Chapter 4
Solutions
to simple boundary and initial value problems
4.3 Spherically symmetric solution to quasistatic 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.Â
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 Â (per unit mass) (if any)
4. Prescribed boundary tractions Â 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 CauchyGreen deformation tensor Â and the stress field Â satisfying the following equations:
Â Displacementstrain
relation Â Â Â Â Â Â Â
Â Incompressibility
condition
Â Â Stressstrain
relation
where Â is the Cauchy stress tensor,Â Â 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 .
Â Equilibrium
Equation Â
Â Traction
boundary conditions Â on parts of the boundary where tractions are
known.
Â Displacement
boundary conditions Â 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.Â 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 Â and ,
and act in the radial direction only).Â
The solution is most conveniently expressed using a
sphericalpolar coordinate system, illustrated in the figure.Â For a finite deformation problem, we need a
way to characterize the position of material particles in both the undeformed
and deformed solid.Â To do this, we let
Â 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 sphericalpolar coordinates .Â One way to describe the deformation would
be to specify each of the deformed coordinates Â 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 Â associated with the position of material
points in the undeformed solid, or, if more convenient, in a basis Â associated with material points in the
deformed solid.Â For spherically
symmetric deformations the two bases are identical Â consequently, we can write
Â Position vector in the undeformed solidÂ Â Â Â Â Â
Â Position vector in the deformed solidÂ Â Â Â Â Â
Â Displacement
vector
The
stress, deformation gradient and deformation tensors tensors (written as
components in Â ) have the form
and
furthermore must satisfy Â .
For
spherical symmetry, the governing equations reduce to
Â Strain Displacement Relations
Â Incompressibility condition
Â StressStrain
relations
Â Equilibrium Equations
Â 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
Â Before deformation, the sphere has inner
radius A and outer radius B
Â After deformation, the sphere has inner
radius a and outer radius b
Â The solid is made from an incompressible
MooneyRivlin solid, with strain energy potential
Â No body forces act on the sphere
Â The inner surface r=a is subjected to pressure
Â 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 Â are related by
Provided the pressure is not too large (see below), the
preceding two equations can be solved for Â and Â 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
positionÂ 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
The variation of the internal radius of the spherical shell
with applied pressure is plotted in the figure, for Â (a representative value for a typical
rubber).Â For comparison, the linear
elastic solution (obtained by setting Â and Â in the formulas given in section 4.1.4) is
also shown.Â 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).Â This isÂ
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 ,
Â and B/A=3.Â The radial stress remains
close to the linear elastic solution even in the large deformation
regime.Â The hoop stress distribution
is significantly altered as the deformation increases, however.
Â Â Â Â Â Â Â Â Â Â Â Â
Â Â
Derivation
1. Integrate the incompressibility
condition from the inner radius of the sphere to some arbitrary point R
2. Note that Â by definition, and Â since the point at R=A moves to r=a after
deformation.Â 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 CauchyGreen tensor follow as
4. The stresses follow from the stressstrain equation
as
5. Substituting these stresses into the equilibrium
equation leads to the following differential equation for
6. After substituting for Â and ,
and expressing R in terms of r, this equation can be integrated and
simplified to see that
7. The boundary conditions require
that Â on (r=a,R=A),
while Â on (r=b,R=B),
which requires
where Â and .Â The expression that relates Â and Â to the pressure follows by subtracting the
first equation from the second.Â Â
Adding the two equations gives the expression for C.
8. Finally, the hoop stress follows by
noting that, from (4)
