Displacement Gradient as a Unitless Quantity
Displacement gradient is considered as a unitless quantity due to the following reasons:

Definition of Displacement Gradient
- Displacement gradient is defined as the rate of change of displacement with respect to position in a material body.
- Mathematically, it is represented as the gradient of the displacement vector field.

Physical Interpretation
- Displacement itself is a vector quantity that describes the change in position of a point in space.
- When we take the gradient of displacement, we are essentially looking at how the displacement changes in different directions at a given point.
- Since displacement is measured in units of length (e.g., meters), the gradient of displacement will have units of length per length, which simplifies to a unitless quantity.

Unitless Nature
- The displacement gradient is a ratio of two lengths, representing the change in displacement along different axes.
- As the numerator and denominator have the same units of length, they cancel out, leaving the displacement gradient as a unitless quantity.
- This unitless nature allows for a clear understanding of how the displacement varies within a material body without being influenced by any specific units of measurement.
Therefore, displacement gradient is considered a unitless quantity as it provides a pure indication of the change in displacement within a material body without any dimensional units affecting its interpretation.