3.3 Analysis of Deformation
89
To extend this analysis to a general coordinate system, these relations are
expressed in tensor form. To set the stage, we first note that the cross product of the
vectors a and b yields a vector of magnitude equal to the area of the parallelogram
with sides defined by these vectors and direction normal to this area. Therefore, we
can define an area vector in the form
A = a × b = n A,
(3.80)
with A = |A| being the area and n the unit normal to the area. In addition, the
volume V of the parallelepiped with edges defined by the vectors a, b, and c is
given by the vector triple product
V = (a × b) · c = det [[a], [b], [c]] ,
(3.81)
where [a], [b], and [c] are column vectors.
With this background, we now consider a rectangular element in an undeformed
body defined by the mutually orthogonal line elements dR 1 , dR 2 , and dR 3 . After
deformation, the element becomes a parallelepiped defined by the vectors
dr i = F · dR i .
(3.82)
The undeformed and deformed volumes of the element, respectively, are
dV
0
= (dR 1 × dR 2 ) · dR 3
= det [[dR 1 ], [dR 2 ], [dR 3 ]]
dV = (dr 1 × dr 2 ) · dr 3
= det
[F ij ][dR 1 ], [F ij ][dR 2 ], [F ij ][dR 3 ]
= det[F ij ] det [[dR 1 ], [dR 2 ], [dR 3 ]] ,
(3.83)
since det(A · B) = det A det B. Therefore, the volume ratio is given by
J =
dV
dV 0 = det F.
(3.84)
By volume, soft biological tissues are mostly water and, therefore, are often
treated as incompressible materials. In this case, J = 1 (dV = dV 0 ). This
assumption is reasonable if the water is relatively immobile, such as the water within
cells. Soft tissues, however, also contain extracellular water that can flow through
the spaces between cells. During deformation, fluid can flow into or out from the
tissue, causing the tissue as a whole to gain or lose volume. Fluid also can flow
from one region to another, causing local increases or decreases in volume. In these
instances, the bulk material is effectively compressible, and it can be analyzed using
89
To extend this analysis to a general coordinate system, these relations are
expressed in tensor form. To set the stage, we first note that the cross product of the
vectors a and b yields a vector of magnitude equal to the area of the parallelogram
with sides defined by these vectors and direction normal to this area. Therefore, we
can define an area vector in the form
A = a × b = n A,
(3.80)
with A = |A| being the area and n the unit normal to the area. In addition, the
volume V of the parallelepiped with edges defined by the vectors a, b, and c is
given by the vector triple product
V = (a × b) · c = det [[a], [b], [c]] ,
(3.81)
where [a], [b], and [c] are column vectors.
With this background, we now consider a rectangular element in an undeformed
body defined by the mutually orthogonal line elements dR 1 , dR 2 , and dR 3 . After
deformation, the element becomes a parallelepiped defined by the vectors
dr i = F · dR i .
(3.82)
The undeformed and deformed volumes of the element, respectively, are
dV
0
= (dR 1 × dR 2 ) · dR 3
= det [[dR 1 ], [dR 2 ], [dR 3 ]]
dV = (dr 1 × dr 2 ) · dr 3
= det
[F ij ][dR 1 ], [F ij ][dR 2 ], [F ij ][dR 3 ]
= det[F ij ] det [[dR 1 ], [dR 2 ], [dR 3 ]] ,
(3.83)
since det(A · B) = det A det B. Therefore, the volume ratio is given by
J =
dV
dV 0 = det F.
(3.84)
By volume, soft biological tissues are mostly water and, therefore, are often
treated as incompressible materials. In this case, J = 1 (dV = dV 0 ). This
assumption is reasonable if the water is relatively immobile, such as the water within
cells. Soft tissues, however, also contain extracellular water that can flow through
the spaces between cells. During deformation, fluid can flow into or out from the
tissue, causing the tissue as a whole to gain or lose volume. Fluid also can flow
from one region to another, causing local increases or decreases in volume. In these
instances, the bulk material is effectively compressible, and it can be analyzed using