# Part III Chapter 7. Elasticity

This chapter covers general models for computing and interpreting elastic properties. The techniques used can map out the microstructure-property relationships for the elastic properties of composite and porous materials.

This section discusses the 3-D digital-image-based finite element algorithm for computing the linear elastic properties of random (or non-random) materials. Further computational details, and actual software, can be found in Part III Chapter 2, which contains a manual for finite element and finite difference algorithms applied to digital images.
This section presents a study of how well the physical properties of porous Vycor glass can be modelled using a 2D-3D reconstruction of the microstructure from a TEM image. Physical properties considered include: fluid permeability, electrical and thermal conductivity, water vapor adsorption, drying shrinkage at all relative humidities, surface area, and elastic moduli.
This section applies some of the elastic techniques discussed in Chapter 3 to reconstructed models of a Tungsten-Silver composite material. Finite element results for the effective moduli are used to judge between models of the material.

This section computes the elastic moduli for various general models of random porous materials and relates them to effective medium theories of various kinds and existing elastic moduli data for ceramics.

This section computes the elastic moduli for various general models of random open-cell cellular solids and relates them to various simple analytical theories and existing elastic moduli data for similar materials.

This section computes the elastic moduli for various general models of random closed-cell cellular solids and relates them to various simple analytical theories and existing elastic moduli data for similar materials.

This section develops a differential effective medium theory for the linear elastic properties for materials with composite inclusions. An example of this is concrete, where the composite inclusion is the aggregate plus the surrounding interfacial transition zone. The theory is then tested with accurate finite element simulations of random systems. The agreement is quite good.

This section studies the elasticity properties of simple models, in 2-D and 3-D, of porous materials whose solid phase is made up of elongated particles (crystals, rods, etc.). Discussion is given of how to properly isotropically average the results of simulations.

A finite element method is used to study the elastic properties of random three- dimensional porous materials with highly interconnected pores. It is shown that the Young's modulus is practically independent of the Poisson's ratio of the solid phase over the entire solid volume fraction range. Also, the behavior of the porous Poisson's ratio vs. the solid Poisson's ratio appears to imply that information in the dilute (small porosity) limit can affect behavior in the percolation threshold limit.

Elastic property-porosity relationships are derived directly from microtomographic images. This is illustrated for a suite of four samples of Fontainebleau sandstone with porosities ranging from 7.5% to 22%. A finite-element method is used to derive the elastic properties of digitized images. By estimating and minimizing several sources of numerical error, very accurate predictions of properties are derived in excellent agreement with experimental measurements over a wide range of the porosity. We compare the numerical predictions to various empirical, effective medium and rigorous approximations used to relate the elastic properties of rocks to porosity under different saturation conditions.

### Go to Appendix III-1: Digital images of porous materials and computer modelling: Description of a digital toolkit Go back to Part III Chapter 6: Fluid flow

References

(1) E.J. Garboczi and A.R. Day, Journal of Physics and Mechanics of Solids 43, 1349-1362 (1995).
(2) D.P. Bentz, E.J. Garboczi, and D.A. Quenard, Mod. and Sim. in Mater. Sci. and Eng. 6, 211-236 (1998).
(3) A.P. Roberts and E.J. Garboczi, Journal of the Mechanics and Physics of Solids 47, 2029-2055 (1999).
(4) A.P. Roberts and E.J. Garboczi, Journal of the American Ceramic Society, 83 (12) 3041-3048, (2000).
(5) A.P. Roberts and E.J. Garboczi, J. Mech. Phys. Solids, 50 (1), 33-55 (2002).
(6) A.P. Roberts and E.J. Garboczi, Acta Materiala, 49 (2) 189-197 (2000).
(7) E.J. Garboczi and J.G. Berryman, Mechanics of Materials, 33 (8) 455-470 (2001).
(8) S. Meille and E.J. Garboczi, Mod. Sim. Mater. Sci. and Eng. 9 (5) 371-390 (2001).
(9) A.P. Roberts and E.J. Garboczi, Proc. Royal Society, 458 (2021), 1033-1054 (2002).
(10) C.H. Arns, M.A. Knackstedt, W. V. Pinczewski, E.J. Garboczi, Geophysics, 67 (5), 1396-1405 (2002).