Summary: Solid Mechanics

Table of Contents
1. Metal's Property, Stress and Strain
2. Stress
3. Types of Stresses
4. Strain
5. Types of Strains
6. True Stress and True Strain
7. Stress-Strain Relationship
8. Comparison: Engineering vs True Stress-Strain
9. Hooke's Law
10. Properties of Materials
11. Engineering Stress-Strain Curve (ductile material)
12. Elongation
View more

Metal's Property, Stress and Strain

Stress

• The internal resisting force per unit area in a material under load. Stress = Force / Area. Unit: Pascal (Pa) = N/m2. Practical multiples: 1 kPa = 10^3 Pa, 1 MPa = 10^6 Pa = 1 N/mm2, 1 GPa = 10^9 Pa.

Types of Stresses

• Normal stress - force normal to area (σn = Fn / A).
• Shear stress - tangential force on area.
• Bulk stress - inward normal force per area (volumetric effect).

Strain

• Deformation measure: relative displacement between particles divided by reference length. Dimensionless.

Types of Strains

• Normal strain - change in length / original length (εn = δl / l). Two kinds: longitudinal (along force) and lateral (perpendicular). Poisson's ratio μ = -(lateral strain / longitudinal strain).
• Shear strain - tangential displacement / original normal length. Angle measured in radians. (εt = δt / f = γ (radians)). Volume is unchanged by pure shear.
• Bulk (volumetric) strain - -(change in volume) / original volume (εB = -δV / V).
• Stress and strain are tensor quantities (have magnitude and direction).

True Stress and True Strain

• True stress = load / instantaneous cross-sectional area. For engineering stress σ and engineering strain ε: σT = σ(1 + ε).
• True strain is defined using successive values of length (Lo = original length, L = successive lengths as it changes).
• During plastic deformation the specimen volume is assumed constant.

Stress-Strain Relationship

• Brittle materials: almost no yield point or necking; little change in rate of strain before fracture.
• Elastic behavior: loading to within elastic limit returns along same path - strain disappears on unloading.
• Plastic behavior: loading beyond elastic limit produces permanent (residual) strain on unloading.

Comparison: Engineering vs True Stress-Strain

• True curve (flow curve) is based on instantaneous dimensions and shows actual deformation behavior.
• Engineering curve uses original area; apparent stress drops after necking, while true stress increases after necking due to decreasing area.
• Flow curve in uniform plastic region: σT = K (εT)^n, where K = strength coefficient and n = strain hardening exponent. n = 0 → perfectly plastic, n = 1 → elastic solid; for most metals 0.1 < n < 0.5.

Hooke's Law

• Stress is proportional to strain within elastic limit: σ = E ε. Shear relation: ζ = G γ.
• E = modulus of elasticity (resistance to elastic strain). G = shear modulus (modulus of rigidity).

Properties of Materials

• Elasticity - ability to return to original shape after load removal.
• Plasticity - permanent deformation beyond elastic limit.
• Ductility - ability to undergo large longitudinal deformation (can be drawn out).
• Brittleness - lack of ductility; fractures without significant plastic deformation.
• Malleability - ability to be extended in all directions (hammered or rolled into sheets) without rupture.
• Toughness - ability to absorb energy without fracture.
• Hardness - resistance to indentation or abrasion (tested by Brinell hardness test).
• Strength - ability to resist fracture under load.

Engineering Stress-Strain Curve (ductile material)

• Curve starts at origin (no initial stress or strain).
• Up to point A: Hooke's law holds (limit of proportionality).
• Point B: elastic limit. Between B and D apparent stress can fall to a lower yield point D (due to area change); true stress continues to rise to upper yield point C.
• From point E onward strain hardening raises strength until ultimate point F is reached.

Elongation

• Prismatic bar under axial force P: total elongation δ = P L / (A E).
• Composite bar of different segments: total elongation equals sum of elongations of segments (each using δ = PL / AE for its length and area).
• Tapered and conical bars: specific formulas apply for varying cross section (treated by integration or standard expressions).
• Elongation due to self-weight: uniform rod of length L and weight W → δ = W L / (2 A E). For rod with weight per unit length ω → δ = ω L^2 / (2 E A).