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

$\frac{\sigma }{{\sigma }_{\max }}=\frac{{\Delta }_n}{{\delta }_n}\exp \left(1-\frac{{\Delta }_n}{{\delta }_n}\right)\frac{U}{{\Delta }_n}=1+\frac{\sigma }{{\sigma }_{\max }}\Lambda$

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

3.13.1.2.        Plot graphs of $\sigma /{\sigma }_{\max }$ 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.