1 Stress

The stress distribution inside a body is given by:

τ (x, y, z) =

−P + ρgy 0 0

0 −P + ρgy 0

0 0 −P + ρgy

, (1)

where P, ρ and g are constants.

1. What is the distribution of the stress vector on the six faces of the block pictured here? Compute

and draw.

2. Find the total resultant force acting on the face located at y = 0, and the face located at x = 0.

2 Spherical stress and strain

1. Detailing your calculations using indicial notation, and starting from Hooke’s law and the law of

static equilibrium linking external volume forces (weight, EM forces. . . ) and the stress tensor,

derive the following equation:

fi + µ∂2

j ui + (λ + µ)∂i∂juj = 0, (2)

where µ and λ are the Lamé parameters, µ being the shear modulus. Note that the index in ∂

2

j

is considered repeated in the sense of Einstein’s convention.

2. Rewrite the equation above in vector, or dyadic, form, i.e. with ~f, ∇~ . . . Even though the

derivation using indicial notation was done with Cartesian coordinates in mind, the vector form

of Navier’s equation in valid in any coordinate system.

1

PHY454H1 Problem Set #2 Due 1 March 2017

3. We now consider a spherical problem, in which the external volume forces are all radial ( ~f = frˆ),

the only dependence is on the radial coordinate r and no other, and in which the only displacement

is also in the radial direction (~u = u(r)ˆr).

Show that:

f + (2µ + λ)

d

dr

1

r

2

d(r

2u)

dr

= 0. (3)

You may find some useful formulae on the Wikipedia page “Del in cylindrical and spherical

coordinates’’:

https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates#Del_formula.

4. We now compute the stresses and strains of a spherical shell under pressure. We assume that the

deformations due to the external volume forces are negligible when compared to those induced

by these pressure forces. Show that:

u = Ar + B/r2

, with A and B two constants to be determined later. (4)

5. The diagonal terms of the strain tensor in spherical coordinates (which are the only non-zero

ones) are, under our symmetry assumptions:

Urr =

du

dr

and Utt =

u

r

, (5)

where the index ‘t’ refers to the two tangential directions. The strain is the same in each direction

in an isotropic medium under isotropic contraints such as ours.

Compute Urr and Utt (retain A and B for now), and briefly explain why Utt 6= 0: what does it

term mean in terms of the deformation of a solid element? (2 lines, or a quick sketch).

6. Hooke’s law is still valid in spherical coordinates:

τrr = 2µUrr + λ(Urr + 2Utt) and τtt = 2µUtt + λ(Urr + 2Utt). (6)

Compute τrr and τtt in terms of (µ, K), with K the bulk modulus, instead of (µ, λ).

7. Let a and b the inner and outer radii, respectively, of our spherical shell, and Pa and Pb the

corresponding pressure. Write down the two boundary conditions these two pressures correspond

to, and use them to show that:

u =

a

3

b

3

b

3 − a

3

Pa/b3 − Pb/a3

3K

r −

Pb − Pa

4µr2

. (7)

8. Numerical applications: a spherical submarine of inner radius a = 75 cm, with a cast steel wall

(K = 200 GPa, µ = 75 GPa) that is 10 cm thick, dives to depths d = 10 m, 1 km and 11 km.

By how much does the inner radius shrink at each depth? Use Pa = 1.0 × 105 Pa for both the

inner pressure and the atmospheric pressure, g = 9.8 m2

s

−1 and ρocean ≈ 1030 kg m−3

.

3 Fluid kinematics

All questions are independent.

1. We consider a steady, axisymmetric flow of a compressible fluid of mass density. Under these

assumptions, the mass conservation equation in cylindrical coordinates is:

1

r

∂(ρrvr)

∂r +

∂(ρvz)

∂z = 0. (8)

Define a stream function, different than the one we saw in class, such that this equation of

continuity is automatically satisfied.

2

PHY454H1 Problem Set #2 Due 1 March 2017

2. Consider the following components of a two-dimensional, unsteady flow:

vx =

x

1 + t

and vy =

2y

2 + t

. (9)

Compute the streamlines of this flow and draw a couple of them at a couple different times (4

total then).

3. What field lines are obtained by integrating the coupled ODEs

dx

dt

= vx(~x, t) and dy

dt

= vy(~x, t)? (10)

Calculate these field line for eqns. (9), and draw a couple.

3