Foundations of Engineering Mechanics Series Editors: V.I Babitsky, J Wittenburg Foundations of Engineering Mechanics Series Editors: Vladimir I Babitsky, Loughborough University, UK Jens Wittenburg, Karlsruhe University, Germany Further volumes of this series can be found on our homepage: springer.com Slivker, V.I Mechanics of Structural Elements, 2007 ISBN 978-3-540-44718-4 Kolpakov, A.G Stressed Composite Structures, 2004 ISBN 3-540-40790-1 Elsoufiev, S.A Strength Analysis in Geomechanics, 2007 ISBN 978-3-540-37052-9 Shorr, B.F The Wave Finite Element Method, 2004 ISBN 3-540-41638-2 Awrejcewicz, J., Krysko, V.A., Krysko, A.V Thermo-Dynamics of Plates and Shells, 2007 ISBN 978-3-540-37261-8 Svetlitsky, V.A Engineering Vibration Analysis - Worked Problems 1, 2004 ISBN 3-540-20658-2 Babitsky, V.I., Shipilov, A Resonant Robotic Systems, 2003 ISBN 3-540-00334-7 Wittbrodt, E., Adamiec-Wojcik, I., Le xuan Anh, Wojciech, S Dynamics of Flexible Multibody Systems, 2006 Dynamics of Mechanical Systems with Coulomb Friction, 2003 ISBN 3-540-32351-1 ISBN 3-540-00654-0 Aleynikov, S.M Nagaev, R.F Spatial Contact Problems in Geotechnics, Dynamics of Synchronising Systems, 2006 2003 ISBN 3-540-25138-3 ISBN 3-540-44195-6 Skubov, D.Y., Khodzhaev, K.S Non-Linear Electromechanics, 2006 ISBN 3-540-25139-1 Feodosiev, V.I., Advanced Stress and Stability Analysis Worked Examples, 2005 ISBN 3-540-23935-9 Lurie, A.I Theory of Elasticity, 2005 ISBN 3-540-24556-1 Sosnovskiy, L.A., TRIBO-FATIGUE · Wear-Fatigue Damage and its Prediction, 2005 ISBN 3-540-23153-6 Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I (Eds.) Asymptotical Mechanics of Thin-Walled Structures, 2004 ISBN 3-540-40876-2 Ginevsky, A.S., Vlasov, Y.V., Karavosov, R.K Acoustic Control of Turbulent Jets, 2004 ISBN 3-540-20143-2, (Continued after index) Neimark, J.I Mathematical Models in Natural Science and Engineering, 2003 ISBN 3-540-43680-4 Perelmuter, A.V., Slivker, V.I Numerical Structural Analysis, 2003 ISBN 3-540-00628-1 Lurie, A.I., Analytical Mechanics, 2002 ISBN 3-540-42982-4 Manevitch, L.I., Andrianov, I.V., Oshmyan, V.G Mechanics of Periodically Heterogeneous Structures, 2002 ISBN 3-540-41630-7 Babitsky, V.I., Krupenin, V.L Vibration of Strongly Nonlinear Discontinuous Systems, 2001 ISBN 3-540-41447-9 Landa, P.S Regular and Chaotic Oscillations, 2001 ISBN 3-540-41001-5 Serguey A Elsoufiev Strength Analysis in Geomechanics With 158 Figures and 11 Tables Series Editors: V.I Babitsky Department of Mechanical Engineering Loughborough University Loughborough LE11 3TU, Leicestershire United Kingdom J Wittenburg Institut f uă r Technische Mechanik Universităat Karlsruhe (TH) Kaiserstraòe 12 76128 Karlsruhe Germany Author: Serguey A Elsoufiev Vierhausstr 27 44807 Bochum, Germany ISSN print edition: 1612-1384 ISBN-10: 3-540-37052-8 ISBN-13: 978-3-540-37052-9 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006931488 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable to prosecution under German Copyright Law Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use A EX package Typesetting:Data conversion by the author\and SPi using Springer LT Cover-Design: deblik, Berlin Printed on acid-free paper SPIN: 11744382 62/3100/SPi - Foreword It is hardly possible to find a single rheological law for all the soils However, they have mechanical properties (elasticity, plasticity, creep, damage, etc.) that are met in some special sciences, and basic equations of these disciplines can be applied to earth structures This way is taken in this book It represents the results that can be used as a base for computations in many fields of the Geomechanics in its wide sense Deformation and fracture of many objects include a row of important effects that must be taken into account Some of them can be considered in the rheological law that, however, must be simple enough to solve the problems for real objects On the base of experiments and some theoretical investigations the constitutive equations that take into account large strains, a non-linear unsteady creep, an influence of a stress state type, an initial anisotropy and a damage are introduced The test results show that they can be used first of all to finding ultimate state of structures – for a wide variety of monotonous loadings when equivalent strain does not diminish, and include some interrupted, step-wise and even cycling changes of stresses When the influence of time is negligible the basic expressions become the constitutive equations of the plasticity theory generalized here At limit values of the exponent of a hardening law the last ones give the Hooke’s and the Prandtl’s diagrams Together with the basic relations of continuum mechanics they are used to describe the deformation of many objects Any of its stage can be taken as maximum allowable one but it is more convenient to predict a failure according to the criterion of infinite strains rate at the beginning of unstable deformation The method reveals the influence of the form and dimensions of the structure on its ultimate state that are not considered by classical approaches Certainly it is hardly possible to solve any real problem without some assumptions of geometrical type Here the tasks are distinguished as antiplane (longitudinal shear), plane and axisymmetric problems This allows to consider a fracture of many real structures The results are represented by relations that can be applied directly and a computer is used (if necessary) on a final stage of calculations The method can be realized not only in VI Foreword Geomechanics but also in other branches of industry and science The whole approach takes into account five types of non-linearity (three physical and two geometrical) and contains some new ideas, for example, the consideration of the fracture as a process, the difference between the body and the element of a material which only deforms and fails because it is in a structure, the simplicity of some non-linear computations against linear ones (ideal plasticity versus the Hooke’s law, unsteady creep instead of a steady one, etc.), the independence of maximum critical strain for brittle materials on the types of structure and stress state, an advantage of deformation theories before flow ones and others All this does not deny the classical methods that are also used in the book which is addressed to students, scientists and engineers who are busy with strength problems Preface The solution of complex problems of strength in many branches of industry and science is impossible without a knowledge of fracture processes Last 50 years demonstrated a great interest to these problems that was stimulated by their immense practical importance Exact methods of solution aimed at finding fields of stresses and strains based on theories of elasticity, plasticity, creep, etc and a rough appreciation of strength provide different results and this discrepancy can be explained by the fact that the fracture is a complex problem at the intersection of physics of solids, mechanics of media and material sciences Real materials contain many defects of different form and dimensions beginning from submicroscopic ones to big pores and main cracks Because of that the use of physical theories for a quantitative appreciation of real structures can be considered by us as of little perspective For technical applications the concept of fracture in terms of methods of continuum mechanics plays an important role We shall distinguish between the strength of a material (considered as an element of it – a cube, for example) and that of structures, which include also samples (of a material) of a different kind We shall also distinguish between various types of fracture: ductile (plastic at big residual strains), brittle (at small changes of a bodies’ dimensions) and due to a development of main cracks (splits) Here we will not use the usual approach to strength computation when the distribution of stresses are found by methods of continuum mechanics and then hypotheses of strength are applied to the most dangerous points Instead, we consider the fracture as a process developing in time according to constitutive equations taking into account large strains of unsteady creep and damage (development of internal defects) Any stage of the structures’ deformation can be supposed as a dangerous one and hence the condition of maximum allowable strains can be used But more convenient is the application of a criterion of an infinite strain rate at the moment of beginning of unstable deformation This approach gives critical strains and the time in a natural way When the influence of the latter is small, ultimate loads may be also VIII Preface found Now we show how this idea is applied to structures made of different materials, mainly soils The first (introductory) chapter begins with a description of the role of engineering geological investigations It is underlined that foundations should not be considered separately from structures Then the components of geomechanics are listed as well as the main tasks of the Soil Mechanics Its short history is given The description of soil properties by methods of mechanics is represented The idea is introduced that the failure of a structure is a process, the study of which can describe its final stage Among the examples of this are: stability of a ring under external pressure and of a bar under compression and torsion (they are represented as particular cases of the common approach to the stability of bars); elementary theory of crack propagation; the ultimate state of structures made of ideal plastic materials; the simplest theory of retaining wall; a long-time strength according to a criterion of infinite elongations and that of their rate The properties of introduced non-linear equations for unsteady creep with damage as well as a method of determination of creep and fracture parameters from tests in tension, compression and bending are given (as particular cases of an eccentric compression of a bar) In order to apply the methods of Chap to real objects we must introduce main equations for a complex stress state that is made in Chap The stresses and stress tensor are introduced They are linked by three equilibrium equations and hence the problem is statically indeterminate To solve the task, displacements and strains are introduced The latters are linked by compatibility equations The consideration of rheological laws begins with the Hooke’s equations and their generalization for non-linear steady and unsteady creep is given The last option includes a damage parameter Then basic expressions for anisotropic materials are considered The case of transversally isotropic plate is described in detail It is shown that the great influence of anisotropy on rheology of the body in three options of isotropy and loading planes interposition takes place Since the problem for general case cannot be solved even for simple bodies some geometrical hypotheses are introduced For anti-plane deformation we have five equations for five unknowns and the task can be solved easily The transition to polar coordinates is given For a plane problem we have eight equations for eight unknowns A very useful and unknown from the literature combination of static equations is received The basic expressions for axi-symmetric problem are given For spherical coordinates a useful combination of equilibrium laws is also derived It is not possible to give all the elastic solutions of geo-technical problems They are widely represented in the literature But some of them are included in Chap for an understanding of further non-linear results We begin with longitudinal shear which, due to the use of complex variables, opens the way to solution of similar plane tasks The convenience of the approach is based on the opportunity to apply a conformal transformation when the results for simple figures (circle or semi-plane) can be applied to compound sections The displacement of a strip, deformation of a massif with a circular hole and Preface IX a brittle rupture of a body with a crack are considered The plane deformation of a wedge under an one-sided load, concentrated force in its apex and pressed by inclined plates is also studied The use of complex variables is demonstrated on the task of compression of a massif with a circular hole General relations for a semi-plane under a vertical load are applied to the cases of the crack in tension and a constant displacement under a punch In a similar way relations for transversal shear are used, and critical stresses are found Among the axi-symmetric problems a sphere, cylinder and cone under internal and external pressures are investigated The generalization of the Boussinesq’s problem includes determination of stresses and displacements under loads uniformly distributed in a circle and rectangle Some approximate approaches for a computation of the settling are also considered Among them the layer-by-layer summation and with the help of the so-called equivalent layer Short information on bending of thin plates and their ultimate state is described As a conclusion, relations for displacements and stresses caused by a circular crack in tension are given Many materials demonstrate at loading a yielding part of the stress-strain diagram and their ultimate state can be found according to the Prandtl’s and the Coulomb’s laws which are considered in Chap 4, devoted to the ultimate state of elastic-plastic structures The investigations and natural observations show that the method can be also applied to brittle fracture This approach is simpler than the consequent elastic one, and many problems can be solved on the basis of static equations and the yielding condition, for example, the torsion problem, which is used for the determination of a shear strength of many materials including soils The rigorous solutions for the problems of cracks and plastic zones near punch edges at longitudinal shear are given Elastic-plastic deformation and failure of a slope under vertical loads are studied among the plane problems The rigorous solution of a massif compressed by inclined plates for particular cases of soil pressure on a retaining wall and flow of the earth between two foundations is given Engineering relations for wedge penetration and a load-bearing capacity of a piles sheet are also presented The introduced theory of slip lines opens the way to finding the ultimate state of structures by a construction of plastic fields The investigated penetration of the wedge gives in a particular case the ultimate load for punch pressure in a medium and that with a crack in tension A similar procedure for soils is reduced to ultimate state of a slope and the second critical load on foundation Interaction of a soil with a retaining wall, stability of footings and different methods of slope stability appreciation are also given The ultimate state of thick-walled structures under internal and external pressures and compression of a cylinder by rough plates are considered among axi-symmetric problems A solution to a problem of flow of a material within a cone, its penetration in a soil and load-bearing capacity of a circular pile are of a high practical value Many materials demonstrate a non-linear stress-strain behaviour from the beginning of a loading, which is accompanied as a rule by creep and damage This case is studied in Chap devoted to the ultimate state of structures 218 M Appendix P,MN 0.20 0.15 0.10 0.05 k m.10−3 Fig M.2 Results of experiments on quasi-static loading Table M.1 Data of quasi-static loading No a b f 18.2 30 7,5 12.5 0.143 0.024 0.153 0.136 0.138 C,MN/m MN/m 51.5 35 36 3ß 38 287 281 298 254 264 press scale and deformation κ – from two indicators The simultaneous record of P and κ at cycling loading was fulfilled by special electronic apparatus 90-16.2 To make the tests in liquid the device was situated in a cylinder The results of tests at quasi-static loading at different environment and α are given in Fig M.2 (solid lines for air at α = 0.331, – for lubricant with graphite at α = 0.331, broken line – for air at α = 0.262, interrupted by points lines – for oil at α = 0.262 and – for oil at α = 0.296) and in Table M.1 under the same numbers The character of graphs P(κ) which have the form of triangles with upper sides responding to loading and vertical as well as lower straight lines – to unloading shows an opportunity of theoretical prediction of deformation characteristics of the device At the increase of upper or lower rings diameter Db (Fig M.1,a) on value ∆Db the system becomes shorter on ∆κb = 0.5∆Db cot α Similarly for other rings we have ∆κs = ∆Ds cot α As a result the whole shortening of the system of n rings is (M.1) κ = (∆Db + (n − 2)∆Ds ) cot α M Appendix 219 When diameter D changes its value on ∆D the circumferential stresses appear They can be replaced by radial forces q on the unit length of the perimeter (Fig M.1, b) as q = 2EA∆Di /Di (i = b, s) (M.2) where E is modulus of elasticity, A – area of cross-section of a ring (shaded in Fig M.1) Making a sum of projections of forces on axis x in Fig M.1, a we have for upper or lower and middle rings respectively − (P/πDb ) cos α + qb sin α + f(qb cos α + (P/πDb ) sin α) = 0, − (P/πDs ) cos α + 0.5qs sin α + f(0.5qs cos α + (P/πDs ) sin α) = (M.3) Here friction force T is found from the Coulomb’s law T = fN, f – coefficient of friction and normal component N is computed according to the equilibrium equation Replacing in (M.3) values qb , qs by relations (M.2) and putting values ∆Db , ∆Ds into (M.1) we find after transformations P = Cκ tan(α + ϕsign(dκ/dt)) tan α (M.4) where sign (dκ/dt) reflects the difference of link between P, κ at loading and unloading, ϕ – angle of friction and C is a rigidity factor which at Ds = Db is given by expression C = πEA/(n − 1.5)D (M.5) When a, b are tangents of angles between straight lines and axis κ in Fig M.2 at loading and unloading respectively, friction coefficient and rigidity factor can be found as f − 2(a + b)/(a − b) sin 2α + = 0, C2 − 0.5C(a + b)(cot2 α − 1) − ab cot2 α = (M.6) In the solutions we must take before square root sign minus for the first relation and plus – for the second one Values a, b are also given in Table M.1 and we can see that f-values are near to their meanings in reference books and mean value of C = 277 MN/m coincides with that for tested reams at n = 7, E = 2x105 MN/m , A = 3.15x10−4 m2 , D = 0.13 m according to (M.5) The analysis of the test results on quasi-static loading gives an opportunity to suppose two work types of the device at cycling loading – with mutual slip of rings and without it (as a piece of tube) with rigidity coefficients much more than follows from (M.4) as C tan(α ± ϕ) tan α 220 M Appendix P, kN 10 A 40 20 0.4 11 B 12 0.8 −3 1.2 k,m x 10 Fig M.3 Function P(κ) at cycling loading Table M.2 Data of cycling loading No of regime I II III loop in figure 1-4 5-8 9-12 0-A – B-0 frequency in Hc 21 21 21 statics P1 P2 kN 25 35 55 any 5 ψ test 2.86 0.84 0.87 0-32 theory 2.70 1.18 0.72 0-32 The latter regime takes place if at the beginning of loading the minimum value of force in a cycle is more than force P* responding to the beginning of rings slip at unloading For P* we have from (M.4) P∗ = P1 (tan(σ + ϕ))/ tan(σ − ϕ) (M.7) where P1 is maximum load in a cycle In Fig M.3 as an example the test results for rings ream with α = 0.262 in oil at frequency 21 Hc in regimes I III of Table M.2 together with broken straight lines of loading OA and unloading OB at quasi-static deformation are given It is seen some difference of form and position of hysteresis loops at quasi-static and quick loadings It is biggest for the regime with maximum of amplitude that may be explained by the divergence from the Coulomb’s law with growth of a velocity of mutual displacement of sliding surfaces M Appendix 221 Now we find absorption coefficient ψ as the ratio of hysteresis loop area to potential energy 0.5Cκa of the system where κa is amplitude of the deformation Theoretical values of ψ are computed according to following from (M.4) relation ψ = 8((P1 + P2 )(1 + f ) tan α − (P1 − P2 )f(1 + tan2 α))f sin α/((P1 − P2 ) ×(1 + f ) tan α − (P1 + P2 )f(1 + tan2 α))(1 − f tan2 α)) ×(1 − f tan2 α)) cos2 α where P2 is minimum force in a cycle In conclusion we must notice that the difference between quick and slow deformations of the ream is not big Using the data on quasi-static loading we can predict the character of the reams work including their dissipative losses at cycling loading The received results may be used for explanation of behavior of other elastic-plastic systems including some soils N Appendix Investigation of Gas Penetration in Polymers and Rubbers Since polymers and rubbers are often used as shells and membranes (see Sects 6.24, 6.2.5.) it is necessary to study gas penetration through them Herewith such investigation is made (it was fulfilled by the author in Leningrad State University) The process of gas penetration in materials is often considered as successive phenomena of absorption and diffusion that go into direction of decreasing concentration gradient Unsteady state of diffusion gas stream in a material is described by differential equation similar to (1.12) ∂C/∂t = D∂ C/∂x2 (N.1) Here D is diffusion coefficient, C – concentration of gas, x – coordinate in the direction of its movement Steady state of diffusion stream subdues to relation q = −D∂C/∂x (N.2) where q – quantity of gas penetrating through unit surface area of a material in unit time It is supposed that the gas absorption in polymers is described by the Henri’s law C = βp (N.3) Here β – absorption coefficient, p – gas pressure With consideration of (N.3) expression (N.2) may be represented in form Q = Dβ∆pAt/h (N.4) 224 N Appendix where Q is a volume of gas that is diffused through a sample of material with thickness h and area A during time t at difference of pressures before and after the plate ∆p If we define the penetration factor r as gas volume which goes through unit area af cross-section in second at unit gradient of pressure that is r = Qh/At∆p (N.5) constants r, D, β will be linked by relation r = Dβ (N.6) Hence the gas penetration may be fully characterized by three constants – of diffusion D, absorption β and penetration r They can be found experimentally by different methods One of them allows to determine all the constants simultaneously from one test as follows For the plate at initial and border conditions C = Co at t = 0, ≤ x ≤ h, C = C1 at x = 0, C = C2 at x = h for all t we have the following solution of (N.1) ∞ C(x, t) = C1 + (C2 − C1 )x/h + (2/π) ((C2 cos n ˜n − C1 )/n) sin(nπx/h) n=1 ∞ × exp(−Dn2 π2 t/h2 ) + (4Co /π) (2m + 1)−1 sin((2m + 1)πx/h) m=0 × exp(−D(2m + 1)2 π2 t/d2 ) (N.7) The gas stream through unit area of the plate cross-section into volume Vo is determined by expression Vo ∂C/∂t = −D∂C/∂x x=h (N.8) Differentiating (N.7) by x and putting the result at x = h into (N.8) we receive after integration from to t following relation for gas volume Q which went through the plate at time t ∞ Q = βA((p1 − p2 )Dt/h + (h/π2 ) (p1 − p2 cos πn)(1 − exp(−Dn2 π2 t/h2 )) n=1 ∞ (cos(2m + 1)π)(1 − exp(−D(2m + 1)2 π2 t/h2 )) +4po m=0 (N.9) N Appendix 225 where po is the pressure at initial gas concentration Co in the plate, p1 and p2 – pressures before and after the plate At big t (N.9) becomes Q = βA((p1 − p2 )Dt/h − (p1 + 2p2 )h/6 + po h/2) (N.10) If Q = we can find the “time of lagging” (at unsteady penetration) t1 as t1 = h2 ((p1 + 2p2 − 3po )/6D(p1 − p2 ) (N.11) At small p2 and po comparatively to p1 we receive D = h2 /6t1 (N.12) So, when we have experimental dependence Q(t) for unsteady and steady state of diffusion gas stream we can find with the help of relations (N.5), (N.12) and (N.6) constants r, D and β For the experiment a plant was created which gives an opportunity to test material plates with thickness 0.1 cm and working area 110 cm2 Duration of a test is determined by the time when the diffusion velocity becomes constant and it is equaled usually to 2t1 The volume of a gas for computation is found according to the Clapeyron’s relation Q = 3.6HΣ∆Qi /T (N.13) Q cm3 30 20 10 50 100 150 200 t1 Fig N.1 Diagrams Q(t) 250 t, h 226 N Appendix Table N.1 Data of tests No material ∆p cm2 number of samples h cm rx1010 cm4 /sN Dx107 cm2 /s βx103 cm2 /N rubber rubber polyvinilchlorid polypropilen polyethilen polycarbonat 0.5 0.5 0.5 5 5 0.22 0.30 0.23 0.22 0.31 0.27 5.76 0.41 4.26 1.36 0.08 0.47 1.64 0.57 1.29 0.52 0.92 0.22 3.54 0.78 3.33 2.65 1.36 1.93 where ∆Qi is the gas volume at considered time, H – atmospheric pressure in cm of a mercury pillar, T – temperature of in ◦ K The diagrams Q(t) for materials are given in Fig N.1 and computed according to relations (N.5), (N.12), (N.6) constants - in Table N.1 The analysis of the results (here only a part of them is represented) shows that the penetration of the materials falls with a growth of pressure difference It is also seen that the fall of r takes place due to a decrease of D and β The change of penetration should be taken into account at a construction of rubber and polymer structures Elsoufiev Sergiy Alexeevich, Vierhausstr.27,44807 Bochum Germany Elsoufiev@gmx.de, elsoufie@et.rub.de Curriculum Vitae ELSOUFIEV Serguey (ELSUF’EV Sergiy) was born in November 1935 in Omsk (Siberia, Russia) From February 1997 lives with his wife and a stepson constantly in Germany Graduated with a distinction from the Hydrotechnical Faculty of Leningrad Polytechnic Institute in 1959 (the diploma project was presented in English) Worked as an engineer at building sites In 1964 received his master’s degree on the experimental bases of the Plasticity Theory In 1985 defended his doctorate thesis in which he grounded a new scientific direction on strength computation of structures made of plastic metals, polymers, rubbers, soils etc On the theme published 135 works, 29 of them in English and 15 his works are re-edited in USA The book “Geomechanics” is edited in Odessa State University From 1964 to 1970 worked as a scientist in the Polymer Strength Laboratory at Leningrad State University, then as assistant, associate professor and professor in Leningrad Institutes (Polytechnic and Water transport) From 1987 till 1997 - as a professor, the Head of the Strength of Materials Department in Odessa State Academy of Civil Engineering and Architecture Delivered lectures on the Strength of Materials and Structural Mechanics (3 years to foreign students in English), Theory of Elasticity, Plasticity and Creep, Applied Mechanics which combines very closely the Mechanism Theory, Machines Parts and Strength of Materials, and additionally in Leningrad and Odessa Universities, as well as in special institute - the Fracture Mechanics a unique and original discipline All the courses above are published (partly in English) Supervises the post-graduate students For many years is a member of master and doctorate Councils on different branches of Mechanics Is a member of GAMM (Society for Applied Mathematics and Mechanics) 228 Curriculum Vitae Speaks Russian, English, German, Polish, understands French, Ukranian Has driving license Healthy, is busy with sport (jogging, cycling, swimming) Neither drinks, nor smokes The contact address: Elsoufiev S., Vierhausstr.27, 44807 Bochum BRD (Germany) Tel 0049(0)2349586213 E-Mail: Elsoufiev@gmx.de and elsoufie@et.rub.de Index absorption coefficient, 221, 223 absolute rigidity, 37, 182 active pressure, 22 acumulations of solid particles, 1, angle of divergence, 105 angle of internal friction, 17 angle of repose, 22 angle of rotation, 29 anisotropic materials, 36 anisotropic factors, 36 antiplane problem, 40 apex, 58, 140 asymptotic approach, 66 axisymmetric problem, 40, 44, 72, 149 base of structure, beam on elastic foundation, 2, 61 bearing capacity, 94, 123, 139, 144, 154 beck form, 67 biggest main strain, 36 boundary conditions, 40 Boussinsq’s problem, 74 brittle fractures, 11 byharmonic equation, 43 capillaries in earth, 12 choice of structures places, centroidal field, 98 centroids, 18, 98 circular hole, 63, 99 circular pile, 123, 154 circular punch, 75 coefficient of friction, 22, 219 coefficient of stability, 110 coherence pressure, 104 coherent soil, 104 cohesiveness, combination of equilibrium equations, 43, 45 compatibility equations, 33, 42, 45 complex variables, 47, 62 compression of cylinder, 119, 179 compulsory flow, 56 condensation of soil, conformal transformations, 48 consolidation degree, 13 constant displacement in transversal shear, 70 constitutive laws, 34 core Coulomb’s law, 23, 217 crack in tension, 65, 71, 102 crack mechanics, 16 crack propagation, 17, 51, 66, 126 creep phenomenon, 24 critical stress, 17 cupillaries in soil, 12 cycling loadings, 9, 217 cycloids, 93 cylindrical rigidity, 15 cylindrical surface of slip, 114 damage parameter, 4, 26 Darcy’s law, 11 dangerous surface, 115 decomposition of complex variables, 49 deep shear, 110 230 Index deflection of beam, 28, 61 delta of amplitude, 145 derivative of complex variables, 47 deviators, 34, 35, 36 differential static equations, 33 disintegration of rocks, disperse soils, 10 ductile materials, 25 eccentric compression, 26 effective stress, 35 effective strain, 35, 120 elastic bed, 61 elastic foundation, 62 energy balance, 70 engineering geology, equations of Caushy-Riemann, 47 equivalent layer, 109 equivalent stress, 36 equivalent strain, 36 expenditure of water, 13 factor of stability, 110 failure of slope, 88 feeble matrix, 36 filtration consolidation, 11 filtration of water, 12 filtration pressure, 112 finite strains, 33 first boundary problem, 63 ferst critical load, 60 flow within cone, 121 foundations, 1, 55, 129 Fracture mechanics, 52 fracture toughness, 51 free energy, 127 friction coefficient, 22, 219 frozen earth, full elliptic integral, 77 gamma-function, 127 gas penetration, 223 generalization of Boussinesq’s solution, 149 generalization of Flamant’s problem, 128 generalized Hooke’s law, 34 geological schemes and cross-sections, gradient of function, 85 Griffith’s relation, 67 Gvozdev’s theorems, 3, 98 hardening, 17 heap of sand, 85 Hoff’s method, 161 homogeneity of soil, Hooke’s law, hydraulic gradient, 12 hydrodynamic pressure, 112 hydrotechnical structures, 79 hypotheses of geometrical character, 40 ideal plasticity, 35, 43 imaginary part of complex variable, 48 inclined crack in tension, 71 incompressible body, 36 independence of integral J, 149 influence of stress state type, 36 integral static equations, 54 intensity factor, 51 invariants of tensor, 32 isotropy plane, 37 kinematic hypotheses, 144 laminated soils, 36 landslide stability, 116 Laplace’s operator, 45 L’Hopital’s rule, 201 leaned slopes, 116 limit of brittle strength load-bearing capacity, 123, 139, 154 loading plane, 37 lock, 144 Lode’s parameters, 36 longitudinal shear, 40 longtime strength, 24 loss of stability, 14 main axes, 96 main stresses, 32 maximum shearing stress, 33 maximum strain, 36 media mechanics, 31 mean stress, 34 membranes, 176 method of corner points, 195 mixed boundary problem, 40 Index mixed fracture, 166 mode of fracture, 21 modulus of elasticity, modulus of shear, 34 Mohr’s circle, 22 moment of inertia, 15 moment of torsion, 81 natural pressure, Neper’s number, 13, 216 orthotropic directions, 36 parallel plates, 57 passive pressure, 22 penetration of gas, 223 penetration of wedge, 94, 139 perfect plastic body, 86 permeability, phases of soil state, place of prop structure, 1, 116, 117 plane deformation for Hooke’s Law, 42 plane shear, 110 plane stress state, 42 plastic hinges, 18 plasticity theory methods, play-wood, 37 Poisson’s ratio, polar coordinates, 41 porosity, potential function, 146 power law, 25 Prandtl’s solution, 94 primary creep, 24 principle of Saint-Venant, propagate, 17 propagation of crack, 146 rate of soil’s stabilisation, 10 rays of fracture, 19 real part of complex variable, 47 reliability degree, repose angle, 22 residial stress, resultant of soil pressure, 21, 108, 110 retaining wall, 2, 22, 55 rheological equations, 34 Rice approach, 86 231 rigolith, rock, role of the bed, 62 second ultimate load, 108 sediments, 1, semi-infinite solid, 74 semi-plane under vertical load, 64 semi-space, 74 shearing displacements, sine of amplitude, 145 skeleton’s volume, skid, 144 sliding support, 144 slip arc, 111 slip element, 96 slip lines, 7, 90, 93 slopes stability, 2, 111 slope under one-sided load, 131 sluice, 144 solid core, solution in series, 64 specific coherence, 111 specific gravity, 12 specific weight, spherical coordinates, 45 squizing water out, 10 stability of footings, stable crack, 66 statically determinate problem, 41, 43 statically indeterminate problem, 33 stick condition, 54 stratified soil, 36 strength condition for crack, 67 stress intensity coefficient, 67 stresses under rectangles, 75 stretching energy, 51 subtangent, 182 sucker with holes, 11 summation “layer by layer”, 78 superposition method, 74 surface of stretching, 17 tension under hydrostatic pressure, 161 tensor of strains, 33 tensor of stresses, 31 Terzaghi-Gersewanov model, 11 torus of revolution, 174 transversally isotropic plate, 36 232 Index transversal shear, 69, 148 true strain, 25 wedge pressed by inclined plates, 52 wedge under concentrated force, 58 wedge under one-sided load, 53 ultimate plastic state theory, 17 ultimate plasticity, 182 unstable deformation, yielding demand, 117 yielding point in shear, 35 volume of heap, 85 Zhoukovski’s relation, 48 Foundations of Engineering Mechanics Series Editors: Vladimir I Babitsky, Loughborough University, UK Jens Wittenburg, Karlsruhe University, Germany Further volumes of this series can be found on our homepage: springer.com (Continued from page ii) Alfutov, N.A Stability of Elastic Structures, 2000 ISBN 3-540-65700-2 Astashev, V.K., Babitsky, V.I., Kolovsky, M.Z., Birkett, N Dynamics and Control of Machines, 2000 ISBN 3-540-63722-2 Kolovsky, M.Z., Evgrafov, A.N., Semenov Y.A, Slousch, A.V Advanced Theory of Mechanisms and Machines, 2000 ISBN 3-540-67168-4 ... the settling in time can be represented by Fig 1.8 Curve corresponds to sands in which settling happens fast since the resistance to squeezing a water out is small Case takes place in disperse soils... soil strength and stability including their pressure on retaining walls Investigation of structures strength problems in different phases of deformation For the solution of these tasks two main... Properties of Soils and Other Materials 23 σy x σx τ σ Ψ y Fig 1.21 Stress state near retaining wall t t 2Ψ s sr O s = sq c j s1 = sx sm s3 = sy Fig 1.22 Mohr? ?s circle or σy /σx = (1 + sin ϕ) /(1 − sin