Muddy Card Responses Lecture M20 12/1/2003
There were a lot of different matrices presented today. How can I keep them all straight.
I
certainly don’t expect you to memorize all of this. You should be aware that they all stem
from the definitions of the engineering elastic constants – which are all defined in terms of
simple loading (i.e one component of stress only). This means that the compliance form
(known stress leading to resulting strain) is the most straightforward to follow directly, and I
would expect you to be able to explain where this comes from, if not actually remember the
terms in it.
Ê
1

n

n
0
0
0
ˆ
Ê
e
x
ˆ
Á
E
E
E
˜
Ê
s
x
ˆ
Á
˜
Á

n
1

n
0
0
0
˜
Á
˜
Á
e
y
E
E
Á
s
y
˜
˜
Á

E
n

n
1
˜
Á
e
z
˜
Á
E
0
0
0
˜
Á
s
z
˜
E
E
=
Á
˜
Á
1
Á
g
zy
0
0
0
0
0
˜
Á
Á
t
zy
˜
˜
Á
G
˜
˜
Á
g
zx
˜
Á
0
0
0
0
1
0
˜
Á
t
zx
˜
Á
˜
Á
˜
Ë
g
xy
¯
Á
Á
0
G
1
˜
˜
Ë
t
xy
¯
Ë
0
0
0
0
G
¯
I do expect you to be aware of the fact that anisotropic materials exist and to have some
physical understanding of why there are different elastic properties for a composite material
in particular.
Connection between tensor and engineering terms was confusing.
Fair enough. The most
straightforward route to understanding this is by understanding what the terms in the
compliance matrix mean (known stress to unknown strain – given by matrix in previous mud
response). These connect directly to the definitions of the Engineering elastic constants, and
how they are defined. Once you understand this, it is a small step to convert the engineering
notation stress and strain to tensor stress and strain.
What exactly are
l
and
m
in the 6x6 matrix?
They are just convenient groupings of E and
n
.
All we have done is invert the 6x6 compliance matrix (see previous two questions) to allow
us to go from known strain to unknown stress.
E
m
=
=
G
2 1
+
n
)
(
n
E
l
=
(
1
+
n
)(
1

2
n
)
What exactly are
l
and
m
? Starting to understand tensors….
See previous response
regarding
l
and
m
? I am glad that you are gaining confidence with tensors.
They are not
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really that frightening once you become familiar with them  they are just a convenient way
of representing several simultaneous linear equations.
What exactly are
l
and
m
?
See above.
How are are
l
and
m
calculated?
See above
What physically is G?
It is the shear modulus. It links shear stress to shear strain, defined in
engineering notation.
I am wondering if the matrices make understanding the concepts any easier, because they
have so many zeros.
I guess this depends on one’s point of view. I agree that there are a lot
of terms, and if you had to memorize them it might be confusing. However, the nice aspect
to using a matrix presentation is that you can see quite readily how different terms link stress
to strain in different directions – and that you can relate that back to your physical
understanding of the engineering elastic constants.
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 Winter '12
 CharlesColeman
 Materials Science, Aeronautics, Astronautics, Shear Stress, Shear strain, Transverse isotropy

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