Properties of Materials | Strength of Materials (SOM) - Mechanical Engineering PDF Download

Metal's Property, Stress and Strain

Stress and strain is the first topic in Strength of Materials. It introduces how materials respond internally when external forces are applied, the types of stresses and strains, and material properties needed for safe design and analysis.

Stress

When a component is subjected to an external load, internal forces develop that resist the applied load. Stress at a point is defined as the internal resisting force per unit area acting on a material. Stress is a tensor quantity; in simple one-dimensional cases it is treated as a scalar.

Formula: stress = force / area

Notation and unit: Stress is usually denoted by σ and has SI unit N/m² or Pascal (Pa). Force (F) is in newton (N) and area (A) in m².

• 1 kPa = 10³ Pa = 10³ N/m² (kPa = kilopascal)
• 1 MPa = 10⁶ Pa = 10⁶ N/m² = 1 N/mm² (MPa = megapascal)
• 1 GPa = 10⁹ Pa (GPa = gigapascal)

Types of Stress

1. Normal stress - acts perpendicular to the cross section; caused by axial tensile or compressive forces. The average normal stress is σ_n = F_n / A.
2. Shear stress - acts tangential to the cross section; caused by transverse forces or torsion. Average shear stress is τ = V / A (V = shear force).
3. Bulk (hydrostatic) stress - equal compressive stress acting in all directions (e.g., pressure). Volumetric stress is associated with hydrostatic pressure p.

Strain

Strain is the measure of deformation produced by stress. It expresses the relative displacement between particles in the body referred to an original length. Strain is dimensionless (ratio of lengths) and is often expressed in microstrain (10⁻⁶) for small deformations.

Types of Strain

1. Normal (axial) strain- the ratio of change in length to original length in the direction of the force.

   ε_n = δl / l

   Normal strain is dimensionless. It occurs as two components:

   • Longitudinal strain - in the direction of applied axial load.
   • Lateral (transverse) strain - perpendicular to the applied load (change in diameter or breadth).
   • Poisson's ratio μ is defined as the negative ratio of lateral strain to longitudinal strain:
   • μ = - (lateral strain) / (longitudinal strain)

2. Shear strain- measured by the change in angle between two lines originally at right angles. For small deformations the shear strain γ (in radians) is approximately equal to the tangent of the angular distortion.

   γ = tangential displacement / original normal length

   The angle of distortion is measured in radians. Shear strain does not change volume for small deformations.

3. Volumetric (bulk) strain- ratio of change in volume to original volume.

   ε_B = - ΔV / V

   Bulk strain is associated with hydrostatic stress states and volumetric material properties.

Stress and strain are tensor quantities in the general three-dimensional case; simple formulae above apply for uniaxial or simple states of loading.

True Stress and True Strain

• The true stress is defined as the ratio of the load to the cross section area at any instant.
  (σ_T) = load/Instantaneous area = σ(1 + ε)
  Where σ and ε is the engineering stress and engineering strain respectively.
• True strain (ε_T) (also logarithmic strain) is defined from successive instantaneous lengths. If L₀ is original length and L is current length, true strain is:
• True strain is additive for sequential deformations and more accurate for large plastic deformations.
• During plastic deformation the volume of a metal specimen is often assumed approximately constant (valid for many metals before fracture), so changes in cross-section are related to axial changes in length.

Stress-Strain Relationship

The stress-strain diagram illustrates material response from elastic behaviour through yield, plastic deformation and fracture. Typical curves differ for ductile and brittle materials.

• In figure (a), the specimen is loaded only upto point A, when load is gradually removed the curve follows the same path AO and strain completely disappears. Such a behaviour is known as the elastic behaviour.
• In figure (b), the specimen is loaded upto point B beyond the elastic limit E. When the specimen is gradually loaded the curve follows path BC, resulting in a residual strain OC or permanent strain.

Comparison of Engineering and True Stress-Strain Curves

• True stress-strain (flow) curve reflects instantaneous specimen dimensions and therefore gives a more accurate indication of deformation behaviour, especially in plastic region.
• Engineering stress-strain uses original area and thus underestimates stress after significant cross-sectional change (necking).
• Many metals in uniform plastic deformation follow a power law (empirical):

σ_T = K (ε_T)ⁿ

• K is the strength coefficient.
• n is the strain-hardening exponent.
• n = 0 corresponds to a perfectly plastic solid; n = 1 corresponds to an elastic solid. For most metals 0.1 < n < 0.5.

Hooke's Law

Hooke's law states that, within the elastic limit, stress is proportional to strain.

Normal (axial) form: σ = E ε

Shear stress form: ζ = G γ

• E is the modulus of elasticity (Young's modulus) - resistance to elastic axial deformation.
• G is the shear modulus (modulus of rigidity) - resistance to elastic shear deformation.
• These elastic constants are related by Poisson's ratio μ:

E = 2 G (1 + μ)

• The bulk modulus K (for volumetric loading) is related by:

K = E / [3 (1 - 2 μ)]

Properties of Materials

Important mechanical properties used to judge materials for structural applications are:

• Elasticity - ability of a material to return to original dimensions after removal of load.
• Plasticity - ability to undergo permanent (inelastic) deformation after exceeding the elastic limit.
• Ductility - ability to sustain large plastic deformation in tension; measured by elongation or reduction of area before fracture.
• Brittleness - lack of ductility; brittle materials fracture with little or no plastic deformation.

• Malleability - ability to be compressed or rolled into thin sheets without cracking; a form of plasticity under compressive state.
• Toughness - ability to absorb energy up to fracture; related to both strength and ductility.
• Hardness - resistance to indentation or surface abrasion; common tests include Brinell, Rockwell and Vickers.
• Strength - general term for the capacity to resist load without failure (tensile strength, compressive strength, shear strength, etc.).

Engineering Stress-Strain Curve for a Ductile Material

• The curve starts at the origin indicating no initial stress or strain.
• Up to point A the curve is linear and Hooke's law is obeyed; A is the limit of proportionality.
• Point B is the elastic limit. Beyond B, permanent deformation occurs and unloading leaves residual strain.
• At yielding the specimen may show an upper yield point and a lower yield point; the apparent engineering stress may fall after upper yield until the lower yield point.
• From a certain point work hardening begins (strain hardening) and further strain requires increasing stress until the ultimate tensile strength is reached (point F in some diagrams).

Elongation of Bars under Axial Load

For a prismatic bar of original length L, cross-sectional area A and Young's modulus E, subjected to an axial tensile force P, the axial elongation δ is:

δ = P L / (A E)

Elongation of Composite and Varying Section Bodies

1. Elongation of a composite bar- if the bar consists of n segments with areas A₁, A₂, ..., Aₙ, lengths l₁, l₂, ..., lₙ and (optionally) different moduli E₁, E₂, ..., Eₙ, and the same axial force P acts through them in series, total elongation is the sum of elongations of each segment.

2. Elongation of a tapered bar - when cross-section varies continuously, integrate the elemental elongation δ = ∫ (P dx) / (A(x) E).

Uniformly tapering circular bar

3. Elongation due to self-weight

   (i) For a uniform vertical rod of length L carrying its own weight W (total weight), the extension due to self-weight is

   δ = W L / (2 A E)

   Thus, the deformation under self-weight is half of that produced by an equivalent axial load equal to the rod's weight applied at the free end.

   (ii) If ω is weight per unit length, total extension of length L is

   δ = ω L² / (2 A E)

   (iii) For a conical bar or other non-uniform shapes the extension due to self-weight is obtained by integrating ω(x) dx / (A(x) E) over the length.