Short Answer:

The state of pure shear is a condition in which a material element is subjected to equal and opposite shear stresses on two mutually perpendicular planes, with no normal stresses acting on it. In this state, the material experiences only a change in shape without any change in volume.

In other words, during pure shear, the material tends to deform by sliding one layer over another without elongation or compression. This condition commonly occurs in elements like shafts under torsion, where shear stress is the only type of stress acting within the material.

Detailed Explanation:

State of Pure Shear

The state of pure shear is an important concept in the study of mechanics of materials and stress analysis. It refers to the condition when a material element is under the action of shear stresses that are equal in magnitude but act in opposite directions on perpendicular planes. In this state, there are no normal stresses (no tensile or compressive stresses) acting on the material.

When a body experiences pure shear, the deformation occurs only by angular distortion, meaning that the shape of the body changes but its volume remains constant. The material tends to twist or slide within itself without any elongation or compression.

A simple example of pure shear is the action of a rectangular block under opposite tangential forces applied on its top and bottom surfaces. The block gets deformed in such a way that its shape becomes a parallelogram while its area remains the same.

Explanation of Pure Shear Condition

In the state of pure shear, the stress components act tangentially on the faces of the element. Suppose a small square element is subjected to shear stress on two perpendicular planes. To maintain equilibrium, an equal and opposite shear stress acts on the opposite faces in opposite directions.

Thus, the shear stress system consists of:

Shear stress acting on one plane,
Equal and opposite shear stress acting on the perpendicular plane.

These four shear stresses form a balanced system where no net force or moment acts on the element. Since there are no normal stresses (tensile or compressive), the element is said to be in a pure shear state.

Mathematical Representation

In pure shear, the stresses acting on an element can be represented as:

Here:

and are the normal stresses in the x and y directions (both zero).
is the shear stress acting on the element.

Therefore, only shear stress acts on the material, and the normal stresses are absent.

Example of Pure Shear

One of the most common examples of pure shear is the solid circular shaft under torsion. When a torque is applied to a circular shaft, every cross-section of the shaft experiences a twisting action that produces shear stresses parallel to the surface. In this condition, the shaft is said to be under pure shear stress because the normal stresses are zero.

Other examples include:

A punching operation, where the metal sheet near the punch edge is under pure shear.
Bolts and rivets under transverse loading also experience a state close to pure shear.

Principal Stresses in Pure Shear

Even though pure shear involves no normal stresses in its original orientation, if we analyze the element along the principal planes, normal stresses will appear as principal stresses.

In pure shear:

Here, is the maximum principal stress (tensile) and is the minimum principal stress (compressive).

This means that the state of pure shear can also be considered equivalent to a combination of equal tensile and compressive stresses acting at 45° to the direction of shear.

Characteristics of Pure Shear

Only shear stresses act: There are no normal stresses on the element.
Shape changes but volume remains constant: The material experiences angular deformation.
Balanced stress system: Shear stresses on all sides are equal and opposite, maintaining equilibrium.
Principal stresses exist at 45°: The directions of maximum tensile and compressive stresses are inclined at 45° to the shear planes.
Occurs in torsion and shear tests: Common in shafts, beams, and connections under tangential loading.

Importance of Studying Pure Shear

Understanding pure shear is crucial for mechanical and structural engineers because many components in real-life applications experience such stresses. It helps in:

Designing shafts, bolts, and rivets to resist shear failure.
Predicting material behavior under torsional loads.
Determining the shear modulus and other material properties.
Preventing structural distortions and ensuring safety under twisting or sliding loads.

By studying pure shear, engineers can estimate how much angular distortion a material will undergo before it fails. This is important in mechanical systems where motion and torque are transmitted, such as in engines, turbines, and gear assemblies.

Difference Between Simple Shear and Pure Shear

Although both terms sound similar, simple shear and pure shear are not exactly the same.

In simple shear, one pair of parallel surfaces experiences tangential forces without the balancing shear stress on the perpendicular plane.
In pure shear, equal and opposite shear stresses act on two perpendicular planes, maintaining complete equilibrium.

Therefore, pure shear is a more balanced and ideal condition used in theoretical and analytical studies.

Conclusion

The state of pure shear is a stress condition where equal and opposite shear stresses act on two perpendicular planes without any normal stresses. It causes only shape change, not volume change, and is fundamental in analyzing materials subjected to torsion and tangential forces. Understanding this state is essential for designing safe and efficient mechanical components that can resist twisting and sliding actions without failure.