Page 192 - Introduction to Continuum Mechanics

Components of Stress Tensor 177

Since ^ is the stress vector acting on the plane whose outward normal is e^, it is clear from
Eq. (4.3.3a) that TU is its normal component and TI\ andT 31 are its tangential components.
Similarly, T^ is the normal component on the C2-plane and 7\2, 1*32 are the tangential
components on the same plane, etc.
We note that for each stress component T^, the second index] indicates the plane on which
the stress component acts and the first index indicates the direction of the component; e.g.,
7\2 is the stress component in the direction of ej acting on the plane whose outward normal
is in the direction of 62- We also note that positive normal stresses are also known as tensile
stresses and negative normal stresses as compressive stresses. Tangential stresses are also
known as shearing stresses. Both TI\ and T$i are shearing stress components acting on the
same plane (the e rplane), thus the resultant shearing stress on this plane is given by

the magnitude of this shearing stress is given by

Similarly, on C2-plane

and on C3~plane

From t = Tn, the components of t are related to those of T and n by the equation

Or, in a form more convenient for computations,

Thus, it is clear that if the matrix of T is known, the stress vector t on any inclined plane is
uniquely determined from Eq. (4.3.5b). In other words, the state of stress at a point is
completely characterized by the stress tensor T. Also since T is a second-order tensor, any one
matrix of T determines the other matrices of T, see Section 2B13 of Chapter 2.