1. Using suitable units, the stress at a particular point in a solid is found to be
2 1 −4
𝜎𝑖𝑗 = [ 1 4 0 ]
−4 0 1
Determine the traction vector on a surface with unit normal (cos θ, sin θ, 0), where θ is a
general angle in the range 0 < θ < π. Plot the variation of the magnitude of the traction
vector |𝑇 𝑛 | as a function of θ.
2. Determine the spherical and deviatoric stress tensors for the stress states given in
Problem 1 (above)
3. Show that equilibrium equations are identically satisfied if the stresses are expressed in
the form
𝜎𝑥 =
𝜕 2𝜙
,
𝜕𝑦 2
𝜎𝑦 =
𝜕 2𝜙
𝜕 2𝜙
,
𝜏
=
−
𝑥𝑦
𝜕𝑥 2
𝜕𝑥𝜕𝑦
where 𝜙(x,y) is an arbitrary function. This stress representation will be used later on in
this course to establish a very useful solution scheme for two-dimensional problems.
4. A one-dimensional problem of a prismatic bar (see the following figure) loaded under its
own weight can be modeled by the stress field
𝜎𝑥 = 𝜎𝑥 (𝑥), 𝜎𝑦 = 𝜎𝑧 = 𝜏𝑥𝑦 = 𝜏𝑥𝑧 = 𝜏𝑦𝑧 = 0,
with body forces 𝐹𝑥 = 𝜌𝑔, 𝐹𝑦 = 𝐹𝑧 = 0, where 𝜌 is the mass density and 𝑔 is the local
acceleration of gravity. Using the equilibrium equations, show that the nonzero stress will
be given by 𝜎𝑥 = 𝜌𝑔 (𝑙 − 𝑥), where 𝑙 is the length of the bar.