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5
Fundament al Laws and Equations
5.1
Balance Laws, Field Equations, Constitutive Equations
A number of the fundamental laws of continuum mechanics are expressions
of the conservation of some physical quantity. These
as they are
often called, are applicable to all material continua and result in equations
that must always be satisfied. In this introductory text, we consider only the
conservation laws dealing with mass, linear and angular momentum, and
energy. With respect to energy, we shall first develop a purely mechanical
energy balance and follow that by an energy balance that includes both
mechanical and thermal energies, that is, a statement of the first law of
thermodynamics. In addition to that, the Clausius-Duhem form of the second
law of thermodynamics is covered.
The balance laws are usually formulated in the context of global (integral)
relationships derived by a consideration of the conservation of some prop-
erty of the body as a whole. As explained in Chapter One, the global equa-
tions may then be used to develop associated
balance laws,
which are valid
at all points within the body and on its boundary. For example, we shall
derive the local equations of motion from a global statement of the conser-
vation of linear momentum.
field equations
which reflect the internal constitution of a material,
define specific types of material behavior. They are fundamental in the sense
that they serve as the starting point for studies in the disciplines of elasticity,
plasticity, and various idealized fluids. These equations are the topic of the
final section of this chapter.
Before we begin a discussion of the global conservation laws, it is useful
to develop expressions for the material derivatives of certain integrals. This
we do in the next section.
Constitutive equations,
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5.2
Material Derivatives of Line, Surface, and Volume Integrals
Let any scalar, vector, or tensor property of the collection of particles occu-
pying the current volume
V
be represented by the integral
∫
()
*
P
(
t
) =
Pt dV
ij
x
,
(5.2-1)
ij…
L
V
P ij *
where represents the distribution of the property per unit volume and
has continuous derivatives as necessary. The material derivative of this prop-
erty is given in both spatial and material forms, using Eq 4.11-8, by
d
dt
Pt dV d
dt
∫
∫
[
]
˙
() =
() =
(
)
*
*
o
Pt
x
,
P
x X
,
t t JdV
,
ij
L
ij
L
ij
L
o
V
V
Since
is a fixed volume in the referential configuration, the differentiation
and integration commute, and the differentiation can be performed inside
the integral sign. Thus, from Eq 4.11-6, using the notation
V
°
[]
•
to indicate
differentiation with respect to time,
[
]
(
)
∫
•
∫
˙
˙
(
)
*
o
*
*
o
Pt J dV PJ PJ dV
ij
X
,
=
L
ij
LL
ij
o
o
V
V
(
)
∫
˙ *
*
o
=
P + v P
JdV
ij
L
k,k ij
L
o
V
and converting back to the spatial formulation
[
]
∫
˙
˙
() =
()
()
*
*
Pt
Pt v Pt dV
x
,
x
,
(5.2-2)
ij
L
ij
L
k,k ij
L
V
With the help of the material derivative operator given in Eq 4.5-5, this
equation may be written (we omit listing the independent variables
x
and
t
for convenience),
*
*
∂
P
t
∂
∂
P
x
∫
˙
() =
ij
L
ij
L
*
Pt
+
v
+v P dV
ij
L
k
k,k ij
L
∂
V
k
*
∂
P
t
(
)
∫
ij
L
*
=
+
vP , dV
k j k
L
∂
V
809236709.001.png 809236709.002.png 809236709.003.png
 
809236709.004.png 809236709.005.png 809236709.006.png
 
which upon application of the divergence theorem becomes
*
∂
P
t
∫
∫
˙
() =
ij
L
*
Pt
dV + v P n dS
(5.2-3)
ij
L
k
ij
L
k
∂
V
S
This equation gives the time rate of change of the property
P
as the sum of
ij
the amount created in the volume
V
, plus the amount entering through the
bounding surface
.
Time derivatives of integrals over material surfaces and material curves
may also be derived in an analogous fashion. First, we consider a tensorial
property
S
, and is often spoken of as the
transport theorem
Q
of the particles which make up the current surface
S
, as given by
ij
∫
∫
() =
() =
()
*
*
Qt
Qt dS
x
,
Qt n dS
x
,
(5.2-4)
ij
L
ij
L
p
ij
L
p
S
S
()
where is the distribution of the property over the surface. From
Eq 4.11-7, we have in Eulerian form (again omitting the variables
*
Qt
x
,
ij
L
x
and
t
),
(
)
∫
∫
˙
˙ *
() =
*
*
Q
t
Q + v Q
dS
−
Q v dS
ij
L
ij
L
k,k ij
L
p
ij
L
q,p
q
S
S
[
]
(
)
∫
˙ *
*
*
=
Q+vQ
δ
−
Qv S
(5.2-5)
ij
L
k,k ij
L
pq
ij
L
q,p
q
S
Similarly, for properties of particles lying on the spatial curve
C
and
expressed by the line integral
∫
() =
()
*
Rt
R t x
x
,
(5.2-6)
ij
L
ij
L
p
C
we have, using Eq 4.11-1,
∫
∫
˙
˙ *
() =
*
R t
Rdx
+
vRdx
ij
L
ij
L
p
p,q ij
L
q
C
C
(
)
∫
˙ *
*
=
R
δ
+
v R
dx
(5.2-7)
ij
L
pq
p,q ij
L
q
C
809236709.007.png 809236709.008.png
5.3
Conservation of Mass, Continuity Equation
Every material body, as well as every portion of such a body is endowed
with a non-negative, scalar measure, called the
of the body or of the
portion under consideration. Physically, the mass is associated with the
inertia property of the body, that is, its tendency to resist a change in motion.
The measure of mass may be a function of the space variables and time. If
mass
∆
m
is the mass of a small volume
∆
V
in the current configuration, and if we
assume that
∆
m
is absolutely continuous, the limit
∆
∆
m
V
ρ =
(5.3-1)
lim
∆
V
→
0
defines the scalar field
ρ
=
ρ
(
x
,
t
) called the
mass density
of the body for that
configuration at time
t
. Therefore, the mass
m
of the entire body is given by
∫
()
m
=
ρ
x ,
t dV
(5.3-2)
V
In the same way, we define the mass of the body in the referential (initial)
configuration in terms of the density field
ρ
=
ρ
(
X
,
t
) by the integral
0
0
∫
(
)
o
m
=
ρ o
X ,
t dV
(5.3-3)
o
V
asserts that the mass of a body, or of any
portion of the body, is invariant under motion, that is, remains constant in
every configuration. Thus, the material derivative of Eq 5.3-2 is zero,
The law of
conservation of mass
d
dt
∫
()
˙
m
=
ρ
x
,
tdV
= 0
(5.3-4)
V
*
which upon application of Eq 5.2-2 with
P ij L
≡ ρ
becomes
(
)
∫
˙
˙
m
=
ρρ
+
v dV
i,i
= 0
(5.3-5)
V
is an arbitrary part of the continuum, the integrand here must
vanish, resulting in
and since
V
˙
ρρ
+
v i,i
= 0
(5.3-6)
809236709.009.png 809236709.010.png 809236709.011.png 809236709.012.png 809236709.013.png 809236709.014.png
 
which is known as the
continuity equation
in Eulerian form. But the material
derivative of
ρ
can be written as
∂ρ
∂
∂ρ
∂
˙
ρ
=+
t
v
i
x
i
so that Eq 5.3-6 may be rewritten in the alternative forms
∂ρ
∂
∂ρ
∂
+
v
+
ρ
v
= 0
(5.3-7
a
)
i
i,i
t
x
i
or
∂ρ
∂
+ (
)
ρ
v i i
= 0
(5.3-7
b
)
t
,
If the density of the individual particles is constant so that
= 0, the
ρ
˙
material is said to be
incompressible,
and thus it follows from Eq 5.3-6 that
v
= 0
or
div
v
= 0
(5.3-8)
i,i
for incompressible media.
Since the law of conservation of mass requires the mass to be the same in
all configurations, we may derive the continuity equation from a comparison
of the expressions for
in the referential and current configurations. There-
fore, if we equate Eqs 5.3-2 and 5.3-3,
m
∫
∫
()
(
)
o
m
=
ρ
x ,
t dV
=
ρ 0 X
, tdV
(5.3-9)
o
V
V
and, noting that for the motion
x
=
x
(
X
,
t
), we have
∫
[
]
∫
(
)
(
)
o
ρ
xX
, ttdV
=
ρ
X
, tJdV
o
V
V
Now if we substitute the right-hand side of this equation for the left-hand
side of Eq 5.3-9 and collect terms,
∫
[
]
(
)
(
)
o
ρ
XJ X
,
t
−
ρ
,
t
dV
= 0
o
o
V
But
V
°
is arbitrary, and so in the material description
ρ
J
=
ρ
(5.3-10
a
)
o
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