Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

deformation vary. This dependency is summarized in a load–deformation diagram,
such as the one shown in Figure 17.3. The shape of the diagram depends
upon the particular material being considered.
Now consider an arbitrary value of load P 1 between zero and the maximum
value P. At the load P 1 , the deformation of the bar is δ 1 . From this state,
an additional load increment dP will produce an increment of deformation dδ.
During this incremental deformation, the load P 1 will also move, and in so
doing, it will perform work equal to dW = P 1 dδ. This work is shown in Figure 17.3
by the darkly shaded area beneath the load–deformation curve. The total work
done by the load as it increases in magnitude from zero to P can be determined
by summing together all such infinitely small increments:
W
=
∫
δ
0
Pdδ
The shaded area underneath the load–deformation curve represents the total work
done. When the load–deformation diagram is linear (Figure 17.4), the work done
by P is
W
= 1 2
Pδ
(17.1)
P
dP
P 1
P
O
δ 1
FIGURE 17.3 Load–deformation diagram.
P
δ
dδ
δ
which is simply the area under the diagram.
P
U= W=
1
2
Pδ
Strain Energy
As external load P is applied to the bar, work is performed, and energy is expended. Since
this work is performed by an external load, it is typically referred to as external work. The
load causes the bar to deform, and in the process, it produces strains in the bar. The principle
of conservation of energy asserts that energy in a closed system is never created or destroyed;
rather, it is only transformed from one state to another. So, where does the energy expended
by the work of the external load P go? The answer is that it is transformed into internal energy
stored in the strains of the bar. The energy absorbed by the bar during the loading process is
termed strain energy. In other words, strain energy is the energy that is stored in a material
body as a consequence of the body’s deformation. Provided that no energy is lost in the form
of heat, the strain energy U is equal in magnitude to the external work W:
∫
1
U = W = Pdδ
While external work may be either a positive or a negative quantity, strain energy is
always a positive quantity.
An examination of Equation (17.2) reveals that work and energy are expressed in
the same units—that is, the product of force and distance. In the SI, the unit of work
and energy is the joule (J), which is equal to 1 N ⋅ m. In U.S. customary units, work and
energy may be expressed in units of lb ⋅ ft., lb ⋅ in., kip ⋅ ft, or kip ⋅ in.
Because of its stored energy, the bar in Figure 17.2 is capable of doing work as
it returns to its undeformed configuration after the load is removed. If the elastic limit
is not exceeded, the bar will return to its original length. If the elastic limit is exceeded,
as illustrated in Figure 17.5, a residual strain will remain after the load is removed.
The total strain energy is always the area under the load–deformation curve (area
OABCDO); however, only the elastic strain energy (the triangular area BCD) can be
recovered. The other portion of the area under the curve (area OABDO) represents the
strain energy that is spent in permanently deforming the material. This energy dissipates
in the form of heat.
δ
0
(17.2)
P
O
O
δ
FIGURE 17.4 Linear load–
deformation diagram.
Strain energy is sometimes
referred to as internal work to
distinguish it from the external
work done by the load.
Inelastic strain
energy
A
D
FIGURE 17.5 Elastic and inelastic
strain energy.
C
B
δ
δ
Elastic strain
energy
717