Introduction to Continuum Mechanics 3 Episode 14 pdf
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...
Introduction to Continuum Mechanics 3 Episode 2 doc
... <
2B 13
-
lb
)
T
2l
T
22
T
23
~
Ql2
Qn
0 32
T
21
T
22
T
23
Qzi
Ql2
Q 23
TII
T^
2
7 *33
^ 13
0 23
033
T
31
T
32
T
33
031 0 32 033
or
PartB
Symmetric
and
Antisymmetric
Tensors
35
This
is the
... have
//jr
j2
11
=cos(e
1
,ei)=cos30°=—,
(2i2
=cos
(
e
i»
e
2)
=cosl2
0
0
= ,
(2i3
=
cos(e
1
,e
3
)=cos90
0
=G
^2i=cos(e2,ei)=cos60
0
=...
Introduction to Continuum Mechanics 3 Episode 3 ppsx
... lines
of
every particle
in a
continuum
can be
described
by a
vector equation
of the
Fig.
3. 1
form
where
x =
x^i
+*2
e
2
+JC
3
e
3
*
s
tne
position vector
at
time
t for the
particle
... Eq.
(2D3.17)
we
have
(v)Components
ofdiv
T
Using
the
definition
of the
divergence
of a
tensor,
Eq.
(2C4 .3) , with
the
vector
a
equal
to
the
unit
base
vector
e...
Introduction to Continuum Mechanics 3 Episode 4 docx
... from
Eq.
(3. 23. 4) ,
we
have
That
is
That
is
similarly,
Kinematics
of a
Continuum
133
thus,
for
this material element
(d)
For
dX
=
dS^
and
dX
=
dS
2
e
2
Example
3. 23. 3
Show
that
... dsp
where
m
and n are
unit
vectors
having
an
angle
of ft
between them, then
Eq.
(3. 23. 4) gives
That
is
Similarly
Kinematics
of a
Continuum
1 23
(b)At
the
deform...
Introduction to Continuum Mechanics 3 Episode 5 docx
... 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. 30.8d).
3. 75.
Verify
Eq.( 3. 30.9b)
and
(3. 30.9d).
3. 76. ... (3. 30.9d).
3. 76. Derive
Eqs. (3. 30.10).
3. 77. Using
Eqs. (3. 30.10)
derive Eqs. (3. 30.12a)
and
(3. 30.12d).
3. 78.
Verify
Eqs.
(3. 30. 13
a) and
(3....
Introduction to Continuum Mechanics 3 Episode 6 docx
... vector
on
the
left
end
face
x\
~
0.
4.12.
For any
stress state
T., we
define
the
deviatoric
stress
S to be
Stress
Power
2 03
where
Div
denotes
the
divergence with respect
to ... normal stress acting.
4 .6.
For the
following state
of
stress
findTn'
and
T
1
^'
where
e
x
'
is
in
the
direction
of
ej
+
2e
2
+
3e
3
and
63 & apos;is
i...
Introduction to Continuum Mechanics 3 Episode 7 potx
... Solid
2 27
Example
5 .3. 1
Find
the
components
of
stress
at a
point
if the
strain matrix
is
and the
material
is
steel
with
A =
119.2
GPa
( 17 .3
xl()
b
psi)
and
p
=
79 .2
GPa
(11.5xl0
6
psi).
Solution.
... we
have
a
unit normal vector
n
=
(l/a)(x2*2
+
X
3* $)-
Therefore,
the
surface traction
on the
lateral surface
Substituting
from
Eqs.
(5. 13. 3)...
Introduction to Continuum Mechanics 3 Episode 8 pps
... respect
to
Si
and 52
planes
automatically ensures
the
symmetry
with
respect
to the
53
plane.
(c)
All
planes
that
are
perpendicular
to the
3
plane have their
normals
parallel
to ... — 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)
... because
both
Eqs.
(5 .33 .9)
and
(5 .33 .11)
have
the
same
function
f.
We
note that
the
special case
where
a is a
constant,
is
called
a
Hookean Solid.
5 ,34
Consti...
Introduction to Continuum Mechanics 3 Episode 10 pptx
... Eq.
(6 .3. 5),
we
have
Newtonian
Viscous
Fluid
37 1
In
the
following
sections,
we
restrict ourselves
to the
study
of
laminar
flows
only.
It is
therefore
to be
understood that
... upper
limit
can be as
high
as
100 ,000.
Newtonian
Viscous
Fluid
38 3
6.17
Dissipation Functions
for
Newtonian Fluids
The
rate
of
work done
P by the
stress vectors
and the...
Introduction to Continuum Mechanics 3 Episode 11 pptx
... which
the
gate
will
start
to
fall.
63. The
liquids
in the
U-tube shown
in
Fig.P6 .3
is in
equilibrium. Find
h^
as a
function
of
Pi'P2»P3»
hi
an
d
^3-
The
liquids
are
immiscible.
6.4. ... Problems
4 23
630 . Referring
to
Problem 6.29, consider
a
pipe
having
an
elliptic cross section given
by
y
2
/a
2
+
z
2
/
b
2
=
1.
Assuming
that
find
A and
B...
Introduction to Continuum Mechanics 3 Episode 12 potx
... [xxdivT],
we
have
OJtn
Example
7.2 .3
Referring
to
Example 7.2.2, show that
the
total power (rate
of
work
done)
by the
stress
vector
on S is
given
by,
436
Integral
Formulation
of
... Eq.
(7.7.8)
to the
surface
and
body force terms.
7.8
Control
Volume
Fixed
with
respect
to a
Moving
Frame
If
a
control volume
is
chosen
to be
fixed
with
respect
t...
Introduction to Continuum Mechanics 3 Episode 13 pptx
... Section 3. 13,
Eq.
(3. 13. 6a),
that
1 T
where
D
=
:r[Vv
+
(Vv)
] is the
rate
of
deformation
tensor.
Thus,
£
Next,
from
Eq.
(8.10.4),
Non-Newtonian
Fluids
489
Example 8.9 .3
Consider
... are
similar
to
those
of the
deformation tensors using
a
fixed
reference time. [See
Chapter
3,
Section
3. 18
to
3. 29].
Indeed
by
polar decomposition theorem (Se...
Introduction to Continuum Mechanics 3 Episode 14 pdf
... example into
Eq.
(8.19.4),
we
obtain
544
Answers
2B31. Eigenvector
of
T
isn,
Eigenvector
of T is
r^
(c) No, the
first
invariants
are not
equal
Non-Newtonian
Fluids
5 13
Since
... be
shown
to be
objective stress rates. [See Prob.
23]
These
are
called
the
Oldroyd
upper
convected
derivatives.
and
note
that
one can
derive
Non-Newtonian
Fluids
531
Thus,
fr...
Introduction to Continuum Mechanics 3 Episode 15 pptx
... material,
32 2
Creep
function,
466
volume change,
146
Current configuration
as
reference con- Finite deformation tensor,
121,128, 134 ,
figuration,
476
136 , 138 ,141 ,151 ,1 53, 155 -156 ,206 ,31 8 -32 1
in
... 508
Dissipation functions,
38 3
First
Piola
Kirchhoff
stress tensor,
202
Divergence theorem,
430
Flow
Dual
vector,
36 ,94
channel
flow,
37 2,5 23
Dummy...