Page 189 - Introduction to Continuum Mechanics
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174 Stress
Fig. 4.1
Next, let S be a surface (instead of a plane) passing the point P. Let AF be the resultant force
on a small area AS on the surface 5. The Cauchy stress vector at P on 5 is defined as
We now state the following principle, known as the Cauehy's stress principle: The stress vector
at any given place and time has a common value on all parts of material having a common
tangent plane at P and lying on the same side of it. In other words, if n is the unit outward
normal (i.e., a vector of unit length pointing outward away from the material) to the tangent
plane, then
where the scalar t denotes time.
In the following section, we shall show from Newton's second law that this dependence on
n can be expressed as
where T is a linear transformation.
4.2 Stress Tensor
According to Eq. (4.1.4) of the previous section, the stress vector on a plane passing through
a given spatial point x at a given time t depends only on the unit normal vector n to the plane.
Thus, let T be the transformation such that
