The beam-bending method was first applied (when I was working at DuPont Co.) to silica gels, which proved too soft for conventional permeametry. The method is extremely easy, rapid, and accurate - and it yields viscoelastic properties as well as permeability. After coming to Princeton, I decided to extend the method to cement. The theory needed to be modified to take account of the compressibility of the liquid and solid phases. In her PhD thesis work, Wilasa Vichit-Vadakan tested the method on porous Vycor® glass, then on cement paste. The theory was further extended to apply to rectangular samples with transverse isotropy (such as shale). John Valenza successfully tested the theory on homogeneous plates of cement paste.
The principle of the method is simple. When a beam of material is bent, the top (or, concave side) is compressed and the bottom (convex side) is stretched. If the beam is a saturated porous material, then the liquid in the pores is also compressed in the upper half and stretched in the lower half of the beam. Consequently, the liquid tends to flow toward regions of lower pressure, according to Darcy’s law. Liquid flows through the bar from top to bottom, and also flows out of the top of the beam into the surrounding bath, and from the bath into the lower half. As the pressure equilibrates, the force needed to sustain a constant deflection decreases. The theory provides an analytical solution for the kinetics of relaxation of this force, so the permeability can be obtained by fitting the theory to the data.
If the solid phase is elastic, then when the relaxation of the pore pressure is complete, the force relaxes to a constant value that is related to Young’s modulus. In this case, the relaxation of the force is entirely due to hydrodynamic relaxation of the pore pressure. On the other hand, if the solid is viscoelastic, then the stress in the solid phase also relaxes, and the stress relaxation function can be measured. In this case, there are two relaxation processes: hydrodynamic relaxation of pore pressure and viscoelastic relaxation of the solid phase. The kinetics can be distinguished, if the characteristic times for the processes differ by at least an order of magnitude. Most conveniently, there is an inflection if the curve of force versus time, and a detailed analysis shows that the hydrodynamic and viscoelastic processes can be accurately separated whenever the inflection can be distinguished.
Relevant papers:
Application to gels:
“Bending of Gel Beams: method of characterizing mechanical properties and permeability”, G.W. Scherer, J. Non-Cryst. Solids, 142 [1-2] (1992) 18-35
“Relaxation of a Viscoelastic Gel Bar: I. Theory”, G.W. Scherer, J. Sol-Gel Sci. Tech. 1 (1994) 169-175
“Relaxation of a Viscoelastic Gel Bar: II. Silica Gel”, G.W. Scherer, J. Sol-Gel Sci. Tech. 2 [1/2/3] (1994) 199-204
“Influence of viscoelasticity and permeability on the stress response of silica gel”, G.W. Scherer, Langmuir 12 [5] (1996) 1109-1116
This paper shows how to correct for indentation of the gel by the pushrod:
“Bending of gel beams: Effect of deflection rate and Hertzian indentation”, G.W. Scherer, J. Non-Cryst. Solids 201 (1996) 1-25
This paper shows how viscoelasticity in silica gel depends on the chemistry of the pore liquid:
“Viscoelasticity and permeability of silica gels”, G.W. Scherer, Faraday Disc. 101 (1995) 225-234;287-291
This paper shows that hydrodynamic relaxation is much slower when there is no exchange of pore lqiuid with the bath:
“Bending of a gel rod with an impermeable surface”, G.W. Scherer, J. Non-Cryst. Solids 204 [1] (1996) 73-77
“Permeability and structure of resorcinol-formaldehyde gels”, G.W. Scherer, C. Alviso, R. Pekala, and J. Gross, pp. 497-503 in Microporous and Macroporous Materials, eds. R.F. Lobo, J.S. Beck, S.L. Suib, D.R. Corbin, M.E. Davis, L.E. Iton, and S.I. Zones, MRS Symp. Proc. Vol. 431 (Mater. Res. Soc., Pittsburgh, PA, 1996)
This study follows the change in properties as the gel is progressively dried:
“Effect of drying on properties of silica gel”, G.W. Scherer, J. Non-Cryst. Solids, 215 [2,3] (1997) 155-168
“Bulk properties of a cyanogel network: Toward an understanding of the elastic, mechanical, and physical processes associated with sol-gel processing of cyanide-bridged gel systems”, S.L. Sharp, A.B. Bocarsly, and G.W. Scherer, Chemistry of Materials, 10 [3] (1998) 825-832
Application to rigid materials:
“Structure and properties of gels”, G.W. Scherer, Cement Concr. Res. 29 [8] (1999) 1149-1157
“Measuring Permeability of Rigid Materials by a Beam-Bending Method: I. Theory”, G.W. Scherer, J. Am. Ceram. Soc. 83 [9] (2000) 2231-2239; Erratum, J. Am. Ceram. Soc. 87 [8] (2004) 1612-1613
“Measuring Permeability of Rigid Materials by a Beam-Bending Method: II. Porous Vycor”, W. Vichit-Vadakan and G.W. Scherer, J. Am. Ceram. Soc. 83 [9] (2000) 2240-2245; Erratum, J. Am. Ceram. Soc. 87 [8] (2004) 1614
“Beam-bending method for permeability and creep characterization of cement paste and mortar”, W. Vichit-Vadakan and G.W. Scherer, pp. 27-32 in Creep, Shrinkage and Durability Mechanics of Concrete and Other Quasi-Brittle Materials, eds. F.-J. Ulm, Z.P. Bazant, & F.H. Wittmann (Elsevier, Amsterdam, 2001)
“Measuring Permeability of Rigid Materials by a Beam-Bending Method: III. Cement Paste”, W. Vichit-Vadakan and G.W. Scherer, J. Am. Ceram. Soc. 85 [6] 1537–44 (2002) ; Erratum, J. Am. Ceram. Soc. 87 [8] (2004) 1615
This paper provides excellent data for the viscoelastic stress relaxation function for young paste:
“Measuring permeability and stress relaxation of young cement paste by beam-bending”, W. Vichit-Vadakan and G.W. Scherer, Cement Concrete Res. 33 (2003) 1925-1932
“Characterization of Saturated Porous Bodies”, G.W. Scherer, Concr. Sci. Eng. 37 [265] (2004) 21-30
“Measuring Permeability of Rigid Materials by a Beam-Bending Method: IV. Transversely Isotropic Plate”, G.W. Scherer, J. Am. Ceram. Soc. 87 [8] (2004) 1517-1524
“Measuring Permeability of Rigid Materials by a Beam-Bending Method: V. Isotropic Rectangular Plates of Cement Paste”, J.J. Valenza II and G.W. Scherer, J. Am. Ceram. Soc. 87 [10] (2004) 1927-1931
This paper compares the permeability found by beam-bending and thermopermeametry, and demonstrates that the thermal expansion of pore liquid in cement paste is anomalously high:
“Evidence of anomalous thermal expansion of water in cement paste”, J.J. Valenza and G.W. Scherer, Cement Concr. Res. 35 (2005) 57-66
This paper compares the approximate analytical solution used in the preceding papers to an “exact” finite element simulation, and shows the analytical results to be very accurate:
“Finite-Element Analysis of the Bending of a Saturated Beam”, G.W. Scherer, J.H. Prévost, Z. Wang, pp. 890-895 in Poromechanics IV, Proc. Fourth Biot Conf. on Poromechanics, New York, 2009. eds. H.I. Ling, A. Smyth, R. Betti (DEStech Publications, Lancaster, PA, 2009)
This is paper provides a complete finite-element simulation of the beam-bending problem, which shows the analytical results to be very accurate:
“Bending of a Poroelastic Beam with Lateral Diffusion”, G.W. Scherer, J.H. Prévost, and Z.-H. Wang, Int. J. Solids and Structures 46 (2009) 3451–3462
“A Novel Method for Measuring Permeability of Shale”, J. Zhang and G.W. Scher-er, Int. J. Rock Mechanics and Mining 53 (2012) 179–191