'4
Deformation & Strain
�!'4´KJ_T,R'
�!'°J±´_'©¤¨«
�!%I n, '4nv
uA=u
uB=u+(!u/!x)dx
*,
$ #u ' .,
+u + && )) dx / " u
u " uA ,% #x ( ,0
#u
!= B
=
=
dx
#x
dx
°o'±
''4
Displacement & Deformation
!!'
u = x! " x
* '4M:
v = y! " y
*x
w = z! " z
u,v,w: x,y,zj]q=
!!'
!!'°KJT,_}'x±
!!k'
!!"i'
!!'4'°K(~4M}' ±
u}Q T,'°1±
Relative Displacement
!!'°'4''±
A(x,y,z)
B(x1,y1,z1)
I°'4±
!!k'r
II°'45±
A’(x’,y’,z’)
B’(x1’,y1’,z1’)
r’=r+u: A’_©¤¨«
r: A_©¤¨«
r1: B_©¤¨« r1’=r1+u1 : B’_©¤¨«
dr=r1–r
u: A'©¤¨«
u1: B'©¤¨« du=u1–u: T,'©¤¨«
r = ix + jy + kz
r1 = ix1 + jy1 + kz1
u = iu + jv + kw
u1 = iu1 + jv1 + kw1
T,'°2±
Relative Displacement
!!BJ'©¤¨«u1
!u
!u
!u
u1 = u +
dx +
dy +
dz
!x
!y
!z
!!T,'©¤¨«du
du = u1 ! u
=
=
"u
"x
"u
"x j
dx +
dx j
"u
"y
dy +
"u
"z
dz
n? ©¤¨«
en = i! nx + j! ny + k! nz
=i
dx
+j
dy
+k
dz
dr
dr
dr
dr = endr = idx + jdy + kdz
!!T,'; !!'m
!ui
!ui
dui =
dx j
!x j
!x j
dr 2 = dr ! dr
°1±
dr "2 = dr" ! dr"
= (dr + du) ! (dr + du)
= dr ! dr + 2dr ! du + du ! du
dr !2 " dr 2 = 2dr # du + du # du
Strain
!!$S!nn
!nn =
dr " # dr
dr
= (i%nx + j%ny
= %nx %nj
= %ni %nj
&u
&x j
=
en $ du
dr
= en $
du
dr
2(dr ! " dr)dr = 2dr # du
= 2(e ndr) # du = 2dr(en # du)
*
'
&u
&v
&w ,
+ k%nz ) $ )i%nj
+ j%nj
+ k%nj
,
)
&x
&x
&x
(
j
j
j +
+ %ny %nj
&v
&x j
+ %nz %nj
&w
&x j
&ui
&x j
du =
dui
dr
!u
!x j
= #nj
dx j "
!ui
!x j
du
dr
= #nj
!u
!x j
°2±
Strain
!!$S!nn
$ #w #v '
2 #u
2 #v
2 #w
!nn = "nx
+ "ny
+ "nz
+ "ny "nz &&
+ ))
#x
#y
#z
% #y #z (
$ #u #w '
$ #v #u '
)) + "nx "ny && + ))
+ "nz "nx && +
% #z #x (
% #x #y (
!nn = "2nx! xx + "2ny!yy + "2nz! zz + 2"ny "nz!yz
Bbf 1 # "u j "ui &
! ij = %
+
(
%
2 $ "xi "x j ('
+ 2"nz "nx! zx + 2"nx "ny! xy
!ij: ;
= "ni "nj !ij
i=j: $S;
i"j: £>;
°3±
Strain
!!£>!yz
cos !yz = eB " eC
=
jdy + duB kdz + duC
"
(1 + #yy )dy (1 + # zz )dz
1+!xx#1
'
$ du ' $
du
C)
B) &
! && j +
"
k
+
) &
)
dy
dz
%
( %
(
duB duC
+k "
+
"
= j"k + j"
dz
dy
dy dz
duC
! j"
duC
+k "
duB
dz
dy
*w *v
=
+
= 2#yz
*y *z
duB
drB = jdy + duB = (1 + !yy )dyeB
drC = kdz + duC = (1 + ! zz )dzeC
duB
dy
duC
dz
=i
!u
=i
!u
!y
!z
+j
!v
+j
!v
!y
!z
+k
!w
+k
!w
!y
!z
°4±
Strain
!!£>!yz
d/"/2|'d/
cos "yz sin(# / 2 $ "yz ) "/2- &>0
!yz =
=
"/2(~ &<0
2
2
(
1 %#
! '' $ "yz **
2 &2
)
1 % +w +v (
**
= ''
+
2 & +y +z )
!yz: PgR£>
°§¦«±
#yz: .)R£>
#yz=2!yz
§¦«
Stain Tensor
!!x?
T,';du
)1 " !v !u % 1 " !v !u %,
du =
dx +
dy +
dz =
dx + + $$ + '' ( $$ ( ''. dy
!x
!y
!z
!x
+*2 # !x !y & 2 # !x !y &.)1 " !u !w % 1 " !w !u %,
'' ( $$
+ + $$ +
( ''. dz
1 $ "u j "ui '
! ij = &
#
)
+*2 # !z !x & 2 # !x !z &.2 &% "xi "x j )(
= (/ xx dx + / xy dy + / zx dz) + (0dx ( 0 xy dy + 0 zx dz)
!u
!u
)
!
% + ( xx
# du # +
" dv & = ( xy
#
# +
$ dw ' + (
* zx
!u
( xy
(yy
(yz
!u
,
)
( zx . !
% + 0
# dx # +
.
(yz " dy & + 0 xy
.#
# +
( zz . $ dz ' + /0 zx
*
§¦«
/0 xy
0
0yz
,
0 zx . !
%
dx
#
#
.
/0yz " dy &
.#
#
.
$ dz '
0
-
"i§¦«
"i§¦«
Rotation Tensor
!!;}¥¬&
*
!
% , 0
du
#
# ,
" dv & = ) xy
#
# ,
$ dw ' , ()
zx
+
() xy
0
)yz
) zx / !
%
dx
#
#
/
()yz " dy &
/#
#
/
dz
$
'
0
.
"i
!w
!!'4 '
)
!
% + ( xx
# du # +
" dv & = ( xy
#
# +
$ dw ' + (
* zx
( xy
( zx
(yy
(yz
(yz
( zz
,
. ! dx %
#
. #"
&
. # dy #
. $ dz '
-
!y
!yz
="
!v
!z
1 $ "w "v ' "w
)) =
= &&
#
2 % "y "z ( "y
;
Components of Strain
�!$S; �!£>;
!x =
!y =
!z =
"u
�!.)R£> �!PgR£>
"x
"v
"y
"w
"z
! xy =
! yz =
! zx =
"u
"y
"v
"z
"w
"x
6NX;
§¦«
+
+
+
"v
"x
"w
"y
"u
"z
! xy
1 # "u "v &
= %
+
(
2 $ "y "x '
! yz
1 # "v "w &
= %
+
(
2 $ "z "y '
! zx
1 # "w "u &
= %
+
(
2 $ "x "z '
©¤¨«°1±
Strain Vector
!!n?
)
!
% + ( xx
# du # +
" dv & = ( xy
#
# +
$ dw ' + (
* zx
( xy
(yy
(yz
,
( zx . !
%
dx #
#
(yz . " dy &
.#
#
( zz . $ dz '
-
^\AB°=dr±'4'du
du = idu + jdv + kdw o3du
!!n?
^\©¤¨«!!n
du
dv
dw
n
! =
=i
+j
+k
dr
dr
dr
dr
= i!nx + j!ny + k!nz
¢
du
= (i! xx + j! xy + k! zx )
+(i! zx + j!yz + k! zz )
dx
dr
dz
+ (i! xy + j!yy + k!yz )
dr
= "nx! x + "ny! y + "nz! z = "ni! i
dy
dr
! x = i! xx + j! xy + k! zx
! y = i! xy + j!yy + k!yz
! z = i! zx + j!yz + k! zz
©¤¨«°2±
Strain Vector
! xy = ! zx = 0
" xx =
#u
#x
, " xy =
!!x?
^\AX°=dx±'©¤¨«
!u
!v
!w
x
du = i
dx + j
dx + k
dx
!x
!x
!x
!u
!v
!w
dux
=i
+j
+k
dx
!x
!x
!x
= i" xx + j" xy + k" zx = " x
!xox?
!!n?
n
! =
^\'©¤¨«
©¤¨«!!n
du
=
dun
+
dut
dr
dr
dr
= en!nn + et !nt
#v
#x
, " zx =
#w
#x
0C'<
Coordinate Transformation
!!i'?
©¤¨«!!i’
! i " = #i "i! i
! i = e x!ix + ey!iy + e z!iz = e j !ij
= #i "i e j !ij
!i "j " = e j " # !
i"
= e j " # e j $i "i!ij
= $i "i $ j "j !ij
!i’j’:
!i’j’?
;
i,j,kuex,ey,ez
i',j',k'uex',ey',ez'
;0C'<1
°7&±
°1±
Principal Strain
!!k?
©¤¨«
! k = !ek = !(i"kx + j"ky + k"kz )
= i"ki!ix + j"ki!iy + k"ki!iz
i(!ki"ix # !kx") + j(!ki"iy # !ky") + k(!ki"iz # !kz") = 0
!ki"ix # !kx" = 0, !ki"iy # !ky" = 0, !ki"iz # !kz" = 0
!kx (" xx # ") + !ky" xy + !kz" zx = 0
!kx" xy + !ky ("yy # ") + !kz"yz = 0
!kx" zx + !ky"yz + !kz (" zz # ") = 0
W
°2±
dV = dx(1 + ! xx )dy(1 + !yy )dz(1 + ! zz ) " dxdydz
= V #$(1 + ! xx )(1 + !yy )(1 + ! zz ) "1%&
Principal Strain
= V(! xx + !yy + ! zz + !yy! zz + ! zz! xx
+! xx!yy + ! xx!yy! zz )
!!L8?V1
dV
! xx " !
! xy
! zx
! xy
!yy " !
!yz
! zx
!yz
! zz " !
V
=0
J1 = ! xx + !yy + ! zz = !ii
J2 = !yy! zz + ! zz! xx + ! xx!yy
=
1
!!
(
2
ii jj
" !ij !ij
)
J3 = ! xx!yy !zz " ! xx !2yz " !yy!2zx
(! " !1)(! " !2 )(! " !3 ) = 0
J1 = !1 + !2 + !3
J3 = !1!2!3
= ! xx + !yy + ! zz = J1
"!2yz " ! 2zx " !2xy
!3 " J1!2 + J2! " J3 = 0
!1 , !2 , !3
J2 = !1!2 + !2!3 + !3!1
! V(! xx + !yy + ! zz )
"! zz!2xy + 2!yz! zx! xy
= !ij
'n
J1: W²Wa2O
l@1°1±
Compatibility Equations
!"
!u
6
'3
=
!y
!x!y
E¡ "!
! xy
!"
!u
$ xx
!v
=
$
#
!x
!y!x
v
!yz
"
16 #!yy
!"
1# ! u
$
#w
= %%
!x!y
2 $ !x!y
$
$! zz
! zx
%
!!9E¡x
!!;p£|[@}6c
!"
2
xx
2
2
yy
2
2
xy
3
2
3
2
3
2
!3v &
(
+
2(
!y!x '
2
'q1|u, v, w¢G
* !2"
!2"yy
!2" xy
, xx +
=2
2
,, !y 2
!x!y
!x
+ 2
'
$
, ! " xx
! & !"yz !" zx !" xy )
=
#
+
+
,
&
!y
!z )(
,- !y!z !x % !x
xx
!y!z
=
!3u
!x!y!z
! " xy
!2 # !u !v &
%% + ((
=2
!x!z
!x!z $ !y !x '
2
!2"yz
!2 # !v !w &
((
= 2 2 %% +
!x 2
!x $ !z !y '
!2" zx
!2 # !w !u &
%%
=2
+ ((
!x!y
!x!y $ !x !z '
l@1°2±
Compatibility Equations
# !2"
!2"yy
!2" xy
% xx +
=2
2
% !y 2
!x!y
!x
% 2
!2"yz
% ! "yy !2" zz
=2
$ 2 +
2
!y!z
!y
% !z
% !2"
2
2
!
!
"
" zx
xx
% zz +
=
2
% !x 2
!z!x
!z 2
&
'
$
! & !"yz !" zx !" xy )
=
#
+
+
&
!y!z !x % !x
!y
!z )(
'
$
!2"yy
! & !"yz !" zx !" xy )
=
#
+
&
!z!x !y % !x
!y
!z )(
'
$
!2" zz
! & !"yz !" zx !" xy )
=
+
#
&
!x!y !z % !x
!y
!z )(
!2" xx
;pHh¡¡x@
!s
"! JM:}Dy§¦
«z¡~²(~{
? ¢E³²bU ;
|'n¢E²¡}Y
x¢U³k*=w ³
"
$
5
&1
&1
(
"! $ = k '
' &1 4 0 (
# ij %
'# &1 0 4 (%
!s
"! Sd0CZq ';}2µu=5x
+3y, 2µv=3x–Ay, 2µw=3z+2°µ: *=±
b¡x ³'4W'
}Qx¡²Ay |µ
!s
"! '&}2µu=4x–y+3z, 2µv=x+7y,
2µw=–3x+4y+4z°µ: *=±b¡x
³§¦«{§¦«¢
E³
!s
"! ;}k¢*=
! xx = k(x 4 + y 4 + x 2 + y 2 + 5)
!yy = k(x 4 + y 4 + 3x 2 + 3y 2 + 6)
! xy = k(2x 3y + 2xy 3 + 4xy + 5)
! zz = !yz = ! zx = 0
z¡ y&}+#`|y
|¢Ae8³`²¡
J{ k'"i'}0w &
';¢F*³
7®0C'<
Coordinate Transformation of Stress & Strain
�!7
! x ' = ! x cos2 " + ! y sin2 " + 2# xy sin " cos "
(
) (
)
# x ' y ' = # xy cos2 " $ sin2 " + ! y $ ! x sin " cos "
t
ª¯«
�!
! x ' = ! x cos2 " + ! y sin2 " + # xy sin " cos "
(
) (
)
# x ' y ' = # xy cos2 " $ sin2 " + 2 ! y $ ! x sin " cos "
�!
�!? °?
�!³³³
±