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ENGN 1750: Advanced Mechanics of Solids
Homework 7
(85 points total)
1
1. (12 points) Epitaxial thin films: If two single crystals share a plane interface, and if
every boundary atom on one side of the interface is in registry with the lattice site of the
crystal on the other side, then the interface is said to exhibit epitaxial bonding. In most
situations, the two crystals belong to the same crystal class. During epitaxial growth, a
substrate crystal is used as the growth template and the film crystal is deposited atom
by atom at a high temperature such that the film atoms register with the lattice sites
of the substrate. This deposition takes place one atomic layer at a time and the film
becomes a perfect crystal. When the film is identical to the substrate material, we have
homoepitaxy; when the film material differs from the substrate, we have heteroepitaxy.
In heteroepitaxy, the film atomic spacing differs from its natural equilibrium spacing,
creating a lattice strain referred to as the mismatch strain. Assuming the substrate is
very thick compared to the film, the mismatch strain is entirely accommodated by the
film (i.e., the substrate retains the equilibrium spacing of its own lattice).
(a) Calling the film equilibrium atomic spacing af and the substrate equilibrium
atomic spacing as, provide an expression for the in-plane lattice mismatch strain
ǫm. Refer to the film plane as the e1-e2 plane.
(b) Assuming the film to be made from an isotropic material with elastic constants E
and ν, determine expressions for the film stress state and for the through-thickness
film strain ǫ33 as a function of mismatch strain ǫm.
22. (16 points) Consider a thin plate consisting of three bonded elastic layers, shown below:
!!
!!
L
L
B
A
B 3E/2, ν = 1/4, α
E, ν = 1/3, 2α
3E/2, ν = 1/4, α e2
e3
e1
All three layers have the same thickness t with t ≪ L and are made from isotropic,
linear elastic materials. The Young’s modulus, Poisson’s ratio, and coefficient of ther-
mal expansion of Material A are E, 1/3, and 2α, respectively, and those of Material B
are 3E/2, 1/4, and α, respectively.
The plate is unconstrained in all directions and is subjected to a homogeneous tem-
perature increase ∆T . We denote the strain of the plate in the e3-direction as ǫ, and
due to symmetry, the lateral strains of the plate in the e1- and e2-directions are equal
and denoted by ǫlat.
Determine the effective thermal expansion coefficients of the plate in the lateral and
e3-directions,
α¯1 = α¯2 =
ǫlat
∆T
and α¯3 =
ǫ
∆T
,
respectively.
3. (12 points) Consider a solid cylindrical shaft of outer radius R subjected to an axial
tensile load F and a torsional moment T .
(a) Write down the matrix of the stress tensor in a cylindrical coordinate system with
z-axis aligned with the central axis of the shaft.
Suppose the shaft is made from a ductile material with a yield stress in tension of σY
that obeys the von Mises yield criterion.
(b) If the applied axial load F is such that the resulting axial stress is one half of σY,
then at what torque T will the shaft first yield?
Suppose the shaft is made from a brittle material with an ultimate tensile stress of σu.
(c) If the applied axial load F is such that the resulting axial stress is one half of σu,
then at what torque T will the shaft fail?
34. (10 points) Consider a plate of thickness h that separates two chambers that are held
at different temperatures T1 and T2, respectively, where T1 > T2, as shown below.
T1 > T2 T2h
Plate
Chamber 1 Chamber 2
The plate is made from isotropic, ductile material with Young’s modulus E, Poisson’s
ratio ν, thermal expansion coefficient α, and yield stress in tension σY. The boundary
of the plate is constrained so that the strain components within the plane of the plate
– the e2-e3 plane – are zero, but the through-thickness direction – the e1-direction – is
unconstrained. Under steady-state conditions, the temperature varies linearly through
the thickness. The plate is strain-free at an initial temperature of T2. Determine the
value of T1 at which the plate yields by the Tresca yield criterion.
45. (15 points) Consider the problem of a flat elastic substrate being indented by a rigid
sphere, discussed in Abaqus Handout 4 and shown below.
R
F
d
Substrate
Rigid
indenter
Rigid
indenter
Substrate
(a) (b)
4mm
4mm
1mm
The substrate is made of aluminum with E = 70GPa and ν = 0.3, and the spherical
indenter has a radius of R = 1mm.
(a) For the case of the rigid indenter, submit a plot of the calculated force/displacement
relation along with the analytical relation
F =
4
3
E
1− ν2
R1/2d3/2.
Comment on how they compare. What might be the source of any discrepancy?
(b) Now consider the case of a deformable indenter. In your analysis, make your quar-
ter circle indenter deformable, and make it of the same material as the substrate,
as shown below. (You will need to create a new model to do this.)
5Substrate
Deformable
indenter
u1 = 0 u1 = 0
u2 = 0
Apply indenter displacement
1mm
4mm
4mm
Apply the downward displacement to the top surface of the indenter. You will
need to create a set for this top surface in order to obtain RF2 output for all nodes
on this surface, which you will then sum to obtain the total applied force. Submit
a plot of the calculated force/displacement relation along with the analytical
relation for this case,
F =
2
3
E
1− ν2
R1/2d3/2. (1)
How does this compare to that which you calculated for part (a)? Does this make
sense? Also submit a contour plot of the Mises stress on the deformed shape.
