18.16: Shearing Strain

The shearing strain represents a cubic element's angular change when subjected to shearing stress. This type of stress can transform a cube into an oblique parallelepiped without influencing normal strains. The cubic element experiences a significant transformation when exposed solely to shearing stress. Its shape alters from a perfect cube into a rhomboid, clearly demonstrating the effect of shearing strain. The degree of this strain is considered positive if it reduces the angle between the axes.

The relationship between shearing stress and strain can be visualized through a diagram similar to a standard stress-strain chart. However, the yield and ultimate strengths differ. This diagram illustrates Hooke's law in action, with the initial straight line indicating a direct proportionality between shearing stress and strain.

The constant G in Hooke's law for shearing stress is referred to as the modulus of rigidity or shear modulus of the material. This constant is expressed in the same units as shearing stress. Shearing strain can be defined for a range of stresses. By leveraging the principle of superposition, we can derive the generalized form of Hooke's law. This law involves three constants. Two constants are determined experimentally, while the third is obtained computationally.

In a cubic element with a general state of stress, the shearing stresses can deform a cube into an oblique parallelepiped without affecting normal strains. The angular change under shearing stress is the shearing strain.

A cube, when subjected to only shearing stress, morphs into a rhomboid, highlighting the impact of shearing strain.

When the angle between the axes of the cube shrinks, the strain is deemed positive. The shearing stress-strain diagram follows Hooke's law, where the initial straight line indicates proportionality between shearing stress and strain.

The constant G in Hooke's law for shearing is termed the shear modulus or modulus of rigidity of the material. It is expressed in the same units as the shearing stress and is less than half but more than one-third of the modulus of elasticity.

For general stress conditions, a generalized Hooke's law is obtained using the shearing strain relations and Hook's law.

The constants in these relationships are determined experimentally, which can then be used to predict material deformations from various stress combinations.