Short Answer:
Shear stress is defined as the internal resistance developed per unit area of a body when an external tangential (parallel) force acts on it. It tends to cause one layer of the material to slide over an adjacent layer.
Shear strain is the angular deformation produced in a body due to shear stress. It is the ratio of the lateral displacement of one layer to the distance between the layers. In simple terms, shear strain measures the change in shape caused by tangential forces without changing the volume of the body.
Detailed Explanation:
Shear Stress and Shear Strain
When an external load acts tangentially to the surface of a material, it tends to deform the body by sliding one layer over another. This type of deformation is known as shear deformation. The internal resistance offered by the material to this sliding action is known as shear stress, and the measure of the deformation or angular change produced is called shear strain.
Both these quantities are very important in the study of strength of materials because they help engineers understand how materials respond under torsional or tangential forces such as in shafts, rivets, bolts, beams, and connecting rods.
Shear Stress
Definition:
Shear stress is the internal resisting force per unit area of a material when the body is subjected to a tangential or shear load. It resists the tendency of the material’s particles to slide over each other.
Mathematically,
where,
- = Shear stress (N/m² or Pascal, Pa)
- = Tangential (shearing) force acting on the body (N)
- = Area resisting the shear force (m²)
SI Unit:
The SI unit of shear stress is N/m² or Pascal (Pa).
Dimensional Formula:
Explanation of Shear Stress:
Consider a rectangular block fixed at the bottom and subjected to a tangential force on the top surface. The tangential force tries to deform the block such that its top layer moves relative to the fixed bottom layer.
The internal resistance developed per unit area along the layers to oppose this motion is the shear stress.
If the tangential force is small, the deformation is also small, and when the force is removed, the body regains its original shape — this behavior is called elastic shear deformation.
However, if the force exceeds the elastic limit, the deformation becomes permanent, and the material fails in shear.
Examples of Shear Stress in Real Life:
- Cutting Paper with Scissors:
The blades apply opposite tangential forces causing the paper layers to slide past each other. - Riveted or Bolted Joints:
The rivet or bolt experiences shear stress when it resists the relative motion between joined plates. - Torsion in Shafts:
When torque is applied to a circular shaft, shear stress develops along its cross-section. - Beams under Load:
When a transverse load acts on a beam, shear stress develops at every section of the beam. - Soil under Footing:
The soil experiences shear stress beneath a loaded foundation.
Shear Strain
Definition:
Shear strain is the angular deformation produced in a body due to the action of shear stress. It represents the relative displacement of one layer over another in the direction of the applied shear force.
Mathematically,
where,
- = Shear strain (in radians)
- = Tangential displacement or deformation (m)
- = Distance between the fixed and displaced layers (m)
SI Unit:
Shear strain is dimensionless because it is a ratio of two lengths. It is measured in radians, but often expressed as a pure number.
Explanation of Shear Strain:
Referring to the same rectangular block example:
- When a tangential force acts on the top surface, the block deforms into a shape like a parallelogram.
- The top surface slides by a small displacement , while the bottom remains fixed.
- The angular distortion (in radians) between the vertical side and the deformed shape represents the shear strain.
Hence,
For small angles (in the elastic range), the tangent of the angle is nearly equal to the angle itself in radians.
Relationship Between Shear Stress and Shear Strain
In the elastic region, shear stress is directly proportional to shear strain according to Hooke’s Law for shear:
where,
- = Shear stress (N/m²)
- = Shear strain (radians)
- = Modulus of rigidity or shear modulus (N/m²)
Modulus of Rigidity (G) is a material property that measures its ability to resist shear deformation. It is defined as the ratio of shear stress to shear strain.
Thus, a higher value of indicates a stiffer material that resists shear deformation more effectively.
Example Problem
A square block of rubber has a side length of 100 mm and height 50 mm. A tangential force of 200 N acts on its top surface, causing a displacement of 2 mm. Find the shear stress and shear strain.
Given:
, ,
, .
- Shear Stress:
- Shear Strain:
Thus, the shear stress = 20 kPa and shear strain = 0.04 radians.
Applications of Shear Stress and Shear Strain
- Torsion of Shafts:
Used in power transmission where circular shafts experience shear due to torque. - Design of Rivets and Bolts:
Helps determine the maximum shear strength of connecting elements. - Shear in Beams:
Used to analyze shear stress distribution across beam cross-sections. - Material Testing:
Shear tests determine the material’s shear modulus and strength. - Structural Safety:
Ensures that materials and connections do not fail under tangential or twisting loads.
Conclusion
Shear stress is the internal tangential resistance per unit area developed within a material due to an applied shear force, while shear strain is the angular deformation or relative displacement of layers produced due to shear stress. These two are related through the modulus of rigidity (G). Understanding shear stress and shear strain is crucial in designing mechanical components like shafts, bolts, and beams to ensure they can safely withstand tangential or twisting loads without failure.