Mechanical Properties, Stress-Strain, & 3D Stresses

What is strength?

Maximum force

What is ductility?

Ability to stretch or deform without breaking; Maximum strain before fracture

What is elasticity?

Ability to revert back to initial state (Modulus of Elasticity)

What is hardness?

Ability to resist indentation (Rockwell Hardness Testing)

What is toughness?

Total energy absorbed before fracture; Total area under stress-strain curve

What is resilience?

Energy absorbed before plastic deformation; Area under stress-strain curve before plastic deformation

What is Hooke’s Law?

the stress applied is directly proportional to the strain it produces

σ = E ⋅ ε

σ - stress
E - Modulus of Elasticity
ε - strain

What is the equation for true strain (ε)?

ε = ln ( L / L₀ ) = ln ( A₀ / A )

Use volume conservation to convert: A₀ * L₀ = A * L

What is Hollomon’s Equation?

It describes strain hardening in a material.

σ_t = K * εⁿ

K - strength coefficient
n - strain hardening exponent

What is the equation for engineering stress?

σ_e = Force / Area

What is the equation for engineering strain?

e = ( L - L₀ ) / L₀ = ( A₀ - A ) / A

Use volume conservation: A₀ * L₀ = A * L

What are the equations to convert from true stress / strain to engineering stress / strain?

ε = ln ( 1 + e )

σ_t = σ_e * ( 1 + e )

What differentiates engineering stress / strain with true stress / strain?

Engineering uses a constant cross-sectional area whereas true uses the instantaneous cross-sectional area.

Engineering is more for designing in the elastic region. True is more for manufacturing & making plastic deformations.

For strain hardening, how would you know if a material is perfectly plastic / elastic based on the strain hardening exponent?

Perfectly elastic @ n = 1; σ = K * ε = E * ε

Perfectly plastic @ n = 0; σ = K = σ_y

Why does ε = n at necking? Proof?

the material starts necking when the amount of stretching equals its ability to strengthen (strain hardening)

σ_t = K * εⁿ [Hollomon’s Equation]

dσ_t / dε = n * K * ε^n-1 [Derive w.r.t. strain]

K * εⁿ = n * K * ε^n-1 [Based on Considère’s criterion (dσ / dε = σ) states that necking begins when the rate of strain hardening equals the true stress]

ε = n [Divide both sides by K * ε^n-1]

What is annealing?

Heating a material above the recrystallization temperature & cooling down in a furnace to allow grains to reset to rearrange the structure.

What is normalizing?

Cooling in room temperature / open air after heating

What is hot working?

Shaping or deforming a metal at temperatures above its recrystallization point

What are the principal axes and stresses?

Principal axes - direction of principal stresses when shear is not present

Principal stresses - most extreme stresses (σ₁, σ₂, σ₃)

What is the difference between von Mises and Tresca yield criterion?

Tresca yield criterion says yielding begins when the maximum shear stress reaches a critical value.

Von Mises yield criterion says yielding begins when the distortion energy reaches a critical value.