Advanced Solid Mechanics
Assignment - 4: Balance Laws
1. Determine which of the following Cauchy stress fields are possible within
a body at rest assuming that there are no body forces acting on it:

 3 2 2
xy 3 0
−2x y
− 41 y 4 0 ,
(a)
σ =  xy 3
0
0
0


3yz z 2 5y 2
(b)
σ =  z 2 7xz 2x2  ,
5y 2 2x2 9xy


3x + 5y 7x − 3y 0
(c)
σ = 7x − 3y 2x − 7y 0 ,
0
0
0


3x −3y 0

(d)
σ = −7x 7y 0 ,
0
0 0


7x −3x 0
(e)
σ = 7y 3y 0 ,
0
0 0
where the components of the stress are with respect to orthonormal
Cartesian basis and (x, y, z) denote the Cartesian coordinates of a typical material particle in the current configuration of the body.
2. The Cartesian components of the Cauchy stress tensor in a plate at
rest is


−2x2
−7 + 4xy + z
1 + x − 3y
.
0
σ = −7 + 4xy + z 3x2 − 2y 2 + 5z
1 + x − 3y
0
−5 + x + 3y + 3z
1
Find the body force that should act on the plate so that this state of
stress is realizable in the body.
3. A body at rest is subjected to a plane state of stress such that the
non-zero Cartesian components of the Cauchy stress are σxx , σxz and
σzz . Derive the equilibrium equations for this special case. Then, show
that in the absence of body forces the equilibrium equations hold if,
σxx =
∂ 2φ
,
∂z 2
σxz = −
∂ 2φ
,
∂x∂z
σzz =
∂ 2φ
,
∂x2
where φ = φ̄(x, y), for any choice of φ. φ is called the Airy’s stress
potential.
4. Derive form first principles and show that for the plane stress problem in cylindrical polar coordinates with the non-zero cylindrical polar
components of the Cauchy stress being σrr , σrθ and σθθ , as shown in
figure ??, the equilibrium equations in the absence of body forces with
the body in static equilibrium are
∂σrr 1 ∂σθr σrr − σθθ
+
+
= 0,
∂r
r ∂θ
r
∂σrθ 1 ∂σθθ 2σrθ
+
+
= 0.
∂r
r ∂θ
r
5. Is any stress field that satisfies the equilibrium equations realizable in
a given body? Discuss.
2
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Figure 1: Figure for problem 4. Variation of the cylindrical polar components
of the Cauchy stress over an infinitesimal cylindrical wedge, in plane stress
state
3