Force on the Boundary of a Semiinfinite Solid Medium

In this section, the case of a semiinfinite solid with a concentrated force acting on the boundary is introduced. This case was originally solved by Boussinesq (1885). It should be noted that the only difference between this case and the case of a point force in an infinite solid medium is the boundary conditions. Shear stresses vanish on the boundary of the semiinfinite solid. In the following, the concept of a center of compression is introduced. The stress field in a semiinfinite solid with a boundary force can be obtained by superimposing the stress fields from a point force and a series of centers of compression. A center of compression is defined as the combination of three perpendicular pair forces. 

Consider a case of two equal but opposite forces (in the z-direction) positioned a small distance 5 apart in an infinite solid body, as shown in Fig. 2.5. At any point M, the stresses 

the stresses at any point due to the pair forces along the z-direction are given by Eq. (2.35). If the pair forces are relocated along the r-direction but still applied near the origin, the resulting stresses at any point M can be obtained by replacing 4 by (tp — n/2) in Eq. (2.35) as 

To obtain an expression for the normal stress 0 in the tangential direction, consider the equilibrium of an element cut from a sphere by two concentric spherical surfaces of radii R and R + dR and by a circular cone with a small angle dtp, as shown in Fig. 2.6. The force balance along the radial direction gives 

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In this section, an introduction of the general relations of stresses in equilibrium in an infinitely large solid medium is presented, followed by a special application where a concentrated force is acting on a point inside the solid. Also presented is the case of forces on the boundary of a semiinfinite solid medium, which is of importance to the contact of two solid objects. As consequences of the boundary compression, displacements due to the changes of stresses and strains in the region of contact can be linked to the contact force by the Hertzian theory for frictionless contacts and by Mindlin s theory for frictional contacts. For more details on the Hertzian theory for contact, interested readers may refer to books on elasticity [Goldsmith, 1960 Timoshenko and Goodier, 1970 Landau and Lifshitz, 1970]. 

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