Biomedical Engineering Reference
In-Depth Information
11
Local balance of mass, momentum
and energy
11.1
Introduction
A coherent amount of material (a material body or possibly a distinguishable
material fraction) is considered to be a continuum with current volume
V
in three-
dimensional space. In Chapter
8
attention was focussed on the local stress state
(the internal interaction between neighbouring volume elements), in Chapters
9
and
10
on the local kinematics (shape and volume changes of material particles).
To determine the stresses and kinematic variables as a function of the position
in the three-dimensional space, we need a description of the material behaviour,
which will be the subject of subsequent chapters, and we need local balance laws.
In the present chapter the balance of mass (leading to the continuity equation) and
the balance of momentum (leading to the equations of motion) for a continuum
will be formulated. In addition the balance of mechanical power will be derived
based on the balance of momentum.
11.2
The local balance of mass
Let us focus our attention on an infinitesimally small rectangular material element
dV
dxdydz
in the current state, see Fig.
11.1
.
The mass in the current volume element
dV
equals the mass in the refer-
ence configuration of the associated volume element
dV
0
, while the volumes are
related by
=
dV
=
JdV
0
with
J
=
det(
F
) .
(11.1)
It is implicitly assumed that during the transformation from the reference state to
the current deformed state no material of the considered type is created or lost. So
there is no mass exchange with certain other fractions, for example in the form of
a chemical reaction. Based on mass conservation, it can be stated:
ρ
0
dV
0
=
ρ
dV
=
ρ
JdV
0
,
(11.2)
