52
3 Continuum Mechanics and Nonlinear Elasticity
Stress
Strain
A
0
A
F
F
)
b
(
)
a
(
Fig. 3.1 Nonlinearities in solid mechanics. (a) Geometric nonlinearity: stretching a bar causes
cross-section area to decrease from A 0 to A, causing true stress defined by σ = F/A to increase.
(b) Material nonlinearity: nonlinear stress-strain curve
While the assumption of elasticity simplifies matters, there is one major complication we cannot ignore. For the most part, we deal herein with soft tissues, which
normally can (and often do) undergo large (finite) deformations without damage.
Large deformations introduce two types of nonlinearity: geometric nonlinearity
caused by large changes in geometry and material nonlinearity due to nonlinear
stress-strain relations (Fig. 3.1). In contrast, deformation is generally small for hard
tissues, such as bones, teeth, and horns. In this case, the distinction between the
geometries of the undeformed and the deformed object (or body) can be ignored,
and the governing equations are linear.
The equations that govern the mechanical behavior of continua include kinematic
relations which describe the motion of particles; balance principles for mass,
momentum, energy, and entropy; and constitutive relations (e.g., stress-strain
relations) which describe material properties. With the exception of constitutive
relations, the equations are the same for any continuum, whether solid, fluid,
or some combination of solid and fluid. 1 The specific mathematical form of the
constitutive relations must be determined experimentally for each type of material,
and nonlinear constitutive relations can differ considerably even among various
(nearly) elastic materials.
This chapter presents only a relatively brief introduction to nonlinear continuum
mechanics and elasticity. More in-depth treatments can be found elsewhere (Eringen
1962; Green and Zerna 1968; Malvern 1969; Green and Adkins 1970; Chadwick
1976; Eringen 1980; Spencer 1980; Ogden 1984; Truesdell 1991; Truesdell and
Noll 1992; Narasimhan 1993; Rivlin et al. 1997; Holzapfel 2000; Taber 2004;
Chandrasekharaiah and Debnath 2014). Be forewarned, however, that advanced
expositions of the nonlinear theories are usually considerably more complicated
than the one presented here.
1 Since this book focuses primarily on elastic solids, some formulas that may be useful for fluid
mechanics are omitted.
3 Continuum Mechanics and Nonlinear Elasticity
Stress
Strain
A
0
A
F
F
)
b
(
)
a
(
Fig. 3.1 Nonlinearities in solid mechanics. (a) Geometric nonlinearity: stretching a bar causes
cross-section area to decrease from A 0 to A, causing true stress defined by σ = F/A to increase.
(b) Material nonlinearity: nonlinear stress-strain curve
While the assumption of elasticity simplifies matters, there is one major complication we cannot ignore. For the most part, we deal herein with soft tissues, which
normally can (and often do) undergo large (finite) deformations without damage.
Large deformations introduce two types of nonlinearity: geometric nonlinearity
caused by large changes in geometry and material nonlinearity due to nonlinear
stress-strain relations (Fig. 3.1). In contrast, deformation is generally small for hard
tissues, such as bones, teeth, and horns. In this case, the distinction between the
geometries of the undeformed and the deformed object (or body) can be ignored,
and the governing equations are linear.
The equations that govern the mechanical behavior of continua include kinematic
relations which describe the motion of particles; balance principles for mass,
momentum, energy, and entropy; and constitutive relations (e.g., stress-strain
relations) which describe material properties. With the exception of constitutive
relations, the equations are the same for any continuum, whether solid, fluid,
or some combination of solid and fluid. 1 The specific mathematical form of the
constitutive relations must be determined experimentally for each type of material,
and nonlinear constitutive relations can differ considerably even among various
(nearly) elastic materials.
This chapter presents only a relatively brief introduction to nonlinear continuum
mechanics and elasticity. More in-depth treatments can be found elsewhere (Eringen
1962; Green and Zerna 1968; Malvern 1969; Green and Adkins 1970; Chadwick
1976; Eringen 1980; Spencer 1980; Ogden 1984; Truesdell 1991; Truesdell and
Noll 1992; Narasimhan 1993; Rivlin et al. 1997; Holzapfel 2000; Taber 2004;
Chandrasekharaiah and Debnath 2014). Be forewarned, however, that advanced
expositions of the nonlinear theories are usually considerably more complicated
than the one presented here.
1 Since this book focuses primarily on elastic solids, some formulas that may be useful for fluid
mechanics are omitted.