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1.050:
Continuum Strength Models (HW#5)
Due: October 10, 2007
MIT – 1.050 (Engineering Mechanics I)
Fall 2007
Instructor: Markus J. BUEHLER
Team Building and Team Work:
We strongly encourage you to form Homework teams of three
students. Each team only submits one solution for correction. We expect true team work, i.e. one where
everybody contributes equally to the result. This is testified by the team members signing at the end of the
team copy a written declaration that "the undersigned have equally contributed to the homework". Ideally,
each student will work first individually through the homework set. The team then meets and discusses
questions, difficulties and solutions, and eventually, meets with TA or instructor. Important: Specify all
resources you use for your solution.
The following set of exercises is designed to train you in the use of equilibrium and strength
models for continuum systems. For each exercise, show us how you came to your answer and
result. We highly encourage you to make drawings where appropriate.
1.
Stress state and strength criterion
: We consider the next two stress tensors (parameter
p
>
0), given in a Cartesian system of coordinates of basis
(
e
r
1
,
e
r
2
,
e
r
3
)
:
⎡
3
0
0
⎤
⎡
0
−
2
0
⎤
⎢
⎥
⎢
⎥
σ
=
p
0
2
0
;
=
p
−
2
−
3
0
⎢
⎥
⎢
⎥
⎢
0
0
⎥
⎢
⎣
−
1
⎥
⎦
⎣
0
0
3
⎦
For each stress tensor, obtain:
a.
The eigenvalues and eigenvectors.
b.
Display the Mohr circles.
c.
Find the minimal value of
p
at which the material fails for a material governed by a
Tresca strength criterion of cohesion
c
=
1
.
d.
Find the two failure planes (in the
(
e
r
1
,
e
r
2
,
e
r
3
)
basis) and the stress vectors that act on
them.
e.
What is the underlying assumption for the Tresca strength model?
f.
Answer the same questions (a through d) in the case of a material governed by a Mohr
Coulomb criterion of cohesion
c
=
1
and of friction angle
ϕ
=
π
/5.
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Page 2 of 5
2.
A
nano
truss
:
Advances in nanoscience and nanotechnology
*
now enable us to build
structures at molecular scales, with atomistic precision.
An important building block of
many nanostructures are carbon nanotubes (CNTs), a particularly sturdy form of carbon
arranged in a tubular structure.
A picture of a CNT is shown below.
From a mechanical perspective, CNTs are particularly intriguing as they are one of the
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This note was uploaded on 11/29/2011 for the course CIVIL 1.018j taught by Professor Markusbuehler during the Fall '08 term at MIT.
 Fall '08
 MarkusBuehler

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