3.5 Balance Laws
117
Interchanging the dummy indices i and j in this expression yields the forms
ij k σ ij e k = 0
or
jik σ ji e k = − ij k σ ji e k = 0,
where the subscripts of jik in the latter expression have been permuted and the sign
changed according to Eq. (2.6). Finally, adding these relations gives
ij k (σ ij − σ ji )e k = 0,
(3.162)
which yields the expected result σ ij = σ ji or σ T = σ . Thus, in general, the Cauchy
stress tensor is symmetric. Using this anticipated finding, we showed in Sect. 3.4.3
that the second Piola-Kirchhoff stress tensor S also is symmetric, but the first PiolaKirchhoff stress tensor P is not.
3.5.4 Balance of Energy
The first law of thermodynamics states that the total energy in an isolated system
is conserved, i.e., in a system that does not exchange matter or energy with its
surroundings. Here, however, we consider a continuum as an open system. In
addition, we assume that the behavior of the system is dominated by mechanical and
thermal energy. Other types of energy, such as electrical and chemical, are ignored.
Mathematically, the energy balance for a thermomechanical system can be
written in the form
d
dt
(K + U) = P in + Q in ,
(3.163)
where K is the kinetic energy, U is the internal energy, P in is the mechanical power
input, and Q in is the rate of heat input. In words, this equation says that the rate of
change of the total energy, ˙
K + ˙
U , is equal to the rate at which energy is added to
the system, P in + Q in .
Later in this section, the above equation is specialized to a thermomechanical
continuum. But first, it is instructive to consider a simple dynamic system consisting
of a massless spring with stiffness k that is fixed at one end and attached to a mass
m at the other (Fig. 3.20a). The entire system is submerged under water inside a
frictionless box. A force f (t) = f 0 sin ωt applied to the mass causes it to oscillate,
where f 0 is the amplitude and ω the circular frequency. The system is oriented
horizontally, so we do not have to deal with gravity.
Let the spatial coordinate x(t) define the horizontal location (and displacement)
of the mass relative to its position when the spring is undeformed. Since the spring
is massless, the total kinetic energy in the system is
K =
1
2 m ˙
x
2 .
(3.164)
117
Interchanging the dummy indices i and j in this expression yields the forms
ij k σ ij e k = 0
or
jik σ ji e k = − ij k σ ji e k = 0,
where the subscripts of jik in the latter expression have been permuted and the sign
changed according to Eq. (2.6). Finally, adding these relations gives
ij k (σ ij − σ ji )e k = 0,
(3.162)
which yields the expected result σ ij = σ ji or σ T = σ . Thus, in general, the Cauchy
stress tensor is symmetric. Using this anticipated finding, we showed in Sect. 3.4.3
that the second Piola-Kirchhoff stress tensor S also is symmetric, but the first PiolaKirchhoff stress tensor P is not.
3.5.4 Balance of Energy
The first law of thermodynamics states that the total energy in an isolated system
is conserved, i.e., in a system that does not exchange matter or energy with its
surroundings. Here, however, we consider a continuum as an open system. In
addition, we assume that the behavior of the system is dominated by mechanical and
thermal energy. Other types of energy, such as electrical and chemical, are ignored.
Mathematically, the energy balance for a thermomechanical system can be
written in the form
d
dt
(K + U) = P in + Q in ,
(3.163)
where K is the kinetic energy, U is the internal energy, P in is the mechanical power
input, and Q in is the rate of heat input. In words, this equation says that the rate of
change of the total energy, ˙
K + ˙
U , is equal to the rate at which energy is added to
the system, P in + Q in .
Later in this section, the above equation is specialized to a thermomechanical
continuum. But first, it is instructive to consider a simple dynamic system consisting
of a massless spring with stiffness k that is fixed at one end and attached to a mass
m at the other (Fig. 3.20a). The entire system is submerged under water inside a
frictionless box. A force f (t) = f 0 sin ωt applied to the mass causes it to oscillate,
where f 0 is the amplitude and ω the circular frequency. The system is oriented
horizontally, so we do not have to deal with gravity.
Let the spatial coordinate x(t) define the horizontal location (and displacement)
of the mass relative to its position when the spring is undeformed. Since the spring
is massless, the total kinetic energy in the system is
K =
1
2 m ˙
x
2 .
(3.164)