Layered Bodies

Abstract

There are numerous practical applications in which one or both of the contacting bodies have a layered structure. For example, a layer of a different material may be used at the surface of a contacting component in order to enhance surface durability properties or to prevent oxidation or other chemical degradation, surface layers may be developed by chemical action as in the case of oxide or contaminant layers in lubricated systems, manufacturing processes will typically cause differences in the properties of surface layers due to locally high strain rates and temperatures, and biological systems often involve thin layers of cartilage or other soft tissue supported on a harder substrate.

Problems

1. A beam of flexural rigidity EI supported on an elastic foundation of modulus k is indented by a symmetrical rigid wedge of semi-angle $\frac{\pi }{2}\!-\!\alpha$, where $\alpha \!\ll \!1$. Find the maximum force $P_0$ for which contact will be restricted to a single point at the apex of the wedge. Describe in words how you expect the contact configuration to evolve for $P\!>\!P_0$, but do not attempt to solve this problem.

2. Use the boundary conditions (14.25) and the homogeneous solution (14.21) to find the relation between the indenting force P, the indentation depth $\varDelta$ and the contact radius a when a rigid sphere of radius R is pressed into an infinite elastic plate of thickness h supported on an elastic foundation of modulus k. Plot a figure showing P and a as functions of $\varDelta$ in appropriate dimensionless form. Also, find the pressure distribution in the contact area and the magnitude of the force per unit length developed at the edge of the contact area. How does the proportion of P carried in this concentrated force vary with a?

3. A rigid flat punch of equilateral triangular cross section is pressed by a force P into an incompressible layer of thickness h and shear modulus G that is bonded to a rigid foundation. Find an approximate solution for the pressure distribution and the indentation depth $\varDelta$, using the displacement function $\psi \!=\!C(x\!-\!a)(x^2\!-\!3y^2)$ in Eq. (14.43), where C is a constant.

4. The results in Sect. 14.2.3 follow Johnson’s assumption that the contact is frictionless, but the analysis of Sect. 14.2.5 suggests that the friction coefficient required to prevent slip is rather small. Develop a similar analysis under the assumption of no slip by assuming an in-plane displacement of the form

$$ \varvec{u}=z(z-h){\varvec{\nabla }}{\psi }. $$

How much will this increase the indentation force for a flat-ended cylindrical indenter of radius a?

5. The criterion (14.68) implies that for indentation by a flat punch, an arbitrarily small coefficient of friction is sufficient to prevent slip. Show the resulting solution implies the existence of an inwardly directed friction force distributed along the boundary of the contact area and find the magnitude of this force per unit length as a function of the indentation $\varDelta$. If the friction coefficient is actually finite and equal to f, how wide a strip of microslip would you anticipate at this edge.

6. Using the loading strategy of Sect. 12.2.3, find the elastic strain energy $U(a,\varDelta )$ for the bonded incompressible layer indented to a depth $\varDelta$ by a rigid sphere of radius R with contact radius a. The total potential energy is then $\varPi \!=\!U\!-\!\pi a^2\varDelta \gamma$. Find the equilibrium radius a by imposing the condition $\partial \varPi /\partial a\!=\!0$ and hence verify that the resulting traction satisfies the condition (14.73).

7. An elastic layer of thickness h rests on a rigid frictionless half-plane and is loaded by a compressive traction $p_0[1+\cos (\omega x)]$ on the free surface. Use the potential functions (14.89) to find the corresponding normal surface displacements.

8. A normal point force P is applied to the surface of an elastic layer of thickness h bonded to a rigid foundation. Find the double Fourier transform of this loading, using Eq. (14.104)$_1$. Define polar coordinates $(r,\theta )$, $(\rho ,\phi )$ such that

$$ x=r\cos \theta ;\;\;\;y=r\sin \theta ;\;\;\;s=\rho \cos \phi ;\;\;\;t=\rho \sin \phi $$

and use this representation to obtain an integral expression for the axisymmetric normal surface displacement $u_z(r)$ due to the point force.

9. Use the formulation of Sect. 14.5 to determine the normal surface displacements due to the sinusoidal pressure distribution $p(x)\!=\!p_0\cos (\omega x)$ acting on the surface of a half-plane for which $\nu \!=\!1/4$ and $G\!=\!G_0\exp (\lambda z)$ with $\lambda \!>\!0$.