3.6 Constitutive Relations
131
It is important to emphasize again that biological tissues generally exhibit
viscoelastic behavior. If the material making up a spring is viscoelastic, then
the unloading force-displacement curve falls below the loading curve, creating a
hysteresis loop (Fig. 3.23c). The area within the loop represents energy dissipated
as heat, i.e., the difference between the energy input (area under the loading curve)
and the energy recovered (area under the unloading curve).
Experiments have shown that the stress-strain curves for many soft tissues
become repeatable after several loading-unloading cycles. For this reason, Fung
et al. (1979) introduced the concept of pseudoelasticity, whereby a tissue is
assumed to behave approximately as two separate elastic materials—one during
loading and another during unloading, each with its own stress-strain curve. In
practice, this idea is rarely used, if ever. Rather, soft tissues are often treated as
elastic materials to a first approximation, with only the loading or unloading curve
considered. The term “pseudoelasticity” indicates that elasticity is an approximation
whose accuracy decreases as the size of the hysteresis loop increases.
It would be more accurate to conduct a nonlinear viscoelastic analysis, but doing
so can carry a heavy computational cost, because the response depends on the entire
history of loading. 8 In contrast, stress in an elastic solid depends only on the current
deformation. Since the mechanical behavior of a given type of biological tissue is
generally quite variable, the added cost of a viscoelastic analysis often outweighs
the benefits of improved quantitative accuracy.
3.6.1 Hyperelasticity
A hyperelastic material is defined as an elastic material for which the constitutive
equations can be derived from a scalar potential function W , called the strainenergy density function. In this section, we use thermodynamic principles to derive
the appropriate equations. In general, W can be a function of position, but spatial
dependency is not listed explicitly herein.
For large deformation, it is often convenient, especially for numerical calculations, to write the constitutive relations in terms of the second Piola-Kirchhoff
stress tensor S and the Lagrangian strain tensor E. To derive equations that are
thermodynamically consistent, we first combine the energy and entropy relations.
Since loading and unloading an elastic body is a reversible process, equality holds
in Eqs. (3.201), and eliminating r 0 between Eqs. (3.190) and (3.201) 2 yields
ρ 0 (T ˙
η − ˙
u) + S : ˙
E = 0.
(3.203)
In general, the behavior of biological tissues depends to some extent on
temperature T , and experiments should ideally be conducted at body temperature.
8 Similar difficulties are encountered with some theories for remodeling (see Chap. 7).
131
It is important to emphasize again that biological tissues generally exhibit
viscoelastic behavior. If the material making up a spring is viscoelastic, then
the unloading force-displacement curve falls below the loading curve, creating a
hysteresis loop (Fig. 3.23c). The area within the loop represents energy dissipated
as heat, i.e., the difference between the energy input (area under the loading curve)
and the energy recovered (area under the unloading curve).
Experiments have shown that the stress-strain curves for many soft tissues
become repeatable after several loading-unloading cycles. For this reason, Fung
et al. (1979) introduced the concept of pseudoelasticity, whereby a tissue is
assumed to behave approximately as two separate elastic materials—one during
loading and another during unloading, each with its own stress-strain curve. In
practice, this idea is rarely used, if ever. Rather, soft tissues are often treated as
elastic materials to a first approximation, with only the loading or unloading curve
considered. The term “pseudoelasticity” indicates that elasticity is an approximation
whose accuracy decreases as the size of the hysteresis loop increases.
It would be more accurate to conduct a nonlinear viscoelastic analysis, but doing
so can carry a heavy computational cost, because the response depends on the entire
history of loading. 8 In contrast, stress in an elastic solid depends only on the current
deformation. Since the mechanical behavior of a given type of biological tissue is
generally quite variable, the added cost of a viscoelastic analysis often outweighs
the benefits of improved quantitative accuracy.
3.6.1 Hyperelasticity
A hyperelastic material is defined as an elastic material for which the constitutive
equations can be derived from a scalar potential function W , called the strainenergy density function. In this section, we use thermodynamic principles to derive
the appropriate equations. In general, W can be a function of position, but spatial
dependency is not listed explicitly herein.
For large deformation, it is often convenient, especially for numerical calculations, to write the constitutive relations in terms of the second Piola-Kirchhoff
stress tensor S and the Lagrangian strain tensor E. To derive equations that are
thermodynamically consistent, we first combine the energy and entropy relations.
Since loading and unloading an elastic body is a reversible process, equality holds
in Eqs. (3.201), and eliminating r 0 between Eqs. (3.190) and (3.201) 2 yields
ρ 0 (T ˙
η − ˙
u) + S : ˙
E = 0.
(3.203)
In general, the behavior of biological tissues depends to some extent on
temperature T , and experiments should ideally be conducted at body temperature.
8 Similar difficulties are encountered with some theories for remodeling (see Chap. 7).