Stress and Strain

Definitions

1. Stress ($\sigma$):
   Stress is defined as the force per unit area applied to a material:

   $$\text{Stress}=\frac{F}{A}$$

   where:

   • $F$ is the applied force (N),
   • $A$ is the cross-sectional area over which the force is applied (${\text{m}}^2$).

   Stress has SI units of $\text{N}/{\text{m}}^2$ or Pascals (Pa). It quantifies the internal forces within the material caused by external loading^[1].

2. Strain ($ϵ$):
   Strain is the relative deformation of the material and is dimensionless:

   $$\text{Strain}=\frac{\mathrm{\Delta }l}{l_0}$$

   where:

   • $\mathrm{\Delta }l$ is the change in length,
   • $l_0$ is the original length^[2].

Relationship Between Stress and Strain

In the elastic (linear) region of deformation, stress is directly proportional to strain:

where:

• $E$ is Young's modulus, a material constant ($\text{Pa}$), that measures the stiffness of the material^[1:1].

This linear relationship is known as Hooke's Law.

Types of Stress and Strain

1. Tensile Stress and Strain:

   • Tensile stress occurs when a material is stretched (forces act outward).
   • Example: A rod being pulled from both ends experiences tensile stress, and its length increases proportionally^[2:1][3].

2. Compressive Stress and Strain:

   • Compressive stress occurs when a material is compressed (forces act inward).
   • Example: Columns in a building experience compressive stress because of the weight they support^[3:1].

3. Shear Stress and Strain:

   • Shear stress involves forces acting parallel to the material's surface, leading to angular deformation.
   • Shear strain is given by:$$\text{Shear Strain}=\frac{\mathrm{\Delta }x}{h}$$where $\mathrm{\Delta }x$ is the lateral displacement and $h$ is the thickness of the material^[4].

Elastic Moduli

For different types of deformation, the relationship between stress and strain is characterized by different elastic moduli:

1. Young's Modulus ($E$): Tensile/Compressive Stress.
2. Shear Modulus ($G$): Shear Stress.
   The relationship is given by:$$\tau =G\cdot \gamma$$where $\tau$ is shear stress and $\gamma$ is shear strain^[4:1].
3. Bulk Modulus ($K$): Volume Compression.