Stress and Strain Transformation

Plane stress and strain transformation

Plane stress and strain transformation describes how the stress or strain components at a point change when the coordinate axes are rotated about the axis normal to the plane. For a rotation \theta measured from the x-axis, the normal stress and shear stress on the rotated face are \sigma_n = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta and \tau_{nt} = -\frac{\sigma_x-\sigma_y}{2}\sin 2\theta + \tau_{xy}\cos 2\theta. The key geometric rule is that a physical rotation of \theta moves a point through 2\theta on Mohr's circle, so a plane rotated by 45^\circ in the body corresponds to a 90^\circ move on the circle. For example, if the circle is centred at the origin, a plane at 45^\circ from the principal direction carries pure shear with zero normal stress.

The principal stresses are the normal stresses on the planes where shear vanishes. They are \sigma_{1,2} = \sigma_{avg} \pm R, where \sigma_{avg} = \frac{\sigma_x+\sigma_y}{2} and R = \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2} is the radius of Mohr's circle. The maximum in-plane shear stress equals this radius R. It acts on planes inclined at 45^\circ to the principal planes. For a given stress state, R is computed directly from the stress components, as in the simple substitution R = \sqrt{(\sigma_x-\sigma_y)^2/4 + \tau_{xy}^2}.

The normal stress on the maximum shear plane equals the mean stress \frac{\sigma_1+\sigma_2}{2}, not zero. It vanishes only when the mean stress is zero, which is the case of pure shear. Therefore a plane of maximum shear does not in general carry zero normal stress.

For plane stress, the stress normal to the free surface is zero, so the third principal stress is \sigma_3 = 0. The three-dimensional maximum shear stress is then the largest of \frac{\sigma_1-\sigma_2}{2}, \frac{\sigma_1-\sigma_3}{2} and \frac{\sigma_2-\sigma_3}{2}. When the principal stresses are both positive, the largest of these is usually \frac{\sigma_1-\sigma_3}{2}, which is the value to check first.

A state of pure shear has \sigma_x = \sigma_y = 0 with a shear stress \tau. Its principal stresses are +\tau and -\tau, acting on planes at 45^\circ to the axes. Equivalently, pure shear can be described by one tensile and one compressive normal stress on rotated axes. Mohr's circle for pure shear is centred at the origin with radius \tau.

The traction vector on a plane with unit normal \hat{n} is t_i = \sigma_{ij} n_j. This relation gives two scalar equations for the unknown stress components when the traction and the normal are known. Its components must be consistent with the normal stress \sigma_n = \hat{n}\cdot\mathbf{t} and the shear stress along the plane. A typical check is to substitute the solved components back into the equations.

The sum of the principal stresses is an invariant of the stress state. It equals the trace of the stress tensor, \sigma_1+\sigma_2 = \sigma_{xx}+\sigma_{yy}, and it does not change with rotation of axes. Shear entries in the matrix do not enter this sum.