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Section 3.8
Solid Mechanics Part I
Kelly
85
3.8 Properties of the Strain
Stress transformation formulae, principal stresses, stress invariants and formulae for
maximum shear stress were presented in §3.4§3.5.
The strain is very similar to the
stress.
They are both tensors, having nine components, and all the formulae for stress
hold also for the strain.
Formulae for a twodimensional state of strain are given in what
follows.
3.8.1
Strain Transformation Formula
Consider two perpendicular lineelements lying in the coordinate directions
x
and
y
, and
suppose that it is known that the strains are
xy
yy
xx
ε
,
,
, Fig. 3.8.1.
Consider now a
second coordinate system, with axes
y
x
′
′
,
, oriented at angle
θ
to the first system, and
consider lineelements lying along these axes.
It can be shown that the lineelements in
the second system undergo strains according to the following (two dimensional)
strain
transformation equations
:
xy
xx
yy
xy
xy
yy
xx
yy
xy
yy
xx
xx
θε
2
cos
)
(
cos
sin
2
sin
cos
sin
2
sin
sin
cos
2
2
2
2
+
−
=
′
−
+
=
′
+
+
=
′
Strain Transformation Formulae
(3.8.1)
Figure 3.8.1: A rotated coordinate system
Note the similarity between these equations and the stress transformation formulae, Eqns.
3.4.7.
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 Fall '11
 Staff
 Shear Stress, Shear, Strain, Stress

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