In extending the direct method of stress analysis presented in previous modules to geometrically more complex structures, it will be convenient to have available somewhat more general mathematical statements of the kinematic, equilibrium, and constitutive equations; this is the objective of the present chapter. These equations also form the basis for more theoretical methods in stress analysis, as well as for numerical approaches such as the finite element method. We will also seek to introduce some of the notational schemes used widely in the technical literature for such entities as stress and strain. Depending on the specific application, both index and matrix notations can be very convenient; these are described in a separate module.
- 3.3: Tensor Transformations
- One of the most common problems in mechanics of materials involves transformation of axes. For instance, we may know the stresses acting on xy planes, but are really more interested in the stresses acting on planes oriented at, say, 30∘ to the x axis, perhaps because these are close-packed atomic planes on which sliding is prone to occur, or is the angle at which two pieces of lumber are glued together in a "scarf" joint. We seek a means to transform the stresses to these new x′y′ planes.
Thumbnail: Idealized stress in a straight bar with uniform cross-section. (CC BY-SA 3.0; Jorge Stolfi via Wikipedia)