handout 03
TRANSCRIPT
7/26/2019 Handout 03
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H a n d o u t 3
I m p e r i a l C o l l e g e o f S c i e n c e , T e c h n o l o g y & M e d i c i n e
D e p a r t m e n t o f A e r o n a u t i c s
T h i r d Y e a r
E n e r g y M e t h o d s
2 . 4 P r i n c i p l e o f v i r t u a l w o r k
I n t h i s s e c t i o n w e w i l l d e r i v e t h e p r i n c i p l e o f v i r t u a l w o r k f o r a 3 D c a s e . W e w i l l p r o v e
t h a t f o r t h e c o m p a t i b i l i t y e q u a t i o n ( 2 . 1 1 ) t h e r e s u l t i n g g o v e r n i n g e q u a t i o n s a r e e q u i v a l e n t
t o t h e d i e r e n t i a l e q u a t i o n ( 2 . 5 ) a n d t h e f o r c e b o u n d a r y c o n d i t i o n ( 2 . 6 ) .
2 . 4 . 1 V i r t u a l w o r k o f t h e i n t e r n a l f o r c e s
L e t u s r s t a n a l y s e t h e c a s e s o f u n i a x i a l s t r a i n i n g a n d p u r e s h e a r o f a d i e r e n t i a l v o l u m e .
σyy
ydy
v+ y
+σyy
vv dy +
yv
dy x
yyσ
dx
dy
dz z
y
yzσ
σyz
yzε
dy
dx
dz
F o r t h e c a s e o f u n i a x i a l s t r a i n i n g , t h e w o r k o f t h e i n t e r n a l f o r c e
y y
+
@
y y
@ y
d y
d x d z
a c t i n g o n t h e d i e r e n t i a l v o l u m e o v e r t h e c o r r e s p o n d i n g v i r t u a l d i s p l a c e m e n t
@ v
@ y
d y i s
d V
i
=
y y
+
@
y y
@ y
d y
d x d z
@ v
@ y
d y =
y y
@ v
@ y
d x d z d y =
y y
y y
d V
w h e r e
y y
=
@ v
@ y
f o l l o w s f r o m t h e c o m p a t i b i l i t y e q u a t i o n ( 2 . 1 1 ) a p p l i e d t o v i r t u a l d i s p l a c e -
m e n t s a n d v i r t u a l s t r a i n s . F o r t h e c a s e o f p u r e s h e a r , a s i m i l a r r e s u l t e n d s u p i n
{ 2 1 {
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d V
i
=
y z
d x d y
y z
d z =
y z
y z
d V
s o t h a t f o r a g e n e r a l s t r e s s - s t r a i n s t a t e w e m a y d e d u c e t h e f o l l o w i n g i n t e r n a l v i r t u a l w o r k
V
i
=
Z
V
(
x x
x x
+
y y
y y
+
z z
z z
+
x y
x y
+
y z
y z
+
z x
z x
) d V =
Z
V
t
d V ( 2 2 3 )
2 . 4 . 2 V i r t u a l w o r k o f t h e a p p l i e d l o a d s
A s s u m i n g a v i r t u a l d i s p l a c e m e n t v e c t o r e l d
u w e m a y d e d u c e t h e v i r t u a l w o r k d o n e b y
t h e d i s t r i b u t e d v o l u m e a n d s u r f a c e l o a d s f r o m t h e f o l l o w i n g g u r e .
x
s PdS
S u y
u
u S p
z
v P
dV
I t s h o u l d b e n o t e d t h a t t h e v i r t u a l d i s p l a c e m e n t s m u s t b e k i n e m a t i c a l l y a d m i s s i b l e , t h e r e -
f o r e n o v i r t u a l d i s p l a c e m e n t s m a y e x i s t o n t h e p a r t o f t h e s u r f a c e w i t h p r e s c r i b e d d i s -
p l a c e m e n t s S
u
. B y d e n i t i o n , a w o r k d i e r e n t i a l i s a s c a l a r p r o d u c t b e t w e e n a f o r c e a n d
a d i s p l a c e m e n t d i e r e n t i a l , t h e r e f o r e t h e w o r k d o n e b y t h e d i s t r i b u t e d v o l u m e l o a d i n g P
v
o n t h e v i r t u a l ( i . e . d i e r e n t i a l ) d i s p l a c e m e n t ( o v e r d V ) m a y b e s t a t e d a s d V
e
=
u P
v
d V =
u
t
P
v
d V a n d t h e w o r k d o n e b y t h e d i s t r i b u t e d s u r f a c e l o a d i n g P
s
o v e r t h e v i r t u a l d i s -
p l a c e m e n t o f t h e d i e r e n t i a l a r e a o f s u r f a c e d S i s d V
e
=
u
t
P
s
d S . T h e t o t a l e x t e r n a l
v i r t u a l w o r k i s t h e n
V
e
=
Z
V
u
t
P
v
d V +
Z
S
p
u
t
P
s
d S ( 2 2 4 )
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2 . 4 . 3 P r i n c i p l e o f v i r t u a l w o r k a n d d i e r e n t i a l e q u i l i b r i u m e q u a t i o n s
T h e p r i n c i p l e o f v i r t u a l w o r k V
i
; V
e
= 0 f o l l o w s f r o m ( 2 . 2 3 ) a n d ( 2 . 2 4 ) a s
V
i
; V
e
Z
V
t
d V ;
Z
V
u
t
P
v
d V ;
Z
S
p
u
t
P
s
d S = 0 ( 2 2 5 )
I n S e c t i o n 1 . 3 w e c l a i m e d t h a t t h e p r i n c i p l e o f v i r t u a l w o r k i s e q u i v a l e n t t o t h e e q u i l i b r i u m
e q u a t i o n s o f t h e p r o b l e m f o r t h e k n o w n c o m p a t i b i l i t y a n d c o n s t i t u t i v e e q u a t i o n s . W e w i l l
n o w n o t o n l y p r o v e t h i s c l a i m f o r t h e c u r r e n t g e n e r a l 3 D d e f o r m a t i o n s t a t e , b u t a l s o s h o w
t h a t i n o r d e r t o d o s o w e i n f a c t n e e d o n l y t h e c o m p a t i b i l i t y e q u a t i o n s ( t h e c o n s t i t u t i v e
e q u a t i o n s , h o w e v e r , a r e s t i l l n e e d e d t o f u l l y d e n e t h e p r o b l e m ) . W e w i l l n e e d t h e f o l l o w i n g
t w o i m p o r t a n t t r a n s f o r m a t i o n s i n t h e p r o c e s s :
1 ) S u b s t i t u t i n g ( 2 . 1 1 ) i n t o t h e i n t e g r a n d o f t h e i n t e r n a l v i r t u a l w o r k w e o b t a i n
t
=
;
D
t
u
t
=
;
u
t
D
( 2 2 6 )
w h e r e t h e t e r m w i t h i n t h e p a r e n t h e s e s i s ( n o t e t h a t D i s g i v e n i n ( 2 . 4 ) )
h u v w i
2
6
4
@
@ x
0 0
@
@ y
0
@
@ z
0
@
@ y
0
@
@ x
@
@ z
0
0 0
@
@ z
0
@
@ y
@
@ x
3
7
5
= h
@ u
@ x
@ v
@ y
@ w
@ z
@ u
@ y
+
@ v
@ x
@ v
@ z
+
@ w
@ y
@ u
@ z
+
@ w
@ x
i
N o t i n g
t
= h
x x
y y
z z
x y
y z
z x
i , t h e i n t e g r a n d ( 2 . 2 6 ) c a n n o w b e w r i t t e n a s
t
=
@ u
@ x
x x
+
@ v
@ y
y y
+
@ w
@ z
z z
+
@ u
@ y
+
@ v
@ x
x y
+
@ w
@ y
+
@ v
@ z
y z
+
@ w
@ x
+
@ u
@ z
z x
=
@
@ x
( u
x x
+ v
x y
+ w
z x
) +
@
@ y
( u
x y
+ v
y y
+ w
y z
) +
@
@ z
( u
z x
+ v
y z
+ w
z z
)
{ 2 3 {
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; u
@
x x
@ x
+
@
x y
@ y
+
@
z x
@ z
; v
@
x y
@ x
+
@
y y
@ y
+
@
y z
@ z
; w
@
z x
@ x
+
@
y z
@ y
+
@
z z
@ z
= h
@
@ x
@
@ y
@
@ z
i
0
B
B
B
B
B
@
2
4
u 0 0 v 0 w
0 v 0 u w 0
0 0 w 0 v u
3
5
8
>
>
>
>
>
<
>
>
>
>
>
:
x x
y y
z z
x y
y z
z x
9
>
>
>
>
>
=
>
>
>
>
>
1
C
C
C
C
C
A
; h u v w i
0
B
B
B
B
B
@
2
6
4
@
@ x
0 0
@
@ y
0
@
@ z
0
@
@ y
0
@
@ x
@
@ z
0
0 0
@
@ z
0
@
@ y
@
@ x
3
7
5
8
>
>
>
>
>
<
>
>
>
>
>
:
x x
y y
z z
x y
y z
z x
9
>
>
>
>
>
=
>
>
>
>
>
1
C
C
C
C
C
A
I n t r o d u c i n g n o t a t i o n D
u
=
2
4
u 0 0 v 0 w
0 v 0 u w 0
0 0 w 0 v u
3
5
, ( 2 . 2 6 ) t h u s t u r n s o u t t o b e
t
= d i v ( D
u
) ;
u
t
( D ) ( 2 2 7 )
a n d t h e p r i n c i p l e o f v i r t u a l w o r k ( 2 . 2 5 ) b e c o m e s
V
i
; V
e
Z
V
d i v ( D
u
) d V ;
Z
V
u
t
( D + P
v
) d V ;
Z
S
p
u
t
P
s
d S = 0 ( 2 2 8 )
2 ) F o r a n y c o n t i n u o u s v e c t o r e l d a w i t h c o n t i n u o u s p a r t i a l d e r i v a t i v e s d e n e d o n a
3 D d o m a i n V w i t h i n a c l o s e d s u r f a c e S w i t h o u t w a r d n o r m a l n , t h e f o l l o w i n g G a u s s ' s
d i v e r g e n c e t h e o r e m i s v a l i d :
Z
V
d i v a d V =
I
S
n a d S ( 2 2 9 )
T h e r s t t e r m i n ( 2 . 2 8 ) c a n b e t r a n s f o r m e d u s i n g t h e d i v e r g e n c e t h e o r e m a s
Z
V
d i v ( D
u
) d V =
I
S
p
S
u
n ( D
u
) d S =
Z
S
p
n
t
D
u
d S +
Z
S
u
n
t
D
u
d S
| { z }
= 0
( 2 3 0 )
{ 2 4 {
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w h e r e
R
S
u
n
t
D
u
d S v a n i s h e s i d e n t i c a l l y d u e t o t h e k i n e m a t i c a l a d m i s s i b i l i t y o f v i r t u a l
d i s p l a c e m e n t s : o n S
u
t h e d i s p l a c e m e n t s a r e p r e s c r i b e d , h e n c e t h e v i r t u a l d i s p l a c e m e n t s
w i t h i n D
u
o n S
u
a r e a l l z e r o . S u b s t i t u t i n g n
t
D
u
=
u
t
D
n
i n t o e q u a t i o n ( 2 . 3 0 ) t h e n y i e l d s
R
V
d i v ( D
u
) d V =
R
S
p
u
t
D
n
d S a n d t h e p r i n c i p l e o f v i r t u a l w o r k ( 2 . 2 8 ) n a l l y b e c o m e s
V
i
; V
e
;
Z
V
u
t
( D + P
v
) d V +
Z
S
p
u
t
( D
n
; P
s
) d S = 0 ( 2 3 1 )
S i n c e t h i s r e s u l t m u s t b e v a l i d f o r a r b i t r a r y v i r t u a l d i s p l a c e m e n t s
u , t h e f o l l o w i n g d i e r -
e n t i a l e q u i l i b r i u m e q u a t i o n a n d t h e s u r f a c e - l o a d b o u n d a r y c o n d i t i o n a r e o b t a i n e d
D + P
v
= 0 o n V ( 2 3 2 )
D
n
; P
s
= 0 o n S
p
( 2 3 3 )
t h u s p r o v i n g t h a t f o r t h e c o m p a t i b i l i t y e q u a t i o n s ( 2 . 1 1 ) t h e p r i n c i p l e o f v i r t u a l w o r k y i e l d s
t h e d i e r e n t i a l e q u a t i o n o f e q u i l i b r i u m ( 2 . 5 ) a n d t h e f o r c e b o u n d a r y c o n d i t i o n ( 2 . 6 ) .
2 . 5 P r i n c i p l e o f s t a t i o n a r y t o t a l p o t e n t i a l e n e r g y
T h e s a m e r e s u l t c a n a l s o b e o b t a i n e d f r o m t h e p r i n c i p l e o f s t a t i o n a r y t o t a l p o t e n t i a l
e n e r g y . T h e t o t a l p o t e n t i a l e n e r g y f o r t h e p r o b l e m i s
= ; U = ;
Z
V
u
t
P
v
d V ;
Z
S
p
u
t
P
s
d S
w h e r e f o r a l i n e a r e l a s t i c m a t e r i a l = , w i t h g i v e n i n ( 2 . 2 2 ) , a n d =
1
2
R
V
t
d V =
1
2
R
V
t
d V , t h u s g i v i n g
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=
Z
V
1
2
t
; u
t
P
v
d V ;
Z
S
p
u
t
P
s
d S ( 2 3 4 )
V a r y i n g t h e t o t a l p o t e n t i a l e n e r g y ( 2 . 3 4 ) a n d e m p l o y i n g t h e c o n d i t i o n o f s t a t i o n a r i t y y i e l d s
Z
V
;
t
; u
t
P
v
d V ;
Z
S
p
u
t
P
s
d S = 0 ( 2 3 5 )
T h i s e q u a t i o n m a y b e d e a l t w i t h i n t h e s a m e w a y a s ( 2 . 2 5 ) , p r o v i d e d v i r t u a l d i s p l a c e m e n t s
u a n d v i r t u a l s t r a i n s
i n ( 2 . 2 5 ) a n d ( 2 . 2 7 ) a r e s u b s t i t u t e d w i t h t h e v a r i a t i o n s o f t h e
d i s p l a c e m e n t s a n d s t r a i n s u a n d . I n t h i s w a y ( 2 . 3 5 ) t r a n s f o r m s i n t o
Z
V
d i v ( D
u
) d V ;
Z
V
u
t
( D + P
v
) d V ;
Z
S
p
u
t
P
s
d S = 0 ( 2 3 6 )
w i t h D
u
=
2
4
u 0 0 v 0 w
0 v 0 u w 0
0 0 w 0 v u
3
5
. A p p l y i n g t h e d i v e r g e n c e t h e o r e m ( 2 . 2 9 ) t o
( 2 . 3 6 ) , a n d b e a r i n g i n m i n d t h a t t h e d i s p l a c e m e n t s c a n n o t b e v a r i e d o n t h e p a r t S
u
o f t h e
s u r f a c e w i t h p r e s c r i b e d d i s p l a c e m e n t s , n a l l y p r o d u c e s
;
Z
V
u
t
( D + P
v
) d V +
Z
S
p
u
t
( D
n
; P
s
) d S = 0 ( 2 3 7 )
w h i c h i s e q u i v a l e n t t o ( 2 . 3 1 ) o b t a i n e d u s i n g t h e p r i n c i p l e o f v i r t u a l w o r k . E q u a t i o n ( 2 . 3 7 )
m u s t b e s a t i s e d f o r a r b i t r a r y v a r i a t i o n s u o f t h e d i s p l a c e m e n t s , r e s u l t i n g i n
D + P
v
= 0 o n V a n d D
n
; P
s
= 0 o n S
p
( 2 3 8 )
a n d t h u s r e s t o r i n g t h e d i e r e n t i a l e q u a t i o n ( 2 . 5 ) a n d t h e n a t u r a l b o u n d a r y c o n d i t i o n ( 2 . 6 )
i n t h e s a m e w a y a s i n ( 2 . 3 1 ) - ( 2 . 3 3 ) u s i n g t h e p r i n c i p l e o f v i r t u a l w o r k .
{ 2 6 {