4.3 Elastic Strain Energy for Uniaxial Stress
109
strain in the direction x. The average force acting on the element while deformation is
taking place is σ x
dy dz
2
. This average force multiplied by the distance through which
it acts is the work done on the element. For a perfectly elastic body no energy is
dissipated, and the work done on the element is stored as recoverable internal strain
energy. Therefore, the internal elastic strain energy U for an infinitesimal element
subjected to uniaxial stress is
dU =
1
2
σ x dy dz × ε x dx
=
1
2
σ x ε x dx dy dz
Therefore,
dU =
1
2
σ x ε x dV
(4.32)
where
dV = dx dy dz volume of the element.
Thus, the above expression gives the strain energy stored in an elastic body per
unit volume of the material, which is called strain energy density U 0 .
Hence,
dU
dV
= U 0 =
1
2
σ x ε x
(4.33)
The above expression may be graphically interpreted as an area under the inclined
line on the stress–strain diagram as shown in Fig. 4.1b. The area enclosed by the
inclined line and the vertical axis is called the complementary energy. For linearly
elastic materials, the two areas are equal.
4.4 Strain Energy in an Elastic Body
When work is done by an external force on certain systems, their internal geometric
states are altered in such a way that they have the potential to give back equal amounts
of work whenever they are returned to their original configurations. Such systems
are called conservative, and the work done on them is said to be stored in the form of
potential energy. For example, the work done in lifting a weight is said to be stored as
a gravitational potential energy. The work done in deforming an elastic spring is said
109
strain in the direction x. The average force acting on the element while deformation is
taking place is σ x
dy dz
2
. This average force multiplied by the distance through which
it acts is the work done on the element. For a perfectly elastic body no energy is
dissipated, and the work done on the element is stored as recoverable internal strain
energy. Therefore, the internal elastic strain energy U for an infinitesimal element
subjected to uniaxial stress is
dU =
1
2
σ x dy dz × ε x dx
=
1
2
σ x ε x dx dy dz
Therefore,
dU =
1
2
σ x ε x dV
(4.32)
where
dV = dx dy dz volume of the element.
Thus, the above expression gives the strain energy stored in an elastic body per
unit volume of the material, which is called strain energy density U 0 .
Hence,
dU
dV
= U 0 =
1
2
σ x ε x
(4.33)
The above expression may be graphically interpreted as an area under the inclined
line on the stress–strain diagram as shown in Fig. 4.1b. The area enclosed by the
inclined line and the vertical axis is called the complementary energy. For linearly
elastic materials, the two areas are equal.
4.4 Strain Energy in an Elastic Body
When work is done by an external force on certain systems, their internal geometric
states are altered in such a way that they have the potential to give back equal amounts
of work whenever they are returned to their original configurations. Such systems
are called conservative, and the work done on them is said to be stored in the form of
potential energy. For example, the work done in lifting a weight is said to be stored as
a gravitational potential energy. The work done in deforming an elastic spring is said