Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.13.    Constitutive 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 ${\sigma }_{22}=\sigma$ in the blocks, a separation ${\Delta }_{n}$ at the interface, and a displacement U at the surface of the upper block.

3.13.1.1.        Show that the stress and displacement can be expressed in dimensionless form as

$\frac{\sigma }{{\sigma }_{\mathrm{max}}}=\frac{{\Delta }_{n}}{{\delta }_{n}}\mathrm{exp}\left(1-\frac{{\Delta }_{n}}{{\delta }_{n}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{U}{{\Delta }_{n}}=1+\frac{\sigma }{{\sigma }_{\mathrm{max}}}\Lambda$

where $\Lambda =a{\sigma }_{\mathrm{max}}/{\Delta }_{n}E$

3.13.1.2.        Plot graphs of $\sigma /{\sigma }_{\mathrm{max}}$ and ${\Delta }_{n}/{\delta }_{n}$ as functions of $U/{\Delta }_{n}$ for various values of $\Lambda$ (it is easiest to do a parametric plot).   Hence, show that if $\Lambda$ is less than a critical value, the interface separates smoothly under monotonically increasing $U/{\Delta }_{n}$.  In contrast, if $\Lambda$ exceeds the critical value, the interface suddenly snaps apart (with a sudden drop in stress) at a critical value of $U/{\Delta }_{n}$.  (Under decreasing $U/{\Delta }_{n}$ the interface re-adheres, with a similar transition from smooth attachment to sudden snapping at a critical $\Lambda$ ). Give an expression for the critical value of $\Lambda$.

3.13.1.3.        Plot a graph showing the critical displacement $U/{\Delta }_{n}$ at separation and attachment as a function of $\Lambda$.

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 ${V}_{1}$ until the friction force reaches its steady state value.  The sliding speed is then increased instantaneously to a new value ${V}_{2}$.  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 $p=\omega =0$ 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 properties.

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 ${t}_{v}\to \infty$ and ${k}_{e}\to \infty$ in the constitutive equation, for simplicity.  Obtain a governing equation for the rate of change of length of the spring $dx/dt$ in terms of V, M, and the properties of the interface, for the limiting case.   Hence, investigate the stability of steady sliding.