Mindlin R.D., Yang J. An Introduction to the Mathematical Theory of Vibrations of Elastic Plates
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Chapter 1
Elements of the Linear Theory of Elasticity
1.01 Notation
Rectangular coordinates are designated by x
u
x
2
, x
3
or x
h
(/=1, 2, 3)
and components of displacement, referred to these coordinates, by u\, u
2
,
w
3
or
Uj,
(z-1,
2, 3).
The symbols for components of traction, stress and strain are ?,-, Ty
and
Sy,
respectively, where i=\, 2, 3 andy'=l, 2, 3.
The summation convention for repeated indices is employed. Thus,
if
tj
are the components of traction across a surface, at a point where the
components of the outward-drawn unit normal are v„ the relations
between the
tj
and the components of
stress
at that point are
t.
=
vfy ^ v,r,, +
v
2
T
2J
+
v
3
T
3J
(1.011)
i.e., the terms are summed over the repeated index i.
Differentiation with respect to space coordinates is indicated by a
comma followed by an index:
.
=
_9_
dxj
Thus,
the relations between components of strain and displacement are
expressed as
1
2
Mathematical Theory of Vibrations of Elastic Plates
S
ij=^(
u
i,j
+U
j,i)
(1.012)
where it is to be understood that the indices i andy range over 1, 2, 3, i.e.,
(1.012) represents the nine relations
0^^ —
j 22
3«!
3w
2
dx-,
^23
_
^32 -
&,= —
3x,
•"33
^31
_
^13 ~~Z
s -s -I
^12
_
°21
_
»
3w
3
3w
2
3w, 3W
3
\dx
z
3x, j
du
2
3"!
dx, dx-,
(1.013)
V
OX
l
UJV
2
J
Similarly, the components of rotation,
COJJ,
are given by
1
CO,
* = 2
K;-";,.•)
(1.014)
Differentiation with respect to time is indicated by a dot over a
symbol. Thus
3w,
are components of velocity and
u, =-
dt
d\
dt
1
are components of acceleration.
The comma and dot notations and the summation convention are all
employed in writing the stress-equations of motion in the form
T
i),i=pUj
(1.015)
where p is the density and the body-force has been omitted. The scalar
Elements
of the
Linear Theory
of
Elasticity
3
expansion
of
(1.015)
is
37;,
|
dT
2l |
dT
n
9 u,
dx
l
dx
2
dx
3
dt
2
dT,
7
3ZU 37V, 3«,
,*
n
,s\
——
+
——
+
——
=
p—-y-
(1.016)
ox,
ax
2
dx
3
3/
dT
u
, ^23 ,
dT
3i
-
p
*\
dx
{
dx
2
3x
3
dt
2
An abbreviated, indicial notation
is
also employed.
In
this,
a
pair
of
indices ranging over
the
integers
1, 2, 3 is
replaced
by one
index ranging
over
the
integers
1, 2, 3, 4, 5, 6,
according
to the
scheme given
in
Table
1.011.
Table
1.011
ij
or
kl:
porq:
11
1
22
2
33
3
23
32
4
31
13
5
1?
21
6
In
the
case
of
stress,
we
have
[T
D
, i-U
/>
=
1>2,3
T
U
=
\T
-J- A**
(
L017
)
[
T
P
, **J, P
=
4,5,6
or
-'11
_
-M •'23
=
i
32
=
M
722=72
T
3l
=T
n
=T
s
(1.018)
•'33
—
1
3
1
\2 ~
1
2\
=
1
6
Accordingly,
in the
abbreviated notation,
the
scalar stress-equations
of
motion
are
4 Mathematical Theory of Vibrations of Elastic Plates
dx, dx
2
dx
3
' dt
2
d
u-,
dT
6
dT
2
dT.
—- +—- +—-
=
p—f
dx, dx
2
dx
3
dt
dT
5
dT
4
dT
3
d\
—
L
+—-
+
—- = p—j-
dx, dx, dx, dt
(1.019)
In the case of strain, the single-index shears are twice as large as the
double-index shears, but the single- and double-index extensions are the
same:
V i
=
j,
/» = 1,2,3
S
P
>
l
*h ^ = 4,5,6
(1.0110)
or
S,,=S.4*-
^22 — 2
du
1
S
n
= S^
•"33
dx,
IS.
23
-2S„=S
A
=^- + ^
'32"
dx
2
9 x
3
2s -OS -S -lil + ^i
ox
3
d
x,
2
c -2S -S -f^L + i^L
d
x,
d x,
(1.0111)
The notation of the text, for the three-dimensional theory, may be
compared with those employed by Love (1927) , Timoshenko and
Goodier (1951), Sokolnikoff (1946) and the Institute of Radio Engineers
(1949) by referring to Table
1.012.
Notations for the two-dimensional
theory of plates are introduced in Chapter 3.
*A name followed by the year—Jones (1950) or (Jones, 1950)-
indicates an entry in the list of references at the end of the monograph.
Elements of the Linear Theory of Elasticity
5
Table 1.012
Text
X\, Xi, X-i
«1,
U
2
, M
3
S\\, S
2
2, "$33
S
2
3,
^32
Su>
Sii
Sl2,
S
2
\
0)
i
2,CO
l3
,0}
n
h,
h, h
Vl,
V
2
, V
3
T\\, T
2
2, T
3
2
T23,
T32
T31,
T\z
T\2, Tl\
S\, S2, S3
s,
S
5
s
6
T
u
T
2
, T
3
Notations Employed in the Three-Dimensional Theory
Love
x,y,z
w,
v, w
°xx->
^yy>
**zz
/2 c-yz^ /"Z&zy
'l&ZX)
*'
2
&XZ
/26xy, /2Cy
X
m
x
,m
y
,m
z
•A-vi A-vi ^v
I, m, n
•A-xi lyi ^z
£y, I
z
X
z
, Z
x
Y
X
,Xy
@xx> @yyi
^zz
&yzt &zy
&ZX1
&XZ
&xy>
&yx
•A-X)
*y>
^z
Timoskenko
and Goodier
x,y,z
U, V, W
&xi £y> ^z
y
2
y
yz
,
x
Ay
zy
ViYzx,
ViYxz
Viyxy, Viy
yx
ah,
(fy,
a>
z
X,Y,Z
/, w, n
Ox>
&y, &z
Lyzi
*>zy
Tzxs
VJCZ
Txy? *yx
£Jo
£y->
^z
7yz,
Izy
Yzx, 7xz
7xy,
lyx
0x,
<?y,
Oz
Sokolnikoff
X\, X
2
, X3
u
u
u
2
, u
3
en, e
2
2, e
33
^23,
^32
^31,^13
£12,
e
2i
<V}2,G>U,6>2\
V
V V
T
X
,T
2
T,
Vl,
V
2
, V
3
Tll,T22,T33
T23,
T32
T31,Ti3
^12,
^21
«i,
e
2
, e
3
e
4
es
e
6
^1,^2,^3
I.R.E.
^1,-^2,^3
U
u
U
2
, «
3
5*11,
>$22, ^33
•$23,
£32
S31,
5i3
<S"l2, 5*21
-
-
-
^11,
?22, T33
^23,
T32
T
3
\, r
13
T\2, 7*21
S\, S2, S3
s
4
s
5
s
6
Tu T
2
, T
3
6
Mathematical Theory
of
Vibrations
of
Elastic Plates
T
4
T
5
T
6
7
Y
X
z
, Z
x
Yx,Xy
^yz,
^zy
Tzx, Txz
Txy, Tyx
T
4
T
5
*6
T<
T
5
T
6
Oh,
Oh, Oh
tU
x
,tU
y
,tU
z
<*>x,
<%>,
0)z
1.02 Principle of Conservation of Energy
If body-forces and thermal and electromagnetic effects are neglected,
the principle of conservation of energy states that the rate of increase of
energy
of a
body
is
equal
to the
rate
at
which work
is
done
by
surface-tractions. In the linear theory of elasticity the statement takes the
form
\
v
(k
+
u)dV
=
l
s
tj
iij
dS (1.021)
where K and U are the kinetic and internal energy-densities, respectively.
The volume-integral in (1.021) is taken over the volume, V, of the body
and the surface-integral over its surface, S.
Now, the kinetic energy-density is, by assumption,
K
=
^pujUj (1.022)
where p is the density of medium. Hence
K
=
pit j it j (1.023)
If we make use
of
the stress-equations
of
motion (1.015) (which
are
derived from the principle of conservation of linear momentum) (1.023)
may be written as
k
=
T
ij
,u
j
=(T
ij
u
j
).-T
iJ
u
j
,
(1.024)
Now, by the divergence theorem,
Elements of the Linear Theory of Elasticity
7
l
v
(T
ij
u
J
)dV=\
s
v
i
T
ij
u
j
dS
•' (1.025)
= J
5
tj Uj dS
where the last step is based on the definition of stress (1.011).
Inserting (1.022)-(
1.025)
in (1.021), we find
l
v
{u-T
iJ
ii
JJ
)dV
=
0 (1.026)
This relation must hold for all portions of
V;
hence
U =
T
ij
u
jJ
(1.027)
Now, from (1.012) and (1.014),
«,,,=$•,-«„ (1.028)
Also,
from the principle of conservation of angular momentum,
T,j=T
Jt
(1.029)
Thus,
Ty
is symmetric but
coy
is antisymmetric; hence
7].^,..=0 (1.0210)
Accordingly, the principal of conservation of energy takes the form
U
=
T
iJ
S
iJ
(1.0211)
1.03 Hooke's Law
We assume that the internal energy is a function of the six
components of strain:
U
=
U(S
P
\ p
=
l,-,6 (1.031)
as that t/may now be called the strain-energy-density.
The components of strain are functions of the time; hence
8
Mathematical Theory of Vibrations of Elastic Plates
dS„
p
(1.032)
In the abbreviated notation (1.0211) takes the form
U
=
T
P
S
P
(1.033)
Now, the six components of strain are independent. If, in addition, we
assume that the stress is independent of
the
strain-rate, the coefficients of
S
p
in (1.032) and (1.033) may be equated, with the result
T
P
=
dU
dS
P
(1.034)
We now assume that the strain-energy is a homogeneous quadratic
function of the six components of strain:
V=\CPM--
c
pq
C
IP
(1.035)
where the c
pq
are the constants of elasticity. Then, from (1.034) we have,
for r=l,
...,6,
Now
T =
dU 1
dS,
K
Cpg
~ds;
Sq+c
>">~ds;
Sp
;
8S
OS,
P_\0,
P*r
\\,p = r
OS.
OS,
[1,
q
=
r
(1.036)
(1.037)
Hence
T
r
=-(c
rq
S
q
+
c
pr
S
p
)
But/? is a dummy index, so it may be changed to q, giving
T
r
=-(c
rq
+c
qr
)S
q
=c
rq
S
q
(1.038)
(1.039)
Elements of the Linear Theory of Elasticity 9
Finally, replacing r
with/7,
we have Hooke's Law in the familiar form
T
p
=c
pq
S
q
(1.0310)
Also,
from (1.035) and (1.0310),
U
=
-T
p
S
p
(1.0311)
2
Hooke's Law (1.0310) may
be
written,
in the
unabbreviated,
indicial notation, as
T
iJ
=c
ijkl
S
hl
(1.0312)
\
C
ijkl
= C
jikl
= C
ijlk
= C
klij
= C
jilk ~
C
Ikij
= C
klji
= C
lkji)
where the cyu are the same constants as the c
pq
when the pairs of indices
ij and kl are replaced by p and q, respectively, in accordance with Table
1.011.
The
form (1.0312)
can be
obtained from
the
strain-energy-function
C/ =
|c
/>t/
5
lV
5
t/
(1.0313)
through the relation
dU
r,,-— (1.0314)
if, in performing the differentiations, it is assumed that
as,,
--^ = 0,
i*j
(1.0315)
This is because, in unabbreviated notation, the analogue of (1.033) is
'.'
. , . . . v
(1-0316)
"~
•*
11^11
+
^
22^22
+
^33*^33
+
2^23^23
+
^31^31
+
^12^12 /
whereas the analogue of (1.032) is
10
Mathematical Theory of Vibrations of Elastic Plates
SS
U
" hS„ "
3S
22
" 3S„ "
(1.0317)
dS
2J
3
dS
3l
31
dS
l2
Hence, equating coefficients:
dU
T
u=<
t = J
dS,,'
1 W
(1
°
318)
2 35..,.'
Equations (1.0318) may be written in the more convenient form (1.0314)
if the convention (1.0315) is adopted.
1.04 Constants of Elasticity
We shall have occasion to consider only four types of elastic
materials: triclinic, monoclinic, trigonal and isotropic (Mason, 1950, p.
44).
The elastic properties of a triclinic material are represented by the
full array of 21 constants:
7,
=c
n
S
x
+ c
12
S
2
+ c
l3
o
3
+ c
14
S
4
+ c
15
S
5
+
c
16
S
6
T
2
=
c
21
S
{
+ c
22
S
2
+
c
23
S
3
+
c
24
S
4
+
c
25
S
5
+ c
26
S
6
T
3
=
c
31
5,
+
c
32
^ +
c
33
5
3
+ c
34
S
4
+
c
35
5
5
+ c
36
S
6
T
4
= c
41
5, + c
42
S
2
+
c
43
S
3
+ c
44
S
4
+
c
45
5
5
+ c
46
5
6
T
5
=
c
51
S, + c
52
o
2
+
c
53
5
3
+
c
54
5
4
+
c
55
S
5
+
c
56
S
6
^6
=
C
61 ^1
+
C
62 ^2 +
C
63 ^3
+
C
64 ^4 +
C
65 ^5
+
C
66 ^6
The stress-strain relation of a monoclinic crystal involves 13
independent constants of elasticity. When the Xpaxis is the axis of
two-fold symmetry,