Mechanics of Materials Note
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Stress
The stress \( \sigma \) is calculated using the formula:
\[ \sigma = \frac{F}{A} \]
tensile stress \( \sigma_t \)
compressive stress \( \sigma_c \)
shear stress\( \tau \)
strain
The displacement per unit length (dimensionless) is known as strain
Shear Strain
Shear strain (\(\gamma\)) is a measure of the change in angle between two initially perpendicular lines in a material when subjected to shear stress. It is calculated using the formula:
\[ \gamma = \tan(\Delta \phi) = \frac{\text{change in displacement}}{\text{original separation}} \]
Shear strain is a dimensionless quantity and is related to shear stress (\(\tau\)) through the material's shear modulus (\(G\)). The relationship is given by:
\[ \tau = G \cdot \gamma \]
Understanding shear strain is crucial in analyzing the deformation behavior of materials under shear loading conditions.
True Stress and True Strain
In materials engineering, true stress (\(\sigma_{\text{true}}\)) and true strain (\(\varepsilon_{\text{true}}\)) are measures that account for changes in cross-sectional area and length, respectively, as the material undergoes deformation. The formulas are given by:
True Stress (\(\sigma_{\text{true}}\)):
\[ \sigma_{\text{true}} = \frac{F}{A} \cdot \left(1 + \frac{\Delta L}{L}\right) \]
True Strain (\(\varepsilon_{\text{true}}\)):
\[ \varepsilon_{\text{true}} = \ln\left(1 + \frac{\Delta L}{L}\right) \]
These formulations are particularly useful in situations where significant plastic deformation occurs, and the assumption of constant volume is no longer valid.