Section 3.3
Solid Mechanics Part I
Kelly
40
3.3 Internal Stress
The idea of stress considered in §3.1 is not difficult to
conceptualise since objects interacting with other objects are
encountered all around us.
A more difficult concept is the
idea of forces and stresses acting
inside
a material, “within
the interior where neither eye nor experiment can reach” as
Euler put it.
It took many great minds working for centuries
on this question to arrive at the concept of stress we use
today, an idea finally brought to us by Augustin Cauchy,
who presented a paper on the subject to the Academy of
Sciences in Paris, in 1822.
Augustin Cauchy
3.3.1
Cauchy’s Concept of Stress
Uniform Internal Stress
Consider first a long slender block of material subject to equilibrating forces
F
at its ends,
Fig. 3.3.1a.
If the complete block is in equilibrium, then any subdivision of the block
must be in equilibrium also.
By imagining the block to be cut in two, and considering
freebody diagrams of each half, as in Fig. 3.3.1b, one can see that forces
F
must be
acting
within
the block so that each half is in equilibrium.
Thus
external loads create
internal forces
; internal forces represent the action of one part of a material on another
part of the same material.
If the material out of which the block is made is uniform over
this cut, one can take it that a uniform stress
A
F
/
=
σ
acts over this interior surface, Fig.
3.3.1b.
Figure 3.3.1: a slender block of material; (a) under the action of external forces F,
(b) internal normal stress
σ
, (c) internal normal and shear stress
F
F
FF
)
a
()
b
(
)
c
(
F
F
F
F
A
F
=
N
S
F
imaginary
cut
F
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Solid Mechanics Part I
Kelly
41
Note that, if the internal forces were not acting over the internal surfaces, the two half
blocks of Fig. 3.3.1b would fly apart; one can thus regard the internal forces as those
required to maintain material in an uncut state.
If the internal surface is at an incline, as in Fig. 3.3.1c, then the internal force required for
equilibrium will not act normal to the surface.
There will be components of the force
normal and tangential to the surface, and thus both normal (
N
σ
) and shear (
S
)
stresses
must arise.
Thus, even though the material is subjected to a purely normal load, shear
stresses develop.
From Fig. 3.3.2a, the normal and shear stresses arising on an interior surface inclined at
angle
θ
to the horizontal are {
▲
Problem 1}
cos
sin
,
cos
2
A
F
A
F
S
N
=
=
(3.3.1)
Figure 3.3.2: stress on inclined surface; (a) decomposing the force into normal and
shear forces, (b) stress at an internal point
Although stress is associated with surfaces, one can speak of the stress “at a point”.
For
example, consider some point interior to the block, Fig 3.3.2b.
The stress there evidently
depends on which surface through that point is under consideration.
From Eqn. 3.3.1a,
the normal stress at the point is a maximum
A
F
/
when
0
=
and a minimum of zero
when
o
90
=
.
The maximum normal stress arising at a point within a material is of
special significance, for example it is this stress value which often determines whether a
material will fail (“break”) there.
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 Fall '11
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 Force, Strain, Stress

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