Hertz-Mindlin
Learning objectives
- Read Hertz-Mindlin as the elastic modulus of a random dense pack of identical spheres pressed together
- Anchor the numbers: for a quartz pack at the critical porosity and 20 MPa, K about 2.05 GPa and G about 3.02 GPa
- Use the pressure law: the pack moduli grow as pressure to the one-third power, so doubling the pressure raises K by the factor two to the one-third
- Read the coordination-number dependence and see this pack point as the anchor the sand lines are hung from
Two Spheres, Then a Whole Pack
Hertz solved how two elastic spheres deform when pressed together: the harder you push, the wider the flattened contact grows, and the stiffer the pair becomes. Mindlin added the tangential part, how the same contact resists sliding. Put many such spheres into a random dense pack, average over all the contacts, and you get the Hertz-Mindlin moduli: the bulk and shear stiffness of a granular frame at the porosity where the grains first lock into a load-bearing arrangement, the critical porosity . This is the frame with no cement and no compaction beyond simple packing, held together by pressure alone, and it is the single point from which the whole granular story is measured.
The Numbers and the Pressure Law
Feed the model a quartz pack: grain shear modulus 45 GPa, Poisson ratio about 0.06, a coordination number of 9, the critical porosity , and an effective pressure of 20 MPa. It returns GPa and GPa, and with the dry-pack density of about 1.70 g/cc a dry P-wave velocity near 1.89 km/s. Those are the moduli of loose sand: tiny beside the 36.6 GPa of solid quartz, exactly because the load runs through contacts and not through crystal. The striking feature is how they grow with depth. Both moduli scale as the effective pressure to the one-third power, , so the response to burial is blunt: at 10 MPa is 1.63 GPa, at 20 MPa it is 2.05, and at 40 MPa it is 2.585. Doubling the pressure multiplies the modulus by , close to 1.26, no more. A pack does not stiffen in proportion to the load; it stiffens as the cube root of it.
Coordination, and the Anchor Point
The other lever is the coordination number. Hold the pressure at 20 MPa and raise : at the pack gives GPa, at it gives 2.05, and at it gives 2.35. More contacts, more load paths, a stiffer pack, the same physics Part 5.1 argued qualitatively, now with numbers on it. What matters most is what this single point is for. Hertz-Mindlin does not describe a rock across its porosity range; it pins down one end of the range, the modulus of the freshly packed sand at . To turn that anchor into a frame at any lower porosity, you must connect it to the solid mineral at zero porosity. The next two sections draw that connection two different ways, a soft one and a stiff one, and every point on both curves hangs from the Hertz-Mindlin pack computed here.
References
- Mindlin, R. D. (1949). Compliance of elastic bodies in contact. Journal of Applied Mechanics, 16, 259-268.
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.