Problems for Chapter 6
Analytical Techniques and Solutions for Plastic
6.2.1. The figure shows a pressurized cylindrical
cavity. The solid has yield stress in
shear k. The objective of this problem is to calculate
an upper bound to the pressure required to cause plastic collapse in the
Take a volume
preserving radial distribution of velocity as the collapse mechanism. Calculate the strain rate associated with the
Apply the upper
bound theorem to estimate the internal pressure p at collapse. Compare the
result with the exact solution
6.2.2. The figure shows a proposed collapse mechanism for
indentation of a rigid-plastic solid.
Each triangle slides as a rigid block, with velocity discontinuities
across the edges of the triangles.
triangle A moves vertically downwards.
Write down the velocity of triangles B and C
the total internal plastic dissipation, and obtain an upper bound to the force P
Select the angle that minimizes the collapse load.
6.2.3. The figure shows a kinematically admissible velocity
field for an extrusion process. The
velocity of the solid is uniform in each sector, with velocity discontinuities
across each line. The solid has shear
yield stress k.
Assume the ram EF
moves to the left at constant speed V.
Calculate the velocity of the solid in each of the three separate regions of
the solid, and deduce the magnitude of the velocity discontinuity between
the total plastic dissipation and obtain an upper bound to the extrusion force P per unit out-of-plane distance
Select the angle that gives the least upper bound.
6.2.4. The figure shows a kinematically admissible velocity
field for an extrusion process.
Material particles in the annular region ABCD move along radial
lines. There are velocity
discontinuities across the arcs BC and AD.
Assume the ram EF
moves to the left at constant speed V. Use flow continuity to write down the radial
velocity of material particles just inside the arc AD.
Use the fact that
the solid is incompressible to calculate the velocity distribution in ABCD
plastic dissipation, and hence obtain an upper bound to the force P.
The purpose of
this problem is to extend the upper bound theorem to pressure-dependent (frictional)
materials. Consider, in particular, a
solid with a yield criterion and plastic flow rule given by
is a friction coefficient like material
parameter. The solid is subjected to a
traction on its exterior boundary and a body force per unit volume in its interior. The solid collapses for loading ,
where is a scalar multiplier to be deterined.
Show that the
rate of plastic work associated with a plastic strain rate can be computed as
We need to
understand the nature of the plastic dissipation associated with velocity
discontinuities in this material. We can
develop the results for a velocity discontinuity by considering shearing (and
associated dilatation) of a thin layer of material with uniform thickness h as indicated in the figure. Assume
that the strain rate in the layer is homogeneous, and that the surface at has a uniform tangential velocity and normal velocity . Show that (i) the rate of plastic work per
unit area of the layer can be computed as ,
and (ii) to satisfy the plastic flow rule the velocities must be related by . Note that these results are independent of
the layer thickness, and therefore (by letting ) also characterize the dissipation and
kinematic constraint associated with a velocity discontinuity in the solid.
To state the
upper bound theorem for this material we introduce a kinematically admissible
velocity field v, which may have
discontinuities across a set of surfaces in the solid.
Define the strain rate distribution associated with v as
velocity field must satisfy in the interior of the solid, and must satisfy
where denotes a unit vector normal to . Define the plastic dissipation function as
that (i) ,
where denotes the actual velocity field in the solid
at collapse, and (ii)
Hence, show that
an upper bound to the load factor at collapse can be calculated as
6.2.6. As an application of the results derived in the
preceding problem, consider a soil embankment with vertical slope, as shown in
the figure. The soil has mass density and can be idealized as a frictional material
with constitutive equation given in the preceding problem. Using a collapse mechanism consisting of
shearing and dilatation along the line AB
shown in the figure (the angle for the optimal mechanism must be determined),
calculate an upper bound to the admissible height h of the embankment.
6.2.7. The figure shows a statically indeterminate
structure. All bars have cross-sectional
area A, Young’s modulus E and uniaxial tensile yield stress Y.
The solid is subjected to a cyclic load with mean value and amplitude as shown
appropriate distribution of residual stress in the structure, and hence obtain
a lower bound to the shakedown limit for the structure. Show the result as a graph of as a function of
cycles of plastic strain in the structure, and hence obtain an upper bound to
the shakedown limit for the structure.
You should be able to find residual stresses and
plastic strain cycles that make the lower and upper bounds equal, and so demonstrate
that you have found the exact shakedown limit.
and lower bounds to the shakedown limit for a beam subjected to three point
bending as shown in the figure. Assume
the applied load varies cyclically with mean value and amplitude .
The stress state
induced by stretching a large plate containing a cylindrical hole of radius a at the origin is given by
these results to calculate lower and upper bounds to the shakedown limit for
the solid (assume that varies periodically between zero and its