Engineering Strain Basics

Engineering strain is a measure of deformation relative to a material's original size. It is defined as the change in length divided by the original length ($L_0$).

The formula for engineering strain ($\epsilon$) is:

$ \epsilon = \frac{\Delta L}{L_0} = \frac{L_{final} - L_0}{L_0} $

Extension Strain Calculation

The problem states:

• Initial length ($L_0$) = 2 cm.
• The element is extended to twice its initial length.

Calculate the final length after extension ($L_1$):

$L_1 = 2 \times L_0 = 2 \times 2 \text{ cm} = 4 \text{ cm}$.

Calculate the change in length during extension ($\Delta L_{ext}$):

$\Delta L_{ext} = L_1 - L_0 = 4 \text{ cm} - 2 \text{ cm} = 2 \text{ cm}$.

Calculate the engineering strain for the extension phase ($\epsilon_{ext}$):

$ \epsilon_{ext} = \frac{\Delta L_{ext}}{L_0} = \frac{2 \text{ cm}}{2 \text{ cm}} = 1 $

Compression Strain Calculation

After extension, the element's length is $L_1 = 4$ cm.

The element is then compressed back to its initial length ($L_0 = 2$ cm). Let the final length after compression be $L_2$. So, $L_2 = 2$ cm.

Calculate the change in length during compression ($\Delta L_{comp}$):

$\Delta L_{comp} = L_2 - L_1 = 2 \text{ cm} - 4 \text{ cm} = -2 \text{ cm}$.

Calculate the engineering strain for the compression phase ($\epsilon_{comp}$). The reference length (original length for this step) is the length before compression, which is $L_1 = 4$ cm.

$ \epsilon_{comp} = \frac{\Delta L_{comp}}{L_1} = \frac{-2 \text{ cm}}{4 \text{ cm}} = -0.5 $

Final Strain Values

The engineering strains for the extension and compression are 1 and -0.5, respectively.