Problems for Chapter 2
Governing Equations of Solid Mechanics
2.1. Mathematical
Description of Shape Changes in Solids
2.1.1. A thin film of material is deformed in
simple shear during a plate impact experiment, as shown in the figure.
2.1.1.1.
Write
down expressions for the displacement field in the film, in terms of d and h, expressing your answer
as components in the basis shown.
2.1.1.2.
Calculate
the Lagrange strain tensor associated with the deformation, expressing your
answer as components in the basis shown.
2.1.1.3.
Calculate
the infinitesimal strain tensor for the deformation, expressing your answer as
components in the basis shown.
2.1.1.4.
Find the principal values of the infinitesimal
strain tensor, in terms of d and h
2.1.2. Find a displacement field
corresponding to a uniform infinitesimal strain field . (Don’t make this hard in particular do not use the complicated
approach described in Section 2.1.20.
Instead, think about what kind of function, when differentiated, gives a
constant). Is the displacement unique?
2.1.3. Find a formula for the displacement
field that generates zero infinitesimal
strain.
2.1.4. Find a displacement field that
corresponds to a uniform Lagrange strain tensor . Is the displacement unique? Find a formula for the most general
displacement field that generates a uniform Lagrange strain.
2.1.5.
The displacement
field in a homogeneous, isotropic circular shaft twisted through angle at one end is given by
2.1.5.1.
Calculate the
matrix of components of the deformation gradient tensor
2.1.5.2.
Calculate the
matrix of components of the Lagrange strain tensor. Is the strain tensor a function of ? Why?
2.1.5.3.
Find an
expression for the increase in length of a material fiber of initial length dl,
which is on the outer surface of the cylinder and initially oriented in the direction.
2.1.5.4.
Show that
material fibers initially oriented in the and directions do not change their length.
2.1.5.5.
Calculate the
principal values and directions of the Lagrange strain tensor at the point . Hence, deduce the orientations of the
material fibers that have the greatest and smallest increase in length.
2.1.5.6.
Calculate the
components of the infinitesimal strain tensor.
Show that, for small values of ,
the infinitesimal strain tensor is identical to the Lagrange strain tensor, but
for finite rotations the two measures of deformation differ.
2.1.5.7.
Use the
infinitesimal strain tensor to obtain estimates for the lengths of material
fibers initially oriented with the three basis vectors. Where is the error in
this estimate greatest? How large can be before the error in this estimate reaches
10%?
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2.1.6. To measure the in-plane deformation of a sheet of
metal during a forming process, your managers place three small hardness
indentations on the sheet. Using a
travelling microscope, they determine that the initial lengths of the sides of
the triangle formed by the three indents are 1cm, 1cm, 1.414cm, as shown in the
picture below. After deformation, the
sides have lengths 1.5cm, 2.0cm and 2.8cm.
Your managers would like to use this information to determine the inplane components
of the Lagrange strain tensor.
Unfortunately, being business economics graduates, they are unable to do
this.
2.1.6.1.
Explain how the
measurements can be used to determine and do the calculation.
2.1.6.2.
Is it possible to
determine the deformation gradient from the measurements provided? Why?
If not, what additional measurements would be required to determine the
deformation gradient?
2.1.7. To track the deformation in a slowly moving glacier,
three survey stations are installed in the shape of an equilateral triangle,
spaced 100m apart, as shown in the picture.
After a suitable period of time, the spacing between the three stations
is measured again, and found to be 90m, 110m and 120m, as shown in the
figure. Assuming that the deformation of
the glacier is homogeneous over the region spanned by the survey stations,
please compute the components of the Lagrange strain tensor associated with
this deformation, expressing your answer as components in the basis shown.
2.1.8. Compose a limerick that will help you to remember the
distinction between engineering shear strains and the formal (mathematical)
definition of shear strain.
2.1.9. A rigid body motion is a nonzero
displacement field that does not distort any infinitesimal volume element
within a solid. Thus, a rigid body
displacement induces no strain, and hence no stress, in the solid. The deformation corresponding to a 3D rigid
rotation about an axis through the origin is
where R must satisfy ,
det(R)>0.
2.1.9.1.
Show
that the Lagrange strain associated with this deformation is zero.
2.1.9.2.
As
a specific example, consider the deformation
This is the displacement field caused
by rotating a solid through an angle about the axis.
Find the deformation gradient for this displacement field, and show that
the deformation gradient tensor is orthogonal, as predicted above. Show also that the infinitesimal strain
tensor for this displacement field is not generally zero, but is of order if is small.
2.1.9.3.
If
the displacements are small, we can find a simpler representation for a rigid
body displacement. Consider a deformation
of the form
Here is a vector with magnitude <<1, which
represents an infinitesimal rotation about an axis parallel to . Show that the infinitesimal strain tensor
associated with this displacement is always zero. Show further that the Lagrange strain
associated with this displacement field is
This is not, in general, zero. It is small if all .
2.1.10. The formula for the deformation due to
a rotation through an angle about an axis parallel to a unit vector n that passes through the origin is
2.1.10.1.
Calculate
the components of corresponding deformation gradient
2.1.10.2.
Verify
that the deformation gradient satisfies
2.1.10.3.
Find
the components of the inverse of the deformation gradient
2.1.10.4.
Verify that both
the Lagrange strain tensor and the Eulerian strain tensor are zero for this
deformation. What does this tell you
about the distorsion of the material?
2.1.10.5.
Calculate the
Jacobian of the deformation gradient.
What does this tell you about volume changes associated with the
deformation?
2.1.11. In Section 2.1.6 it was stated that the Eulerian
strain tensor can be used to relate the length of a material
fiber in a deformable solid before and after deformation, using the formula
where are the components of a unit vector parallel
to the material fiber after deformation.
Derive this result.
2.1.12. Suppose that you have measured the
Lagrange strain tensor for a deformation, and wish to calculate the Eulerian
strain tensor. On purely physical
grounds, do you think this is possible, without calculating the deformation
gradient? If so, find a formula relating
Lagrange strain to Eulerian strain .
2.1.13. Repeat problem 2.1.6, but instead of
calculating the Lagrange strain tensor, find the components of the Eulerian
strain tensor (you can do this directly, or use the results
of problem 2.1.12, or both)
2.1.14. Repeat problem 2.1.7, but instead of
calculating the Lagrange strain tensor, find the components of the Eulerian
strain tensor (you can do this directly, or use the results
for problem 2.1.12, or both)
2.1.15. The Lagrange strain tensor can be used
to calculate the change in angle between any two material fibers in a solid as
the solid is deformed. In this problem
you will calculate the formula that can be used to do this. To this end, consider two infinitesimal
material fibers in the undeformed solid, which are characterized by vectors with
components and ,
where and are two unit vectors. Recall that the angle between
and before deformation can be calculated from . Let and represent the two material fibers after
deformation. Show that the angle between
and can be calculated from the formula
2.1.16. Suppose that a solid is subjected to a
sequence of two homogeneous deformations (i) a rigid rotation R, followed by (ii) an arbitrary
homogeneous deformation F. Taking the original configuration as
reference, find formulas for the following deformation measures for the final
configuration of the solid, in terms of F
and R:
2.1.16.1.
The
deformation gradient
2.1.16.2.
The
Left and Right Cauchy-Green deformation tensors
2.1.16.3.
The
Lagrange strain
2.1.16.4.
The
Eulerian strain.
2.1.17. Repeat problem 2.1.16, but this time
assume that the sequence of the two deformations is reversed, i.e. the solid is
first subjected to an arbitrary homogeneous deformation F, and is subsequently subjected to a rigid rotation R.
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2.1.18. A spherical shell (see the figure) is
made from an incompressible material. In
its undeformed state, the inner and outer radii of the shell are . After deformation, the new values are . The deformation in the shell can be described
(in Cartesian components) by the equation
2.1.18.1.
Calculate the components of the deformation
gradient tensor
2.1.18.2.
Verify
that the deformation is volume preserving
2.1.18.3.
Find
the deformed length of an infinitesimal radial line that has initial length ,
expressed as a function of R
2.1.18.4.
Find
the deformed length of an infinitesimal circumferential line that has initial
length ,
expressed as a function of R
2.1.18.5.
Using
the results of 2.1.18.3, 2.1.18.4, find the principal stretches for the
deformation.
2.1.18.6.
Find
the inverse of the deformation gradient, expressed as a function
of . It is best to do this by working out a
formula that enables you to calculate in terms of and and differentiate the result rather than to
attempt to invert the result of 10.1.
2.1.19. Suppose that the spherical shell
described in Problem 2.1.18 is continuously expanding (visualize a balloon
being inflated). The rate of expansion
can be characterized by the velocity of the surface that lies at R=A in the undeformed cylinder.
2.1.19.1.
Calculate
the velocity field in the sphere as a function of
2.1.19.2.
Calculate
the velocity field as a function of (there is a long, obvious way to do this and a
quick, subtle way)
2.1.19.3.
Calculate
the time derivative of the deformation gradient tensor calculated in 2.1.18.1.
2.1.19.4.
Calculate
the components of the velocity gradient by differentiating the result of 2.1.19.1
2.1.19.5.
Calculate the
components of the velocity gradient using the results of 2.1.19.3 and 2.1.18.6
2.1.19.6.
Calculate
the stretch rate tensor . Verify that the result represents a volume
preserving stretch rate field.
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2.1.20. Repeat Problem 2.1.18.1, 2.1.18.6 and
all of 2.1.19, but this time solve the problem using spherical-polar
coordinates, using the various formulas for vector and tensor operations given
in Appendix E. In this case, you may
assume that a point with position in the undeformed solid has position vector
after deformation.
2.1.21. An initially straight beam is bent
into a circle with radius R as shown
in the figure. Material fibers that are
perpendicular to the axis of the undeformed beam are assumed to remain
perpendicular to the axis after deformation, and the beam’s thickness and the length
of its axis are assumed to be unchanged.
Under these conditions the
deformation can be described as
where, as usual x
is the position of a material particle in the undeformed beam, and y is the position of the same particle
after deformation.
2.1.21.1.
Calculate
the deformation gradient field in the beam, expressing your answer as a
function of ,
and as components in the basis shown.
2.1.21.2.
Calculate
the Lagrange strain field in the beam.
2.1.21.3.
Calculate
the infinitesimal strain field in the beam.
2.1.21.4.
Compare
the values of Lagrange strain and infinitesimal strain for two points that lie
at and . Explain briefly the physical origin of the
difference between the two strain measures at each point. Recommend maximum allowable values of h/R and L/R for use of the infinitesimal strain measure in modeling beam
deflections.
2.1.21.5.
Calculate
the deformed length of an infinitesimal material fiber that has length and orientation in the undeformed beam. Express your answer as a function of .
2.1.21.6.
Calculate
the change in length of an infinitesimal material fiber that has length and orientation in the undeformed beam.
2.1.21.7.
Show
that the two material fibers described in 2.1.21.5 and 2.1.21.6 remain mutually
perpendicular after deformation. Is
this true for all material fibers
that are mutually perpendicular in the undeformed solid?
2.1.21.8.
Find
the components in the basis of the Left and Right stretch tensors and as well as the rotation tensor for this deformation. You should be able to write down and R by inspection, without needing to wade through the
laborious general process outlined in Section 2.1.13. The results can then be used to calculate .
2.1.21.9.
Find
the principal directions of as well as the principal stretches. You should be able to write these down using
your physical intuition without doing any tedious calculations.
2.1.21.10. Let be a basis in which is parallel to the axis of the deformed beam,
as shown in the figure. Write down the
components of each of the unit vectors in the basis . Hence, compute the transformation matrix that is used to transform tensor components
from to .
2.1.21.11. Find the components of the deformation
gradient tensor, Lagrange strain tensor, as well as and in the basis .
2.1.21.12. Find the principal directions of expressed as components in the basis . Again, you should be able to simply write
down this result.
2.1.22.
A sheet of
material is subjected to a two dimensional homogeneous deformation of the form
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where are constants.
Suppose that a circle of unit radius is drawn on the
undeformed sheet. This circle is
distorted to a smooth curve on the deformed sheet. Show that the distorted circle is an ellipse,
with semi-axes that are parallel to the principal directions of the left
stretch tensor V, and that the
lengths of the semi-axes of the ellipse are equal to the principal stretches
for the deformation. There are many
different ways to approach this calculation some are very involved. The simplest way is probably to assume that
the principal directions of V subtend
an angle to the basis as shown in the figure, write the polar
decomposition in terms of principal stretches and ,
and then show that (where is on the unit circle) describes an ellipse.
2.1.23. A solid is subjected to a rigid rotation so that a
unit vector a in the undeformed
solid is rotated to a new orientation b. Find a rotation tensor R that is consistent with this deformation, in terms of the
components of a and b.
Is the rotation tensor unique? If
not, find the most general formula for the rotation tensor.
2.1.24. In a plate impact experiment, a thin film of material
with thickness h is subjected to a
homogeneous shear deformation by displacing the upper surface of the film
horizontally with a speed v.
2.1.24.1.
Write down the
velocity field in the film
2.1.24.2.
Calculate the
velocity gradient, the stretch rate and the spin rate
2.1.24.3.
Calculate the
instantaneous angular velocity of a material fiber parallel to the direction in the film
2.1.24.4.
Calculate the
instantaneous angular velocity of a material fiber parallel to
2.1.24.5.
Calculate the
stretch rates for the material fibers in 22.3 and 22.4
2.1.24.6.
What is the
direction of the material fiber with the greatest angular velocity? What is the direction of the material fiber
with the greatest stretch rate?
2.1.25. The velocity field due to a rigid rotation about an axis through
the origin can be characterized by a skew tensor or an angular velocity vector defined so that
Find a formula relating the components of and .
(One way to approach this problem is to calculate a formula for W by taking the time derivative of
Rodriguez formula see Sect 2.1.1).
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2.1.26. A single crystal deforms by shearing on a single
active slip system as illustrated in the figure. The crystal is loaded so that the slip
direction and normal to the slip plane maintain a constant direction during the
deformation
2.1.26.1.
Show that the deformation gradient can be
expressed in terms of the components of the slip direction and the normal to the slip plane m as where denotes the shear, as illustrated in the
figure.
2.1.26.2.
Suppose shearing
proceeds at some rate . At the instant when ,
calculate (i) the velocity gradient tensor; (ii) the stretch rate tensor and
(iii) the spin tensor associated with the deformation.
2.1.27.
The properties of
many rubbers and foams are specified by functions of the following invariants
of the left Cauchy-Green deformation tensor .
Invariants
of a tensor are defined in Appendix B they are functions of the components of a
tensor that are independent of the choice of basis.
2.1.27.1.
Verify that are invariants. The simplest way to do this is to show that are unchanged during a change of basis.
2.1.27.2.
In order to
calculate stress-strain relations for these materials, it is necessary to
evaluate derivatives of the invariants.
Show that
2.1.28.
The infinitesimal
strain field in a long cylinder containing a hole at its center is given by
2.1.28.1.
Show that the
strain field satisfies the equations of compatibility.
2.1.28.2.
Show that the
strain field is consistent with a displacement field of the form ,
where . Note that although the strain field is
compatible, the displacement field is multiple
valued i.e. the displacements are not equal at and ,
which supposedly represent the same point in the solid. Surprisingly, displacement fields like this
do exist in solids they are caused by dislocations in a
crystal. These are discussed in more
detail in Sections 5.3.4
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2.1.29. The figure shows a test designed to measure the
response of a polymer to large shear strains.
The sample is a hollow cylinder with internal radius and external radius . The inside diameter is bonded to a fixed
rigid cylinder. The external diameter is
bonded inside a rigid tube, which is rotated through an angle . Assume that the specimen deforms as indicated
in the figure, i.e. (a) cylindrical sections remain cylindrical; (b) no point
in the specimen moves in the axial or radial directions; (c) that a cylindrical element of material at radius rotates through angle about the axis of the specimen. Take the
undeformed configuration as reference. Let denote the cylindrical-polar coordinates of a
material point in the reference configuration, and let be cylindrical-polar basis vectors at . Let denote the coordinates of this point in the
deformed configuration, and let by cylindrical-polar basis vectors located at .
2.1.29.1.
Write down
expressions for in terms of (this constitutes the deformation mapping)
2.1.29.2.
Let P denote the
material point at at in the reference configuration. Write down the
reference position vector X of P,
expressing your answer as components in the basis .
2.1.29.3.
Write down the
deformed position vector x of P,
expressing your answer in terms of and basis vectors .
2.1.29.4.
Find the
components of the deformation gradient tensor F in . (Recall that the gradient operator in
cylindrical-polar coordinates is ;
recall also that )
2.1.29.5.
Show that the
deformation gradient can be decomposed into a sequence of a simple shear followed by a rigid rotation through angle about the direction R. In this case the simple shear deformation
will have the form
where
is to be determined.
2.1.29.6.
Find the
components of F in .
2.1.29.7.
Verify that the
deformation is volume preserving (i.e. check the value of J=det(F))
2.1.29.8.
Find the
components of the right Cauchy-Green deformation tensors in
2.1.29.9.
Find the
components of the left Cauchy-Green deformation tensor in
2.1.29.10. Find in .
2.1.29.11. Find the principal values of the stretch tensor U
2.1.29.12. Write down the velocity field v in terms of in the basis
2.1.29.13. Calculate the spatial velocity gradient L in the basis