Strain Transformations
(plane strain)
t
1
y
dst
x
dsn
n
dy / dsn = n y
dx / dsn = nx
1 u u
ent = n + t
2 st sn
1 u
u
= n
+t
2 st
sn
un displacement in n direction
ut displacement in t direction
1 u
u
ent = n
+t
2 st
sn
u u dx u dy
=
+
sn x dsn y d

Fatigue Life Evaluation S-N Curve (alternating stress amplitude (Sa) versus number of cycles (Nf) to failure)
800
Sa = a (MPa)
600
e= Se endurance limit
400
103
104
105
106
107
108
Nf , cycles to failure Some materials, such as steel, show an endurance li

Local equilibrium (biaxial state of stress) y
yy +
yy y
dy
dx
yx +
dy
yx y
dy
x z
xx
xy
dz
yy
x
yx
xx dx x xy xy + dx x
xx +
F
=0
xx + xx dx dydz xx dydz dx yx + yx + dy dxdz yx dxdz = 0 y
xx yx + =0 x y
similarly
F
x
y
=0
xy
+
yy y
=0
Equa

We found that stresses and strains transform according to the same rules,
namely
T
t (y')
= l l
[ ] [ ] [ ][ ]
T
[e] = [l ] [e][l ]
n (x')
z
where [ l ] is the matrix of direction cosines
nx
[l ] = n y
nz
tx
ty
tz
vx l11 l12
v y = l21 l22
vz l31 l32
l1

Equilibrium equation in polar coordinates
(plane stress)
d
2
+
r
( r + dr ) d
r +
d
2
r d
d
2
d
r
d
rr
r
F
r
r
d
r
dr
d
2
r +
rr +
rr
dr
r
r
=0
rr
d
dr ( r + dr ) d dz rr rd dz r drdz cos
rr +
r
2
d
d
drdz sin
d drdz sin
+
=0
2

EM 424: 3-D Stresses
1 Stresses in Three Dimensions
The traction vector The basic quantity that we use to characterize how external forces distribute themselves within a deformable body is the traction vector. This vector is defined as the force/unit area

2-D stresses, traction vector
2-D (biaxial) state of stress
n
t
yy
yx
nt
xy
xx
y
nn
xx
xy
yx
x
z
yy
nt
nn
normal and tangential
stress components
Ty( n )
Tx( n )
traction vector
components
n , unit normal
Ty( n )
1
xx
nx
(n)
dy ds
xy
Fx = 0
Tx
dx
yx
ny
y

Strains
(2-D deformation -plane strain)
y
x
deformed body
undeformed body
dsy
xy
dy
ds dx
normal exx = x
dx
strains
ds y dy
eyy =
dy
dsx
dx
engineering
shear strain
tensor
shear strain
xy =
exy =
2
xy
xy
2
state of strain (plane strain)
exx
e] =
[
ey

Theories of (static) Failure Consider biaxial (plane stress) case first:
Basic idea is that if some combination of the principal stresses gets too large, the material will fail. We can think of the "safe" stresses as defining some region in terms of princ