Chapter 4
Solutions to simple boundary and initial
value problems
4.3 Spherically symmetric solution to quasi-static large
strain elasticity problems
4.3.1 Summary of governing equations of finite
elasticity in Cartesian components
This
section is intended to illustrate the nature of solutions to elasticity
problems with large shape changes.
We are given the following
information
1.
The geometry of
the solid
2. A constitutive law for the material (i.e. the hyperelastic
strain energy potential)
3.
The body force
density (per unit mass) (if any)
4.
Prescribed
boundary tractions and/or boundary displacements
To simplify the problem we
will assume
The
solid is stress free in its undeformed configuration;
Temperature
changes during deformation are neglected;
The
solid is incompressible
With
these assumptions, we wish to calculate the displacement field ,
the left Cauchy-Green deformation tensor and the stress field satisfying the following equations:
Displacementstrain relation
Incompressibility
condition
Stressstrain relation
where is the Cauchy stress tensor, is the strain energy potential for the elastic
solid, p is the hydrostatic part of
the stress (which must be determined as part of the solution) and .
Equilibrium
Equation
Traction
boundary conditions on parts of the boundary where tractions are
known.
Displacement
boundary conditions on parts of the boundary where
displacements are known.
4.3.2 Simplified
equations for incompressible spherically symmetric solids
A representative spherically symmetric problem is illustrated
in the picture. We consider a hollow,
spherical solid, which is subjected to spherically symmetric loading (i.e.
internal body forces, as well as tractions or displacements applied to the
surface, are independent of and ,
and act in the radial direction only).
The solution is most conveniently expressed using a
spherical-polar coordinate system, illustrated in the figure. For a finite deformation problem, we need a
way to characterize the position of material particles in both the undeformed
and deformed solid. To do this, we let identify a material particle in the undeformed
solid. The coordinates of the same point in the deformed solid is identified by
a new set of spherical-polar co-ordinates . One way to describe the deformation would be
to specify each of the deformed coordinates in terms of the reference coordinates .
For a spherically symmetric deformation, points only move radially, so that
In finite deformation problems vectors and tensors can be
expressed as components in a basis associated with the position of material
points in the undeformed solid, or, if more convenient, in a basis associated with material points in the
deformed solid. For spherically
symmetric deformations the two bases are identical consequently, we can write
Position vector in the undeformed solid
Position vector in the deformed solid
Displacement
vector
The
stress, deformation gradient and deformation tensors tensors (written as
components in ) have the form
and
furthermore must satisfy .
For
spherical symmetry, the governing equations reduce to
Strain Displacement Relations
Incompressibility condition
StressStrain relations
Equilibrium Equations
Boundary Conditions
Prescribed Displacements
Prescribed Tractions
4.3.3 Pressurized
hollow sphere made from an incompressible rubber
As an
example, consider a pressurized hollow rubber shell, as shown in the picture. Assume
that
Before deformation, the sphere has inner
radius A and outer radius B
After deformation, the sphere has inner radius
a and outer radius b
The solid is made from an incompressible
Mooney-Rivlin solid, with strain energy potential
No body forces act on the sphere
The inner surface r=a is subjected to pressure
The outer surface r=b is subjected to pressure
The
deformed radii a,b of the inner and
outer surfaces of the spherical shell are related to the pressure by
where ,
,
and are related by
Provided the pressure is not too large (see below), the
preceding two equations can be solved for and given the pressure and properties of the shell
(for graphing purposes, it is better to assume a value for ,
calculate the corresponding ,
and then determine the pressure).
The
position r of a material particle after deformation is related to its
position R before deformation by
The
deformation tensor distribution in the sphere is
The Cauchy
stress in the sphere is

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The variation of the internal radius of the spherical shell
with applied pressure is plotted in the figure, for (a representative value for a typical
rubber). For comparison, the linear
elastic solution (obtained by setting and in the formulas given in section 4.1.4) is
also shown. Note that:
1. The small strain solution is accurate
for
2. The relationship between pressure and
displacement is nonlinear in the large deformation regime.
3. As the internal radius of the sphere
increases, the pressure reaches a maximum, and thereafter decreases (this will
be familiar behavior to anyone who has inflated a balloon). This is
because the wall thickness of the shell decreases as the sphere expands.
The stress distribution for various displacements in the
shell is plotted in the figures below, for ,
and B/A=3. The radial stress remains
close to the linear elastic solution even in the large deformation regime. The hoop stress distribution is significantly
altered as the deformation increases, however.

Derivation
1. Integrate the incompressibility
condition from the inner radius of the sphere to some arbitrary point R
2. Note that by definition, and since the point at R=A moves to r=a after
deformation. This gives the relationship
between the position r of a point in
the deformed solid and its position R
before deformation
3.
The components of
the Cauchy-Green tensor follow as
4.
The stresses
follow from the stress-strain equation as
5.
Substituting
these stresses into the equilibrium equation leads to the following
differential equation for
6.
After
substituting for and ,
and expressing R in terms of r, this equation can be integrated and
simplified to see that
7.
The
boundary conditions require that on (r=a,R=A),
while on (r=b,R=B),
which requires
where and . The expression that relates and to the pressure follows by subtracting the
first equation from the second. Adding
the two equations gives the expression for C.
8. Finally, the hoop stress follows by
noting that, from (4)