196
4 Problems in Soft Tissue Biomechanics
The block is bent into a circular shape with planes originally normal to the Y -axis
becoming planes normal to the curved surfaces of the bent block (Fig. 4.16b). 6 The
length of the block in the Z-direction is held fixed, and the inner and outer radii of
curvature in the deformed block are r 1 and r 2 , respectively. If r 1 is given, compute
the Cauchy stresses in the block and determine the applied loads required to produce
the specified deformation.
4.7.2 Analysis
Kinematics For problems in the nonlinear theory of elasticity, two separate sets of
coordinates are used to map points before and after deformation. In the problems
considered thus far, both coordinate systems have been of the same type, i.e., both
Cartesian, cylindrical, or spherical. However, this is not required, and sometimes it
is convenient to use one type of coordinate system for the undeformed configuration
and another for the deformed configuration. Such is the case here, and we choose
Cartesian coordinates (X, Y, Z) to locate points in the undeformed block, which is
rectangular, and cylindrical polar coordinates (r, θ, z) for the deformed block, which
has a cylindrical shape (Fig. 4.16).
According to the specified deformation, points at all values of X initially lying
along an arbitrary normal to the Y -axis (Y = constant) would have the same polar
angle θ in the deformed configuration (see Fig. 4.16), and clearly the deformation
does not depend on Z. Therefore, θ is a function of Y alone. Also, since a line X =
constant deforms into a circle centered at the origin, r is independent of Y (and Z).
Hence, we assume the deformation is defined by the mapping
r = r(X),
θ = θ(Y ),
z = Z,
(4.110)
where r(X) and θ(Y ) are functions to be determined. Since the deformed block is a
section of a hollow cylinder, the position vectors to a point in the undeformed and
deformed states are given by
R = Xe X + Y e Y + Ze Z
r = r e r + z e z ,
(4.111)
where the base vectors are related by
e r = e X cos θ + e Y sin θ
e θ = −e X sin θ + e Y cos θ
e z = e Z .
(4.112)
6 Rotating rigid plates attached at Y = ±b can produce this deformation.
4 Problems in Soft Tissue Biomechanics
The block is bent into a circular shape with planes originally normal to the Y -axis
becoming planes normal to the curved surfaces of the bent block (Fig. 4.16b). 6 The
length of the block in the Z-direction is held fixed, and the inner and outer radii of
curvature in the deformed block are r 1 and r 2 , respectively. If r 1 is given, compute
the Cauchy stresses in the block and determine the applied loads required to produce
the specified deformation.
4.7.2 Analysis
Kinematics For problems in the nonlinear theory of elasticity, two separate sets of
coordinates are used to map points before and after deformation. In the problems
considered thus far, both coordinate systems have been of the same type, i.e., both
Cartesian, cylindrical, or spherical. However, this is not required, and sometimes it
is convenient to use one type of coordinate system for the undeformed configuration
and another for the deformed configuration. Such is the case here, and we choose
Cartesian coordinates (X, Y, Z) to locate points in the undeformed block, which is
rectangular, and cylindrical polar coordinates (r, θ, z) for the deformed block, which
has a cylindrical shape (Fig. 4.16).
According to the specified deformation, points at all values of X initially lying
along an arbitrary normal to the Y -axis (Y = constant) would have the same polar
angle θ in the deformed configuration (see Fig. 4.16), and clearly the deformation
does not depend on Z. Therefore, θ is a function of Y alone. Also, since a line X =
constant deforms into a circle centered at the origin, r is independent of Y (and Z).
Hence, we assume the deformation is defined by the mapping
r = r(X),
θ = θ(Y ),
z = Z,
(4.110)
where r(X) and θ(Y ) are functions to be determined. Since the deformed block is a
section of a hollow cylinder, the position vectors to a point in the undeformed and
deformed states are given by
R = Xe X + Y e Y + Ze Z
r = r e r + z e z ,
(4.111)
where the base vectors are related by
e r = e X cos θ + e Y sin θ
e θ = −e X sin θ + e Y cos θ
e z = e Z .
(4.112)
6 Rotating rigid plates attached at Y = ±b can produce this deformation.