A solid is subjected to stresses as shown by the arrows in the figure below. Indicate the indices for each of the stress components, and whether the stresses should be positive or negative.
Suppose that the stress tensor field in a body is given by
Find the body force distribution required to maintain equilibrium. (, , and are in meters). Show units.
The equation of equilibrium is
The required body forces are (in MN/m)
Find the surface tractions at the internal point in the body on an internal surface with a surface normal .
The surface traction is given by
The stress at point is
The traction vector is (after converting into a unit normal)
Find the hydrostatic and deviatoric stress at the point .
The hydrostatic stress is given by
In this case,
The deviatoric stress is given by
Find the principal stresses at this point.
The principal stresses can be found using the equation
where is a principal stress. In expanded form,
Substituting the values of stress into the above equation,
Thus, the first possible value of MPa. Also,
The principal stresses are (in MPa)
Find the principal direction corresponding to the intermediate principal stress.
The directions of the principal stresses can be found using the equation
For the principal direction (corresponding to the principal stress , we have,
gives us and
Taking squares of both sides,
So we get,
The direction corresponding to the intermediate principal stress is
The symmetry of the stress tensor can be derived from a certain balance principle. Name the principle and write it down in index notation.
The balance principle is
In index notation