JNTU B.Tech III MECHANICAL WORKING OF METALS Previous Question Paper (September - 2021 - R18)

Strain at a point is defined as the state of deformation in the immediate vicinity of a specific point within a body. Since a body undergoes non-uniform deformation, the displacement varies from point to point. To define strain at a point, consider a small element of volume surrounding the point. The strain is characterized by the change in length of line elements and the change in angle between them.

Mathematically, it is represented by the strain tensor, which consists of six independent components: three normal strains (εx, εy, εz) representing elongation or contraction along the coordinate axes, and three shear strains (γxy, γyz, γzx) representing the change in angles between the axes. For a small displacement field u, v, w, the normal strain in the x-direction is defined as εx = ∂u/∂x, and the shear strain in the xy-plane is γxy = ∂u/∂y + ∂v/∂x. As the volume element approaches zero, these derivatives define the strain state precisely at that point.

Mohr's circle is a graphical representation of the transformation equations for plane stress. For a 2D stress state with normal stresses σx, σy and shear stress τxy, the circle is plotted on a graph where the horizontal axis represents normal stress (σ) and the vertical axis represents shear stress (τ).

The center of the circle is located at ( (σx + σy)/2, 0 ) on the σ-axis, and the radius R is calculated as:

The circle represents all possible stress states on planes oriented at different angles. Any point on the circumference (σ', τ') corresponds to the normal and shear stresses acting on a plane rotated by an angle θ from the original x-axis, where the angle on the circle is 2θ. The principal stresses σ1 and σ2 are found at the intersection of the circle with the σ-axis, where τ = 0.

Example: Consider an element with σx = 100 MPa, σy = 40 MPa, and τxy = 30 MPa. The center is at (100+40)/2 = 70 MPa. The radius is R = √[((100-40)/2)² + 30²] = √[30² + 30²] = 42.43 MPa. The principal stresses are σ1 = 70 + 42.43 = 112.43 MPa and σ2 = 70 - 42.43 = 27.57 MPa.