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CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
1 Updated: 9/2/2014
MECHANICS OF SOLIDS:
DISPLACEMENTS – STRAINS
2-D
u: displacement along x; u(x,y)
v: displacement along y; v(x,y)
HTensor of strains:
yyyx
xyxx
'''2
1
2
1
2
1
'''
'''
BADDABx
v
y
u
AD
ADDA
y
v
AD
ADDA
AB
ABBA
x
u
AB
ABBA
yxxy
yy
xx
Distortion of ABCD yxxyBADDAB 22
dxx
udy
y
uu
dyy
uu
dyy
vv
dyy
vdx
x
vv
dyy
vdy
D
D
C
C
dy
A
B
dxx
vv
v
Au
x
y
dxx
udx
dxx
uu
dx
y
u
dyy
vdy
dyy
u
DA
DD
x
v
dxx
udx
dxx
v
BA
BB
'''
'''tan
'''
'''tan
B
''B
''D
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
2 Updated: 9/2/2014
WHY IS “H” A TENSOR?
Define:
Tensor of changes of displacements:
y
v
y
u
x
v
x
u
is a tensor because: (due to definition of tensor)
T
v
u
y
xvu
y
x
T: symbol for transpose of a vector, matrix
So it can be easily shown that
)1(2
T
H
Thus H, being the sum of two tensors, is also a tensor.
By using (1) it can be easily shown that THH or that:
H: symmetric tensor
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
3 Updated: 9/2/2014
Change in “volume” (area) of ABCD:
)(
)(111
))(()1)()(1)((
))(())((
))((
))((
2
yyxx
yyxxyyxx
yyxx
V
OVV
ADABADAB
ADABDABAVVV
DABAV
ADABV
So: )(HtrV
Vyyxx
tr: trace of tensor
volumetric strain: )(HtrV
Vyyxx
Note: trace of tensor does not change with coordinate transformation. What is the physical
meaning of this for tr(H)?
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
4 Updated: 9/2/2014
CHANGE OF COORDINATES SYSTEM:
According to tensor analysis:
)2(TRTRT
where R is the matrix of direction cosines ,...)('xx
cossin
sincos
yyxy
yxxxR
For example axisxwithaxisyofanglexy
cos
For vectors:
v
uR
v
u
Prime ( ′ ) corresponds to the rotated system:
vuv
vuu
cossin
sincos
y
ox
y
x
x
o
y
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
5 Updated: 9/2/2014
(2) may also be written (for H):
cossin
sincos
cossin
sincos
yyyx
xyxx
yyxy
yxxx
Principal coordinates system is defined as the one at which
0 xyyx (3)
It can be shown that P
(=angle by which the original system has to rotate, positive in the
counter-clockwise direction, so that it becomes a principal coordinates system) is given as:
yyxx
xy
P
22tan (4)
The corresponding strains are called the principal strains
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
6 Updated: 9/2/2014
EXAMPLE: PURE SHEAR
0,
xy
yyxxyx
xyyx
yyxxyx 0
45
,
Distortion of 1111
DCBA = 2
yy
x
x
D
D
1D
1D
A
A 1A
1A
B
B
1B
1B
C
C
1C1
C
45
45
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
7 Updated: 9/2/2014
MOHR’S CIRCLE FOR STRAINS:
PPyx , principal axes
P
yy
P
xx , principal strains
y
Py
x
Px
P
yy
P
xx
xy
xx
P
yy
P
xx
yy
2
xy yyxx ,
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
8 Updated: 9/2/2014
STRAINS IN 3-D
333231
232221
131211
H
yxDin 2,12
i
j
j
i
ijx
u
x
u
2
1
yxxxvuuuDin 2121
,,,2
iiHtr
332211)(
(according to Einstein’s notation)
3
3
33
2
2
22
1
1
11,,
x
u
x
u
x
u
jiforjiij
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
9 Updated: 9/2/2014
ELEMENTARY FORCES – STRESSES – 2D
ij = stress applying on plane normal to axis i, with direction parallel to axis j
yyxx , normal stresses
yxxy , shear stresses
y
x
dy
dx
dy
dFxx
xx
yy
yx
xy
xx
dx
dFyyyy
dy
dFxy
xy
dx
dFyx
yx
y
x
dy
dx
yydF
yFd
yxdF
xFd
xxdF
xydF
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
10 Updated: 9/2/2014
TENSOR OF STRESSES (2-D)
2221
1211
yyyx
xyxx
ij
Laws of tensors hold!
EQUILIBRIUM OF STRESSES
(5)
(6a)
(6b)
y
dyxx
xy
yx yy
dx
x
dxx
xy
xy
dxx
xx
xx
dyy
yy
yy
dyy
yx
yx
Body force vector per
unit “volume”
B
B
Y
GX
Moments about 0G xyyxxy
0x
F 0
X
yx
yxxx
0yF 0
Y
yx
yyxy
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
11 Updated: 9/2/2014
TENSOR OF STRESSES (3-D):
333231
232221
131211
ij
jijiij
; (Due to equilibrium of moments)
;33
332211 iip
hydrostatic tension (7)
EQUILIBRIUM OF STRESSES (3-D):
0
i
j
jiB
x
(8)
Also written as: 0, ijji B
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
12 Updated: 9/2/2014
BOUNDARY TRACTIONS (2-D)
Boundary tractions are defined as:
ds
dFp x
x (x force per unit “area” {arc length in 2-D})
ds
dFp
y
y (y force per unit “area”)
Balance of forces on triangle ABC:
Along x:
(9a)
Along y:
(9b)
yxnn , Components of unit normal vector, n
, to body boundary.
Also:
cos
sin
y
x
n
n
Q: What about the body forces on the triangle?
y
dy
xdx
ydF Fd
n
xdF
yx yy
A C
B
ds
n
1 nds
dxn
y
ds
dyn
x
),(yx
dFdFFd
is force acting on the boundary of the body
(over BC)
xy
xx
xyxxxdFdxdy
yxyyydFdydx
yyxxxxxnnp
yyyxxyynnp
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
13 Updated: 9/2/2014
BOUNDARY TRACTIONS (3-D):
Boundary tractions are defined as:
dS
dFp
dS
dFp
dS
dFp 3
3
2
2
1
1,,
Balance of forces on pyramid ABCD:
3332321313
3232221212
3132121111
nnnp
nnnp
nnnp
or:
3
2
1
333231
232221
131211
3
2
1
n
n
n
p
p
p
(10)
3x
1x
B
A
D
C
21
22
23
),,(321
dFdFdFFd
2x
(Force acting on the
boundary of the body,
over BCD)
1dxAC
3dxAB
2dxAD
dSBCD
dxdxABD
BCD
ABDn
BCD
ABDn
BCD
ABDn
n
)(
2
1)(
)(
)(
)(
)(
)(
)(
1
23
3
2
1
),,( 321 nnnn
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
14 Updated: 9/2/2014
CONSTITUTIVE RELATIONS:
ijijij 2 (11)
332211
ii
:, Lamé’s constants
ij
Kronecker delta ( 1,;0 ijij
ji for ji )
211
,)1(2
EE
G
G = Shear Modulus
E = Modulus of Elasticity (Young’s Modulus)
= Poisson’s ratio
(11) can also be written as:
ijijij
E
211 (12)
Inverting (12) we get:
ijijijE
pE
13
(13)
3/3/332211
ii
p
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
15 Updated: 9/2/2014
DEVIATORIC STRESSES AND STRAINS:
They are defined as:
03
iiijijij
ee
0iiijijij
sps
Then (11) becomes:
Kp
Gesijij
2 (14)
K= Bulk Modulus
)21(33
2
EK
ALTERNATIVE EXPRESSIONS FOR EQUATION (13):
33221111
1
E
33112222
1
E (15)
22113333
1
E
1313131313
2323232323
1212121212
/2
/2
/2
G
G
G
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
16 Updated: 9/2/2014
SPECIAL CASES:
PURE SHEAR:
Then from (15) we have:
G
121323332211,0,0
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
17 Updated: 9/2/2014
PURE (AXIAL) TENSION
0
0
132312
3322
From (15) we have:
11
11
33
11
11
22
132312
11
110,
E
E
E
Definition of Poisson’s ratio:
11
33
11
22
Change in sectional area: E
11
3322
2
Change in volume of specimen: 1133221121
Eii
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
18 Updated: 9/2/2014
PLATE STRETCHING (2-D):
0,0231333
122211,, functions of
21, xx
Then from (15) we get:
G
E
E
2
1
1
1212
112222
221111
(16)
0221133
E
01323
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
19 Updated: 9/2/2014
PLANE STRAIN (2-D):
0,0231333
122211, functions of
21, xx
From (15) we get:
1212
2313
1122
2
22
2211
2
11
2
01
1
1
1
G
E
E
(17)
Note: Equation (17) can be brought to the form of equation (16) by putting:
1
1 2E
E
22113333
10
E
0221133
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
20 Updated: 9/2/2014
INVERSION OF (16) & (17)
(a) Plate Stretching: 033
1212
2211222
2211211
21
1
G
E
E
(16a)
(b) Plane Strain: 033
1212
221122
221111
2
1211
1211
G
E
E
(17a)
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
21 Updated: 9/2/2014
COMPATIBILITY OF STRAINS (2-D):
yxxy
xyyyxx
2
2
2
2
2
2 (18)
(18) valid
COMPATIBILITY OF STRESSES (2-D)
PLATE STRETCHING 033
From (16)
12;/2
EGGxyxy
So (18) yxxy
xy
xxyyyyxx
2
2
2
2
2
12 (19)
From equations (6a) and (6b)
062
2
2
x
X
yxxa
x
yxxx
062
22
y
Y
yyxb
y
yyxy
2
3
2
2
yx
u
yx
u xxxx
2
3
2
2
xy
v
xy
v yyyy
yx
v
xy
u
yxx
v
y
u xyxy
2
3
2
32
22
1
xxyyyy
yyxxxx
E
E
1
1
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
22 Updated: 9/2/2014
So (19) becomes:
y
Y
x
X
yxxy
yyxx
xxyyyyxx 2
2
2
2
2
2
2
2
1
(20)
For 0YX (negligible body forces):
02
2
2
2
yyxx
yx or 02 p (21)
and the equilibrium equations become:
0
yx
xyxx
(21a); 0
yx
yyxy
(21b)
Defining: (22)
(21a), (21b) are satisfied automatically
y
Y
yyx
x
X
xyx
yyxy
xxxy
2
22
2
22
012
2
2
2
y
Y
x
X
yxyyxx
yx
F
x
F
y
Fxyyyxx
2
2
2
2
2
,,
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
23 Updated: 9/2/2014
In order for (21) to be also satisfied we must have:
02
2
2
2
2
2
2
2
y
F
x
F
yx
or 04 F or 024
4
22
4
4
4
y
F
yx
F
x
F (23)
F = stress function of Airy
F = biharmonic function (satisfies 23)
vu
yxFbaequationscb
xyyyxxxyyyxx ,,,,,
),()9&9.(.23
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
24 Updated: 9/2/2014
EQUILIBRIUM EQUATIONS IN TERMS OF DISPLACEMENTS:
(2-D, PLATE STRETCHING)
0
X
yx
xyxx
(6a)
0
Y
yx
yyyx
(6b)
Equ. (16a):
yx
v
y
uE
y
x
v
yx
uE
x
x
v
y
uE
x
v
y
uG
y
v
yx
uE
yy
v
x
uE
yx
v
x
uE
xy
v
x
uE
x
v
y
u
y
v
x
u
E
GE
xy
xy
xy
yy
yy
xx
xx
xyyyxx
yyxxyy
xyxyyyxxxx
2
2
2
2
22
2
22
22
2
2
2
22
2
2
12
12
12
11
11
2
1,,
1
21
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
25 Updated: 9/2/2014
0
12
1
12 2
2
22
2
2
2
EX
yx
v
y
u
yx
v
x
u
01
22
22
2
2
2
22
2
2
G
X
x
u
yx
v
x
u
y
u
yx
v
x
u
01
1 2
2
2
2
G
X
yx
v
x
uu
Thus (6a) becomes:
0
121
2
2
22
2
2
2
X
yx
v
y
uE
yx
v
x
uE
(24a)
Similarly (6b) becomes:
01
1 2
2
2
2
G
Y
yx
u
y
vv
(24b)
Equations (24a) and (24b) are the equilibrium equations for u, v in the case of plate stretching.
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
26 Updated: 9/2/2014
GOING FROM PLATE-STRETCHING TO PLANE-STRAIN AND VICE-VERSA
EQUILIBRIUM EQUATIONS IN TERMS OF DISPLACEMENTS:
(2-D, PLANE STRAIN)
Put 1/ in (24a) & (24b) to get:
021
1 2
2
2
2
G
X
yx
v
x
uu
(25a)
021
1 2
2
2
2
G
Y
yx
u
y
vv
(25b)
Equations (25a) and (25b) are the equilibrium equations for u, v in the case of plane strain
CE380P. 4 – BOUNDARY ELEMENT METHODS – Review of Mechanics of Solids © S.A. Kinnas 2014
27 Updated: 9/2/2014
BOUNDARY CONDITIONS (B.C.s)
According to (9a), (9b)
yyxxxxxnnp
(9a)
yyyxxyynnp
(9b)
Natural b.c.s involve yx
, of u, v
Essential b.c.s involve u, v
y
A
B
n yn
xn
px, py known
(natural b.c.s)
u, v known
(essential b.c.s)
x
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