3.7 Boundary Value Problems
143
anisotropic elastic material has 21 independent constants. Isotropic, transversely
isotropic, and orthotropic materials have two, five, and nine constants, respectively.
Experimental Considerations
Determining nonlinear constitutive relations from first principles and microstructural considerations has met with limited success. In continuum mechanics, therefore, it is a common practice to develop phenomenological (empirical) equations,
which are based on mathematical fitting of experimental stress-strain data. This
approach involves choosing a specific functional form for W , and then finding
unknown parameters by fitting data by trial and error or using a formal optimization
procedure (Humphrey 2002). Even at the continuum level, this is not a trivial
problem. In mechanobiology, the difficulties are exacerbated by temporal changes
in properties that occur during development, adaptation, and disease.
Ultimately, any function chosen for W must be judged by its ability to fit the
data. If a particular form fits some but not all data, it still may be useful, however.
Technology typically limits data acquisition to certain ranges of deformation, as
well as certain environmental conditions. Inside these ranges, computed results are
expected to be reasonably reliable, but outside these ranges significant error can
occur.
Most material testing of soft tissues is done in vitro and involves relatively simple
geometries and loading protocols. These include uniaxial extension of bar-shaped
samples (e.g., skeletal muscle), biaxial tests of thin rectangular samples (e.g., skin
and heart muscle), and combined extension, inflation, and torsion of tubular samples
(e.g., blood vessels). These problems will be examined in the next chapter.
Currently, some effort also is being devoted to determining material properties
in vivo. For example, magnetic resonance imaging (MRI) can be used to acquire
detailed 3D geometry, fiber orientation (diffusion tensor MRI), and deformation
(tagged MRI) for a beating human heart. This information can then be used to
develop patient-specific finite-element models to determine material properties,
as well as wall stress, for individual patients (Wang et al. 2015). With future
technological advances, such models could ultimately help physicians track how
the heart heals and adapts following myocardial infarction.
3.7 Boundary Value Problems
The governing equations, along with appropriate boundary and initial conditions,
define a boundary value problem in continuum mechanics. For convenience, the
fundamental equations for hyperelasticity are listed below in tensor form, valid
for any coordinate system, as well as indicial form for Cartesian and principal
coordinates. Spatial and material forms of the equations are included.
143
anisotropic elastic material has 21 independent constants. Isotropic, transversely
isotropic, and orthotropic materials have two, five, and nine constants, respectively.
Experimental Considerations
Determining nonlinear constitutive relations from first principles and microstructural considerations has met with limited success. In continuum mechanics, therefore, it is a common practice to develop phenomenological (empirical) equations,
which are based on mathematical fitting of experimental stress-strain data. This
approach involves choosing a specific functional form for W , and then finding
unknown parameters by fitting data by trial and error or using a formal optimization
procedure (Humphrey 2002). Even at the continuum level, this is not a trivial
problem. In mechanobiology, the difficulties are exacerbated by temporal changes
in properties that occur during development, adaptation, and disease.
Ultimately, any function chosen for W must be judged by its ability to fit the
data. If a particular form fits some but not all data, it still may be useful, however.
Technology typically limits data acquisition to certain ranges of deformation, as
well as certain environmental conditions. Inside these ranges, computed results are
expected to be reasonably reliable, but outside these ranges significant error can
occur.
Most material testing of soft tissues is done in vitro and involves relatively simple
geometries and loading protocols. These include uniaxial extension of bar-shaped
samples (e.g., skeletal muscle), biaxial tests of thin rectangular samples (e.g., skin
and heart muscle), and combined extension, inflation, and torsion of tubular samples
(e.g., blood vessels). These problems will be examined in the next chapter.
Currently, some effort also is being devoted to determining material properties
in vivo. For example, magnetic resonance imaging (MRI) can be used to acquire
detailed 3D geometry, fiber orientation (diffusion tensor MRI), and deformation
(tagged MRI) for a beating human heart. This information can then be used to
develop patient-specific finite-element models to determine material properties,
as well as wall stress, for individual patients (Wang et al. 2015). With future
technological advances, such models could ultimately help physicians track how
the heart heals and adapts following myocardial infarction.
3.7 Boundary Value Problems
The governing equations, along with appropriate boundary and initial conditions,
define a boundary value problem in continuum mechanics. For convenience, the
fundamental equations for hyperelasticity are listed below in tensor form, valid
for any coordinate system, as well as indicial form for Cartesian and principal
coordinates. Spatial and material forms of the equations are included.