Chapter 2: Elastic Stress & Strain Relationships

Elastic Stress & Strain Relationships begins by defining the state of stress through normal and shear components, establishing the symmetry of shearing stresses which simplifies the description from nine to six independent variables. The discussion advances to stress transformations on oblique planes for both two-dimensional plane stress and complex three-dimensional states, introducing the concept of principal stresses—the maximum and minimum normal stresses acting on planes where shear stress is absent. To visualize these transformations, the sources detail the graphical utility of Mohr's Circle for both 2D and 3D systems, which represents all possible stress states within a body. A significant portion is dedicated to the stress tensor, utilizing index notation and Einstein summation to express how stress components transform across coordinate systems while maintaining invariant properties like the trace of the stress matrix. Similarly, the nature of strain is analyzed through displacement, distinguishing between volume-altering dilatation and shape-changing distortion. These concepts culminate in the generalized Hooke’s Law, which links stress and strain via essential elastic constants such as Young’s modulus, Poisson’s ratio, the shear modulus, and the bulk modulus. The chapter also explores how total stress and strain are partitioned into hydrostatic components, which affect volume, and deviatoric components, which drive shape changes. Advanced topics include the calculation of strain energy density, the impact of crystallographic anisotropy on elastic properties across various crystal systems, and the critical role of stress concentrations caused by geometric discontinuities like holes or notches. Finally, it introduces the Finite Element Method (FEM) as a powerful computational tool for determining stresses and deflections in complex structural geometries by dividing them into small, discrete elements connected at nodes.