3.6 Constitutive Relations
135
3.6.3 Strain-Energy Density Function
Mechanical properties for a hyperelastic (or pseudoelastic) material are defined by
its strain-energy density function. Once W (E) is determined experimentally, the
constitutive relations are provided by Eqs. (3.212). Determining W for a nonlinear
material is no trivial matter, however. This is true even for a relatively simple
material, such as synthetic rubber. And for soft biological tissues, the task is
considerably more difficult.
In most cases, the mathematical form for W is assumed a priori in terms of a set
of unknown parameters. These parameters are then found by matching theoretical
to experimental results. The form chosen for W should be based on microstructural
considerations and is constrained by fundamental postulates based on physical and
intuitive arguments. Briefly, these postulates include the following:
1. Coordinate Invariance. The form of W must be independent of the coordinate
system. This requirement is satisfied by assuming W depends on the tensor
deformation variable E or F.
2. Local Action. The strain energy at a given point in a body depends only on
the local deformation at that point, not on deformation occurring elsewhere in
the body. One simplification follows from the fact that orthogonal curvilinear
coordinate axes become approximately straight within the neighborhood of a
point, and so we can focus on expressing W in terms of Cartesian strain
components. If a problem is formulated in curvilinear coordinates, the Cartesian
components can simply be replaced by their counterparts in the curvilinear
system.
3. Objectivity (frame indifference). The form of W must be invariant relative to
the motion of an observer (and attached reference frame). In other words, the
strain-energy function should not depend on rigid-body translation or rotation.
Since the deformation gradient tensor includes information on rigid-body rotation, expressing W in terms of F must satisfy certain restrictions. Strain, however,
excludes all rigid-body motion, so any function of E automatically satisfies this
postulate.
4. Physical Admissibility. Constitutive equations must be consistent with the
fundamental balance laws (mass, momentum, energy, and entropy). This consideration led us to associate W with the free energy in Eq. (3.206). In addition,
W and the associated stresses must vanish in the absence of deformation, i.e.,
W = 0 and S = 0 for E = 0 or F = Q, with Q being a rotation tensor.
5. Material Symmetry. Constitutive equations must be invariant under certain rigid
transformations of material frames that depend on symmetries inherent in the
material. For example, the form of W for a material composed of uniaxial fibers
embedded in an isotropic matrix must be consistent with symmetry relative to
the fiber axis. In this regard, it is important to note that material symmetries
are typically defined in the unloaded, zero-stress configuration. Symmetries can
change as deformation becomes large, e.g., a material may become stiffer in
directions of large strain.
135
3.6.3 Strain-Energy Density Function
Mechanical properties for a hyperelastic (or pseudoelastic) material are defined by
its strain-energy density function. Once W (E) is determined experimentally, the
constitutive relations are provided by Eqs. (3.212). Determining W for a nonlinear
material is no trivial matter, however. This is true even for a relatively simple
material, such as synthetic rubber. And for soft biological tissues, the task is
considerably more difficult.
In most cases, the mathematical form for W is assumed a priori in terms of a set
of unknown parameters. These parameters are then found by matching theoretical
to experimental results. The form chosen for W should be based on microstructural
considerations and is constrained by fundamental postulates based on physical and
intuitive arguments. Briefly, these postulates include the following:
1. Coordinate Invariance. The form of W must be independent of the coordinate
system. This requirement is satisfied by assuming W depends on the tensor
deformation variable E or F.
2. Local Action. The strain energy at a given point in a body depends only on
the local deformation at that point, not on deformation occurring elsewhere in
the body. One simplification follows from the fact that orthogonal curvilinear
coordinate axes become approximately straight within the neighborhood of a
point, and so we can focus on expressing W in terms of Cartesian strain
components. If a problem is formulated in curvilinear coordinates, the Cartesian
components can simply be replaced by their counterparts in the curvilinear
system.
3. Objectivity (frame indifference). The form of W must be invariant relative to
the motion of an observer (and attached reference frame). In other words, the
strain-energy function should not depend on rigid-body translation or rotation.
Since the deformation gradient tensor includes information on rigid-body rotation, expressing W in terms of F must satisfy certain restrictions. Strain, however,
excludes all rigid-body motion, so any function of E automatically satisfies this
postulate.
4. Physical Admissibility. Constitutive equations must be consistent with the
fundamental balance laws (mass, momentum, energy, and entropy). This consideration led us to associate W with the free energy in Eq. (3.206). In addition,
W and the associated stresses must vanish in the absence of deformation, i.e.,
W = 0 and S = 0 for E = 0 or F = Q, with Q being a rotation tensor.
5. Material Symmetry. Constitutive equations must be invariant under certain rigid
transformations of material frames that depend on symmetries inherent in the
material. For example, the form of W for a material composed of uniaxial fibers
embedded in an isotropic matrix must be consistent with symmetry relative to
the fiber axis. In this regard, it is important to note that material symmetries
are typically defined in the unloaded, zero-stress configuration. Symmetries can
change as deformation becomes large, e.g., a material may become stiffer in
directions of large strain.