**Author details**

Auckpath Sawangsuriya *Bureau of Road Research and Development Department of Highways, Thailand* 

#### **11. References**


Atkinson, J. H. and Sallfors, G. (1991). "Experimental determination of stress-strain-time characteristics in laboratory and in situ tests" *Proceedings of the 10th European Conference on Soil Mechanics and Foundation Engineering*, Florence, Italy, 915-956.

190 Wave Processes in Classical and New Solids

quality control process during construction.

*Bureau of Road Research and Development Department of Highways, Thailand* 

*Soils and Foundations*, Vol. 13, No. 1, pp. 77-95.

**Author details** 

**11. References** 

1015.

537.

pp. 1445-1460.

NC, Vol. 1, pp. 69-92.

Rankine Lecture, pp. 487-508.

Auckpath Sawangsuriya

The use of bender elements to generate and receive shear waves in geomaterials has become a very robust technique in geoengineering design and analysis and has been widely adopted for determining and monitoring stiffness of geomaterials both in the laboratory and field. However as with any other wave propagation techniques, the interpretation of bender element-collected data is controlled by wave characteristics, boundary conditions, and properties of the medium. Guidelines for designing test geometries and interpretation measured data from the wave propagation experiments are summarized herein. A bender element test has been employed for a variety of geoengineering applications, i.e., the mechanistic based design development, the long-term performance monitoring as well as

Acar, Y. B. and EL-Tahir, E. A. (1986), "Low Strain Dynamic Properties of Artificially Cemented Sand," *Journal of Geotechnical Engineering*, ASCE, Vol. 112, No. 11, pp. 1001-

Afifi, S. S. and Richart, F. E., Jr. (1973), "Stress-History Effects on Shear Modulus of Soils,"

Afifi, S. S. and Woods, R. D. (1971), "Long-Term Pressure Effects on Shear Modulus of Soils," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 97, No. SM10,

Anderson, D. G. and Stokoe, K. H., II (1978), "Shear Modulus: A Time-Dependent Soil Property," *Dynamic Geotechnical Testing, ASTM STP 654*, Philadelphia, PA, pp. 66-90. Anderson, D. G. and Woods, R. D. (1975), "Comparison of Field and Laboratory Shear Moduli," *Proceedings of the Conference on In Situ Measurement of Soil Properties*, Raleigh,

Anderson, D. G. and Woods, R. D. (1976), "Time-Dependent Increase in Shear Modulus of Clay," *Journal of the Geotechnical Engineering Division*, ASCE, Vol. 102, No. GT5, pp. 525-

Arroyo, M., Wood, D.M., Greening, P.D., Medina, L., and Rio, J. (2006). "Effect of sample size on bender-based axial Go measurements" *Geotechnique*, Vol.56, No.1, 39-52. Athanasopoulos, G. A. (1981), "Time Effects on Low-Amplitude Shear Modulus of Cohesive

Atkinson, J. H. (2000), "Non-Linear Soil Stiffness in Routine Design," *Geotechnique*, 40th

Soils," *Report UMEE 81R1*, The University of Michigan, MI.


Dobry, R. (1989), "Some Basic Aspects of Soil Liquefaction during Earthquakes," *Earthquake Hazards and the Design of Constructed Facilities in the Eastern United States*, Annual of the New York Academy of Sciences, No. 558, pp. 172-182.

Wave Propagation Methods for Determining Stiffness of Geomaterials 193

Georgiannou, V. N., Rampello, S., and Silvestri, F. (1991), "Static and Dynamic Measurement of Undrained Stiffness of National Overconsolidated Clays," *Proceedings of the 10th*

Goto, S., Tatsuoka, F., Shibuya, S., and Sato, T. (1991), "A Simple Gauge for Local Strain Measurements in the Laboratory," *Soils and Foundations*, Vol. 31, No. 1, pp. 169-180. Greening, P.D. and Nash, D.F.T. (2004). "Frequency domain determination of Go using

Gupta, S., Ranaivoson, A., Edil, T.B., Benson, C.H., and Sawangsuriya, A. (2007), *Pavement design using unsaturated soil technology*, Report No. MN/RC-2007-11, Minnesota

Hardin, B .O. and Black, W. L. (1966), "Sand Stiffness under Various Triaxial Stress," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 92, No. SM2, pp. 27-42. Hardin, B .O. and Richart, F. E., Jr. (1963), "Elastic Wave Velocities in Granular Soils," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 89, No. SM1, pp. 33-65. Hardin, B. O. (1978), "The Nature of Stress-Strain Behavior of Soils," *Proceedings of the Geotechnical Engineering Division Specialty Conference on Earthquake Engineering and Soil* 

Hardin, B. O. and Black, W. L. (1968), "Vibration Modulus of Normally Consolidated Clay," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 94, No. SM2, pp. 353-

Hardin, B. O. and Drnevich, V. P. (1972), "Shear Modulus and Damping in Soils: Design Equations and Curves," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol.

Hardin, B. O. and Music, J. (1965), "Apparatus for Vibration during the Triaxial Test," *Symposium on Instrumentation and Apparatus for Soils and Rocks, ASTM STP 392*, pp. 55-

Hardin, B. O., (1970), "Suggested Methods of Test for Shear Modulus and Damping of Soils

Hill, J. J., Kurdziel, J. M., Nelson, C. R., Nystrom, J. A., and Sondag, M. (1999), "MnDOT Overload Field Tests of Standard and SIDD RCP Installations," *Transportation Research* 

Hoar, R. J. and Stokoe, K. H, II (1978), "Generation and Measurement of Shear Waves In Situ," *Dynamic Geotechnical Testing, ASTM STP 654*, Philadelphia, PA, pp. 3-29. Hryciw, R. D. and Thomann, T. G. (1993), "Stress-History-Based Model for Ge of Cohesionless Soils," *Journal of Geotechnical Engineering*, ASCE, Vol. 119, No. 7, pp. 1073-

Humboldt Mfg. Co. (1999), *Humboldt Soil Stiffness Gauge (GeoGauge) User Guide: Version 3.3*,

Humboldt Mfg. Co. (2000a), *Test Results: Evaluation of the Humboldt GeoGauge on Soil-Fly Ash-*

Humboldt Mfg. Co. (2000b), *Test Results: Evaluation of the Humboldt GeoGauge on New Mexico* 

by the Resonant Column," *ASTM STP 479*, Philadelphia, PA, pp. 516-529.

*Board 78th Annual Meeting*, Washington, D.C. (CD-ROM)

*European Conference Soil Mechanics*, Florence, Vol. 1, pp. 91-96.

Department of Transportation, MN.

369.

74.

1093.

Norridge, IL.

*Cement Mixtures*, Norridge, IL.

*Route 44*, Norridge, IL.

98, No. SM7, pp. 667-692.

*Dynamics*, ASCE, Pasadena, CA, Vol. 1, pp. 1-90.

bender elements" *Geotechnical Testing Journal*, Vol.27, No.3, 288-294.


Georgiannou, V. N., Rampello, S., and Silvestri, F. (1991), "Static and Dynamic Measurement of Undrained Stiffness of National Overconsolidated Clays," *Proceedings of the 10th European Conference Soil Mechanics*, Florence, Vol. 1, pp. 91-96.

192 Wave Processes in Classical and New Solids

University of Kentucky.

Philadelphia, PA, pp. 91-125.

Pasadena, CA, pp. 394-409.

94, ASCE, Amherst, MA, pp. 365-376.

Highway Administration, Vol. 61, No. 4.

307-314.

243-255.

Dobry, R. (1989), "Some Basic Aspects of Soil Liquefaction during Earthquakes," *Earthquake Hazards and the Design of Constructed Facilities in the Eastern United States*, Annual of the

Drnevich, V. P. (1977), "The Resonant Column Test," *Report to the U.S. Army Corps of Engineers*, Soil Mechanics Series, No. 23, Department of Civil Engineering, The

Drnevich, V. P. (1985), "Recent Developments in Resonant Column Testing," *Richart Commemorative Lectures, Proceedings of the ASCE Specialty Session*, Detroit, MI, pp. 79-107. Drnevich, V. P., Hall, J. R., and Richart, F. E. (1967), "Effects of Amplitude of Vibration on the Shear Modulus of Sand," *Proceedings of the International Symposium on Wave Propagation and Dynamic Properties of Earth Materials*, Albuquerque, NM, pp. 189-199. Drnevich, V. P., Hardin, B. O., and Shippy, D. J. (1978), "Modulus and Damping of Soils by the Resonant-Column Method," *Dynamic Geotechnical Testing, ASTM STP 654*,

Duffy, J. and Mindlin, R. D. (1957), "Stress-Strain Relations of a Granular Medium," *Journal* 

Dyvik, R. and Madshus, C. (1985), "Lab Measurements of Gmax Using Bender Elements," *Proceedings of the Geotechnical Engineering Division: Advances in the Art of Testing Soil* 

Edil, T. B. and Luh, G.-F. (1978), "Dynamic Modulus and Damping Relationships for Sands," *Proceedings on the Specialty Conference on Earthquake Engineering and Soil Dynamics*, ASCE,

Fam, M. and Santamarina, J. C. (1995), "Study of Geoprocesses with Complementary Wave Measurements in an Oedometer," *Geotechnical Testing Journal*, ASTM, Vol. 18, No. 3, pp.

Fam, M., Santamarina, J. C., and Dusseault, M. (1998), "Wave-Based Monitoring Processes in Granular Salt," *Journal of the Environmental and Engineering Geophysics*, Vol. 3, pp. 15-26. Fernandez, A. and Santamarina, J. C. (2001), "Effect of Cementation on the Small-Strain Parameters of Sands," *Canadian Geotechnical Journal*, Vol. 38, No. 1, pp. 191-199. Fiedler, S. A., Main, M., and DiMillio, A. F. (2000), "In-Place Stiffness and Modulus Measurements," *Proceedings of Sessions of ASCE Specialty Conference on Performance Confirmation of Constructed Geotechnical Facilities*, Geotechnical Special Publication, No.

Fiedler, S. A., Nelson, C. R., Berkman, E. F. and DiMillio, A. F. (1998), "Soil Stiffness Gauge for Soil Compaction Control," *Public Roads*, U.S. Department of Transportation, Federal

Fioravante, V. and Capoferri, R. (2001), "On the Use of Multi-Directional Piezoelectric Transducers in Triaxial Testings," *Geotechnical Testing Journal*, ASTM, Vol. 24, No. 3, pp.

Frost J. D. and Burns, S. E. (2003), "In Situ Subsurface Characterization," *The Civil Engineering Handbook,* 2nd Edition by W. F. Chen and J. Y. Richard, CRC Press LLC.

New York Academy of Sciences, No. 558, pp. 172-182.

*of the Applied Mechanics*, Vol. 24, No. 4, pp. 585-593.

*Under Cyclic Conditions*, ASCE, Detroit, MI, pp. 186-196.


Inci, G., Yesiller, N., and Kagawa, T. (2003), "Experimental Investigation of Dynamic Response of Compacted Clayey Soils," *Geotechnical Testing Journal*, ASTM, Vol. 26, No. 2, pp. 125-141.

Wave Propagation Methods for Determining Stiffness of Geomaterials 195

Kokusho, T. (1987), "In Situ Dynamic Soil Properties and their Evaluation," *Proceedings of the 8th Asian Regional Conference on Soil Mechanics and Foundation Engineering*, Kyoto, Japan,

Kokusho, T., Yoshida, Y., and Esashi, Y. (1982), "Dynamic Properties of Soft Clays for Wide

Kramer, S. L. (1996), *Geotechnical Earthquake Engineering*, Prentice-Hall, Inc., Upper Saddle

Kuribayashi, E., Iwasaki, T., Tatsuoka, F., and Horiuchi, S. (1975), "Effects of Particle Characteristics on Dynamic Deformational Properties of Soils," *Proceedings of the 5th Asian Regional Conference on Soil Mechanics and Foundation Engineering*, Bangalore, India,

Lade, P. V. and Overton, D. D. (1989), "Cementation Effects in Frictional Materials," *Journal* 

Lawrence, F. V., Jr. (1963), "Propagation Velocity of Ultrasonic Wave Through Sand," *MIT Research Report R63-8*, Massachusetts Institute of Technology, Cambridge, MA. Lawrence, F.V. (1963). Propagation of ultrasonic waves through sand, Research report R63-

Lawrence, F.V. (1965). Ultrasonic shear wave velocity in sand and clay, Research report R65-

Lee, J.-S. and Santamarina, J. C. (2005). "Bender elements: performance and signal interpretation" *Journal of Geotechnical and Geoenvironmental Engineering*, ASCE, Vol.131,

Lenke, L. R., McKeen, R. G., and Grush, M. (2001), "Evaluation of a Mechanical Stiffness Gauge for Compaction Control of Granular Media," *Report NM99MSC-07.2*, New

Lenke, L. R., McKeen, R. G., and Grush, M. (2003), "Laboratory Evaluation of the GeoGauge for Compaction Control," *Transportation Research Record*, No. 1849, Washington, D.C.,

Lewis, M. D. (1990), "A Laboratory Study of the Effect of Stress State on the Elastic Moduli

Lo Presti, D. C. F., Jamiolkowski, M., Pallara, O., Cavallararo, A., and Pedroni, S. (1997), "Shear Modulus and Damping of Soils," *Geotechnique*, Vol. 47, No. 3, pp. 603-617. Lo Presti, D. C. F., Pallara, O., Lancellota, R., Armandi, M., and Maniscalco, R. (1993), "Monotonic and Cyclic Loading Behavior of Two Sands at Small Strains," *Geotechnical* 

Lo Presti, D. C. F., Shibuya, S., and Rix, G. J. (2001). "Innovative in soil testing" *Pre-failure Deformation Characteristics of Geomaterials*. Jamiolkowski, H., Lancellota, R. H., and

Mair, R. J. (1993), "Developments in Geotechnical Engineering Research: Application to Tunnels and Deep Excavations," Unwin Memorial Lecture 1992, *Proceedings of the* 

Mexico State Highway and Transportation Department, Albuquerque, NM.

of Sand," *Ph.D. Thesis*, The University of Texas at Austin, Austin, TX.

*Testing Journal*, ASTM, Vol. 16, No. 4, pp. 409-424.

LoPresti, D. C. F., Eds. Swets and Zeitlinger, Lisse, 1027-1076.

*Institution of Civil Engineers*-*Civil Engineering*, Vol. 97, No. 1, pp. 27-41.

Strain Range," *Soils and Foundations*, Vol. 22, No. 4, pp. 1-18.

*of Geotechnical Engineering*, ASCE, Vol. 115, pp. 1373-1387.

05, Soil publication No.17, MIT, Cambridge.

Vol. 2, pp. 215-435.

River, NJ.

pp. 361-367.

8, MIT, Cambridge.

No.9, 1063-1070.

pp. 20-30.


Kokusho, T. (1987), "In Situ Dynamic Soil Properties and their Evaluation," *Proceedings of the 8th Asian Regional Conference on Soil Mechanics and Foundation Engineering*, Kyoto, Japan, Vol. 2, pp. 215-435.

194 Wave Processes in Classical and New Solids

2, pp. 125-141.

Inci, G., Yesiller, N., and Kagawa, T. (2003), "Experimental Investigation of Dynamic Response of Compacted Clayey Soils," *Geotechnical Testing Journal*, ASTM, Vol. 26, No.

Isenhower, W. M. (1980), "Torsional Simple Shear/Resonant Column Properties of San Francisco Bay Mud," *M.S. Thesis*, The University of Texas at Austin, Austin, TX. Isenhower, W. M. and Stokoe, K. H., II (1981), "Strain-Rate Dependent Shear Modulus of San Francisco Bay Mud," *Proceedings of the International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamic*, St. Louis, MO, Vol. 2, pp. 597-602. Ishibashi, I., and Zhang, X. (1993). "Unified dynamic shear moduli and damping ratios of

Ishihara, K. (1996), *Soil Behavior in Earthquake Geotechnics*, Oxford University Press, Inc., NY. Ismail and Rammah (2005). "A new setup for measuring Go during laboratory compaction"

Ismail, M.A., Sharma, S.S., and Fahey, M. (2005). "A small true triaxial apparatus with wave

Iwasaki, T. and Tatsuoka, F. (1977), "Effects of Grain Size and Grading on Dynamic Shear

Iwasaki, T., Tatsuoka, F., and Takagi, Y. (1978), "Shear Moduli of Sands under Cyclic

Jamiolkowski, M., Lancellotta, R., Lo Presti, D. C. F., and Pallera, O. (1994), "Stiffness of Toyoura Sand at Small and Intermediate Strain," *Proceedings of the 13th International* 

Jardine, R. J., Potts, D. M., Fourie, A. B., and Burland, J. B. (1986), "Studies of the Influence of Non-Linear Stress-Strain Characteristics in Soil-Structure Interaction," *Geotechnique*, Vol.

Jovicic, V. (1997). The measurement and interpretation on small strain stiffness of soils.

Jovičić, V. and Coop, M. R. (1998), "The Measurement of Stiffness Anisotropy in Clays with Bender Element Tests in the Triaxial Apparatus," *Geotechnical Testing Journal*, ASTM,

Jovicic, V., Coop, M.R., and Simic, M. (1996). "Objective criteria for determining Gmax from

Kawaguchi, T., Mitachi, T., Shibuya, S. (2001). "Evaluation of shear wave travel time in laboratory bender element test" *Proceedings of the 15th International Conference on Soil* 

Kim, D.-S. and Stokoe, K. H., II (1992). "Characterization of resilient modulus of compacted subgrade soils using resonant column and torsional shear tests" *Transportation Research* 

Kokusho, T. (1980), "Cyclic Triaxial Test of Dynamic Soil Properties for Wide Strain Range,"

bender elements tests" *Geotechnique*, Vol.46, No.2, 357-362.

*Mechanics and Geotechnical Engineering*, Istanbul, Turkey, 155-158.

*Conference on Soil Mechanics and Foundation Engineering*, New Delhi, pp. 169-172. Jardine, R. J. (1992), "Some Observations on the Kinetic Nature of Soil Stiffness," *Soils and* 

velocity measurement" *Geotechnical Testing Journal*, Vol.28, No.2, 1-10.

Torsional Shear Loading," *Soils and Foundations*, Vol. 18, No. 1, pp. 39-50.

Moduli of Sands," *Soils and Foundations*, Vol. 17. No. 3, pp. 19-35.

sand and clay" *Soils and Foundations*, Vol.33, No.1, 182-191.

*Geotechnical Testing Journal*, Vol.29, No.4, 1-9.

*Foundations*, Vol. 32, No. 2, pp. 111-124.

*Record*, No.1369, Washington, D.C., 83-91.

*Soils and Foundations*, Vol. 20, pp. 45-60.

36, No. 3, pp. 377-396.

Vol. 21, No. 1, pp. 3-10.

Ph.D. thesis, City University.


Mancuso, C., Vassallo, R., and d'Onofrio, A. (2002), "Small Strain Behavior of a Silty Sand in Controlled-Suction Resonant Column-Torsional Shear Tests," *Canadian Geotechnical Journal*, Vol. 39, No. 1, pp. 22-31.

Wave Propagation Methods for Determining Stiffness of Geomaterials 197

Pennington, D.S. (1999). The anisotropic small strain stiffness of Cambridge Gault clay.

Ray, R. P. and Woods, R. D. (1988), "Modulus and Damping Due to Uniform and Variable Cyclic Loading," *Journal of Geotechnical and Geoenvironmental Engineering*, ASCE, Vol.

Richart, F. E., Hall, J. R., and Woods, R. D. (1970). *Vibrations of soils and Foundations*. Prentice

Richart, F.E. (1977), "Dynamic Stress-Strain Relations for Soils," *Proceedings of the 9th International Conference on Soil Mechanics and Foundation Engineering*, Tokyo, Vol. 2, pp.

Rix, G. J. and Stokoe, K. H., II (1989), "Stiffness Profiling of Pavement Subgrades,"

Robertson, P. K., Campanella, R. G., Gillespie, D., and Rice, A. (1986), "Seismic CPT to Measure In Situ Shear Wave Velocity," *Journal of Geotechnical Engineering*, ASCE, Vol.

Roesler, S. K. (1979), "Anisotropic Shear Modulus due to Stress Anisotropy," *Journal of the* 

Saada, A. S. (1988), "Hollow Cylinder Torsional Devices: Their Advantages and Limitation," *Advanced Triaxial Testing of Soil and Rock, ASTM STP 977*, Philadelphia, PA. pp. 766-779. Sanchez-Salinero, I., Roesset, J. M., and Stokoe, K. H., II (1986). Analytical studies of body wave propagation and attenuation. Report GR 86-15, University of Texas, Austin, TX. Sanchez-Salinero, I., Roesset, J. M., Shao, K., Stokoe, K. H., II, and Rix, G. J. (1987), "Analytical Evaluation of Variables Affecting Surface Wave Testing of Pavements,"

Santamarina, J. C., Klein, K. A., and Fam, M. A. (2001), *Soils and Waves*, John Wiley & Sons,

Sargand, S. M. (2001), "Direct Measurement of Backfill Stiffness for Installation and Design of Buried Pipes," *Ohio Research Institute for Transportation and the Environment*, Ohio

Sargand, S. M., Edwards, W. F., and Salimath, S. (2000), "Evaluation of Soil Stiffness via Non-Destructive Testing," *Ohio Research Institute for Transportation and the Environment*,

Sawangsuriya, A., Biringen, E., Fratta, D., Bosscher, P. J. and Edil, T. B. (2006). "Dimensionless limits for the collection and interpretation of wave propagation data in soils" *GeoShanghai Conference, Site and Geomaterial Characterization*, ASCE, Geotechnical

Sawangsuriya, A., Bosscher, P. J., and Edil, T. B. (2002), "Laboratory Evaluation of the Soil Stiffness Gauge," *Transportation Research Record*, No. 1808, Washington, D.C., pp. 30-37. Sawangsuriya, A., Bosscher, P. J., and Edil, T. B. (2005). Alternative testing techniques for modulus of pavement bases and subgrades. *Proceedings of the 13th Annual Great Lakes Geotechnical and Geoenvironmental Engineering Conference, Geotechnical Applications for Transportation Infrastructure*, ASCE, Geotechnical Practice Publication, No.3, Milwaukee,

*Transportation Research Record*, No. 1235, Washington, D.C., pp. 1-9.

*Geotechnical Engineering Division*, ASCE, Vol. 105, No. GT7, pp. 871-880.

*Transportation Research Record 1136*, Washington, D.C., pp. 132-144.

Special Publication, No.149, Shanghai, China, 160-166.

Ph.D. Thesis, University of Bristol.

114, No. 8, pp. 861-876.

Hall, Englewood Cliffs.

112, No. 8, pp. 71-803.

Chichester, UK.

University, OH.

WI, 108-121.

Ohio University, OH.

605-612.


Pennington, D.S. (1999). The anisotropic small strain stiffness of Cambridge Gault clay. Ph.D. Thesis, University of Bristol.

196 Wave Processes in Classical and New Solids

Vol.143, 31-42.

D.C.

NJ.

*Journal*, Vol. 39, No. 1, pp. 22-31.

*Histories*, Bali, Indonesia, pp. 27-48.

122, No. 4, pp. 302-308.

D.C., pp. 3-10.

MN.

Austin, TX.

Texas at Austin, Austin, TX.

1654, Washington, D.C., pp. 50-60.

Mancuso, C., Vassallo, R., and d'Onofrio, A. (2002), "Small Strain Behavior of a Silty Sand in Controlled-Suction Resonant Column-Torsional Shear Tests," *Canadian Geotechnical* 

Matthews, M. C., Clayton, C. R. I., and Own, Y. (2000) "The use of field geophysical techniques to determine geotechnical stiffness parameters" *Geotechnical Engineering*,

Mayne, P. W. (2001), "Stress-Strain-Strength-Flow Parameters from Enhanced In Situ Tests," *Proceedings of the International Conference on In Situ Measurement of Soil Properties and Case* 

Mayne, P. W., Christopher, B. R., and DeJong, J. (2001), *Manual on Subsurface Investigations*, National Highway Institute Publication No. FHWA NHI-01-031, FHWA, Washington,

Mitchell, J. K. and Soga, K. (2005), *Fundamentals of Soil Behavior*, John Wiley and Sons, Inc.,

Morris, D. V. (1990), "Automatic Feedback System for Resonant Column Testing,"

Nacci, V. A. and Taylor, K. J. (1967), "Influence of Clay Structure on Elastic Wave Velocities," *Proceedings of the International Symposium on Wave Propagation and Dynamic* 

Nakagawa, K., Soga, K., and Mitchell, J. K. (1996), "Pulse Transmission System for Measuring Wave Propagation in Soils," *Journal of Geotechnical Engineering*, ASCE, Vol.

Nazarian, S. and Stokoe, K. H., II (1987), "In Situ Determination of Elastic Moduli of Pavements Systems by Special-Analysis-of-Surface-Waves Method (Theoretical Aspects)," *Research Report 437-2*, Center for Transportation Research, The University of

Nazarian, S., Yuan, D., and Arellano, M. (2002), "Quality Management of Base and Subgrade Materials with Seismic Methods," *Transportation Research Record*, No. 1786, Washington,

Nazarian, S., Yuan, D., and Baker, M. R. (1994), "Automation of SASW Method," *Dynamic* 

Nazarian, S., Yuan, D., and Tandon, V. (1999), "Structural Field Testing of Flexible Pavement Layers with Seismic Methods for Quality Control," *Transportation Research Record*, No.

Nelson, C. R. and Sondag, M. (1999), "Comparison of the Humboldt GeoGauge with In-Place Quasi-Static Plate Load Tests," *CNA Consulting Engineers Report*, Minneapolis,

Ni, S. H. (1987), "Dynamic Properties of Sand under True Triaxial Stress States from Resonant Column and Torsion Shear Tests," *Ph.D. Thesis*, The University of Texas at

Ohara, S. and Matsuda, H. (1988), "Study on the Settlement of Saturated Clay Layer Induced

*Geotechnical Testing Journal*, ASTM, Vol. 13, No. 1, pp. 16-23.

*Properties of Earth Materials*, Albuquerque, New Mexico, pp. 491-502.

*Geotechnical Testing II, ASTM STP 1213*, Philadelphia, PA, pp. 88-100.

by Cyclic Shear," *Soils and Foundations*, Vol. 28, pp. 103-113.


Sawangsuriya, A., Edil, T. B., and Benson, C. H. (2009a). "Effect of suction on resilient modulus of compacted fine-grained subgrade soils" *Transportation Research Record 2101*, TRB, National Research Council, Washington, D.C., 82-87.

Wave Propagation Methods for Determining Stiffness of Geomaterials 199

Shibuya, S. and Tanaka, H. (1996), "Estimate of Elastic Shear Modulus in Holocene Soil

Shibuya, S., Hwang, S. C., and Mitachi, T. (1997), "Elastic Shear Modulus of Soft Clays from Shear Wave Velocity Measurement," *Geotechnique*, Vol. 47, No. 3, pp. 593-601. Shibuya, S., Mitachi, T., Fukuda, F., and Degoshi, T. (1995), "Strain Rate Effect on Modulus and Damping of Normally Consolidated Clay," *Geotechnical Testing Journal*, ASTM, Vol.

Shibuya, S., Tatsuoka, S., Teachavorasinskun, S., Kong, X. J., Abe, F., Kim, Y. S., and Park, C. S. (1992), "Elastic Deformation Properties of Geomaterials," *Soils and Foundations*, Vol.

Shirley, D.J. (1978). "An improved shear wave transducer" J. Acoust. Soc. Am., 63, No.5,

Shirley, D.J. and Hampton, L.D. (1978). "Shear–wave measurements in laboratory

Silver, M. L. and Seed, H. B. (1971), "Deformation Characteristics of Sands under Cyclic Loading," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 97, No. SM8,

Silvestri, F. (1991), "Stress-Strain Behaviour of Natural Soils by Means of Cyclic/Dynamic Torsional Shear Tests," *Experimental Characterization and Modelling of Soils and Soft Rocks*,

Souto, A., Hartikainen, J., and Özüdoğru, K. (1994), "Measurement of Dynamic Parameters of Road Pavement Materials by the Bender Element and Resonant Column Tests,"

Stokoe, K. H., II and Richart, F. E., Jr. (1973), "In Situ and Laboratory Shear Wave Velocities," *Proceedings of the 8th International Conference on Soil Mechanics and Foundation* 

Stokoe, K. H., II and Woods, R. D. (1972), "In Situ Shear Wave Velocity by Cross-Hole Method," *Journal of the Soil Mechanics and Foundations Division*, ASCE, Vol. 98, No. SM5,

Stokoe, K. H., II, Hwang, S. K., Lee, J. N.-K., and Andrus, R. D. (1995), "Effects of Various Parameters on the Stiffness and Damping of Soils at Small to Medium Strains," *Pre-*

Stokoe, K. H., II, Lee, S. H. H., and Knox, D. P. (1985), "Shear Moduli Measurement under True Triaxial Stresses," *Proceedings of the Geotechnical Engineering Division: Advances in* 

Sukolrat, J. (2007). Destructuration of Bothkennar clay, Ph.D. Thesis, Department of Civil

Tatsuoka, F. and Shibuya, S. (1991), "Deformation Characteristics of Soils and Rocks from Field and Laboratory Tests," *Proceedings of the 9th Asian Regional Conference on Soil* 

*failure Deformation of Geomaterials*, Balkema, Rotterdam, Vol. 2, pp. 785-816.

*the Art of Testing Soil Under Cyclic Conditions*, ASCE, Detroit, MI, pp. 166-185.

*Mechanics and Foundation Engineering*, Bangkok, Thailand, Vol. 2, pp. 101-170. Thomann, T. G. and Hryciw, R. D. (1990), "Laboratory Measurement of Small Strain Shear Modulus under Ko Conditions," *Geotechnical Testing Journal*, ASTM, Vol. 13, No. 2, pp.

Deposits," *Soils and Foundations*, Vol. 36, No. 4, pp. 45-55.

sediments" J. Acoust. Soc. Am., Vol.63, No.2, 607-613.

*Engineering*, Vol. 1, Part 2, Moscow, U.S.S.R., pp. 403-409.

The University of Napoli Federico, pp. 7-73.

*Geotechnique*, Vol. 44, No. 3, pp. 519-526.

Engineering, University of Bristol, UK.

18, No. 3, pp. 365-375.

32, No. 3, pp. 26-46.

1643-1645.

pp. 1081-1098.

pp. 443-460.

97-105.


Shibuya, S. and Tanaka, H. (1996), "Estimate of Elastic Shear Modulus in Holocene Soil Deposits," *Soils and Foundations*, Vol. 36, No. 4, pp. 45-55.

198 Wave Processes in Classical and New Solids

Sawangsuriya, A., Edil, T. B., and Benson, C. H. (2009a). "Effect of suction on resilient modulus of compacted fine-grained subgrade soils" *Transportation Research Record 2101*,

Sawangsuriya, A., Edil, T. B., and Bosscher, P. J. (2003), "Relationship between Soil Stiffness Gauge Modulus and Other Test Moduli for Granular Soils," *Transportation Research* 

Sawangsuriya, A., Edil, T. B., and Bosscher, P. J. (2004), "Assessing Small-Strain Stiffness of Soils Using the SSG," *Proceedings of the 15th Southeast Asia Geotechnical Conference*,

Sawangsuriya, A., Edil, T. B., and Bosscher, P. J. (2008a). "Modulus-suction-moisture relationship for compacted soils" *Canadian Geotechnical Journal*, Vol.45, No.7, 973-983. Sawangsuriya, A., Edil, T. B., and Bosscher, P. J. (2009b). "Modulus-suction-moisture relationship for compacted soils in postcompaction state" *Journal of Geotechnical and* 

Sawangsuriya, A., Fall, M., and Fratta, D. (2008b). "Wave-based techniques for evaluating elastic modulus and Poisson's ratio of laboratory compacted lateritic soils" *Geotechnical* 

Sawangsuriya, A., Fratta, D., Bosscher, P. J., and Edil, T. B. (2007a). "S-wave velocity-stress power relationship: packing and contact behavior of sand specimens" *Geo-Denver Conference, Advances in Measurement and Modeling of Soil Behavior*, ASCE, Geotechnical

Sawangsuriya, A., Fratta, D., Edil, T. B., and Bosscher, P. J. (2007b). "Small-strain shear modulus anisotropy of compacted soils using bender elements" *60th Canadian Geotechnical Conference & 8th Joint CGS/IAH-CNC Groundwater Conference*, Ottawa,

Sawangsuriya, A., Sukolrat, J., and Jotisankasa, A. (2008c), "Innovative testing methods for determining stiffness of geomaterials," *Seminar of Highway Engineering: Best Practices in* 

Seed, H. B. and Idriss, I. M. (1970), "Soil Moduli and Damping Factors for Dynamic Response Analyses," *Report EERC 70-10*, Earthquake Engineering Research Center,

Sharma, P. V. (1997), *Environmental and Engineering Geophysics*, Cambridge University Press,

Sheeran, D. E., Baker, W. H., and Krizek, R. J. (1967), "Experimental Study of Pulse Velocities in Compacted Soils," *Highway Research Record*, No. 177, Washington, D.C., pp.

Shibata, T. and Soelarno, D. S. (1975)," Stress-Strain Characteristics of Sands under Cyclic Loading," *Proceedings of the Japan Society of Civil Engineering*, No. 239, pp. 57-65. (in

*Highway Engineering*, Department of Highways, Bangkok, Thailand, pp. 589-609. Saxena, S. K., Avramidis, A. S., and Reddy, K. R. (1988), "Dynamic Moduli and Damping Ratios for Cemented Sands at Low Strains," *Canadian Geotechnical Journal*, Vol. 25, No. 2,

TRB, National Research Council, Washington, D.C., 82-87.

*Geoenvironmental Engineering*, Vol.135, No.10, 1390-1403.

*and Geological Engineering*, Vol.26, No.5, 567-578.

Special Publication, No.173, Denver, CO, 1-10.

University of California, Berkeley, CA.

Canada

pp. 353-368.

Cambridge, UK.

226-238.

Japanese)

*Record*, No. 1849, Washington, D.C., pp. 3-10.

Bangkok, Thailand, pp. 101-106.


Trudeau, P. J., Whitman, R. V., and Christian, J. T. (1974), "Shear Wave Velocity and Modulus of a Marine Clay," *Journal of the Boston Society of Civil Engineers*, Vol. 61, No. 1, pp. 12-25.

**Chapter 0**

**Chapter 8**

**Velocity-Stress Equations for Wave Propagation**

Conventionally, the second-order elastodynamics equation has been employed to model

where *ρ* is the density of the medium, **w** the displacement, and *c*[4] the fourth-order stiffness tensor [2]. Equation (1.1) has been derived based on the equation of motion in conjunction with the elastic constitutive equation. Equation (1.1) has been solved by the finite-difference methods, e.g., [21], and the time-domain finite-element methods, e.g., [33], for propagating waves, and the frequency-domain finite-element methods, e.g., [6], for normal mode analysis

Alternatively, one can use a modern numerical method [11, 12] to solve the first-order velocity-stress equations to obtain the transient solution of wave propagation in elastic solids. For example, Virieux [25] modeled waves in earth crust by using a finite-difference method to solve the velocity-stress equations. LeVeque [12, 13] simulated stress waves in isotropic, elastic solids by using a upwind scheme. Shorr [22] developed a specialized formulation of the finite-element method to simulate propagating waves. Käser and Dumbser [10] used the discontinuous Galerkin method [20] to model waves in anisotropic earth crust. Yu et al. [32] simulated non-linear stress-wave propagation in hypo-elastic media by using the space-time Conservation Element and Solution Element (CESE) method [4]. In the setting of the second-order wave equation, Eq. (1.1), ample theoretical analyses about anisotropic elasticity can be found in the literature, e.g., [2, 24]. In general, mathematical properties of the first-order velocity-stress equations must be similar to that of the second-order wave equation Eq. (1.1). However, only limited discussions, e.g., Puente et al. [18] and Yang et al. [30], about

the eigen structure of the velocity-stress equations can be found in the literature.

cited.

The objective of the present chapter is to analyze the mathematical properties of the first-order velocity-stress equations. In particular, their connection to the conventional second-order

and reproduction in any medium, provided the original work is properly cited.

©2012 Yu et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

*c*[4] ∇**w** 

, (1.1)

*<sup>∂</sup>t*<sup>2</sup> <sup>=</sup> ∇ ·

*ρ ∂*2**w**

**in Anisotropic Elastic Media**

Sheng-Tao John Yu, Yung-Yu Chen and Lixiang Yang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50512

**1. Introduction**

of standing waves.

waves in elastic solids:

