Material Engineering - Characteristics & Manufacture
Introduction to Material Engineering
Learning Outcomes
- Define engineering stress and engineering strain.
- State Hooke’s law.
- Define Poisson’s ratio.
- Determine modulus of elasticity, yield strength, and estimate percent elongation from a stress-strain diagram.
- Understand manipulation of density and modulus of hybrid materials.
Key Concepts
- Stress causes strain.
- Resilience in humans is coping with stress without strain; in materials, it's elastic modulus.
- Stress is applied load; strain is the material's shape change response.
- Strain depends on material, stress magnitude, and loading mode.
Stiffness and Strength
- Stiffness: Resistance to elastic shape change (returns to original shape when stress is removed).
- Strength: Resistance to permanent distortion or failure.
- Stress and strain are stimulus and response, not material properties.
- Stiffness (elastic modulus E) and strength (elastic limit σy or tensile strength σ{ts}) are material properties.
- Density \rho and modulus are microstructure-insensitive; strength and toughness are microstructure-sensitive.
Density
- Density is mass per unit volume, measured in \frac{kg}{m^3}.
- Double weighing is the best measure of density.
Modes of Loading
- Axial tension.
- Compression.
- Axial tension/compression.
- Torsion.
- Bi-axial tension or compression.
Stress
- \sigma = \frac{F}{A}
- Tensile stress: Force applied normal to surface (positive F = tension, negative F = compression).
- Shear stress: Force applied parallel to surface.
- Pressure: Equally applied tensile and compressive forces.
- Units: MPa (Mega Pascal, 10^6 N/m^2).
Strain
- Strain is dimensionless (ratio of two lengths).
- Tensile stress lengthens (tensile strain +).
- Compressive stress shortens (compressive strain -).
- Tensile strain: \epsilon = \frac{L-L0}{L0}
- Shear strain: \gamma = \frac{w}{L_0}
- Volumetric strain: \frac{V-V0}{V0}
Tensile Test
- Typical stress-strain behavior shows elastic region, yield strength \sigma_y, tensile strength TS, and fracture point F.
- Necking acts as stress concentrator.
Stress-Strain Curves
- Initial linear part is Hooke’s law (elastic), strain is recoverable.
- Stresses above elastic limit cause permanent deformation (ductile) or brittle fracture.
- E is Young’s modulus in \sigma = E \epsilon.
- G is Shear modulus in \tau = G \gamma.
- K is Bulk modulus in p = K \Delta.
- Units for moduli: GPa (10^9 N/m^2).
Poisson’s Ratio
- \nu = - \frac{\epsilon_t}{\epsilon}
- Typically ~1/3.
- For isotropic material: G = \frac{E}{2(1+\nu)} and K = \frac{E}{3(1-2\nu)}.
- For elastomers, \nu \approx 1/2 when G = \frac{1}{3}E and K >> E.
Hooke’s Law in 3D
- \epsilon1 = \frac{\sigma1}{E}, \epsilon2 = \epsilon3 = -\nu \epsilon1 = -\nu \frac{\sigma1}{E}
- Constrained compression: \epsilon1 = \frac{\sigma1}{E}(1-\nu^2)
- \frac{\sigma1}{\epsilon1} = \frac{E}{1-\nu^2}
Elastic Energy
- Work per unit volume: dW = \frac{FdL}{AL} = \sigma d\epsilon.
- Elastic energy: \int_0^{\sigma^} \sigma d\epsilon = \frac{1}{2} \frac{(\sigma^)^2}{E}.
Stress-Free Strain
- Materials respond to magnetic fields, electrostatic fields, and thermal expansion (\epsilon_T = \alpha \Delta T).
Ductility
- \%EL = \frac{Lf - L0}{L_0} \times 100
- \%RA = \frac{A0 - Af}{A_0} \times 100
Toughness
- Energy to break a unit volume of material.
- Approximated by area under stress-strain curve.
Anisotropy
- Isotropic: Properties are the same in all directions.
- Anisotropic: Properties depend on direction.
Manipulating Modulus and Density
- Hybrid materials: Mixtures of solids for internal structure.
- Composites: Fibers or particles in a matrix (polymer, metal, or ceramic).
- \hat{\rho} = f\rhor + (1-f)\rhom
- EU = fEr + (1-f)E_m
- EL = \frac{Em Er}{fEm + (1-f)E_r}
Foams
- Cellular solids characterized by relative density.
- \frac{\hat{\rho}}{\rho_s} = 3(\frac{t}{L})^2
- \frac{\hat{E}}{Es} = (\frac{\hat{\rho}}{\rhos})^2