Strength of Materials

Introduction to SOM Stress Strain Stress Strain Curve

Strain

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Strain

Strain is the measurement of deformation that occurs in a material when subjected to external forces. It is expressed as the ratio of change in dimensions to original dimensions. This dimensions may be length or area.

Strain is denoted by Greek letter sigma (\(\epsilon\)).

Mathematical representation of strain

Mathematically strain is represented as ratio of change in length to original length. Aslo, we can consider area insted of length.

\[\epsilon = \frac{\Delta L}{L_0}\ = \frac{L - L_0}{L_0}\]

Where:

  • \(L_0\) = Original length
  • \(L\) = Deformed length
  • \(\Delta L\) = Change in length

Unit of Strain

Strain is a dimensilnless quantity, its the ratio of same quantities like, length or area.

But often it expressed as:

  • Ratio of quantities - mm/mm or in/in
  • Percentage - \( \% \epsilon = \epsilon \times 100\)

Types of Stress

The major type of stresses we deal in engineering problems are as follow:

  1. Normal Strain (\(\epsilon\))

    • Normal strain (also called linear strain) describes the change in length relative to the original length.
    • Normal strain is denoted by (\(\epsilon\))

    Normal strain is further divided into two types:

    1. Tensile strain - The elongation of bar due to tensile load / stress.

      • Formula
        \[\epsilon = \frac{\Delta L}{L_0}\ = \frac{L - L_0}{L_0}\]
      • It is represented by a Positive (+) sign.
      • Example - Elongation of cable strain

    2. Compressive Strain - The compression of bar due to compressive load / stress.

      • Formula
        \[\epsilon = \frac{\Delta L}{L_0}\ = \frac{L - L_0}{L_0}\]
      • It is represented by a Negative (-) sign.
      • Example - Reduction in length of bar due to compressive load / stress.

  2. Shear Strain (\(\gamma\))

    • Shear strain represents the angular deformation that occurs when layers of material slide relative to each other.
    • Formula
      \[\gamma = \tan \theta \approx \theta \, (\text{for small angles})\]
    • Shear stress is denoted by (\(\gamma\))
    • Example - Change in shape and not volume of chease cube due to shear load.

  3. Volumetric Stress (\(\epsilon_{\text{v}}\))

    • Change in volume relative to the original volume.
    • Formula
      \[\epsilon{\text{v}} = \frac{\Delta V}{V_0}\ = \frac{V - V_0}{V_0}\]

      • Where:

      • \(V_0\) = Original volume
      • \(V\) = Deformed volume
      • \(\Delta V\) = Change in volume
    • Volumetric strain is denoted by (\(\epsilon_{\text{v}}\))
    • For isotropic materials, volumetric strain εᵥ = εₓ + εᵧ + εᵤ (sum of linear strains in three perpendicular directions)
    • Example - Change in volume of chease cube due to load from all directions.

  4. Thermal Strain (\(\epsilon_{\text{th}}\))

    • Strain resulting from temperature changes, causing expansion or contraction.
    • Formula
      \[\sigma = \alpha \cdot \Delta T \]

      • Where:

      • \(\alpha\) = Coefficient of thermal expansion in \(1/^\circ C\)
      • \(\Delta T\) = Temperature change \(C\)
    • Thermal stress is denoted by \(\epsilon\)
    • Example - Railway rails expansion during summer.
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