Stress

Stress is a tensor quantity that describes the mechanical force density (force per unit area) on the complete surface of a domain inside a material body. A stress exists wherever one part of a body exerts a force on neighboring parts. Its orientation is not tied to any particular directions that are intrinsic to the material like, say, crystallographic axes. It is thus distinct from the matter tensors that were discussed in the preceding chapter, all of which have definitive orientations within a crystal or other anisotropic material; it is called a field tensor (and so is strain). The definition of stress depends on the concept of a continuum. Let f(xi) be a single-valued function defined for every point xi in a region. This function is said to be continuous at the point xi if the following holds for all paths of approach of xi to °xi:. . . f(xi) → f(°xi) as xi → °xi (4.1)· . . . Equivalently, for any number ∊, no matter how small, there exists a neighborhood of nonzero radius around the point xi in which: . . . . 〈f(xi) − f(°xi)〉2 < ∊,­ (4.2) . . . for all points xi in that neighborhood. A continuum is an idealized material whose physical attributes are continuous functions of position. Thus neighboring points remain neighbors, and a continuum cannot have gaps or jumps (discontinuities) in its properties. Surfaces bounding gaps or defining discontinuities must be specially treated in continuum mechanics. Examples are surfaces between two fluids of differing density or viscosity, or between solids with different thermal conductivity or elastic properties. Real materials are never continua; they are discontinuous at the atomic scale, and often at larger scales as well. The notion of a continuum is, therefore, only a macroscopic approximation, but it allows useful mathematical approaches to the treatment of real phenomena.