Problems for Chapter 2
Description of internal forces in solids in Solids
2.2.1. A rectangular bar is loaded in a state of uniaxial
tension, as shown in the figure.
Write down the
components of the stress tensor in the bar, using the basis vectors shown.
components of the normal vector to the plane ABCD shown, and hence deduce the
components of the traction vector acting on this plane, expressing your answer
as components in the basis shown, in terms of
normal and tangential tractions acting on the plane shown.
2.2.2. Consider a state of hydrostatic stress . Show that the traction vector acting on any
internal plane in the solid (or, more likely, fluid!) has magnitude p
and direction normal to the plane.
2.2.3. A cylinder of radius R is partially immersed in
a static fluid.
Recall that the
pressure at a depth d in a fluid has magnitude . Write down an expression for the horizontal
and vertical components of traction acting on the surface of the cylinder in
terms of .
Hence compute the
resultant force exerted by the fluid on the cylinder.
2.2.4. The figure shows two designs for a glue joint. The glue will fail if the stress acting
normal to the joint exceeds 60 MPa, or if the shear stress acting parallel to
the plane of the joint exceeds 300 MPa.
the normal and shear stress acting on each joint, in terms of the applied
calculate the value of that will cause each joint to fail.
2.2.5. For the Cauchy stress tensor with components
vector acting on an internal material plane with normal
2.2.6. Show that the hydrostatic stress is invariant under a change of basis i.e. if and denote the components of stress in bases and ,
respectively, show that .
2.2.7. A rigid, cubic solid is immersed in a fluid with mass
Recall that a stationary fluid exerts a compressive pressure of magnitude at depth h.
Write down expressions for the traction vector
exerted by the fluid on each face of the cube.
You might find it convenient to take the origin for your coordinate system
at the center of the cube, and take basis vectors perpendicular to the cube faces.
Calculate the resultant
force due to the tractions acting on the cube, and show that the vertical force
is equal and opposite to the weight of fluid displaced by the cube.
2.2.8. Show that the result of problem 2.2.7 applies to any
arbitrarily shaped solid immersed below the surface of a fluid, i.e. prove that
the resultant force acting on an immersed solid with volume V is ,
where it is assumed that is vertical.
To do this
Let denote the components of a unit vector normal
to the surface of the immersed solid
Write down a
formula for the traction (as a vector) exerted by the fluid on the immersed
traction to calculate the resultant force, and manipulate the result obtain the
2.2.9. A component contains a feature with a 90 degree corner
as shown in the picture. The surfaces
that meet at the corner are not subjected to any loading. List all the stress components that must be
zero at the corner
2.2.10. In this problem we consider further the beam bending
calculation discussed in Problem 2.1.21. Suppose that the beam is made from a
material in which the Material Stress tensor is related to the Lagrange strain
(this can be regarded as representing an elastic
material with zero Poisson’s ratio and shear modulus )
distribution of material stress in the bar, expressing your answer as
components in the basis
distribution of nominal stress in the bar expressing your answer as components
in the basis
distribution of Cauchy stress in the bar expressing your answer as components
in the basis
but express the stresses as components in the basis
distribution of traction on a surface in the beam that has normal in the undeformed beam. Give expressions for the tractions in both and
Show that the
surfaces of the beam that have positions in the undeformed beam are traction free after
resultant moment acting on the ends of the beam.
2.2.11. A solid is subjected to some loading that induces a
Cauchy stress at some point in the solid. The solid and the loading frame are then
rotated together so that the entire solid (as well as the loading frame) is
subjected to a rigid rotation . This
causes the components of the Cauchy stress tensor to change to new values . The goal of this problem is to calculate a
formula relating ,
Let be a unit vector normal to an internal
material plane in the solid before rotation.
After rotation, this vector (which rotates with the solid) is . Write down the formula relating and
Let be the internal traction vector that acts on a
material plane with normal in the solid before application of the rigid
rotation. Let be the traction acting on the same material
plane after rotation. Write down the
formula relating and
the definition of Cauchy stress, find the relationship between ,
2.2.12. Repeat problem 2.2.11, but instead, calculate a
relationship between the components of Nominal stress and before and after the rigid rotation.
2.2.13. Repeat problem 2.2.11, but instead, calculate a
relationship between the components of material stress and before and after the rigid rotation.