MECE3070, Fall 2023
Mechanics of Engineering Materials
Mid-term examination
Wednesday, November 22, 2023
Student name and Signature: Invigilator:
(for invigilator)
Grade: Marker’s name and signature:
(for marker)
… / 100
General instructions:
- There are six questions, with a total of 110 points.
- You are allowed to have a “cheat sheet”: one side of a A4 paper. The
cheat sheet will be collected at the same time of the copies.
- You are allowed to have two draft papers (A4). Raise your hand to ask
for more if you need. Your draft papers will be collected at the same
time of the copies.
1
Exercise 1: True/False [16 points]
True False
1. After yielding is reached, when the load is removed, •
strain returns to zero.
2. An incompressible solid has Poisson ratio equal to 0.5. •
3. Shear modulus G could be written in terms of •
Poisson’s ratio and Young’s modulus.
4. The stress vector applied on a surface is always •
perpendicular to that surface.
5. The direction of maximum shear stress is obtained by •
rotating 45o from the current stress state.
6. The stress tensor is always symmetric. •
7. The trace (summation of diagonal of any stress tensor
and strain tensor) changes when the coordinate system
is rotated.
8. The principal directions of stress tensors are
perpendicular to each other.
2
Exercise 2: Tensile test [20 points]
Aluminum-lithium alloys are used to build certain parts of an airplane
because of their relatively low density and their tunable mechanical
properties via adjusting Li concentration.
Compared to a bar made of pure Al, adding Li into Al will create material
(Al-Li alloy) with higher Young’s modulus and higher yield strength. If we
add Al3Li in the Al-Li alloy, the Young’s modulus of the new material, called
(Al-Li, Al3Li), will not change (compared to Al-Li), but the elastic limit can
increase considerably.
Figure below shows the stress-strain curve resulted from three tensile tests
on the three bars made of Al, Al-Li, and (Al-Li, Al3Li).
a. With the description above, match the number of the tensile test with
the corresponding materials (Al, Al-Li, or (Al-Li, Al3Li)).
[5 points]
Al: Al-Li: Al-Li, Al3Li:
b. Among the material 2 and 3, which one can store more energy per
volume before reaching 𝜖0 . Represent graphically the difference of
elastic energy density between the material 2 and 3. [5 points]
Answer:
3
c. Considering the stress-strain curve in the figure below of a bar made
with the alloy (Al-Li, Al3Li). The yield strength of the material
corresponds to a strain of 0.7. With this graph, deduce the following
properties of the materials:
- Yield strength [5 points]
- Young’s modulus [5 points]
Yield Strength:
Young’s modulus:
4
Exercise 3: Stress tensors [20 points]
Consider the following two stress tensors:
10 10
𝜎
̿̿̿1 = ( )
10 10
And
15 5√3
𝜎
̿̿̿2 = ( )
5√3 5
a. Find the two principal matrices. [10 points]
b. These two tensors represent the same stress state. Explain why. [5
points]
c. Find the rotation angle which transforms the stress tensor 𝜎
̿̿̿1 to 𝜎 2 [5
̿̿̿.
points]
Answer:
a. Two principal matrices:
5
b.
c. Rotation angle:
6
Exercise 4: Strain tensors [20 points]
In an (𝑥, 𝑦), a plate is presented by the points: 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1
The points of the plate undergo the displacement represented by the
following vector:
⃗ (𝑥, 𝑦) = 𝑢𝑥 (𝑥, 𝑦)𝑖̂ + 𝑢𝑦 (𝑥, 𝑦)𝑗̂
𝑢
with:
𝑢𝑥 (𝑥, 𝑦) = 14𝑥 − 5𝑦 𝑢𝑦 (𝑥, 𝑦) = −5𝑥 − 10𝑦
a. Build the strain tensor. [10 points]
b. Find the principal strain. [10 points]
Answer:
a. Strain tensor
7
b. Principal strain:
8
Exercise 5. Thin-wall vessel [18 points]
Consider a cylindrical vessel with closed ends of radius 2 m and thickness of
50 mm to be constructed by filament winding, in which fibers are laid down
at a prescribed helical angle 𝛼. An inner pressure of 3 MPa is imposed.
a. What is the longitudinal normal stress to the fiber if 𝛼 = 60° ? [10
points]
b. If the longitudinal normal stress exceeds 350 MPa, the fiber will break,
what is the maximum inner pressure that vessel can hold? [8 points]
Longitudinal direction
Answer:
a. Longitudinal normal stress:
9
b. Maximum inner pressure:
10
Exercise 6. Yield criteria [16 points]
𝑟
Consider a tube of radius r and thickness 𝑡 (with ≫ 1 , so we can consider
𝑡
the thin-wall vessel condition). At first, the tube first is closed (containing
only air with negligible pressure 𝑝0 ~ 0) and immersed in water at the depth
which is much larger than the radius of the tube, so that we can assume that
the hydrostatic pressure applied to the tube 𝑝1 is constant (Figure (a)), and
that water surrounds all side of the tube.
a. Write the stress tensor at the middle of the tube in the cylindrical
coordinates in terms of 𝑝1 , 𝑟, 𝑡. [5 points]
The yield strength of the material making up the tube is 𝜎𝑦 . Write the
Tresca and Von Mises criteria for this condition in terms of 𝑝1 , 𝑟, 𝑡 and
𝜎𝑦 . [5 points]
b. The tube is used to link two tanks of oil. The pressure at this depth in
the oil tanks is 𝑝2 and is constant. When the gates are open, the tube
is filled with oil, as shown in Figure (b). Express the stress tensor at
the middle of the tube in terms of 𝑝1 , 𝑝2 , 𝑟, 𝑡. [3 points]
Write the Tresca and Von Mises criteria for this condition in terms of
𝑝1 , 𝑝2 , 𝑟, 𝑡 and 𝜎𝑦 . [3 points]
11
a. Closed, empty tube:
Stress tensor:
Tresca criterion:
Von Mises criterion:
b. Open-ended tube:
Stress tensor:
Tresca criterion:
Von Mises criterion:
12
