The Analytical Method in 1 A. P. S. Selvadurai Geomechanics Department of Civil Engineering and Applied Mechanics, This article presents an overview of the application of analytical methods in the theories McGill University, of elasticity, poroelasticity, flow, and transport in porous media and plasticity to the 817 Sherbrooke Street West, solution of boundary value problems and initial boundary value problems of interest to Montreal, QC, H3A 2K6 Canada geomechanics. The paper demonstrates the role of the analytical method in geomechanics e-mail: [email protected] in providing useful results that have practical importance, pedagogic value, and serve as benchmarking tools for calibrating computational methodologies that are ultimately used for solving more complex practical problems in geomechanics. There are 315 references cited in this article. ͓DOI: 10.1115/1.2730845͔ The development of classical soil mechanics and geomechanics bedded structures, stability and failure of soils, flow in porous owes a great deal to the availability and utilization of theoretical media, and consolidation and creep of soils. results in elasticity, plasticity, poroelasticity, and flow and trans- The analytical method has always played an important role not port in porous media. As the discipline evolves to encompass only as a component of the educational enterprise in geomechan- either new areas of application or the use of new theories of geo- ics but also as a tool for the development of concise results of material behavior, the natural tendency is to resort to computa- practical value for preliminary design calculations ͓19–25͔. This tional treatments of the complex problems. This is inevitable, but latter aspect is particularly important to geotechnical engineering at the same time opportunities do exist for the development of since, in most instances, preliminary designs are carried out with analytical approaches to the solution of problems in geomechan- only a limited knowledge of the range of values associated with ics; these would be of considerable help to geotechnical engineers, geotechnical material parameters. particularly in preliminary assessments of the performance of This review aims to outline some seminal classical treatments problems in geotechnical engineering. and recent developments in the application of the analytical method, in particular to the study of problems of interest to geo- 1 Introduction mechanics. The research, largely the purview of geomechanics half a century ago, now extends to a number of other disciplines, The conventional definition of the term “analysis” refers to the including mathematics, physics, materials science, earth physics, resolution or separation of a problem or a task into its elements. In geophysics, particulate media, solid mechanics, biomechanics, this sense all engineering endeavors are examples of analysis. In chemical engineering science, etc. The literature covering these the context of problem solving in engineering, the term analytical areas is extensive and it is difficult to adequately review and docu- solutions takes on a different meaning, one that specifically refers ment all the available analytical developments within the limits of to the use of advanced mathematical procedures for the solution of the present paper. Therefore this review will focus on a limited problems. This raises the question: Is any solution scheme that number of topics of interest to the general theme of the analytical uses mathematical procedures also an analytical solution? There- approach and present a discussion of problems that may be of fore, are computational schemes that use mathematical techniques potential interest to the geomechanics community. Due to limita- also providing analytical solutions? In the strict definition of the tions of space, attention will be restricted to isothermal quasi- term analysis, they are indeed examples of analytical solutions, static problems. There is a wealth of research dealing with the but conventional use of the term “an analytical solution” in past dynamics, thermo-mechanics, and hydro-thermo-mechanics of the and recent times has become synonymous with solution schemes types of problems discussed in this paper. Discussions of other that rely on the exhaustive application of mathematical procedures important methods that revolve around the semi-analytical tech- for the development of a solution to a problem in engineering. It is niques also require a fuller treatment, which cannot be achieved the preoccupation with the latter that distinguishes a very elegant within the context of the present article. These are available in the computational procedure from a purely analytical solution. Soil leading journals and symposia devoted to geomechanics, solid mechanics and geomechanics are excellent examples of disci- mechanics, computational mechanics, materials science, applied plines that have extensively used analytical solutions to full ad- mathematics, and applied mechanics. vantage, not only to provide the foundations of the subject but also to develop a set of usable solutions that, to this day, continues to benefit many areas of application in geotechnical engineering. 2 Elasticity and Geomechanics ͓ ͔ ͓ ͔ The historical treatises by Coulomb 1 , Rankine 2 , the volumes In their recent volume, Davis and Selvadurai ͓26͔ refer to elas- written after the coining of the term soil mechanics, by Terzaghi ticity as the “glue” that holds the governing equations. This is not ͓ ͔ ͓ ͔ ͓ ͔ ͓ ͔ ͓ ͔ 3,4 , Krynine 5 , Taylor 6 , Florin 7 , Tschebotarioff 8 ,Ca- an understatement, particularly when one looks at the typical civil ͓ ͔ quot and Kerisel 9 , and the more recent treatises by many au- engineering curriculum at the undergraduate level, which includes ͓ ͔ ͓ ͔ ͓ ͔ thors including Sokolovskii 10 , Leonards 11 , Nadai 12 , Scott a large collection of subjects in mechanics of solids, structural ͓ ͔ ͓ ͔ ͓ ͔ ͓ ͔ ͓ ͔ 13,14 , Harr 15 , Suklje 16 , Tsytovich 17 , and Bell 18 , mechanics, geomechanics, and advanced stress analysis that rely illustrate the range and depth of the use of analytical methods for on principles that are deeply rooted in the classical theory of the solution of conventional problems in geomechanics and foun- elasticity. The influences of nonlinear approaches are certainly dation engineering. They deal with diverse areas including stress being introduced into curricula, but not at the expense of the ex- distributions in soils, settlement analysis, stress states around em- clusion of the classical approaches. On occasions, particularly in the context of geomechanics, the linear theory has been referred to 1 as a children’s model; this is perhaps through ignorance and a lack Dedicated to Professor A. J. M. Spencer FRS, on the occasion of his 75th Birthday. of appreciation of the history of the subject, its content, and its Transmitted by Assoc. Editor P. Adler. impact on the engineering sciences: The children who were instru- Applied Mechanics Reviews Copyright © 2007 by ASME MAY 2007, Vol. 60 /87 Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/30/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use Fig. 1 Boussinesq’s and Kelvin’s problems mental in developing the theory of elasticity include Euler, James dures for obtaining a solution to Boussinesq’s problem, there is a Bernoulli, Hooke, Young, Lagrange, Poisson, Navier, Cauchy, more direct approach. Selvadurai ͓52͔ recognized the advantages Lamé, Clapyeron, Saint-Venant, Kirchhoff, Mohr, Green, Kelvin, of formulating the problem in spherical coordinates and utilizes Maxwell, Stokes, Rayleigh, Boussinesq, Hertz, Michell, Morera, the properties of Lamé’s strain potential ␸͑R,⌰͒ and Love’s strain Betti, Beltrami, Somigliana, Cerruti, Castigliano, Neuber, Papkov- potential ⌽͑R,⌰͒ that, respectively, satisfy ich, Galerkin, Airy, Lamb, Love, Filon, Southwell, Timoshenko, ͒ ͑ ͒⌰ 2␸͑ ⌰͒ ٌ2ٌ2⌽ٌ͑ Inglis, Mushkhelishvili, Mindlin, Fichera and many others ͑see R, =0; R, =0 1 e.g., Todhunter and Pearson ͓27,28͔, Timoshenko ͓29͔, Goodier where ͑R,⌰͒ are the spherical coordinates and ͓30͔, Truesdell ͓31,32͔, Teodorescu ͓33͔, Volterra and Gaines ͓34͔, 2ץ 1 ץ ⌰ cot ץ 2 2ץ Gurtin ͓35͔, Szabo ͓36͔, and Selvadurai ͓37,38͔, Meleshko and 2͒͑ + + + = 2ٌ ͔͒ ͓ 2⌰ץ R2 ⌰ץ R R2ץ R2 Rץ Selvadurai 39 . Linear elasticity is regarded as one of the more successful theories of mathematical physics. Quite apart from its utility, the study of elasticity should be viewed as part of the These functions can then be used directly to determine the stress classical education process, intended to develop the role of prob- and displacement fields. For example, from the Lamé’s strain po- lem formulation, modelling, and analysis in order to bring prob- tential lem solving to a successful conclusion. ␸ −1␸ ␴ ␸ ␴ ͑ −1␸͒ 2GuR = ,R;2Gu⌰ = R ,⌰; RR = ,RR; R⌰ = R ,R⌰; 2.1 Boussinesq’s Problem. Every geotechnical engineer, ͑ ͒ both practitioner and student, has had the occasion to use Bouss- etc. 3 inesq’s classical problem dealing with the action of a normal force Similar expressions can be obtained from ⌽͑R,⌰͒. We can start at the surface of a traction-free isotropic elastic halfspace region P ͑Fig. 1͒. Determination of the state of stress in an isotropic elastic with Kelvin’s problem for a concentrated force of magnitude K halfspace, subjected to a concentrated force acting normal to a acting at the interior of the elastic infinite space, since, in all of ͓ ͔ these concentrated force problems, there is no