1855_PDF_C02.PDF

(452 KB) Pobierz
2
Essential M athematics
2.1
Scalars, Vectors, and Cartesian Tensors
Learning a discipline’s language is the first step a student takes towards
becoming competent in that discipline. The language of continuum mechan-
ics is the algebra and calculus of
. Here, tensors is the generic name
for those mathematical entities which are used to represent the important
physical quantities of continuum mechanics. Only that category of tensors
known as
tensors
is used in this text, and definitions of these will
be given in the pages that follow. The tensor equations used to develop the
fundamental theory of continuum mechanics may be written in either of two
distinct notations: the
Cartesian tensors
We shall make
use of both notations, employing whichever is more convenient for the
derivation or analysis at hand, but taking care to establish the interrelation-
ships between the two. However, an effort to emphasize indicial notation in
most of the text has been made. This is because an introductory course must
teach indicial notation to students who may have little prior exposure to the
topic.
As it happens, a considerable variety of physical and geometrical quanti-
ties have important roles in continuum mechanics, and fortunately, each of
these may be represented by some form of tensor. For example, such quan-
tities as
symbolic notation,
or the
indicial notation.
may be specified completely by giving their
magnitude, i.e., by stating a numerical value. These quantities are repre-
sented mathematically by
density
and
temperature
scalars,
which are referred to as
zeroth-order tensors.
It should be emphasized that scalars are not constants, but may actually be
functions of position and/or time. Also, the exact numerical value of a scalar
will depend upon the units in which it is expressed. Thus, the temperature
may be given by either 68°F or 20°C at a certain location. As a general rule,
lowercase Greek letters in italic print such as
, etc. will be used as
symbols for scalars in both the indicial and symbolic notations.
Several physical quantities of mechanics such as
α
,
β
,
λ
require
not only an assignment of magnitude, but also a specification of direction
for their complete characterization. As a trivial example, a 20-Newton force
acting vertically at a point is substantially different than a 20-Newton force
force
and
velocity
809236967.024.png 809236967.025.png 809236967.026.png
 
acting horizontally at the point. Quantities possessing such directional prop-
erties are represented by
vectors,
which are
first-order tensors.
Geometrically,
vectors are generally displayed as
having a definite length (the mag-
nitude), a specified orientation (the direction), and also a sense of action as
indicated by the head and the tail of the arrow. Certain quantities in mechan-
ics which are not truly vectors are also portrayed by arrows, for example,
finite rotations. Consequently, in addition to the magnitude and direction
characterization, the complete definition of a vector requires this further
statement: vectors add (and subtract) in accordance with the triangle rule
by which the arrow representing the vector sum of two vectors extends from
the tail of the first component arrow to the head of the second when the
component arrows are arranged “head-to-tail.”
Although vectors are independent of any particular coordinate system, it
is often useful to define a vector in terms of its coordinate components, and
in this respect it is necessary to reference the vector to an appropriate set of
axes. In view of our restriction to Cartesian tensors, we limit ourselves to
consideration of Cartesian coordinate systems for designating the compo-
nents of a vector.
A significant number of physical quantities having important status in con-
tinuum mechanics require mathematical entities of higher order than vectors
for their representation in the hierarchy of tensors. As we shall see, among the
best known of these are the
arrows,
stress tensor
and the
strain tensors.
These particular
tensors are
Third-order
and fourth-order tensors are not uncommon in continuum mechanics, but they
are not nearly as plentiful as second-order tensors. Accordingly, the unqualified
use of the word
second-order tensors,
and are said to have a rank of
two.
tensor
in this text will be interpreted to mean
second-order tensor.
With only a few exceptions, primarily those representing the stress and strain
tensors, we shall denote second-order tensors by uppercase Latin letters in
boldfaced print, a typical example being the tensor
.
Tensors, like vectors, are independent of any coordinate system, but just
as with vectors, when we wish to specify a tensor by its components we are
obliged to refer to a suitable set of reference axes. The precise definitions of
tensors of various order will be given subsequently in terms of the transfor-
mation properties of their components between two related sets of Cartesian
coordinate axes.
T
2.2
Tensor Algebra in Symbolic Notation —
Summation Convention
The three-dimensional physical space of everyday life is the space in which
many of the events of continuum mechanics occur. Mathematically, this
space is known as a Euclidean three-space, and its geometry can be refer-
enced to a system of Cartesian coordinate axes. In some instances, higher
809236967.001.png
FIGURE 2.1A
Unit vectors in the coordinate directions
x
,
x
, and
x
.
1
2
3
FIGURE 2.1B
Rectangular components of the vector
v
.
order dimension spaces play integral roles in continuum topics. Because a
scalar has only a single component, it will have the same value in every
system of axes, but the components of vectors and tensors will have different
component values, in general, for each set of axes.
In order to represent vectors and tensors in component form, we introduce
in our physical space a right-handed system of rectangular Cartesian axes
e 1
e 2
Ox
, and identify with these axes the triad of unit base vectors , ,
shown in Figure 2.1A . All unit vectors in this text will be written with a
caret placed above the boldfaced symbol. Due to the mutual perpendicularity
of these base vectors, they form an orthogonal basis; furthermore, because
they are unit vectors, the basis is said to be orthonormal. In terms of this
basis, an arbitrary vector
x
x
1
2
3
e 3
v
is given in component form by
3
=++ = = ∑
ˆ
ˆ
ˆ
ˆ
ve e e
v
v
v
v ii
e
(2.2-1)
11
22
33
i
1
This vector and its coordinate components are pictured in Figure 2.1B . For
the symbolic description, vectors will usually be given by lowercase Latin
letters in boldfaced print, with the vector magnitude denoted by the same
letter. Thus
.
At this juncture of our discussion it is helpful to introduce a notational
device called the
v
is the magnitude of
v
summation convention
that will greatly simplify the writing
809236967.002.png 809236967.003.png 809236967.004.png 809236967.005.png
of the equations of continuum mechanics. Stated briefly, we agree that when-
ever a subscript appears exactly
in a given term, that subscript will
take on the values 1, 2, 3 successively, and the resulting terms summed. For
example, using this scheme, we may now write Eq 2.2-1 in the simple form
twice
ˆ
ve
=
v ii
(2.2-2)
and delete entirely the summation symbol
. For Cartesian tensors, only
subscripts are required on the components; for general tensors, both sub-
scripts and superscripts are used. The summed subscripts are called
Σ
dummy
e
indices
since it is immaterial which particular letter is used. Thus, is
completely equivalent to , or to , when the summation convention
is used. A word of caution, however: no subscript may appear more than
twice, but as we shall soon see, more than one pair of dummy indices may
appear in a given term. Note also that the summation convention may
involve subscripts from both the unit vectors and the scalar coefficients.
v jj
e
e
v ii
v kk
Example 2.2-1
Without regard for their meaning as far as mechanics is concerned, expand
the following expressions according to the summation convention:
e
e
e
(a)
uvw
ii j j
(b)
Tv
ij
(c)
Tv
ii
i
j
j
j
Solution:
(a) Summing first on
i
, and then on
j,
(
)
(
)
ˆ
ˆ
ˆ
ˆ
uvw
e
=
uv
+
u v
+
u v
w
e
+
w
e
+
w
e
ii j j
11
22
33
1
1
2
2
3
3
(b) Summing on
i
, then on
j
and collecting terms on the unit vectors,
ˆ
ˆ
ˆ
ˆ
Tv
e
=
T v
e
+
T v
e
+
T v
e
ij
i
j
11
j
j
22
j
j
33
j
j
(
)
(
)
(
)
ˆ
ˆ
ˆ
=
Tv
+
Tv
+
Tv
e
+
Tv
+
Tv
+
Tv
e
+
Tv
+
Tv
+
Tv
e
11
1
21
2
31
3
1
12
1
22
2
32
3
2
13
1
23
2
33
3
3
(c) Summing on
i
, then on
j
,
(
)
(
)
v ˆ
ˆ
ˆ
ˆ
T
e
=++
T
T
T
v
e
++
v
e
v
e
ii
j
j
11
22
33
1
1
2
2
3
3
Note the similarity between (a) and (c).
With the above background in place we now list, using symbolic notation,
several useful definitions from vector/tensor algebra.
809236967.006.png
 
809236967.007.png
 
809236967.008.png
 
809236967.009.png 809236967.010.png 809236967.011.png 809236967.012.png 809236967.013.png
1.
Addition of vectors:
(
)
ˆ
ˆ
w
=
u
+
v
or
wu
e
=+
v
e
(2.2-3)
ii
i
i
i
2.
Multiplication:
(a) of a vector by a scalar:
ˆ
λλ
v
=
v ii
e
(2.2-4)
(b) dot (scalar) product of two vectors:
uv vu
⋅=⋅=
uv cos
θ
(2.2-5)
where is the smaller angle between the two vectors when drawn from a
common origin.
θ
KRONECKER DELTA
e i
From Eq 2.2-5 for the base vectors
(
i
= 1,2,3)
1
0
if numerical value of = numerical value of
if numerical value of
i
j
ˆˆ
ee
i
⋅
=
j
i
≠
numerical value of
j
Therefore, if we introduce the Kronecker delta defined by
1
0
if numerical value of = numerical value of
if numerical value of
i
j
δ ij =
i
≠
numerical value of
j
we see that
(
)
ˆˆ
ee
i
⋅
=
δ
ij
,
=
123
,
,
(2.2-6)
j
ij
Also, note that by the summation convention,
δδδδ δ
ii
==++ =++=
11
111 3
jj
22
33
and, furthermore, we call attention to the substitution property of the Kro-
necker delta by expanding (summing on
j
) the expression
ˆ
ˆ
ˆ
ˆ
δ
e
=
δ
e
+
δ
e
+
δ
e
ij
j
i
11
i
22
i
33
809236967.014.png 809236967.015.png 809236967.016.png 809236967.017.png 809236967.018.png 809236967.019.png 809236967.020.png 809236967.021.png 809236967.022.png 809236967.023.png

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika:

Zgłoś jeśli naruszono regulamin