Biomedical Engineering Reference
In-Depth Information
4 Mathematical Preliminaries
4.1 Continuum Mechanics
The presence of a continuum assumes that the length scale of microstructures is
infinitesimally small in comparison to the macroscale, and that the deformation
gradient, and thus strain and stress, can be defined uniquely at every point within
the domain. This latter point implies a homogeneous deformation map in which an
infinitesimal line element dX in the reference configuration is mapped to the
current configuration dx:
dx ¼ F dX þ X 0 ;
ð 1 Þ
in which dX is an infinitesimal material line element in the reference configuration,
dx is the deformed version of the material line element, and X 0 represents a rigid
body translation vector. F is the nonsymmetric deformation gradient:
F ¼ ox
ð 2 Þ
oX :
A number of second order strain measurement tensors are computed from the
deformation gradient, including the right Cauchy deformation tensor (C), the
Green-Lagrange strain (E) and the engineering or infinitesimal strain (e):
C ¼ F T F ;
ð 3 Þ
E ¼ 1
2
ð
C 1
Þ ;
ð 4 Þ
h
i :
e ¼ 1
2
Þ T
ð
F 1
Þþ F 1
ð
ð 5 Þ
The engineering strain is used extensively for linear elasticity, but is generally
of limited use for the finite deformations seen in biological tissues. A useful
concept in the study of aligned collagenous tissue is the notion of a unit vector to
describe the fiber direction, which is denoted a 0 in the reference configuration. The
fiber vector is rotated and stretched by the deformation gradient, ka ¼ F a 0 ,
where k is the fiber stretch. The concept of strain invariants is of particular
importance in biosolid mechanics, since they provide an objective measure of
strain that is invariant to rotation and rigid body motion [ 108 , 208 ].
4.2 Continuum Based Constitutive Models
In order to compute a stress from the aforementioned strain measures, a consti-
tutive model is required. In the case of linear elasticity, this constitutive model
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