Principal stresses - Mohr’s circle in 3D - Strain tensor - Principal strains

raffleescargatoireMechanics

Jul 18, 2012 (5 years and 28 days ago)

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Prof. Ramesh Singh
Outline
•Principal stresses
•Mohr’s circle in 3D
•Strain tensor
•Principal strains
Prof. Ramesh Singh
Principal Stresses in 3D
•3-D Stresses can be represented by in usual
notation
We will use a concept from continuum mechanics










zyzxz
yzyxy
xzxyx
σττ
τστ
ττσ
Tn
r
)
=⋅
σ
Stress Tensor
Unit normal vector
Traction vector
Force/area
Prof. Ramesh Singh
Principal Stresses
Prof. Ramesh Singh
Principal Stresses in 3D




















=










=






























=




















++=
n
m
l
n
m
l
n
m
l
T
T
T
n
m
l
knjmiln
zyzxz
yzyxy
xzxyx
z
y
x
zyzxz
yzyxy
xzxyx
σ
σ
σ
σ
σττ
τστ
ττσ
σττ
τστ
ττσ
00
00
00
ˆ
ˆˆ
ˆ
Prof. Ramesh Singh
Principal Stresses in 3D
0
0
0
0
=













=























σσττ
τσστ
ττσσ
σσττ
τσστ
ττσσ
zyzxz
yzyxy
xzxyx
zyzxz
yzyxy
xzxyx
n
m
l
Prof. Ramesh Singh
3D Stress –Principal Stresses
32
123
0III
σσσ

+−=
The three principal stresses are obtained as the
three real roots of the following equation:
where
1
222
2
222
3
2
xyz
xyxzyzxyxzyz
xyzxyxzyzxyzyxzzxy
I
I
I
σ
σσ
σσσσσστττ
σ
σστττστστστ
=++
=++−−−
=+−−−
I
1
, I
2
,
and
I
3
are known as stress invariantsas
they do not change in value when the axes are
rotated to new positions.
Prof. Ramesh Singh
Principal Stress













28000
0200240
02400
In[3]:=
Det@880
−σ,

240,0
<,
8−
240,200
−σ,0
<,
80,0,

280
−σ
<<D
Out[3]=
16128000
+
113600
σ−
80
σ
2
?V
3
In[2]:=
Solve@
Det@880
−σ
,

240,0
<
,
8−
240,200
−σ,0
<,
80,0,
−280
−σ<<D
m
0,
σD
Out[2]=
88σ→−
280
<,
8σ→−
160<,
8σ→360<<
In[1]:=
Eigenvalues@880,
−240,0
<,
8−
240,200,0
<,
80,0,

280<<D
Out[1]=
8360,
−280,
−160<
Prof. Ramesh Singh
Principal Stresses in 3-D
Prof. Ramesh Singh
Linear Strains
∆x
∆u
Linear strain formulation:
z
w
y
v
x
u
x
u
zyx
x


=


=


=


=
εεε
ε
;;
as, drepresente becan it limits Taking
Prof. Ramesh Singh
Shear Strain
()










+


=










+


=+==


≈≈


≈≈
+=−=
x
v
y
u
x
v
y
u
y
u
x
v
xyxy
xy
2
1
2
1
2
1
2
1
tan
tan
2
21
22
11
21
θθγε
θθ
θθ
θθψ
π
γ
∆x
∆y
∆u
∆v
Prof. Ramesh Singh
Strain Tensor




























+










+












+














+










+












+




=










=
z
w
x
w
z
u
x
w
z
u
y
w
z
v
y
v
x
v
y
u
x
w
z
u
x
v
y
u
x
u
zyzxz
yzyxy
xzxyx
ji
2
1
2
1
2
1
2
1
2
1
2
1
,
εεε
εεε
εεε
ε
Prof. Ramesh Singh
Strain Transformation
www.efunda.com
xyxy
γε
2
1
=
Prof. Ramesh Singh
Mohr’s Circle for Strain
xyxy
γε
2
1
=
Prof. Ramesh Singh
Principal Strains
xyxy
where
γε
2
1
,
=