## Part 10 - Coefficient of friction: crust / mantle interface

Friction is the resistance to motion that takes place when two bodies are moved relative to each other. Thus for a body to be moved relative to another body, the resistive frictional force that acts between the contacting surfaces has to be overcome by the motive or pushing force. The ratio of the moving force (F) to the force exerted by the weight at contact surface is defined as the coefficient of friction, denoted by the Greek letter (μ). The coefficient of friction is taken into account in all engineering applications involving the forces associated with the relative movement between two surfaces.

The estimation of the magnitude of the forces acting on an element of crust (see section 8 and Appendix 2) allows for the calculation of the coefficient of friction (μ) at the crust / mantle interface. The force diagram required for this calculation is depicted in Fig.14. The determination of the coefficient of friction will thus yield an indication of the material and topography at the crust/mantle interface.

Fig. 14. The coefficient of friction μ is defined as the Ratio of the moving force (F) to the downward force (W) exerted by the weight. Thus μ = F/W.

Let F be the ‘moving force’ on the 1 m x 1 m x 1000 m element of Earth and

W= the ‘vertical downward force’ exerted by the weight of the above element of Earth.

As the ratio of the height of the Andes to the distance between the western edge of the South American shelf to the Cordillera Real is c.1:250 the inclined plane presented by the Nazca plate can be considered as being essentially flat.

If we take the radius of eccentricity to be 1 Km then from eq. (2) as above we have

F= 7.44 x 107 N and

W=2.8x106x 9.807 N= 2.745 x107 N.

Hence μ= F/W            = 7.44x107 / 2.745 x107 = 2.710.

Alternatively if the radius of eccentricity is taken to be 0.5km then μ =1.35

Fig 15 shows the relationship in between μ (coefficient of friction) and the calculated force (F) and the corresponding radius of eccentricity. As these calculated values of the coefficient of friction are not unrealistic, they may be compared with tabled values for common substances. The following values were obtained from the Machinery Handbook 22 Edition. Diamond /Diamond = 0.1: Glass/Glass = 0.9;Iron/Iron = 1; Brick / Wood = 0.6; Brake Material / Cast Iron=0.4: and Solids/Rubber =1- 4

Fig.15 above. Relationship between the force (Newtons) needed to move a 1km x 1m element of crust and the coefficient of friction at the crust / mantle interface.

The extensive research programme 29 (Morrow and Lockner) on the porosity, density and strength measurements on cored rock samples collected from the Hayward Fault region in Northern California showed some very interesting results. In general terms the coefficients of friction of the different samples (sandstone, basalt, fine and coarse grained gabbros, and keratophyre), which were calculated, from the fracture angles which occurred under the applied axial loading were shown to range between 0.5 and 0.9.

However at low applied loads of 32 and 64 MPa (to simulate depths of 2 and 4 km) the μ values for the coarse gabro, basalt and keratophyre, varied between 1 and 1.5. These values compare favourably with the generally accepted range of values of 0.5-0.7 for brick on brick and concrete on dry rock.

Over a continental size area local variations in Isostatic equilibrium will give a very much coarser surface areas in contact than would be obtained with laboratory prepared samples simulating conditions of use. This coarser profile would yield a slightly higher value of the calculated value of the coefficient of friction (μ) between the continental crust and mantle as compared to a standard brick: brick or brick: rock interface.

It is thus reasonable to assume that the slightly higher values of  μ obtained by calculation from the force considerations will take into account the distorted interface topography at the crust/mantle interface. These distortions would be due in part to the Isostatic considerations as shown on Fig. 6 and discussed in Section 7. Taking this into account there is close agreement between the author’s calculated values and those obtained under laboratory conditions 29

As movement is taking place, the work done in overcoming the resistive frictional force will cause heat to be generated. No attempt is made at this stage to determine the magnitude of the heat generation.