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For a three-dimensional problem there will be a set of six
compatibility
equations, which
in matrix form, using the compatibility functions
R
ij
, appear as [Leipholz, 1968, p. 75]
.
R
xx
R
yy
R
zz
...
x
y
z
...
0
∂
2
z
∂
2
y
0
0
−
∂
y
∂
z
.
z
x
∂
0
∂
0
−
∂
x
∂
z
0
.
−
∂
x
∂
y
∂
2
y
∂
2
x
0
0
0
=
...
...
...
.
...
...
...
=
0
(1.28)
.
−
0
0
−
∂
x
∂
y
1
2
∂
2
z
1
2
∂
y
∂
z
1
2
∂
z
∂
x
R
xy
R
xz
R
yz
γ
xy
γ
xz
γ
yz
.
1
2
1
2
2
y
1
2
0
−
∂
x
∂
z
0
∂
y
∂
z
−
∂
∂
x
∂
y
.
1
2
1
2
1
2
−
∂
y
∂
z
0
0
∂
x
∂
z
∂
x
∂
y
−
∂
2
x
R
=
D
1
=
0
(1.29)
x
y
z
γ
yz
γ
zx
γ
xy
]
T
and the variables
R
ij
are re-
ordered similarly,
D
1
will be a symmetric matrix. As we will see, there are methods other
than using equations such as Eqs. (1.27) and (1.28) to assure that the displacements are
single-valued and continuous. In fact, most matrix methods of approximate structural
analysis avoid Eq. (1.28) and deal directly with the displacements, rather than with the
strains.
The homogeneous equations of Eq. (1.28) can be shown to be related to each other. It
can be verified readily that [Leipholz, 1968, p. 76]
R
ij, j
=
If the strain vector is reordered as
=
[
0
,
or using the operator matrix
D
of Eq. (1.22),
D
T
R
=
0
(1.30)
and
R
xy
=
R
yx
,R
xz
=
R
zx
,R
yz
=
R
zy
where
[
R
xx
R
yy
R
zz
R
xy
R
xz
R
yz
]
T
Thus, the six equations of Eq. (1.28) are reduced to the three independent equations of
Eq. (1.30). Note that Eq. (1.30) has the same form as the homogeneous equations of equi-
librium. For this formulation in Cartesian coordinates, these relations merely express that
the partial derivatives can be interchanged, e.g.,
∂
x
∂
y
∂
z
f
R
=
(
x, y, z
)
=
∂
z
∂
y
∂
x
f
(
x, y, z
)
(1.31)
which can be considered as a basic statement of compatibility. In curvilinear coordinates
embedded in an Euclidean
7
space, the compatibility conditions are related to the so-called
Bianchi
8
identities.
1.3
Material Laws
The material law or constitutive equation provides a relationship between the stresses
σ
and the strains
. In the linear, elastic range of a homogeneous solid, the material law is
usually called
Hooke's
9
law
. A material is said to be
isotropic
if the material properties at a
point are independent of orientation. In the case of an element of isotropic material with
7
Euclid of Alexandria (c. 300 B.C.) was an early Greek philosopher-mathematician, perhaps the most celebrated
mathematician ever. So little is known of his life that he has often been mistaken for another Euclid, a philosopher
who studied under Socrates. It is presumed that the mathematician Euclid studied with the students of Plato or
perhaps at Plato's academy. He wrote the thirteen topics
Elements
which influenced the study of geometry for
centuries.
8
Luigi Bianchi (1856-1928) was Italy's most resourceful 19th century mathematician dealing with differential
geometry.
9
Robert Hooke (1635-1702) was an English physicist who devised numerous mechanics experiments. In 1662,
he became curator of experiments of the Royal Society, and in 1664, he became professor of geometry in
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