Volume 16
Issue 3
Civil Engineering
JOURNAL OF
POLISH
AGRICULTURAL
UNIVERSITIES
Available Online: http://www.ejpau.media.pl/volume16/issue3/art-03.html
DETERMINATION OF POISSON’S RATIO BY MEANS OF RESONANT COLUMN TESTS
Wojciech Sas^{1}, Katarzyna Gabryś^{1}, Alojzy Szymański^{2}
^{1} Laboratory - Water Centre,
Warsaw University of Life Sciences - SGGW, Poland
^{2} Department of Geotechnical Engineering,
Warsaw University of Life Sciences - SGGW, Poland
In comparison with other basic mechanical properties of soils, Poisson's ratio is regarded as an elastic constant considerably underestimated. However, there is a significant number of diverse areas in which a proper knowledge of Poisson's ratio or even estimation of its value is required [6]. This article contains: definition of Poisson's ratio with distinction between static and dynamic, the values of Poisson's ratio for various materials, simple formulas how to receive this property as well as its applications in soil mechanics. Moreover, two techniques, an uniaxial loading test and resonant column test, as these which help us to determine the value of Poisson's ratio are briefly presented here. A special emphasis was placed on the second procedure of finding Poisson's ratio, which means measurements of wave velocities, longitudinal and shear one, from seismic data. On the basis of the latter methods, laboratory experiments were performed and some exemplary results were included. The obtained results display the received values of Poisson's ratio and the relationships between Poisson's ratio, elastic modules as well as velocities of elastic waves.
Key words: Poisson’s ratio, resonant column tests, natural cohesive soils.
INTRODUCTION
In the last forty years, a significant amount of research has been carried out to better understand the mechanical reaction of soils to dynamic excitations. A variety of laboratory techniques were used for these studies e.g. cyclic torsional shear tests, cyclic direct simple shear tests, cyclic triaxial tests and resonant column tests. They allowed researchers to measure above all strain amplitude and frequency of excitation on soil behavior [11].
For proper seismic response analysis as well as soil modeling program, appropriate evaluation of dynamic soils properties is essential [15]. Two principal parameters in the soil dynamic characteristic are following: dynamic shear modulus and damping ratio. These parameters are required to set up the Hardin-Drnevich (1972) [7] model, which describes the stress-strain relationship [11].
To evaluate dynamic properties of soils the resonant column test is applied. The basic principle of this kind of test is to vibrate a cylindrical soil sample in an elemental mode of vibration: torsion or flexure. Historically it has been used to estimate small-strain shear modulus (Gmax), small-strain material damping (Dmin) as well as the relationship between shear modulus (G), material damping (D) and shear strain (g) in soils and rocks. Resonant column tests are performed to better understand a mechanisms affecting stiffness [8].
One of the advantages of these tests is the possibility of determination of Poisson's ratio, so called dynamic Poisson's ratio (nd). In ASTM E 132 [2] the value of Poisson's ratio (static, ns) is obtained from strains resulting from uniaxial stress only. Whereas here, in resonant column apparatus, the knowledge of shear wave velocity (Vs) and longitudinal wave velocity in a bounded medium (Vrod) is essential. The importance of this mechanical parameter has not been appreciated as much as it deserves since the value of it reported for rocks and soils vary in a narrow range. The use of typical or even approximate values in most soil mechanics applications does not create any significant problems. Poisson's ratio plays an undeniably important role in the elastic deformation of soils subjected to static or dynamic stresses. Moreover, its effect becomes known in a wide variety of rock and soil engineering applications, consequently, information about various aspects of Poisson's ratio can be truly beneficial [6].
This article aims to show how Poisson's ratio with the aid of resonant column apparatus can be obtained, especially in the range of small strains. First, the definition of this interesting parameter is presented and later its importance in mechanics is emphasized. Then, the laboratory equipment useful for determination of Poisson's ratio is summarized, with the particular accent put on the resonant column. Also, some exemplary results are included, which display the received values of Poisson's ratio and the relationships between Poisson's ratio, elastic modules as well as velocities of elastic waves.
POISSON'S RATIO IN SOIL MECHANICS
Numerous definitions of Poisson's ratio can be found in the literature, but many of them miss completeness. Plainly, Poisson's ratio (n) is the negative of ratio of transversal strain to the axial strain in an elastic material, which is subjected to an uniaxial stress [11]. When a sample object is squeezed or stretched to a contraction or an extensions in the direction of applied load, it agrees with a contraction/extension in a perpendicular direction to the applied load. Poisson's ratio is exactly the ratio between these two quantities. In mechanics of deformable bodies exists so called the "Poisson's effect". This phenomenon is, when a material compressed in one direction, it tends to expand in perpendicular directions to the direction of compression. Poisson's ratio is a degree of the Poisson's effect; is the ratio of the percent of expansion divided by the percent of compression. In a converse manner, if the material is stretched rather than compressed, it tends mostly to contract in the directions transverse to this one of stretching. And here, Poisson's ratio will be the ratio of relative contraction to relative stretching, with the same value as above. Sometimes however, a sample can actually shrink in the transverse course while compressed, or expand when stretched. This situation will yield a negative value of Poisson's ratio [12].
Isotropic, stable, linear elastic materials have the value of Poisson's ratio from the range of -1,0 to 0,5 because of the requirement that shear modulus (G), Young's modulus (E) and bulk modulus (K) have positives values. For most materials n varies between 0,0 and 0,5. It is generally assumed that a perfectly incompressible material, which is elastically deformed at small strains, has Poisson's ratio exactly 0,5. Steel and rigid polymers, usually while used within their design limits, before yield, show values of n about 0,3, raising up to 0,5 for post-yield deformation. Poisson's ratio for rubber is nearly 0,5, however for cork close to 0. There are some substances with negative Poisson's ratio; when they are stretched in one course, whereas they become thicker in perpendicular. What is more, there are some anisotropic materials with one or even more Poisson's ratio above value 0,5 in some directions [3, 13, 14].
Poisson's ratio, assuming stretching or compressing the material along the axial ‘x’ direction, can be easily calculated from the formula:
where: n – resulting Poisson's ratio,
etrans – transverse strain (for
axial tension negative, for axial compression positive),
eaxial – axial
strain (for axial tension positive, for axial compression negative).
In Table 1 the values of Poisson's ratio for some significant materials are listed.
Table 1. Poisson's ratio values for different materials [23] |
For elastic and isotropic soils, some parameters that depend, among others, on Poisson's ratio can be expressed as follows:
where: E – Young's modulus,
G – shear modulus,
K – bulk modulus.
Also, the ratio of wave velocities, shear (VS) to longitudinal (VP), in an isotropic solid with an infinite extent, can be formulated:
where: nd – dynamic Poisson's ratio, which may be different than that one received from static tests.
Additionally, the ratio (a) of wave velocities, Rayleigh (VR) to shear (VS), depends only on the value of Poisson's ratio of the substance. It can be calculated from the following equation [6]:
Generally, Poisson's ratio is a mechanical property that plays a significant role in the deformation of elastic materials. It can be used to predict, for example, the geomechanical behaviour during the drilling process of wells, as well as the following recovery processes. Sand production, well instability and hydraulic fracturing are widely affected by strength parameters, which consequently might relate to its magnitude [10]. The volume changes of reservoir caused by the injection or the production, the subsequent uplift of the surface or the subsidence could be substantial [20]. Numerical modelling requires the value of Poisson's ratio as input in order to forecast their influence on surface installations and wells. Therefore, it is essential to make reasonable predictions with the aim of obtain the successful oilfield operations.
LABORATORY TESTS
In the main, there are two methods to determine Poisson's ratio. The first one is an uniaxial loading test. Thanks to this research, the negative of the ratio of radial strain to axial strain can be computed, which is described as static Poisson's ratio (ns). The second way of defining Poisson's ratio is by measurement of wave velocities, longitudinal and shear one, from seismic data, and then calculation of dynamic Poisson's ratio (nd). Determination of n by means of wave velocity has more advantages compared to the first technique. Above all, velocity measurements are believed to be non-destructive to the specimen. Furthermore, the plenty of dynamic Poisson's ratio methods reduces the possibility of risk of incorrect predictions or inferences from data points. The value of dynamic Poisson's ratio, however, might be considerably different from the value of static Poisson's ratio [20, 21].
Calculation of Poisson's ratio, as well as elastic modulus, during the uniaxial compression can be made from measurements of axial deformations and strains, and either radial deformations or circumferential strains. Because of the inherent natural variability of soils and rocks, it is necessary for a number of replicate tests to be performed. The minimum number of tests is usually at least 5, although it has been shown that 10 tests would be required to be fully confident of knowing the properties of a material with a coefficient of variation of 20%.
In an uniaxial strain test (UXE) the lateral confinement is continuously adjusted to maintain zero circumferential strain as the axial load is increased. Alternatively, the sample is placed in a "rigid" cylinder, which prevents lateral displacements. This test is typically applied to soils, as a replacement for an unconfined compression test, that is inappropriate for most soils; they have almost no strength without some confinement. UXE is a very useful test for characterizing rocks and concrete, especially if a cap type model is used for the constitutive model [18].
The method of resonant column test was developed in 1930s by Japanese engineer K. Iida [5]. It became worldwide popular since 1950s. Firstly, Ishimoto and Iida [5] elaborated on both theory and a device for resonant column tests on soils, in which loading frequency at maximum response was employed to calculate elastic properties of examined materials. Generally, this technique consists in application of cyclic force to a soil sample at various frequencies. The specimen's dynamic response to used force is measured in terms of acceleration and/or velocity. Velocity and acceleration only at high frequencies are large enough to be assessed, in order to obtain a precise measurement of small deformations. Modifications of loading frequency allow to receive the variation of amplification in amplitude of response and create the graph that shows differences in amplitudes against frequency [19].
The resonant column apparatus applied by the authors of this paper in their laboratory researches (fig. 1) was developed by a British company, i.e. GDS Instruments Ltd., as an example of Hardin-Drnevich device, projected in configuration "fixed-free". A single soil sample is located in the triaxial chamber and is fixed at the bottom. At the top, the specimen is subjected to two types of vibrations: torsional and flexural. These vibrations are caused by the electromagnetic drive system, which is composed by the set of four magnets and four coils. By applying the idea of wave propagation to torsion or flexure of a solid cylinder with a top mass, a theoretical value of resonant (natural) frequency is detected. By equating the theoretical and experimental resonant frequencies, for example, the shear modulus (G) can be determined [4, 17].
Fig. 1. Photography of the GDS Resonant Column Apparatus |
The resonant column technique can be used with success as well for the analysis of dynamic Poisson's ratio. To define nd, seismic wave measurements are needed. In this paper, two types of this kind of measurements are presented: shear waves (S) and longitudinal waves (P), narrowly longitudinal wave velocity in a bounded medium (Vrod). Shear wave velocity and longitudinal wave velocity in a bounded medium are related among them by the Poisson's effect, which is analytically expressed by dynamic Poisson's ratio, according to the equation [5]:
Longitudinal wave velocity in a bounded medium is defined by the following calculation:
where: Eflex – Young's modulus for flexural excitation, estimated based on
the geometric
properties of the specimen and apparatus with the measurement
of the resonant
frequency of flexural vibration,
r – bulk density.
Shear wave velocity is determined from:
where: f – natural frequency of sample as found from the resonant column test,
l – sample length,
b – parameter depends on quotient of mass polar moment of inertia
of soil specimen
(I) and mass polar moment of inertia of resonant column drive
system (I0).
EXPERIMENTAL RESULTS
The experimental program was carried out in natural cohesive soils from the Warsaw area, taken from the test side located near the planned express route (S2), between its two nodes "Konotopa-Lotnisko". The testing material was selected carefully considering the uniformity of the soils structure, its physical properties and its double-phase. Grain size distribution of the samples is shown in Fig. 2 and the essential basic physical properties in Table 2.
Fig. 2. Soil grain size distribution curve for analysed specimens |
Table 2. Basic physical properties of analysed specimens |
Based on equation (9) with previously found values of resonant frequency from the resonant column test, shear wave velocities of three tested specimens were calculated. Samples varied from each other with the value of mean effective stress (p') maintained during the whole experimental procedure. For the first one p' was equal to 45 kPa, for the second p'=85 kPa, for the last p'=100 kPa. Assuming that Vrod=8,25E+05 m/s and using the formula (7), requested values of dynamic Poisson's ratio were deliberated. Tables 3, 4 and 5 indicate not only the results of determination of VS and nd, but also the specimens' dynamic stiffness, represented here by shear modulus (G) and Young's modulus (E), as well as shearing strain (γ) and P-wave velocity (VP).
Table 3. Dynamic Poisson's
ratio computed for the first sample |
Table 4. Dynamic Poisson's
ratio computed for the second sample |
Table 5. Dynamic Poisson's ratio computed for the third sample |
Analysing all of Tables, it is simply noticeable, that the values of n are comparable with the literature results (e.g. Amaral et al. 2011) and fit into the typical averages for sand and clay pointed out in the Table 1. Furthermore, the increase of wave amplitude causes the proper raise in the value of Poisson's ratio (Fig. 3). A similar trend of changes can be observed with regard to P-wave velocity (Fig. 4), while the opposite is the case, when n is a function of S-wave velocity (Fig. 5), as expected due to Eqn. (5). Herein, the increase of shear wave velocity induces the decease of Poisson's ratio. For dynamic properties of examined materials and their relationship with Poisson's ratio two figures were created (Fig. 6 & 7). These figures show the course of the surface illustrating the mathematical model, selected from a number of models analyzed in the program Table Curve 3D v4.0. Chosen models have the following forms:
where: a=391.04416; b=-0.0072874831; c=-2.8318636; d=-1.6540359; e=-875.37544;
f=8.8770674e-06; g=0.0030992826; h=0.33988048; i=474.92468; j=-0.00095073044;
k=1.2740366
where: a=823.554; b=-0.055844629; c=-6530.1298; d=16305.993; e=-12847.447;
f=-6.0596239-05; g=-8.2772918; h=21.984833; i=-18.843508
The functions (10) and (11) indicate some relationship between elastic modulus (shear modulus and Young's modulus) and mean effective stress (p') as well as Poisson's ratio (n). The coefficient of determination (R2) for both models amounts 0.98 for function (10) and 0.99 for (11), adjusted R2 has the value 0.96 for model (10) 0.99 for model (11).
As the last, association between n and γ is presented (Fig. 8). Modifications in the value of shearing strain cause changes in the value of Poisson's ratio.
Fig. 3. Change of Poisson's ratio as a function of wave amplitude |
Fig. 4. Change of Poisson's ratio as a function of P-wave velocity |
Fig. 5. Change of Poisson's ratio as a function of S-wave velocity |
Fig. 6. The course of the surface by the model (10) showing the relationship between shear modulus, Poisson's ratio and mean effective stress |
Fig. 7. The course of the surface by the model (11) showing the relationship between Young's modulus, Poisson's ratio and mean effective stress |
Fig. 8. Poisson's ratio as a function of shearing strain |
SUMMARY AND CONCLUSIONS
This article is dedicated to the methods of determination of Poisson's ratio, which is considered as an important mechanical property. Conventionally, there are two techniques to define this parameter and both are shortly described here. Greater emphasis was placed on the second procedure of finding n, which means measurements of wave velocities, longitudinal and shear one, from seismic data. Resonant column tests were used as the latter method of computing Poisson's ratio. By means of simple formulas and the knowledge about resonant frequency of samples, shear wave velocity and longitudinal wave velocity in a bounded medium it was possible to estimate the desired parameter.
Calculations were based on the studies carried on natural cohesive soils from Warsaw area. Results are summarized in tables as well as on graphs.
From the data presented in this paper the following conclusions may be drawn:
It is possible to estimate fair values of Poisson's ratio using resonant column technique. Poisson's ratio can be correlated with other soil's parameters. For analysed materials, Poisson's ratio varies directly with wave amplitude and its velocities as well as with shearing strain. To estimate elastic modulus (shear modulus or Young's modulus) the empirical formulas proposed by the authors can be used. The suggested models are based on the knowledge of two parameters: mean effective stress and Poisson's ratio. This observation raises the importance of Poisson's ratio and thus extends its applicability.
The method of determination of Poisson's ratio proposed in the article obviously requires further researches and verification of the obtained results. In order to find some practical application in geotechnics, it seems to be necessary a conversion of dynamic Poisson's ratio to static one.
REFERENCES
- Amaral M.F., Fonseca A.V., Carvalho J., Consoli N.C., 2011. Dynamic Poisson ratio analysis. Proceedings of the 15th European Conference on Soil mechanics and Geotechnical Engineering, IOS Press, Vol. 1, 115–120.
- ASTM E 132 Tensile Compression Poisson Ratio Test Equipment | Test Resources, ASTM Mechanical testing Standards.
- Fung Y.C., 1965. Foundations of solid mechanics. Englewood Cliff, NJ: Prentice-Hall.
- Gabryś K., Szymański A., 2012. Badania parametrów odkształceniowych gruntów spoistych w kolumnie rezonansowej [Research of deformation parameters of cohesive soils in resonant column] Inżynieria Morska i Geotechnika, nr 4/2012, 324–327 [In Polish].
- GDS Instruments Ltd., 2010. Product Help sheets.
- Gercek H., 2006. Poisson's ratio values for rocks. International Journal of Rock Mechanics and Mining Sciences, Elsevier, Vol. 44, 1–13.
- Hardin B.O., Drnevich V.P., 1972. Shear Modulus and Damping in Soils: Design Equations and Curves. Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98(7), 667–692.
- Kalinski M.E., Thummaluru M.S.R., 2004. A New Free-Free Resonant Column Device for Measurement of Gmax and Dmin at Higher Confining Stresses.Geotechnical testing Journal, Vol. 28, No. 2, 180–187.
- Kolsky H., 1963. Stress waves in solids. New York: Dover Publications.
- Kumar J., 1976. The effect of Poisson's ratio on rock properties. The 51st Annual Fall technical Conference and Exhibition of the Society of Petroleum Engineering of AIME, SPE 6094.
- Lai C.G., Pallara O., Lo Presti D.C.F., Turco E., 2001. Low-strain stiffness and material damping ratio coupling in soils. Advanced Laboratory Stress-Strain Testing Geomaterials. Swets&Zeitlinger Publishers: 265–274.
- Lakes R., 1987. Foam Structures with a Negative Poisson's Ratio. Science 27, Vol. 235, No. 4792, 1038–1040.
- Lee H.S., Kim J., 2009. Determination of Viscoelastic Poisson's Ratio and Creep Compliance from the Indirect Tension Test. Journal of Materials in Civil Engineering. ASCE(2009), 21, 8 (416).
- Love A.E.H., 1944. A treatise on the mathematical theory of elasticity, 4th edn. New York: Dover Publications.
- Malvern L.E., 1969. Introduction to the mechanics of continuous medium. Englewood Cliffs, NJ: Prentice-Hall.
- Rayhani M.H.T., Naggar M.H.El., 2008. Dynamic Properties of Soft Clay and Loose sand from Seismic Centrifuge Tests. Springer (2008) 26, 593–602.
- Sas W., Gabryś K., Szymański A., 2012. Analiza sztywności gruntów spoistych przy wykorzystaniu kolumny rezonansowej [Analysis of stiffness of cohesive soils with use of resonant column] Inżynieria Morska i Geotechnika, nr 4/2012, 370–375 [in Polish].
- Schwer L., 2001. Laboratory Tests for Characterizing Geomaterials. Draft-03Dec01.
- Towhata I., 2008. Geotechnical Earthquake Engineering. Springer.
- Tutuncu A.N., Sharma M.M., 1992. Relating static and ultrasonic laboratory measurements to acoustic log measurements in tight gas sand: The 67th SPE Annual technical Conference and Exhibition, SPE 24689.
- Wang Zhijing, Nur A., 2000. Seismic and acoustic velocities in reservoir rocks, recent developments. Geophysics reprint series.
- Zhang J.J., Bentley L.R., 2005. Factors determining Poisson's ratio. CREWES Research Report, Vol. 17, 1–15.
- http://www.engineeringtoolbox.com
Accepted for print: 20.08.2013
Wojciech Sas
Laboratory - Water Centre,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: wojciech_sas@sggw.pl
Katarzyna Gabryś
Laboratory - Water Centre,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: katarzyna_gabrys@sggw.pl
Alojzy Szymański
Department of Geotechnical Engineering,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: alojzy_szymanski@sggw.pl
Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.