Stress

Stress
Surface traction
The forces acting in the interior of a continuous body .Suppose that part of a body β occupies a region R which has surface φ, n a unit vector directed along the outwardnormal to φ at P, and δS the area of an element of φ which contains P. we assume that φ and R possess any necessary smoothness and continuity properties. It is also assumed that on the surface elementwith area δS , the material outside R exerts a force
δp = t^(n) * δS
On the material inside R. the force δp is called the surface force and t^(n) the mean surface traction transmitted acrossthe element of area δS from the outside to the inside of R.
Clearly t^(n) will depend on the position of P and the direction of n. δS -> 0 , t^(n) tends to a finite limit which isindependent of the shape of the element with area δS.
t^(n) = Lim δp/ δS , δS->0
Call t^(n) the surface traction at the point P on the surface with normal n.
It is important to remember that ingeneral t^(n) does not coincide in direction with n .

Components of stress
At P, there is a vector t^(n) associated with each direction through P. in Particular, given a system ofrectangular Cartesian coordinates with base vectors ei. There is such a vector associated with the direction of each of the base vectors. Lets t1 be the surface traction associated with the direction of e1,from the positive to the negative side. Surface traction vectors t2 and t3 are similarly defined in relation to the direction of e2 n and e3.
Now resolve the vectors t1,t2 and t3 intocomponents in the coordinate system with base vectors ei, as follows:
T1= T11e1 + T12e2 + T13e3
T2= T21e1 + T22e2 + T23e3
T3= T31e1 + T32e2 + T33e3

These equations may be written in matrixfrom as:
T1 (t11 t12 t13) (e1)
T2 = (t21 t22 t23) (e2)
T3 (t31 t32 t33)(e3)
Or, using the summation convention, as
Tij = Tijej (I,j =1,2,3)
Since ei * ej = δij , it...