Problem Set 4: Mechanical properties of materials

Due Nov 5, 2024; 11:59pm. Upload onto Canvas

118 pts, scaled to 100 pts.

Problem 1: 6061 aluminum (10 pt)

6061 is one of the most common aluminum alloys. A polycrystalline (isotropic) sample of a
cylindrical piece of 6061 aluminum is stressed in compression along the two flat faces of the
cylinder. The original and final diameters are 20.000 and 20.025 mm, and its final length is 51.05
mm. You may use the following values: Young’s Modulus (E): 69 GPa; Shear Modulus (G): 26
GPa.

a. Compute the Poisson ratio and the original length of the material (4 pt)
b. Compute the stress applied to achieve this deformation (3 pt)
c. Compute the fraction change in volume (Δ𝑉/𝑉) upon this stress (3 pt).
Problem 2: Lennard-Jones (10 pt)

A Lennard-Jones solid has a potential curve V(R) that can be described as

𝜎 !" 𝜎 #
𝑉(𝑟) = 4 𝜖 +, . − , . 0
𝑟 𝑟
a. Plot the potential curve V(r). Let 𝜖 = 0.9 𝑒𝑉 and 𝜎 = 3.4 Å (3 pt)
b. Calculate the equilibrium bond energy and bond length at 0 K (3 pt)
c. Estimate the Young’s Modulus of the bond at 0K (4 pt)
Problem 3: Bulk Modulus (14 pt)

The elastic bulk modulus (K) of a solid is defined as its stiffness under hydrostatic pressure,
wherein 𝜎!! = 𝜎"" = 𝜎$$ = −𝑃, and −𝑃 is the compressive pressure from all directions. Imagine
this as putting a material within a fluid of very high pressure, such as the bottom of the ocean.
Δ𝑉
𝑃 = −𝐾 9 :
𝑉
%&
Here, is the volumetric strain caused by the hydrostatic pressure P. Note that the symbols V
&
and 𝜎 are not related to the ones in problem 3.
a. Show that the volumetric strain is related to the axial stress. Assume that 𝜖 is small,
much less than 1. For this problem, you may assume that the solid is a rectangular prism
with side lengths l1, l2, and l3, but this solution is true in general (5 pt)
Δ𝑉
= 𝜖!! + 𝜖"" + 𝜖$$
𝑉
b. Express K in terms of the Young’s Modulus E and Poisson’s Ratio 𝜈. Assume the
material is isotropic (3 pt).
%&
c. What is the value of 𝜈 for an incompressible solid where = 0 (3 pt)
&
d. Prove that the Poisson’s ratio must be less than or equal to ½ (3 pt)
Problem 4: Toughness (10 pt)

Estimate the toughness of this material from the stress-strain curve. Show that the units for
toughness are given by J m-3 (10 pt)
Problem 5: Shear Stress and strain (15 pt)

A metallic object (gray) is rigidly attached to a wall (blue). The gray metal has lateral
dimensions of 1 cm in all three directions. Assume that the object is isotropic and does not bend.

Fa : 2,000 N applied

3 Fa : 2,000 N applied

2
1 Fb : weight of 100 kg object
Fb : weight of 100 kg object

a. Force Fa equals 2,000 N. What is the stress applied on the metal object? Is this stress
compressive, tensile, or shear (3 pt).
b. Force Fb comes from a rigid beam attached to the right side of the gray object. This beam
weights 100 kg. What is the shear stress from this object? Assume that g = 9.8 m s-2 (3 pt)
c. Write the vector containing the values for the six stresses on the object, which include
both the normal and shear stresses (3 pt)
𝜎!!
⎡𝜎"" ⎤
⎢𝜎 ⎥
⎢ $$ ⎥
⎢𝜎!" ⎥
⎢𝜎!$ ⎥
⎣𝜎"$ ⎦

d. The bulk modulus is measured to be 50 GPa and the poisson ratio is 0.33. What is the
shear modulus? Recall that this object is isotropic (3 pt)
e. Using the compliance matrix for isotropic materials, compute the six strains (3 pt)
𝜖!!
⎡𝜖"" ⎤
⎢𝜖 ⎥
⎢ $$ ⎥
⎢𝜖!" ⎥
⎢𝜖!$ ⎥
⎣𝜖"$ ⎦
Problem 6: FCC Slip plane (10 pt)

Consider a single crystal of silver (FCC) oriented such that a tensile stress is applied along a [001]
direction. Slip occurs on a (111) plane and in a [101] direction.

a. Compute the two angles 𝜙! (angle between tensile stress direction and normal of the slip
plane) and 𝜙" (angle between tensile stress direction and the slip direction). Hint: the angle
between two vectors can be defined using the dot product |𝑎||𝑏| cos(𝜙) = 𝑎 ⋅ 𝑏 (5 pt)
b. If slip is initiated at an applied tensile stress of 1.1 MPa, compute the critical resolved shear
stress (5 pt)
Problem 7: Hall-Petch equation (10 pt)

The yield point for an iron with an average grain diameter of 10 µm is 230 MPa. At a grain
diameter of 6 µm, the yield point increases to 275 MPa. Using the Hall-Petch equation and a
computational plotting tool, plot the grain yield strength as a function of the grain diameter. Do
not sketch the plot.

Problem 8: Strain Hardening (10 pt)

Two previously undeformed specimen of the same metal are to be plastically deformed by
reducing their cross-sectional area. One has a circular cross-section and the other has a
rectangular one. What is the degree of cold working for each material, and which material will
have a higher yield stress?
Problem 9: Strain Engineering for Electronics (14 pt)

Strain is a powerful method to tune material properties; one example is to improve the mobility
of Si in transistors. Recent work by Nguyen Vu and John Heron studied the use of strain to tune
the Magnetic Anisotropy of Thulium Iron Garnet (TmIG). To do so, they grew the film on
different single-crystal substrates such that the TmIG will have the same lattice constant as the
substrate in two dimensions. One substrate Gd3Sc2Ga3O12 (GSGG) has a lattice constant of 12.56
Å. A relaxed (unstrained) TmIG has a lattice constant of 12.35 Å. Both materials are cubic in the
unstrained (relaxed) state. An exaggerated view of this strain is shown below.

TmIG
a= 12.35 Å Strained TmIG film
a3

a1

GSGG substrate GSGG substrate

a. If GSGG substrate is cubic and the TmIG film is strained, what is the effective crystal
system of the strained TmIG lattice? The seven crystal systems are triclinic, monoclinic,
orthorhombic, tetragonal, trigonal, hexagonal, and cubic (2 pt)
b. Assume that the lattice constant in the a1 and a2 direction of the TmIG film equals that of
the GSGG substrate. What is the strain 𝜖!! and 𝜖"" on the TmIG film? Is the strain tensile
or compressive (4 pt)?
c. Find 𝜎!! and 𝜎"" when the Young’s Modulus E = 100 GPa and the poisson ratio equals 0.25
(5 pt)
d. Find the strain 𝜖$$ and the lattice constant a3 in the out-of-plane direction when 𝜎$$ = 0.
Assume that the tensile or compressive strain induced by the lattice directly strains the
individual unit cells (3 pt)
Problem 10: Strengthening Mechanisms (15 pt)

For each problem, identify the material with the higher yield stress. Write one or two sentences
describing the reason, and show your work when needed. Assume the samples are otherwise
identical except for the properties indicated (3 pt each). You may consult a periodic table for this
problem.

a. 1) 99% Fe, 1% Ni solid solution

b. 1) A 98% Fe, 2% C solid solution

c. 1) Cylinder Al cold-worked from 30 mm to 25 mm diameter cross-section

d. 1) Al with a dislocation density of 106 mm

e. 1) Fe with 10 µm grain size