Problems for Chapter 3
Constitutive Models: Relations between
Stress and Strain
Laws for Contacting Surfaces and Interfaces
3.13.1. The figure shows two elastic blocks with Young’s
modulus E that are bonded together at
an interface. The interface can be
characterized using the reversible constitutive law described in Section
3.13.1. The top block is subjected to
tractions which induce a uniaxial stress in the blocks, a separation at the interface, and a displacement U at the surface of the upper block.
Show that the
stress and displacement can be expressed in dimensionless form as
Plot graphs of and as functions of for various values of (it is easiest to do a parametric plot). Hence, show that if is less than a critical value, the interface
separates smoothly under monotonically increasing . In contrast, if exceeds the critical value, the interface
suddenly snaps apart (with a sudden drop in stress) at a critical value of . (Under decreasing the interface re-adheres, with a similar
transition from smooth attachment to sudden snapping at a critical ). Give an expression for the critical value
Plot a graph
showing the critical displacement at separation and attachment as a function of .
3.13.2. Two rigid surfaces slide against one another under an
applied shear force T and a normal force N. The interface may be characterized using the
rate and state dependent friction law described in Section 3.13.2. The blocks
slide at speed until the friction force reaches its steady
state value. The sliding speed is then
increased instantaneously to a new value . Calculate an expression for the variation of
the friction force T as a function of
the distance slid d.
3.13.3. A rigid block with length L slides over a flat rigid surface at constant speed V under an applied shear force T and
a normal force N. The contacting surfaces may be idealized
using the rate and state dependent friction law described in Section
3.13.2. The state variables for any point on the stationary surface that
lies ahead of the rigid block.
Calculate the shear force T as
a function of the length of the block, the sliding speed, and relevant material
3.13.4. A rigid block with mass M is pulled over a flat surface by a spring with stiffness k.
The end of the spring is pulled at a steady speed V. The contacting surface may
be idealized using the rate and state dependent friction law described in
Section 3.13.2. Take and in the constitutive equation, for
simplicity. Obtain a governing equation
for the rate of change of length of the spring in terms of V, M, and the properties
of the interface, for the limiting case.
Hence, investigate the stability of steady sliding.