23.9 : Three-Dimensional Analysis of Strain

Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal stress axes are orthogonal, representing directions where the stress does not induce shear within the material.

Mohr's circle is an essential tool in strain analysis. It provides a graphical representation of the stress states at a point and evaluates strain components when the element rotates around a principal axis, such as the n-axis. This analysis focuses on plane strain transformations, where strains at the origin of the n-axis are zero, simplifying the determination of maximum and minimum strains depicted on opposite sides of Mohr's circle.

In scenarios like thin plates under plane stress, the n-axis becomes a principal stress axis. Here, the principal strain along the n-axis directly correlates with the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps pinpoint the locations and magnitudes of maximum shearing strain, which are crucial for predicting material behavior under load and ensuring the safety and reliability of structural designs.