Chapter 4 Continuum Formulations of Conservation Laws

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Chapter 4 Continuum Formulations of Conservation
Laws
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4.1 The conservation principles Conservation
principles will be cast here in the form of a
balance or accounting statement for the time rate
of change of a quantity in a system. On the plus
side of the accountants ledger are the amount of
the quantity coming into the system and the
quantity produced within the system. On the minus
side are the amount of the quantity leaving the
system and the quantity consumed within the
system. The system will be either a fixed
material system consisting always of the same set
of particles or a fixed spatial (continuum)
volume through which material is passing. The
quantity conserved will be either
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mass, linear momentum, angular momentum or
energy. The statement of conservation of mass is
usually the statement that mass cannot be created
or destroyed. The conservation of momentum is
usually stated in the form of Newtons second
law the sum of the forces acting on an object is
equal to the product of the mass of the object
and the acceleration of the object. The
conservation of angular momentum is the statement
that the time rate of change of angular momentum
must equal the sum of the applied moments. The
conservation of energy is
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the requirement that the time rate of change of
the sum of all the kinetic and internal energies
must equal the mechanical power and heat power
supplied to the object. 4.2 The conservation of
mass The total mass M at time t of an object O is
given by where ?(x, t) is the
mass density at the
place x within the object at the time t. The
statement of mass conservation for the object O
is that the M does not change with time
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The material time derivative may be interchanged
with the integration over the object O since a
fixed material volume is identified as the object,
Then, using the relationship relating the time
rate of change in the volume to the present size
of the volume, , it
follows that
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The next step in the development of this
continuum representation of the conservation of
mass is to employ the argument that the integral
equation over the object O may be replaced by the
condition that the integrand in the integral
equation be identically zero, thus
. The argument that is used to go from
the equation before last to the last equation is
an argument that will be employed three more
times in this chapter. The argument is that any
part or subvolume of an object O may also be
considered as an object and the equation before
last also holds for that sub-object (figure). In
this figure an object and a portion of an object
that
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may be considered as an object itself are
illustrated. The argument for the transition from
the equation before last to the last equation is
as follows suppose it is not true that the
integrand is not zero everywhere (i.e., suppose
the transition from the equation before last to
the last equation is not true). If that is the
case then there must exist domains of the object
in which the integrand is positive and other
domains in which the integrand is negative so
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that when the integration is accomplished over
the entire object, the sum is zero. If that is
the case consider a domain of the object in which
the integrand is always positive (or negative).
Let this domain be an object. For the object
chosen in this way the integral on the left hand
side of the equation before last cannot be zero.
This conclusion contradicts the equation before
last because (the equation before last must be
zero). It may therefore be concluded that the
requirement that the integral be zero for an
object and all sub-objects that can be formed
from it means that the integrand must be zero
everywhere in the object. Integral statements
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are global statements because they apply to an
entire object. However the non-integral
requirement is a local, pointwise condition valid
at the typical point (place) in the object. Thus
the transition is from the global to the local or
from the object to the point (or particle) in the
object. Note that and
may be combined to obtain this alternate local
statement of mass conservation
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Another consequence of the conservation of mass
is a simple formula for the material time
derivative of an integral of the Form where k(x,
t) is a physical quantity (temperature,
momentum, etc.) of arbitrary scalar, vector or
tensor character and K is the value of the
density times quantity k(x, t) integrated over
the entire object O. Since the material time rate
of change of r(x, t)dv is zero, it follows that
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4.3 The state of stress at a point Stress is a
mental construct of humans to represent the
internal interactions or internal forces on a
material object. The concept of a stress will
first be introduced as a vector. Consider the
potato-shaped object shown in the figure below.
An imaginary plane ? characterized by its
normal n divides the object into an upper
portion U and a lower portion L.
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Considering L as a free object, the action of U
upon L is statically equivalent to a resultant
force f and couple m. Assuming that the
interaction is distributed across S, each area
element ?A(i) of the intersection of S and the
object may be considered as transmitting a force
?f(i) and a moment ?m(i). The average stress
vector, (P), at the point P acting on the plane
whose normal is n, is defined as the ratio The
quantity is a vector because the force ?f(i) is
a vector and ?A(i) is a scalar. The average
couple stress vector at the point
P acting
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on the plane whose normal is n is defined
analogously by the ratio The stress vector
t(n)(P) acting at the point P on the plane whose
normal is n is defined as the limit of the
average stress vector, , as DA(i) tends
to zero through a sequence of Progressively
smaller areas, DA(1), DA(2), DA(3) ,..., DA(N),
..., all containing the point P,
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When a similar limit is applied to the average
couple stress vector we assume that the limit is
zero In effect we are assuming that the
forces involved are of finite magnitude and that
the moment arm associated with ?m(i) vanishes
as ?A(i) tends to zero. For almost all continuum
theories this assumption is adequate.
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The internal force interaction at a point in an
object is adequately represented by the stress
vector t(n) across the plane whose normal is n.
However, there is a double infinity of distinct
planes with normals n passing through a single
point thus there is a double infinity of
distinct stress vectors acting at each point. The
multitude of stress vectors, t(n) , at a point
are called the state of stress at the point The
totality of vectors t(n)(P) at a fixed point P,
and for all directions n, is called the state of
stress at the point P. The representation of the
state of stress at a point is simplified by
proving that t(n)(P) must be a linear function of
the vector n,
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and this will be done. The coefficients of this
linear relationship will be the stress tensor T.
Thus T will be a linear transformation that
transforms n into t(n) , t(n)(P) T(P)?n. The
proof that T is, in fact, a tensor and the
coefficient of a linear transformation will also
be provided below. However, even though it has
not yet been proved, T will be referred to as a
tensor. The stress tensor T has components
relative to an orthonormal basis that are the
elements of the matrix
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The component Tij of the stress tensor is the
component of the stress vector t(n) t(j) acting
on the plane whose normal n is in the ej
direction, n ej, projected in the ej
direction,Tij ei?t(j). The nine components of
the stress tensor represented above are therefore
the e1, e2, and e3 components of the three stress
vectors t(i) , t(j) and t(k) which act at the
point P, on planes parallel to the three mutually
perpendicular coordinate planes. This is
illustrated in the figure on the next slide. The
components T11, T22, T33 are called normal
stresses, and the remaining components T12, T23,
T13, T21, T32, T31 are called shearing stresses.
Each of these
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components has the dimension of force per unit
area. Recall that stress was defined as the
force per unit area that the upper portion U
exerts on the lower portion L. From this
definition, it follows that if the exterior
normal of the object is in the positive
coordinate direction, then positive normal and
shear stresses will also be in
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the positive coordinate direction. If the
exterior normal of the object is in the negative
coordinate direction however, positive normal and
shear stresses will point in the negative
coordinate direction. The rule for the signs of
stress components is as follows the stress on a
plane is positive if it points in a positive
direction on a positive plane, or in a negative
direction on a negative plane. Otherwise it is
negative.
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It will now be shown, following an 1822 result of
Cauchy, that the nine components of stress are
sufficient to characterize the entire state of
stress at a point. Specifically, it will be shown
that the state of stress at a point P is
completely determined if the stress vectors
associated with three mutually perpendicular
planes are known at P and are continuous in a
neighborhood of P. The stress vector t(n) acting
on any plane whose normal is n is given by t(n)
T?n, or
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This means that the stress tensor T can be
considered as a linear transformation that
transforms the unit normal n into the stress
vector t(n) acting on the plane whose normal is
n.
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To prove this result we consider the tetrahedral
element of an object shown in figure below as a
free object and apply Newton's second law to the
force system acting on the tetrahedron. The
tetrahedron is selected in such a way that the
stress vectors acting on the mutually orthogonal
faces are the stress vectors acting on the
coordinate planes. Recall that we have
represented the components of the stress vectors
acting on the planes whose normals
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are e1, e2 and e3 by the components of the stress
tensor. The three scalar equations of Newton's
second law will suffice for the determination of
the unknown components of t(n) acting on the
fourth face. For simplicity we will only derive
the result in the e1 direction. We let A be the
area of the inclined face with normal n, and h
the perpendicular distance from P to the inclined
face. The mean value of t(n) will be denoted and
defined by
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where, as a consequence of the mean value
theorem, the point Q lies inside A. Analogously
defined mean values of the components of T over
their respective areas will be denoted by .
The reason for requiring that the stress
components be continuous of position in a
neighborhood of P is to insure that the mean
values of the stress components actually occur at
certain points always within the corresponding
areas.
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Since the area of the inclined face of the
tetrahedron may be represented by the vector An,
where A is the magnitude of the area and n is the
normal to the plane containing the area, the
areas of the orthogonal faces are each given by
An1, An2, An3. The fact that the areas of four
faces of a tetrahedron, where three of the faces
of the tetrahedron are orthogonal, are A, An1,
An2 and An3 is a result from solid geometry.
Summing forces in the e1 direction and setting
the result equal to the mass times the
acceleration of the tetrahedron we find that
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where Ah/3 is the volume of the tetrahedron, d1
is the action-at-a-distance force (e.g., gravity)
in the e1 direction and the acceleration of
the tetrahedron in that direction. The next step
in this development is to cancel A throughout the
previous equation and allow the plane whose
normal is n to approach P, causing the volume of
the tetrahedron to vanish as h tends to zero.
Before doing this, note that since Q must always
lie on A, as h tends to zero, by the mean value
theorem
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Canceling A throughout the equation and taking
the limiting process as h tends to zero, noting
that the object force and the acceleration vanish
as the volume of the tetrahedron vanishes, it
follows from that Repeating this analysis for
the e2 and e3 directions the desired result, t(n)
T?n, is established. Thus we have shown that
the double infinity of possible stress vectors
t(n) , which constitute a state of stress at a
point in a object, can be completely
characterized by the nine components of T. These
nine components are
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simply the three components of three different
stress vectors, one acting on each of the
coordinate planes of a reference frame. Thus, in
the matrix of tensor components the first row
consists of the components of the stress vector
acting on a plane whose normal is in the e1
direction. A similar interpretation applies to
the second and third rows. When the meaning is
not obscured, we will drop the subscript (n) in
the equation t(n) T?n and write it as t T?n,
with it being understood that the particular t
depends upon n. The normal stress on a plane is
then given by t ?n n?T?n and the shear stress
in a
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direction m lying in the plane whose normal is n,
m?n 0, is given by t ?m m?T?n n?T?m. Note
that if a vector m is introduced that reverses
the direction of m, m - m then the associated
shear stress is given by t ?m m ?T?n n?T?m
- -t ?m. This shows that, if the unit vector m
is reversed in direction, the opposite value of
the shear stress is obtained. Note also that if
the unit vector n is reversed in direction, the
opposite value of the shear stress is obtained.
If both the unit vectors n and m are reversed in
direction, the sign of the shear stress is
unchanged. These conclusions are all consistent
with the definition of the sign of the shear
stress.
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A short calculation will show that the stress
matrix T is a tensor. Recall that, in order for T
to be a tensor, its components in one coordinate
system had to be related to the components in
another coordinate system by To show that the T
in the relationship, t T?n, has the tensor
property, the equation t T?n is specified in
the Latin coordinate system, t(L) T(L)?n(L).
Then, using the vector transformation law for t
and n, t(L) Q?t(G) and n(L) Q?n(G),
respectively, the expression t(L) T(L)?n(L) is
then rewritten as Q?t(G) T(L)?Q ? n(G) , or
t(G) QT?T(L)?Q ? n(G).
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Finally, it can be noted from t(G) QT?T(L)?Q ?
n(G) that since t(G) T(G)?n(G) , it follows
that T(G) QT?T(L)?Q . Since this is the
transformation rule for a tensor, T is a
tensor. Example Determine the stress tensor
representing the state of stress at a typical
point in the uniform bar subjected to a uniform
tensile stress (figure on the next slide). The
applied tensile stress is of magnitude s and it
is assumed that the stress state is the same at
all points of the bar. Determine the stress
vector t acting on the plane
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whose normal is n, where n is given by n cosq
e2 sinq e3. Determine the normal stress on the
plane whose normal is n and the shear stress in
the direction m, m?n 0, m - sinq e2 cosq
e3, on the plane whose normal is n. Solution
The components of the stress tensor T at a
typical point in the bar and relative to the
coordinate system shown in the figure , are
given by
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thus the only non-zero component of the stress
tensor is T33. The stress vector t acting on the
plane whose normal is n is then given by
thus this vector has only one non-zero
component, namely, t3 s sin q. The normal
stress on the plane whose normal is n is given by
tn s sin q2. The shear stress on the plane
whose normal is n in the direction m, mn 0, m
- sinq e2 cosq e3 is given by tm s cosq
sinq
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Note that one could choose m sinq e2 - cosq e3
where m - m, then tm -s?sinqcosq ???-tm
and the direction of the shear stress is
reversed. From these results it is seen that
when the normal to the plane n coincides with the
e3 direction (q p/2), the stress component t3
is equal to s as one would expect, and when the
normal to the plane n coincides with the e2
direction, the stress component t2 is zero. The
shear stress tm s cosq sin q has maxima of
s/2 at q p/4 and 3 p/4.
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4.4 The stress equations of motion In continuum
mechanics the stress equations of motion are the
most useful form of the principles of balance of
linear and angular momentum. The stress equations
of motion are statements of Newton's second law
(i.e. that force is equal to mass times
acceleration) written in terms of stress. The
forces that act on the object in the figure on
the next slide are the surface traction t(x, t),
which acts at each boundary point, and the
action-at-a-distance force rd, which represents
forces such as gravity and the effect of
electromagnetic forces on charges within the
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object. For example, at the surface of the earth,
in the absence of electromagnetic forces, d
-ge2, where e2 is a positive unit normal to the
surface of the earth and g is the acceleration of
gravity at the earth's surface. The total force
?F acting on the object is given by
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where ?O is the surface of the object O. The
total moment about the origin of the coordinate
system illustrated is where x is a position
vector from the origin. The linear momentum p and
the angular momentum H of the object in the
figure are written as the following integrals
over the object O The balance of linear
momentum requires that the sum of the applied
forces equals the time
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rate of change of the linear momentum, and the
balance of angular momentum requires that the sum
of the applied moments equals the time rate of
change of the angular momentum. In computing the
time rates of change of the integrals above we
note that the object O is material so that the
material time derivative may be taken inside the
integral sign and applied directly to the
integrand with the density excluded. The material
time derivative applied to the integrals for the
momentum yields
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where the term in the square brackets vanishes
because 0. Equating to SF and
to SM as required by the conservation of
linear and angular momentum, respectively, we
obtain These integral forms of the balance
of linear and angular momentum are the global
forms of these principles. The global forms are
stronger statements of these balance principles
than are the point forms that we will now derive.
We say that the point forms are weaker because it
must be assumed that the stress is continuously
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differentiable and that and rd are
continuous everywhere in the object in order to
obtain the point forms from the global forms. The
point form of the balance of linear momentum is
obtained from the following sequence of
substitutions and manipulations ?
?
?
?
at each point in the object O. In the next
paragraph the conservation of angular momentum is
used to show that T is symmetric.
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The arguments to show that the stress tensor T is
symmetric are algebraically simpler if we replace
the statement of the conservation of angular
momentum by equivalent requirement that the
skew-symmetric part of Z, where
be zero. The sequence of steps applied to the
conservation of linear momentum are are now
applied to the expression for Z. The point form
of is obtained by first substituting into the
surface integral in the expression for Z,
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This result is simplified by observing that ??x
1 and collecting all the remaining integrals
containing x ? together, thus
The second integral above is exactly zero because
its integrand contains the point form statement
of linear momentum conservation thus
?
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The final form of the stress equations of motion
is thus
This local statement of Newtons second law
retains aspects of the original. The mass times
acceleration is represented by density times
acceleration on the left-hand side. The sum of
the forces is represented on the right-hand side
by the gradient of the stress tensor and the
action-at-a-distance force. The expanded scalar
version of the stress equations of motion are
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where the symmetry of the stress tensor is
expressed in the subscripted indices. For a
two-dimensional motion the stress equations of
motion reduce to
Example The stress tensor in an object is given
by
where ci, i 1, ,5, are constants. This same
object is subjected to an action-at a-distance
force d with components 0, -g, 0. Determine the
components of the acceleration vector of this
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object. Solution Substitution of the given
stress tensor T and the action-at-a-distance
force d into the stress equations of motion
yields the fact that all the components of the
acceleration are zero
4.5 The conservation of energy The conservation
of energy is therefore viewed here more as a
basic method of science rather than as a basic
fact in the sense that the charge of an electron
is a scientific fact. The conservation of energy
is viewed here as a method of checking energetic
interactions and
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discovering new energies. Whenever one approaches
a new scientific problem, one tries to select or
invent energies such that, by setting their sum
equal to a constant, some aspect of the physical
phenomenon is correctly described. In the
continuum theories, the known energies will
include kinetic energy, heat energy, chemical
energy, electromagnetic energy and so forth. The
total energy E of a system consists of the sum of
all the energies we choose to recognize or define
and the remainder of the total energy of a system
is said to be the internal energy U of the
system. That is to say, all the energies that are
not singled out and explicitly
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defined are placed in the category of internal
energy. The total energy E of an object consists
of a kinetic energy,
and an internal energy U, E K U where U
consists of all energies except kinetic. The
principle of conservation of energy is the
statement that total energy of an object is
constant. It is more convenient to reformulate
the conservation of energy as a balance of rates
the rate of increase of the total energy of an
object is equal to the rate of energy flux into
the object. The flux of energy into a object
occurs in two
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ways, firstly through the mechanical power P of
the surface tractions and action-at-a-distance
forces and, secondly, through a direct flow of
heat Q into the object. With these definitions
and conventions established, the conservation of
energy may be written in the form
where P Q is the rate of energy supply. This is
a global statement of energy conservation and we
will need a point form of the principle in
continuum mechanics applications. In the point
form representation all the variables will be
intensive in the conventional thermodynamic use
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of that word. In thermodynamics an extensive
variable is a variable that is additive over the
system, e.g. volume or mass, and an intensive
variable is a variable that is not additive over
the system, e.g., pressure or temperature. Thus
density or specific volume (the reciprocal of
density) is the intensive variable associated
with the normally extensive variable mass. The
internal energy U, an extensive variable, is
represented in terms of the specific internal
energy e, an intensive variable, by the following
volume integral
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Integral representations of the mechanical power
P and the non-mechanical or heat power Q supplied
to a object O are necessary in order to convert
the global form of the energy conservation
principle to a point form. Heat is transferred
into the object at a rate q per unit area the
vector q is called the heat flux vector. The
negative sign is associated with q because of the
long-standing tradition in thermodynamics that
heat coming out of a system is positive while
heat going into a system is negative. The
internal sources of heat such as chemical
reactions and radiation are represented by the
scalar field r per unit mass. Using these
representations the total
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non-mechanical power supplied to an object may be
written as the sum of a surface integral and a
volume integral,
The mechanical power P delivered to the object is
represented in integral form by
where t is the surface traction acting on the
surface of the object O, d is the
action-at-a-distance force and v is the velocity
vector. The terms t vda and ? d vdv both
represent the rate at which mechanical work is
done on the
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object, t vda is the rate of work of surface
forces and ? d vdv is the rate of work of
action-at-a-distance forces. Substitution of t
Tn, into this expression and subsequent
application of the divergence theorem to the
surface integral in the resulting expression
yields
where L is tensor of velocity gradients. This
result may be further reduced by using the stress
equations of motion to replace ??T rd by
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Two more manipulations of this expression for P
will be performed. Using L D W, it follows
then that TL TD TW, but the TW is zero
because T is symmetric by and W is
skew-symmetric, hence TL TD. The second
manipulation is to observe that the first
integral in is the material time rate of change
of the kinetic energy K,
With these two changes, the
integral expression for P now has the
form
54
This formula shows that the total mechanical
power supplied to the object is equal to the time
rate of change of kinetic energy plus an integral
representing power involved in deforming the
object. The point form statement of the
principle of energy conservation will now be
obtained by placing the integral representations
for U, Q and P, into the global statement thus
?
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A formula for the quasi-static work done during a
loading of an object The desired new formula
relates to the work done rather that to the rate
of doing work or power. The work is given
by where W is the mechanical work delivered to
the object in a quasi-static loading, t is the
surface traction acting on the surface of the
object O, d is the action-at-a-distance force and
u is the displacement vector. The terms t ? uda
and ?? d ? udv both represent the mechanical work
done on the object, t ? uda is the work of
surface forces and ?? d ? udv is the work of
action-at-a-distance forces. Substitution
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of t T ? n, and subsequent application of the
divergence theorem to the surface integral in the
resulting expression yields
This result may be further reduced by using the
stress equations of motion in the case when
0 to replace ??T rd by, thus
57
Recall from that (??u)T may be decomposed into a
symmetric part E and a skew-symmetric part Y by
(??u)T E Y. It follows then that T(??u)
TE TY but the TY is zero because T is
symmetric and Y is skew-symmetric, hence T(??u)
TE. The work done on the object is then given
by
This means that the local work done is TE.
This result will be of interest in the
consideration of elastic objects.