4.1
Continuum Mechanics
In computational sciences, deformable objects are often modeled as continuous (three di
mensional) objects. For the description of their behavior, the three quantities
displacement
,
strain
and
stress
play a major role. In one dimensional problems, these quantities are all one
dimensional and have intuitive interpretations. Let us consider a beam with cross sectional
area
A
as shown in Fig. 4.1. When a force
f
n
is applied in the direction of the beam perpen
dicular to the cross section, the beam with original length
l
expands by
Δ
l
. These quantities
are related via Hooke’s law as
f
n
A
=
E
Δ
l
l
.
(4.1)
The constant of proportionality
E
is Young’s modulus. For steel E is in the order of
10
11
N
/
m
2
while for rubber it lies between 10
7
and 10
8
N
/
m
2
. The equation states that the
stronger the force per area, the larger the relative elongation
Δ
l
/
l
as expected. Also, the
magnitude of the force per area needed to get a certain relative elongation increases with
increasing
E
so Young’s modulus describes the beam’s
stiffness
. Hooke’s law can be written
in a more compact form as
σ
=
E
ε
,
(4.2)
where
σ
=
f
n
/
A
is the applied
stress
and
ε
=
Δ
l
/
l
the resulting
strain
or the other way
around,
σ
the resulting internal stress due to applied strain
ε
. It follows that the unit of
stress is force per area and that strain has no unit. This is true in three dimensions as well.
A three dimensional deformable object is typically defined by its undeformed shape
(also called equilibrium configuration, original, rest or initial shape) and by a set of material
parameters that define how it deforms under applied forces. If we think of the rest shape as
a continuous connected subset
Ω
of
R
3
, then the coordinates
x
∈
Ω
of a point in the object
are called
material coordinates
of that point.
When forces are applied, the object deforms and a point originally at location
x
(i.e.
with material coordinates
x
) moves to a new location
p
(
x
)
, the
spatial
or
world coordinates
of that point. Since new locations are defined for all material coordinates
x
,
p
(
x
)
is a vector
CHAPTER 4. THE FINITE ELEMENT METHOD
23
field defined on
Ω
. Alternatively, the deformation can also be specified by the
displacement
field, which, in three dimensions, is a vector field
u
(
x
) =
p
(
x
)

x
defined on
Ω
.
4.1.1
Strain
In order to formulate Hooke’s law in three dimensions, we have to find a way to measure
strain, i.e. the relative elongation (or compression) of the material. In the general case, the
strain is not constant inside a deformable body. In a bent beam, the material on the convex
side is stretched while the one on the concave side is compressed. Therefore, strain is a
function of the material coordinate
ε
=
ε
(
x
)
. If the body is not deformed, the strain is zero
so strain must depend on the displacement field
u
(
x
)
. A spatially constant displacement
field describes a pure translation of the object. In this situation, the strain should be zero
as well.
Thus, strain is derived from the
spatial variation
or
spatial derivatives
of the
displacement field. In three dimensions the displacement field has three components
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