Material Mechanics: Stress and Strain

Stress

• Stress is the internal force per unit area within a material due to externally applied forces.
• Measured in Pascals (Pa).
• Formula: σ = F/A

Strain

• Strain is the measure of deformation representing elongation or compression.
• It is dimensionless.
• Formula: € = ΔL/L

Young's Modulus

• Young's Modulus (E) is a material's stiffness, representing the ratio of stress to strain in the elastic region.
• Units are Pascals (Pa).
• Formula: E = σ/ε

Shear Stress

• Shear stress is the force per unit area that causes layers of a material to slide relative to each other.
• Measured in Pascals (Pa).
• Formula: τ = F/A

Shear Modulus

• Shear modulus (G) measures a material's resistance to shearing deformation.
• Related to Young's modulus E and Poisson's ratio ν.
• Units are Pascals (Pa).
• Formula: G = τ/γ or G = E / 2(1 + ν)

Poisson's Ratio

• Poisson's Ratio (ν) is the ratio of lateral strain to axial strain under uniaxial stress.
• It is dimensionless.
• Formula: ν = εlateral / εaxial

Bending Stress

• Bending stress is the stress induced in a material when it is subjected to a bending moment (M).
• Formula: σ = M⋅c/I
  • c: distance from the neutral axis
  • I: moment of inertia

Moment of Inertia

• Moment of inertia (I) is a geometric property indicating a cross-section's resistance to bending and deflection.
• Units are m4.
• Formula:
  • For a rectangle: I = (b⋅h3)/12
  • For a circle: I = π⋅d4/64

Section Modulus

• Section modulus (S) is the ratio of the moment of inertia to the outermost fiber distance.
• Represents the strength of a cross-section to resist bending.
• Units are m3.
• Formula: S = I/c

Torsional Stress in Circular Shafts

• Torsional stress is the shear stress produced in a shaft due to applied torque (T).
• Formula: τ = T⋅r/J
  • r: shaft radius
  • J: polar moment of inertia

Polar Moment of Inertia

• Polar moment of inertia (J) measures a shaft's resistance to torsion.
• Units are m4.
• Formula:
  • For a solid circular shaft: J = π⋅d4/32
  • For a hollow shaft: J = π⋅(D4 - d4) / 32

Angle of Twist

• The angle of twist (θ) is the rotational deformation of a shaft under torque.
• Formula: θ = T⋅L / (G⋅J)
  • L: shaft length

Euler's Critical Load for Buckling

• Euler's critical load is the maximum axial load a column can withstand before it buckles.
• Formula: Pcr = π2⋅E⋅I / (K⋅L)2
  • K: effective length factor
  • L: column length

Circumferential (Hoop) Stress in Thin-Walled Pressure Vessels

• Circumferential stress is the stress around the circumference of a cylindrical pressure vessel.
• Formula: σhoop = P⋅r / t
  • P: internal pressure
  • r: radius
  • t: wall thickness

Volumetric Strain

• Volumetric strain is the change in volume per unit volume in a material under stress.
• It's the sum of strains in all three orthogonal directions.
• Formula: εv = ΔV/V = εx + εy + εz