{\min}}{\sigma_{\max}}$$

$\sigma_m$: mean stress; $\sigma_a$: amplitude; $R$: stress ratio

Steady-State Creep Rate

$$\dot{\varepsilon}_s = K_2 \sigma^n \exp\left(-\frac{Q_c}{RT}\right)$$

$\dot{\varepsilon}_s$: steady creep rate ($\text{s}^{-1}$); $n$: stress exponent; $Q_c$: creep activation energy

Larson-Miller Parameter

$$P = T \left(20 + \log_{10} t_r\right)$$

$T$: absolute temperature in Kelvin; $t_r$: rupture life in hours

SCIENCE OF MATERIALS

How to Use This Problem Suite for Maximum Midterm Exam Success

The Eight Curricular Chapters

Master Formula Compendium & Physical Constants (Ch 1 – 8)