Essential concepts and memory tricks for mastering Mechanical Properties of Solids

Elasticity and Plasticity Fundamentals

Elasticity: Property of materials to regain original shape after removing deforming force. Perfectly elastic bodies return completely to original form. Plasticity: Permanent deformation remains even after removing force. Real materials show both properties within different limits.

Stress Definition and Types

Stress (σ) = Force per unit area = F/A. Tensile stress: stretching force perpendicular to area. Compressive stress: squeezing force. Shear stress: tangential force parallel to surface. Hydraulic stress: equal normal stress in all directions. Units: N/m² or Pascal.

Strain Definition and Types

Strain (ε) = Fractional change in dimension. Dimensionless quantity. Longitudinal strain = ΔL/L (change in length). Volumetric strain = ΔV/V (change in volume). Shear strain = angular deformation. Strain measures deformation relative to original size.

Hooke's Law and Applications

Within elastic limit, stress ∝ strain. Stress = k × strain, where k is elastic modulus. Linear relationship between stress and strain for small deformations. Foundation for calculating Young's modulus, bulk modulus, and shear modulus. Valid only in elastic region.

Elastic Moduli Types

Young's Modulus (Y): ratio of longitudinal stress to longitudinal strain. Bulk Modulus (K): ratio of volume stress to volume strain. Shear Modulus (G): ratio of shear stress to shear strain. All measure resistance to different types of deformation.

Poisson's Ratio Concept

μ = -(lateral strain)/(longitudinal strain). When material is stretched lengthwise, it contracts sideways. Poisson's ratio quantifies this effect. Typical values: 0 to 0.5. Cork ≈ 0, rubber ≈ 0.5. Most metals: 0.2-0.3. Important for understanding 3D deformation.

Stress-Strain Curve Analysis

Graph shows material behavior under loading. Elastic region: linear, follows Hooke's law. Proportional limit: end of linearity. Elastic limit: maximum stress for complete recovery. Yield point: permanent deformation begins. Ultimate strength: maximum stress before breaking.

Elastic Energy and Applications

Elastic potential energy = ½ × stress × strain × volume = ½σεV. Energy stored in deformed elastic material. Can be recovered when deformation is removed. Applications: springs, shock absorbers, building design. Energy density helps compare materials' energy storage capacity.

All essential mechanical properties formulas with LaTeX equations and simple explanations

Concept Formula Meaning in Simple Words When to Use
Stress (General) \(\sigma = \frac{F}{A}\) Stress equals applied force divided by cross-sectional area For any type of normal stress calculation (tensile or compressive)
Longitudinal Strain \(\varepsilon = \frac{\Delta L}{L}\) Strain equals change in length divided by original length For stretching or compression along length direction
Young's Modulus \(Y = \frac{\sigma}{\varepsilon} = \frac{F/A}{\Delta L/L} = \frac{FL}{A\Delta L}\) Young's modulus equals longitudinal stress divided by longitudinal strain For problems involving stretching or compressing rods, wires, beams
Bulk Modulus \(K = -\frac{\Delta P}{\Delta V/V} = -\frac{\text{Volume stress}}{\text{Volume strain}}\) Bulk modulus relates pressure change to fractional volume change For uniform pressure problems, liquid/gas compression
Shear Modulus \(G = \frac{\tau}{\gamma} = \frac{\text{Shear stress}}{\text{Shear strain}}\) Shear modulus equals shear stress divided by shear strain For twisting, sliding, or angular deformation problems
Shear Stress \(\tau = \frac{F_{tangential}}{A}\) Shear stress equals tangential force divided by area When force acts parallel to surface area
Shear Strain \(\gamma = \frac{\Delta x}{L} = \tan\theta \approx \theta\) Shear strain is angular deformation (in radians for small angles) For small angular deformations in shear problems
Poisson's Ratio \(\mu = -\frac{\text{lateral strain}}{\text{longitudinal strain}} = -\frac{\Delta d/d}{\Delta L/L}\) Poisson's ratio relates sideways contraction to lengthwise extension For 3D deformation analysis, material characterization
Volumetric Strain \(\varepsilon_v = \frac{\Delta V}{V}\) Volumetric strain equals change in volume divided by original volume For volume change calculations under pressure
Elastic Energy Density \(u = \frac{1}{2}\sigma\varepsilon = \frac{1}{2Y}\sigma^2 = \frac{1}{2}Y\varepsilon^2\) Elastic energy per unit volume stored in deformed material For energy calculations in elastic deformation
Total Elastic Energy \(U = \frac{1}{2}\sigma\varepsilon \times V = \frac{1}{2} \times \frac{F^2L}{AY}\) Total elastic energy stored in deformed object For calculating total energy stored in springs, beams
Relation: E, G, K (with Poisson's ratio) \(E = 2G(1+\mu) = 3K(1-2\mu)\) Young's modulus related to shear and bulk modulus via Poisson's ratio To find one modulus when others are known
Relation: E, G, K (general) \(E = \frac{9KG}{3K + G}\) General relationship between all three elastic moduli When Poisson's ratio is unknown but G and K are known
Wire Extension Formula \(\Delta L = \frac{FL}{AY}\) Extension of wire equals force times length divided by area times Young's modulus For calculating extension in wires, rods under tension
Spring Constant Relation \(k = \frac{AY}{L}\) Spring constant relates to material properties and geometry For connecting spring constant to elastic modulus

Systematic approach to solve Mechanical Properties problems efficiently

1

Identify Type of Deformation

Determine if it's tension (stretching), compression (squeezing), shear (sliding), or volume change (pressure). This determines which stress, strain, and modulus formulas to use.

2

Determine What Needs to be Found

Identify the unknown: stress, strain, elastic modulus, force, extension, or energy. This helps choose the correct formula and approach for the problem.

3

Choose Appropriate Elastic Modulus

Use Young's modulus (Y) for length changes, Bulk modulus (K) for volume changes, Shear modulus (G) for angular deformation. Match the modulus to the type of deformation.

4

Calculate Cross-sectional Area Correctly

For circular cross-section: A = πr² or πd²/4. For rectangular: A = length × width. For wire: use diameter, not radius. Double-check area calculations as they're common error sources.

5

Apply Proper Sign Conventions

Tensile stress/strain: positive. Compressive stress/strain: negative. Volume decrease: negative strain. Poisson's ratio: negative because lateral strain opposes longitudinal strain.

6

Convert Units to SI System

Force: Newtons (N), Length: meters (m), Area: m², Stress: Pa (N/m²), Strain: dimensionless. Convert mm to m, kg to N (multiply by g), GPa to Pa (multiply by 10⁹).

7

Use Dimensional Analysis

Check units: [Y] = [σ] = N/m² = Pa, [ε] = dimensionless, [μ] = dimensionless. Verify final answer has correct units. Use unit analysis to catch calculation errors.

8

Check Physical Reasonableness

Verify results make sense: strain should be small for elastic deformation (<1%), stress should be below material's yield strength, Young's modulus should match material type.

Avoid these common errors to improve your exam performance

Mistake How to Avoid
Confusing stress and pressure concepts Stress is internal restoring force per unit area in deformed material. Pressure is external force per unit area applied to surface. Stress develops as response to external pressure or force.
Wrong calculation of cross-sectional area For circular wire: A = πr² = π(d/2)² = πd²/4. For diameter in mm, convert to m first. For rectangular: A = width × thickness. Always check if radius or diameter is given.
Mixing up different types of strain Longitudinal strain = ΔL/L (length change). Volumetric strain = ΔV/V (volume change). Shear strain = tan θ ≈ θ (angular change). Use correct strain for problem type.
Incorrect application of Hooke's law Hooke's law (σ ∝ ε) applies only within elastic limit. Beyond yield point, material shows plastic behavior. Always check if deformation is within elastic range.
Using wrong elastic modulus for problem type Young's modulus: for length changes (tension/compression). Bulk modulus: for volume changes under pressure. Shear modulus: for angular deformation. Match modulus to deformation type.
Unit conversion errors in calculations Convert all units to SI before calculation: mm→m (×10⁻³), GPa→Pa (×10⁹), kg→N (×g). Write units throughout calculation to catch errors. Final check: stress in Pa, strain dimensionless.
Misunderstanding negative signs in Poisson's ratio Poisson's ratio μ = -(lateral strain)/(longitudinal strain). Negative sign because when material stretches lengthwise, it contracts sideways. Don't add extra negative signs.
Confusing elastic limit with proportional limit Proportional limit: end of linear stress-strain relationship (Hooke's law valid). Elastic limit: maximum stress for complete recovery. Elastic limit ≥ proportional limit.
Wrong interpretation of stress-strain curves Linear region: elastic behavior, slope = modulus. Yield point: permanent deformation starts. Ultimate strength: maximum stress. Breaking point: material fails. Each region has different properties.
Forgetting to consider material properties Different materials have different elastic moduli: steel (high Y), rubber (low Y). Check if given values match expected material properties. Metals typically have higher moduli than polymers.

Quick memory aids and essential information for last-minute revision

Quick Stress & Strain Formulas

  • Stress: σ = F/A (force per unit area)
  • Longitudinal strain: ε = ΔL/L
  • Volume strain: εᵥ = ΔV/V
  • Shear strain: γ = tan θ ≈ θ (small angles)

Elastic Moduli Definitions

  • Young's modulus: Y = σ/ε (length changes)
  • Bulk modulus: K = -ΔP/(ΔV/V) (volume changes)
  • Shear modulus: G = τ/γ (angular changes)
  • All measured in Pa (N/m²)

Hooke's Law Applications

  • σ = Yε (within elastic limit)
  • F = kx for springs (k = AY/L)
  • Linear stress-strain relationship
  • Valid only for small deformations

Poisson's Ratio Facts

  • μ = -(lateral strain)/(longitudinal strain)
  • Range: 0 ≤ μ ≤ 0.5
  • Cork ≈ 0, rubber ≈ 0.5
  • Most metals: 0.2-0.3

Stress-Strain Curve Points

  • Proportional limit: end of linear region
  • Elastic limit: max stress for full recovery
  • Yield strength: permanent deformation begins
  • Ultimate strength: maximum stress value

Material Property Comparisons

  • Steel: Y ≈ 200 GPa (high stiffness)
  • Aluminum: Y ≈ 70 GPa (medium stiffness)
  • Rubber: Y ≈ 0.01 GPa (low stiffness)
  • Diamond: Y ≈ 1000 GPa (highest known)

Common Problem Types

  • Wire extension: ΔL = FL/(AY)
  • Elastic energy: U = ½σεV = ½F²L/(AY)
  • Spring problems: k = AY/L
  • Volume compression: ΔV/V = -ΔP/K

Important Constants & Units

  • 1 GPa = 10⁹ Pa
  • 1 MPa = 10⁶ Pa
  • Area of circle: A = πr² = πd²/4
  • g = 9.8 m/s² (for weight calculations)