Introduction to Continuum Mechanics 3 Episode 1 ppsx
Introduction to Continuum Mechanics 3 Episode 1 ppsx
... (2A1.2)
3
etc.
x
Contents
6 .17 Dissipation Functions
for
Newtonian Fluids
38 3
6 .18 Energy Equation
for a
Newtonian Fluid
38 4
6 .19 Vorticity Vector
38 7
6.20 Irrotational Flow
39 0
6. 21
... Spectra
4 73
8.4
Generalized Linear
Maxwell
Fluid
with
a
Continuous Relaxation Spectrum
474
Part
A
Permutation Symbol
7
d
11
=
d
22
= d
33
= 1
d
12
=d
13
=d...
Introduction to Continuum Mechanics 3 Episode 3 ppsx
... of
the
tensor
T.
(b) If
11 1 ,11 2 ,1 13
are the
principal directions, write
[T]
n
(c)
Could
the
following matrix represent
the
tensor
T in
respect
to
some basis?
2B 41.
Do the
previous ... the
motion
of a
continuum
can be
described
either
by
the
pathlines equation
Eq.
(3. 1. 1)
or by its
displacement vector
field
as
given
by Eq.
(3. 5 .1) .
Example...
Introduction to Continuum Mechanics 3 Episode 2 doc
... Q 23
TII
T^
2
7 *33
^ 13
0 23
033
T
31
T
32
T
33
0 31 032 033
or
PartB
Symmetric
and
Antisymmetric
Tensors
35
This
is the
transformation
law for the
components
of a
vector. Thus,
fy ... Eq.
(2B 13 . 1a)
reads
Tn
T'n
T^\
\Q
n
Q
2l
0
31
1
|"r
n
T
n
T
13
\
\Q
n
Q
n
Qi3\
<
2B 13
-
lb
)
T
2l
T
22
T
23
~
Ql2
Qn
032
T
21...
Introduction to Continuum Mechanics 3 Episode 4 docx
... existence
of
single-valued continuous solutions
#1, #2
an
^
U
3
°f
the
six
equation
Eq.
(3. 16 .1)
to Eq.
(3. 16 .6)
are
11 8
Kinematics
of a
Continuum
and
Thus,
the
equation
is
satisfied,
... from
Eq.
(3. 23. 4),
we
have
That
is
That
is
similarly,
Kinematics
of a
Continuum
13 3
thus,
for
this material element
(d)
For
dX
=
dS^
and
dX
=
dS
2
e...
Introduction to Continuum Mechanics 3 Episode 5 docx
... (3. 30.9d).
3. 76. Derive
Eqs. (3. 30 .10 ).
3. 77. Using
Eqs. (3. 30 .10 )
derive Eqs. (3. 30 .12 a)
and
(3. 30 .12 d).
3. 78.
Verify
Eqs.
(3. 30. 13
a) and
(3. 30 .13 d).
Kinematics
of a
Continuum
15 1
where
e
0
/
... respect
to the
basis
at the
reference configuration
3. 73. From
Eqs. (3. 30.4a),
obtain Eqs. (3. 30.5).
3. 74.
Verify
Eq. (3. 30.8b)
and
(3....
Introduction to Continuum Mechanics 3 Episode 6 docx
... distribution
of the
stress
vector acting
on the
square
in
thex\
= 0
plane
with
vertices located
at
(0 ,1, 1),
(0, -1, 1),
(0 ,1, -1) ,
(0, -1, -1) .
(b)
Find
the
total
resultant force
and
... Sect.
3. 12 ,
we
obtained [see
Eq.
(3. 12 .4)],
Since
dx
- F
dX
[
see Eq.
(3. 7.2)],
therefore
Equation
(i) is to be
true
for all
dX,
thus
or,
Using Eqs.
(4 .1...
Introduction to Continuum Mechanics 3 Episode 7 potx
... strength
M
t
,
the
state
of
stress inside
the bar is
given
by
Eq,
(5. 13 . 11 ).
Example
5. 13 . 1
For a
circular
bar of
radius
a in
torsion
(a)
find
the
magnitude
and
location
of the
greatest
normal
... 227
Example
5 .3. 1
Find
the
components
of
stress
at a
point
if the
strain matrix
is
and the
material
is
steel
with
A =
11 9.2
GPa
(17 .3
xl()...
Introduction to Continuum Mechanics 3 Episode 8 pps
... Eqs.
(4.8 .1) ],
(noting that there
is
no
z
dependence).
The
Elastic
Solid
279
From
the
compatibility equations, Eqs. (3. 16 .8), (3. 16 .9)
and
(3. 16 .7),
we
have
Thus,
G(XI
1)
= a
... — 0 is
automatically
satisfied
together with
On the
other
hand, since
Q&
= 1, we
have
This requirement leads
to
That
is,
Similarly,
the
equation
Ci 233
=
0
leads...
Introduction to Continuum Mechanics 3 Episode 9 pptx
... (5 .33 .9)
and
(5 .33 .11 ),
as
those
cor-
responding
to a
change
of
rectangular Cartesian basis, then
we
come
to the
conclusion that
the
constitutive
equation
given
by Eq.
(5 .33 .6)
... stress
field?
5 .15 .
Repeat
Problem 5 .14 , except that
the
displacement components
are
ui
=
kX
2
X
3
,
u
2
=
kX
1
X
3
,
u
3
=
kX
1
X
2t
k =
10 ~
4
5 .16 . R...
Introduction to Continuum Mechanics 3 Episode 10 pptx
... Navier-Stokes equations, Eqs. (6.7.2) yield:
Equations
(6 .12 .1b)
and
(6 .12 .1c)
state that
p
does
not
depend
on*
2
and
*
3
.
If
we
differentiate
Eq.
(6 .12 .
la)
with
respect
to
... given
by Eq.
(6. 13 . 3),
however,
the
driving force
now is
Newtonian
Viscous
Fluid
36 1
These
are
known
as the
Navier-Stokcs
Equations
of
motion
for
incompressible...
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