Material Mechanics: Stress and Strain

Study Notes

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

Description

Test your understanding of key concepts in material mechanics, including stress, strain, and Young's Modulus. This quiz covers essential formulas and relationships that define the mechanical properties of materials. Perfect for students studying engineering or physics!