Materials and Structures Data Book

Cambridge University Engineering Department

Materials and Structures Reference

This document consolidates content from the Materials Data Book and Structures Data Book, eliminating redundant elastic moduli definitions while preserving all material properties and structural analysis content.

Part I: Fundamental Definitions

1.1 Stress and Strain

Nominal stress: σn=FA0\sigma_n = \frac{F}{A_0}

True stress: σt=FA\sigma_t = \frac{F}{A}

Nominal strain: εn=λ−λ0λ0\varepsilon_n = \frac{\lambda - \lambda_0}{\lambda_0}

True strain: εt=ln⁡(λλ0)\varepsilon_t = \ln\left(\frac{\lambda}{\lambda_0}\right)

Poisson’s ratio: ν=−lateral strainlongitudinal strain\nu = -\frac{\text{lateral strain}}{\text{longitudinal strain}}

Young’s modulus: E=dσdεE = \frac{d\sigma}{d\varepsilon}

1.2 Elastic Moduli Relations (Isotropic Solids)

G=E2(1+ν),K=E3(1−2ν)G = \frac{E}{2(1+\nu)}, \qquad K = \frac{E}{3(1-2\nu)}

Approximate values for polycrystalline solids: ν≈13,G≈3E8,K≈E\nu \approx \frac{1}{3}, \quad G \approx \frac{3E}{8}, \quad K \approx E

1.3 Stress Notation

  • σxx\sigma_{xx}: normal stress on the xx-face in the xx-direction
  • τxy\tau_{xy}: shear stress on the xx-face in the yy-direction
  • Tension is positive (except in soil mechanics)

1.4 Strain Definition

εxx=∂u∂x,γxy=∂u∂y+∂v∂x\varepsilon_{xx} = \frac{\partial u}{\partial x}, \qquad \gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}

Part II: Physical Properties of Materials

2.1 Representative Values (Typical)

PropertySteelAl alloyConcreteSoftwoodUnits
Young’s modulus21070309GPa
Shear modulus812613-GPa
Bulk modulus1756914-GPa
Poisson’s ratio0.300.330.15--
Thermal expansion112312-10−610^{-6} K−1^{-1}
Density784027002400-kg m−3^{-3}

2.2 Unidirectional Composites

Parallel modulus: E∥=VfEf+(1−Vf)EmE_{\parallel} = V_f E_f + (1-V_f) E_m

Transverse modulus: 1E⊥=VfEf+1−VfEm\frac{1}{E_{\perp}} = \frac{V_f}{E_f} + \frac{1-V_f}{E_m}

Tensile strength (parallel): σts=Vfσf+(1−Vf)σy,m\sigma_{ts} = V_f \sigma_f + (1-V_f) \sigma_{y,m}

Part III: Constitutive Relations

3.1 Stress-Strain Relations (Isotropic Elastic Solids)

εxx=1E(σxx−νσyy−νσzz)+αΔT\varepsilon_{xx} = \frac{1}{E}\left(\sigma_{xx} - \nu\sigma_{yy} - \nu\sigma_{zz}\right) + \alpha\Delta T

γxy=τxyG\gamma_{xy} = \frac{\tau_{xy}}{G}

Plane stress (σzz=0\sigma_{zz} = 0, ΔT=0\Delta T = 0): σxx=E1−ν2(εxx+νεyy)\sigma_{xx} = \frac{E}{1-\nu^2}\left(\varepsilon_{xx} + \nu\varepsilon_{yy}\right)

3.2 Complementary Shear

τxy=τyx,γxy=γyx\tau_{xy} = \tau_{yx}, \qquad \gamma_{xy} = \gamma_{yx}

3.3 Planar Transformation of Stress

For rotation by angle θ\theta:

σaa=σxxcos⁡2θ+σyysin⁡2θ+2τxysin⁡θcos⁡θ\sigma_{aa} = \sigma_{xx}\cos^2\theta + \sigma_{yy}\sin^2\theta + 2\tau_{xy}\sin\theta\cos\theta

τab=(σyy−σxx)sin⁡θcos⁡θ+τxy(cos⁡2θ−sin⁡2θ)\tau_{ab} = (\sigma_{yy}-\sigma_{xx})\sin\theta\cos\theta + \tau_{xy}(\cos^2\theta - \sin^2\theta)

3.4 Principal Stresses (3D)

Principal stresses are eigenvalues of:

\sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{bmatrix}$$ --- ## Part IV: Yield and Fracture Criteria ### 4.1 Yield Criteria (Isotropic Solids) **Tresca:** $$\max\left(|\sigma_1-\sigma_2|, |\sigma_2-\sigma_3|, |\sigma_3-\sigma_1|\right) = Y$$ **von Mises:** $$(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2 = 2Y^2$$ ### 4.2 Dislocations and Plastic Flow **Force on a dislocation:** $$F = \tau b$$ **Critical shear stress:** $$\tau_y = \frac{cT}{bL}$$ **Relations:** $$k \approx \frac{3}{2}\tau_y, \qquad \sigma_y \approx 2k$$ **Hardness:** $$H \approx 3\sigma_y$$ ### 4.3 Fast Fracture **Stress intensity factor:** $$K = Y\sigma\sqrt{\pi a}$$ **Fracture condition:** $$K = K_{IC}$$ **Energy release rate:** $$G = \frac{K^2}{E}(1-\nu^2) \approx \frac{K^2}{E}$$ **Process zone size:** $$r_p = \frac{1}{2\pi}\left(\frac{K_{IC}}{\sigma_f}\right)^2$$ ### 4.4 Statistics of Fracture (Weibull) $$P_s(V) = \exp\left[-\left(\frac{\sigma}{\sigma_0}\right)^m \frac{V}{V_0}\right]$$ --- ## Part V: Fatigue and Creep ### 5.1 Fatigue **Basquin's law:** $$N_f = C_1 (\Delta\sigma)^{-\alpha}$$ **Coffin-Manson:** $$N_f = C_2 (\Delta\varepsilon_p)^{-\beta}$$ **Goodman relation:** $$\Delta\sigma = \Delta\sigma_0 \left(1 - \frac{\sigma_m}{\sigma_{ts}}\right)$$ **Paris law:** $$\frac{da}{dN} = A(\Delta K)^n$$ ### 5.2 Creep **Power-law creep:** $$\dot{\varepsilon}_{ss} = A\sigma^n \exp\left(-\frac{Q}{RT}\right)$$ --- ## Part VI: Diffusion and Heat Flow ### 6.1 Diffusion $$D = D_0 \exp\left(-\frac{Q}{RT}\right)$$ **Fick's laws:** $$J = -D\frac{dC}{dx}, \qquad \frac{\partial C}{\partial t} = D\frac{\partial^2 C}{\partial x^2}$$ ### 6.2 Heat Flow **Fourier's law:** $$q = -\lambda \frac{dT}{dx}$$ **Transient conduction:** $$\frac{\partial T}{\partial t} = a\frac{\partial^2 T}{\partial x^2}$$ --- ## Part VII: Structural Analysis ### 7.1 Thin-Walled Pressure Vessels (Closed Ends) $$\sigma_h = \frac{pr}{t}, \qquad \sigma_l = \frac{pr}{2t}$$ ### 7.2 Beam Behaviour **Sign convention:** - Positive bending: tension at bottom fibre - Positive shear: clockwise rotation **Compatibility:** $$\kappa = \frac{d\psi}{ds}, \qquad \varepsilon = \kappa y$$ For small deflections: $$\kappa \approx -\frac{d^2v}{dx^2}$$ **Equilibrium:** $$\frac{dM}{dx} = S, \qquad \frac{dS}{dx} = w$$ **Elastic bending:** $$\kappa = \frac{M}{EI}, \qquad \sigma = \frac{My}{I}, \qquad \sigma_{\max} = \frac{M}{Z_e}$$ ### 7.3 Plastic Bending **First yield:** $$M_y = Z_e \sigma_y$$ **Fully plastic:** $$M_p = Z_p \sigma_y$$ ### 7.4 Torsion **Circular shafts:** $$\tau = \frac{Tr}{J}, \qquad \phi = \frac{T}{GJ}$$ **Thin-walled closed sections:** $$q = \frac{T}{2A_e}, \qquad \phi = \frac{T}{4G}\oint\frac{ds}{tA_e}$$ ### 7.5 Euler Buckling $$P_E = \frac{\pi^2 EI}{L^2}$$ $$\sigma_E = \frac{P_E}{A} = \frac{\pi^2 E}{(L/k)^2}$$ ### 7.6 Pin-Jointed Trusses **Modified Maxwell rule:** $$s - m = b + r - Dj$$ ### 7.7 Virtual Work $$\sum W \cdot \delta = \sum P \cdot e$$ ### 7.8 Cables **Cable on curved surface:** $$\frac{T_1}{T_2} = e^{\mu\theta}$$ **Cable with uniform load (small sag):** $$T \approx \frac{wL^2}{2d}$$ --- ## Part VIII: Soil Mechanics ### 8.1 Definitions **Void ratio:** $$e = \frac{V_v}{V_s}$$ **Water content:** $$w = \frac{W_w}{W_s}$$ **Effective stress:** $$\sigma' = \sigma - u$$ ### 8.2 Particle Size Classification - Clay: < 0.002 mm - Silt: 0.002 - 0.06 mm - Sand: 0.06 - 2 mm ### 8.3 Groundwater Seepage $$v = ki$$ ### 8.4 Undrained Strength (Tresca) $$\tau_{\max} = c_u$$ $$\sigma_a = \sigma_v - 2c_u, \qquad \sigma_p = \sigma_v + 2c_u$$ ### 8.5 Drained Strength (Coulomb) $$K_a = \frac{1-\sin\phi'}{1+\sin\phi'}, \qquad K_p = \frac{1+\sin\phi'}{1-\sin\phi'}$$ --- ## Part IX: Reinforced Concrete Design **Design strengths:** $$f_{cd} = \alpha_{cc}\frac{f_{ck}}{1.5}, \qquad f_{yd} = \frac{f_{yk}}{1.15}$$ **Bending capacity (singly reinforced):** $$M = f_{yd} A_s \left(d - \frac{x}{2}\right)$$ **Steel types:** - High-yield steel: $f_{yk} = 500$ MPa - Mild steel: $f_{yk} = 250$ MPa --- ## Part X: Material Classification ### Metals - Ferrous and non-ferrous alloys - Applications: automotive, aerospace, nuclear, structural ### Polymers and Foams - Elastomers, thermoplastics, thermosets, foams ### Composites, Ceramics, Natural Materials - CFRP, GFRP, alumina, SiC, wood, bamboo --- ## Part XI: Phase Diagrams and Heat Treatment ### Binary Phase Diagrams Cu-Ni, Pb-Sn, Fe-C, Al-Cu, Al-Si, Cu-Zn, Cu-Sn, Ti-Al, SiO$_2$-Al$_2$O$_3$ ### Heat Treatment of Steels - TTT diagrams - Jominy end-quench hardenability curves --- ## Sources - Materials Data Book (2011 Edition, revised 2019), Cambridge University Engineering Department - Structures Data Book (2018 Edition), Cambridge University Engineering Department