20.5 : Deformations in a Transverse Cross Section

When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.

As the material stretches, it expands or contracts in orthogonal directions to the load. This phenomenon varies across different sections of the material.

For instance, in the vertical transverse direction, expansions and contractions cancel each other out on what is known as the neutral surface, where there is no longitudinal stress.

However, the material behaves differently in the horizontal transverse direction; the sections bend into circular arcs due to the varied expansions and contractions across the material's thickness. The circular arcs are centered on a point O', with a radius of curvature r'. The radius of curvature of the beam due to bending is centered on a point O, with a radius of curvature r. Bending in both directions is related to the material's Poisson's ratio. The radius of curvature of these arcs, particularly noticeable away from the neutral surface, is inversely proportional to Poisson's ratio.

The curvature of the transverse section of the material, known as anticlastic curvature, reveals how the material bends in directions orthogonal to the primary direction of bending.