Theory of Stress and Strain
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description
Relation between stress and strain with few examples are summarized.
Transcript of Theory of Stress and Strain
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-1
EGM 5653
CHAPTER 2
Theories of Stress and Strain
EGM 5653Advanced Mechanics of Materials
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-2
EGM 5653
Objectives
Definition of stress/strain as tensortress/strain on ar!itrar" planes
#rincipal $%antities& Mohr's circle in 2-D and 3-Dmall displacement theories
train rosettes
Sections
2(1&2(2 tress Definition
2(3 tress tranformation& 2() #rincipal $%anities
2(5 E$%ili!ri%m e$%ation& 2(6 strain in an" direction
2(* strain transformation& 2(+ strain theor"
2(, train meas%rment
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-3
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2.1 Stress at a point
-Lt
A
Ft
A
= =
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NLtN NA
Ft
A
= =
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SLtS SA
Ft
A
= =
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-)
EGM 5653
2.2 Stress Notation (Tensor)
PA
=
xx xy xz
xy yy yz
xz yz zz
T
=
Body force vector is
& &x y zB B B
P
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-5
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2.2 Alternate Stress Notation
PA
=
xx xy xz
xy yy yz
xz yz zz
T
=
xx xy xz
xy yy yz
xz yz zz
T
=
11 12 13
12 22 23
23 23 33
T
=
Dia.onal Normal
0ff-dia.onal-hear
En.( hear 2ensor hear
P
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-6
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2.2 Stress on arbitrary planes
PA
=
44 4 4cos cos cos
44 4
44 4x Y Z
N i j k
li mj nk
N i N j N k
= + +
= + +
= + +
xx xy xz
xy yy yz
xz yz zz
T
=
{ } [ ]{ }t T N=tress on ar!itrar" plane is
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-*
EGM 5653
2.4 Stress Transformation (3-D to another 3-D)
14e
14e
24e
24e
34e
3
4e
Consider to s"stems
{ } { }1 2 3 1 2 3 & & and & &e e e e e e &
{ }ie is o!tained from { }ie%sin. a ri.id !od" rotation(
7e are interested to relate the %nit vectors
{ }ie in from that of 6
(
1 11 1 21 2 31 3
2 12 1 22 1 32 3
3 13 1 23 1 33 3
4 4 4 4
4 4 4 4
4 4 4 4
e Q e Q e Q e
e Q e Q e Q e
e Q e Q e Q e
= + +
= + += + +
7e can see that&
4
cos8 &
4cos8 & 9
9
ij i j
old
Q e e
new=
=ransformation matri: component
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Namas ChandraAdvanced Mechanics of Materials Chapter 2-+
EGM 5653
2.4 Stress Transformation (Vector)
14e
14e
24e
24e
34e
3
4e
&
(
;ector from old to ne s"stem& = >
i mi m
T
a Q a
a Q a
Q Q Qa a
a Q Q Q a
a aQ Q Q
=
=
=
142a e= Find in terms of
