136
3 Continuum Mechanics and Nonlinear Elasticity
In summary, taking W = W (E) satisfies all of the above postulates except the
last one. Below, we discuss material symmetries that are commonly encountered in
soft tissues and list some forms for W used in biomechanics and mechanobiology.
However, many details are omitted; the interested reader is referred to excellent
treatments of this subject by, for example, Ogden (1984), Spencer (1984), Zheng
(1994), and Holzapfel (2000).
Isotropy
In the undeformed configuration, an isotropic material has the same properties
in all directions. Thus, W must be independent of direction. The strain invariants
of Eq. (3.69) have this characteristic, since they are independent of the coordinate
system. For a general isotropic material, therefore, the strain-energy density function
can be written in the form
W = W (I 1 , I 2 , I 3 ).
(3.214)
Some modification is needed if the material is incompressible. Equations (3.56),
(3.84), and (3.69) 3 give
I 3 = det(I + 2E) = det(F
T
· F) = (det F
T )(det F) = (det F)
2
= J
2 ,
(3.215)
which states the obvious fact that volume is invariant relative to a change in
coordinates. With the incompressibility constraint I 3 = J 2 = 1, Eq. (3.214)
reduces to
W = W (I 1 , I 2 ).
(3.216)
Rubber is an example of a relatively elastic, isotropic material that can undergo
large deformations without damage. Most soft tissues are anisotropic, but sheets
of cells (epithelia) can be treated as approximately isotropic on the macro scale if
the cells in the undeformed state are not elongated and aligned in any particular
direction (Fig. 3.25a). Extracellular matrix or cells containing disorganized fibers
also may behave approximately as isotropic materials. In the embryo, prospective
cardiac cells contain disorganized sarcomeres that gradually become denser and
more aligned as the cells differentiate into myocardial (heart muscle) cells. Thus, the
mechanical properties of myocardium change from isotropic to anisotropic during
development.
We now present some specific forms for W that are commonly used for isotropic
materials. The simplest form for an incompressible isotropic material is the neoHookean model
W = c 1 (I 1 − 3),
(3.217)
where c 1 is a positive constant. This form can be derived using statistical thermodynamics to analyze the mechanics of polymer networks (Holzapfel 2000). As
3 Continuum Mechanics and Nonlinear Elasticity
In summary, taking W = W (E) satisfies all of the above postulates except the
last one. Below, we discuss material symmetries that are commonly encountered in
soft tissues and list some forms for W used in biomechanics and mechanobiology.
However, many details are omitted; the interested reader is referred to excellent
treatments of this subject by, for example, Ogden (1984), Spencer (1984), Zheng
(1994), and Holzapfel (2000).
Isotropy
In the undeformed configuration, an isotropic material has the same properties
in all directions. Thus, W must be independent of direction. The strain invariants
of Eq. (3.69) have this characteristic, since they are independent of the coordinate
system. For a general isotropic material, therefore, the strain-energy density function
can be written in the form
W = W (I 1 , I 2 , I 3 ).
(3.214)
Some modification is needed if the material is incompressible. Equations (3.56),
(3.84), and (3.69) 3 give
I 3 = det(I + 2E) = det(F
T
· F) = (det F
T )(det F) = (det F)
2
= J
2 ,
(3.215)
which states the obvious fact that volume is invariant relative to a change in
coordinates. With the incompressibility constraint I 3 = J 2 = 1, Eq. (3.214)
reduces to
W = W (I 1 , I 2 ).
(3.216)
Rubber is an example of a relatively elastic, isotropic material that can undergo
large deformations without damage. Most soft tissues are anisotropic, but sheets
of cells (epithelia) can be treated as approximately isotropic on the macro scale if
the cells in the undeformed state are not elongated and aligned in any particular
direction (Fig. 3.25a). Extracellular matrix or cells containing disorganized fibers
also may behave approximately as isotropic materials. In the embryo, prospective
cardiac cells contain disorganized sarcomeres that gradually become denser and
more aligned as the cells differentiate into myocardial (heart muscle) cells. Thus, the
mechanical properties of myocardium change from isotropic to anisotropic during
development.
We now present some specific forms for W that are commonly used for isotropic
materials. The simplest form for an incompressible isotropic material is the neoHookean model
W = c 1 (I 1 − 3),
(3.217)
where c 1 is a positive constant. This form can be derived using statistical thermodynamics to analyze the mechanics of polymer networks (Holzapfel 2000). As