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 $σ<em>y$ or tensile strength $σ</em>{ts}$) 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: $\epsilon = \frac{L-L<em>0}{L</em>0}$
• Shear strain: $\gamma = \frac{w}{L_0}$
• Volumetric strain: $\frac{V-V<em>0}{V</em>0}$

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

• $\epsilon<em>1 = \frac{\sigma</em>1}{E}$, $\epsilon<em>2 = \epsilon</em>3 = -\nu \epsilon<em>1 = -\nu \frac{\sigma</em>1}{E}$
• Constrained compression: $\epsilon<em>1 = \frac{\sigma</em>1}{E}(1-\nu^2)$
• $\frac{\sigma<em>1}{\epsilon</em>1} = \frac{E}{1-\nu^2}$

Elastic Energy

• Work per unit volume: $dW = \frac{FdL}{AL} = \sigma d\epsilon$.
• Elastic energy: $\int_0^{\sigma^<em>} \sigma d\epsilon = \frac{1}{2} \frac{(\sigma^</em>)^2}{E}$.

Stress-Free Strain

• Materials respond to magnetic fields, electrostatic fields, and thermal expansion ($\epsilon_T = \alpha \Delta T$).

Ductility

• $\%EL = \frac{L<em>f - L</em>0}{L_0} \times 100$
• $\%RA = \frac{A<em>0 - A</em>f}{A_0} \times 100$

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).
• $\hat{\rho} = f\rho<em>r + (1-f)\rho</em>m$
• $E<em>U = fE</em>r + (1-f)E_m$
• $E<em>L = \frac{E</em>m E<em>r}{fE</em>m + (1-f)E_r}$

Foams

• Cellular solids characterized by relative density.
• $\frac{\hat{\rho}}{\rho_s} = 3(\frac{t}{L})^2$
• $\frac{\hat{E}}{E<em>s} = (\frac{\hat{\rho}}{\rho</em>s})^2$