Cauchy strain, also known as engineering strain, is a measure of deformation that represents the relative change in length of a material under stress. It is a fundamental concept in materials science and engineering, used to quantify how much a material stretches or compresses relative to its original length.

The formula for Cauchy strain is:

\[ \varepsilon = \frac{L - L_0}{L_0} \]

Where:

- \( \varepsilon \) is the Cauchy strain (dimensionless)
- \( L \) is the final length of the material
- \( L_0 \) is the initial length of the material

Let's calculate the Cauchy strain for a material that has been stretched:

- Given:
- Initial length (\( L_0 \)) = 100 mm
- Final length (\( L \)) = 105 mm

- Apply the Cauchy strain formula: \[ \varepsilon = \frac{L - L_0}{L_0} \]
- Substitute the known values: \[ \varepsilon = \frac{105 \text{ mm} - 100 \text{ mm}}{100 \text{ mm}} \]
- Perform the calculation: \[ \varepsilon = \frac{5 \text{ mm}}{100 \text{ mm}} = 0.05 \]
- Express the result as a percentage: \[ \varepsilon = 0.05 = 5\% \]

Let's visualize the Cauchy strain in a material:

This diagram illustrates:

- The initial length (\( L_0 \)) of the material (blue line)
- The final length (\( L \)) after stretching (green line)
- The change in length (\( \Delta L \)) (red line)
- The Cauchy strain (\( \varepsilon \)) as the ratio of elongation to initial length

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