Table Of ContentFoundations of Engineering Mechanics
SeriesEditors: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. Kolpakov, A.G.
Mechanics of Structural Elements, 2007 Stressed Composite Structures, 2004
ISBN 978-3-540-44718-4 ISBN 3-540-40790-1
Shorr, B.F.
Elsoufiev, S.A.
The Wave Finite Element Method, 2004
Strength Analysis in Geomechanics, 2007
ISBN 3-540-41638-2
ISBN 978-3-540-37052-9
Svetlitsky, V.A.
Awrejcewicz, J., Krysko, V.A., E.ngineering Vibration Analysis - Worked
Krysko, A.V. Problems 1, 2004
Thermo-Dynamics of Plates and Shells, ISBN 3-540-20658-2
2007
Babitsky, V.I., Shipilov, A.
ISBN 978-3-540-37261-8 Resonant Robotic Systems, 2003
ISBN 3-540-00334-7
Wittbrodt, E., Adamiec-Wojcik, I.,
Wojciech, S. Le xuan Anh,
Dynamics of Flexible Multibody Systems, 2006 Dynamics of Mechanical Systems with
ISBN 3-540-32351-1 Coulomb Friction, 2003
ISBN 3-540-00654-0
Aleynikov, S.M.
Spatial Contact Problems in Geotechnics, Nagaev, R.F.
2006 Dynamics of Synchronising Systems,
ISBN 3-540-25138-3 2003
ISBN 3-540-44195-6
Skubov, D.Y., Khodzhaev, K.S.
Non-Linear Electromechanics, 2006 Neimark, J.I.
ISBN 3-540-25139-1 Mathematical Models in Natural Science and
Feodosiev, V.I., Engineering, 2003
Advanced Stress and Stability Analysis ISBN 3-540-43680-4
Worked Examples, 2005 Perelmuter, A.V., Slivker, V.I.
ISBN 3-540-23935-9 Numerical Structural Analysis, 2003
Lurie, A.I. ISBN 3-540-00628-1
Theory of Elasticity, 2005 Lurie, A.I.,
ISBN 3-540-24556-1 Analytical Mechanics, 2002
Sosnovskiy, L.A., ISBN 3-540-42982-4
TRIBO-FATIGUE · Wear-Fatigue Damage Manevitch, L.I., Andrianov, I.V.,
and its Prediction, 2005 Oshmyan, V.G.
ISBN 3-540-23153-6 Mechanics of Periodically Heterogeneous
Andrianov, I.V., Awrejcewicz, J., Structures, 2002
Manevitch, L.I. (Eds.) ISBN 3-540-41630-7
Asymptotical Mechanics of Thin-Walled Babitsky, V.I., Krupenin, V.L.
Structures, 2004 Vibration of Strongly Nonlinear Discontinuous
ISBN 3-540-40876-2 Systems, 2001
Ginevsky, A.S., Vlasov, Y.V., ISBN 3-540-41447-9
Karavosov, R.K. Landa, P.S.
Acoustic Control of Turbulent Jets, 2004 Regular and Chaotic Oscillations, 2001
ISBN 3-540-20143-2, ISBN 3-540-41001-5
(Continued after index)
Serguey A. Elsoufiev
Strength Analysis in
Geomechanics
With 158 Figures and 11 Tables
SeriesEditors:
V.I.Babitsky J.Wittenburg
DepartmentofMechanicalEngineering Institutfu¨rTechnischeMechanik
LoughboroughUniversity Universit¨at Karlsruhe(TH)
LoughboroughLE113TU,Leicestershire Kaiserstraße12
UnitedKingdom 76128Karlsruhe
Germany
Author:
Serguey A. Elsoufiev
Vierhausstr. 27
44807 Bochum, Germany
ISSNprintedition:1612-1384
ISBN-10:3-540-37052-8
ISBN-13:978-3-540-37052-9 SpringerBerlinHeidelbergNewYork
LibraryofCongressControlNumber:2006931488
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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
canbeappliedtoearthstructures.Thiswayistakeninthisbook.Itrepresents
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 con-
stitutive 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 load-
ings 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 hard-
eninglawthelastonesgivetheHooke’sandthePrandtl’sdiagrams.Together
withthebasicrelationsofcontinuummechanicstheyareusedtodescribethe
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
criterionofinfinitestrainsrateatthebeginningofunstabledeformation.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 anti-
plane (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 neces-
sary) 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
twogeometrical)andcontainssomenewideas,forexample,theconsideration
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 plastic-
ity 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.
Allthisdoesnotdenytheclassicalmethodsthatarealsousedinthebook
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, plastic-
ity, 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 com-
plex problem at the intersection of physics of solids, mechanics of media and
material sciences. Real materials contain many defects of different form and
dimensionsbeginningfromsubmicroscopiconestobigporesandmaincracks.
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 me-
chanics 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).
Herewewillnotusetheusualapproachtostrengthcomputationwhenthe
distributionofstressesarefoundbymethodsofcontinuummechanicsandthen
hypotheses of strength are applied to the most dangerous points. Instead, we
considerthefractureasaprocessdevelopingintimeaccordingtoconstitutive
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 geo-
mechanics are listed as well as the main tasks of the Soil Mechanics. Its short
historyisgiven.Thedescriptionofsoilpropertiesbymethodsofmechanicsis
represented.Theideaisintroducedthatthefailureofastructureisaprocess,
the study of which can describe its final stage. Among the examples of this
are:stabilityofaringunderexternalpressureandofabarundercompression
andtorsion(theyarerepresentedasparticularcasesofthecommonapproach
tothestabilityofbars);elementarytheoryofcrackpropagation;theultimate
state of structures made of ideal plastic materials; the simplest theory of
retainingwall;along-timestrengthaccordingtoacriterionofinfiniteelonga-
tionsandthatoftheirrate.Thepropertiesofintroducednon-linearequations
forunsteadycreepwithdamageaswellasamethodofdeterminationofcreep
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.1 to real objects we must intro-
duce main equations for a complex stress state that is made in Chap.2. The
stresses and stress tensor are introduced. They are linked by three equilib-
rium equations and hence the problem is statically indeterminate. To solve
the task, displacements and strains are introduced. The latters are linked by
compatibilityequations.Theconsiderationofrheologicallawsbeginswiththe
Hooke’sequationsandtheirgeneralizationfornon-linearsteadyandunsteady
creep is given. The last option includes a damage parameter. Then basic ex-
pressions 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
besolvedevenforsimplebodiessomegeometricalhypothesesareintroduced.
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
planeproblem wehaveeight equations for eight unknowns. Averyusefuland
unknown from the literature combination of static equations is received. The
basic expressions for axi-symmetric problem are given. For spherical coordi-
nates a useful combination of equilibrium laws is also derived.
Itisnotpossibletogivealltheelasticsolutionsofgeo-technicalproblems.
They are widely represented in the literature. But some of them are included
in Chap.3 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
abrittleruptureofabodywithacrackareconsidered.Theplanedeformation
ofawedgeunderanone-sidedload,concentratedforceinitsapexandpressed
byinclinedplatesisalsostudied.Theuseofcomplexvariablesisdemonstrated
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 rela-
tionsfortransversalshearareused,andcriticalstressesarefound.Amongthe
axi-symmetricproblemsasphere,cylinderandconeunderinternalandexter-
nalpressuresareinvestigated.ThegeneralizationoftheBoussinesq’sproblem
includes determination of stresses and displacements under loads uniformly
distributedinacircleandrectangle.Someapproximateapproachesforacom-
putation of the settling are also considered. Among them the layer-by-layer
summation and with the help of the so-called equivalent layer. Short infor-
mation on bending of thin plates and their ultimate state is described. As a
conclusion,relationsfordisplacementsandstressescausedbyacircularcrack
in tension are given.
Manymaterialsdemonstrateatloadingayieldingpartofthestress-strain
diagramandtheirultimatestatecanbefoundaccordingtothePrandtl’sand
the Coulomb’s laws which are considered in Chap.4, devoted to the ultimate
stateofelastic-plasticstructures.Theinvestigationsandnaturalobservations
showthatthemethodcanbealsoappliedtobrittlefracture.Thisapproachis
simplerthantheconsequentelasticone,andmanyproblemscanbesolvedon
the basis of static equations and the yielding condition, for example, the tor-
sionproblem,whichisusedforthedeterminationofashearstrengthofmany
materialsincludingsoils.Therigoroussolutionsfortheproblemsofcracksand
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 pen-
etration 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 re-
duced 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-walledstructuresunderinternalandexternalpressuresandcompression
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
asoilandload-bearingcapacityofacircularpileareofahighpracticalvalue.
Manymaterialsdemonstrateanon-linearstress-strainbehaviourfromthe
beginning of a loading, which is accompanied as a rule by creep and damage.
This case is studied in Chap.5 devoted to the ultimate state of structures
X Preface
at small non-linear strains. The rigorous solution for propagation of a crack
and plastic zones near punch edges at anti-plane deformation is given. The
generalizationoftheFlamant’sresultsandtheanalysisofthemarepresented.
Deformation and fracture of a slope under vertical loads are considered in
terms of simple engineering relations. The problem of a wedge pressed by
inclined plates and a flow of a material between them as well as penetration
of a wedge and load-bearing capacity of piles sheet are also discussed. The
problem of the propagation of a crack and plastic zones near punch edges at
tension and compression as well as at transversal shear are also studied. A
load-bearing capacity of sliding supports is investigated. A generalization of
theBoussinesq’sproblemanditspracticalanalysisarefulfilled.Theflowofthe
material within acone,its penetration inamassifandtheload-bearing capa-
city of a circular pile are studied. As a conclusion the fracture of thick-walled
elements (an axi-symmetrically stretched plate with a hole, sphere, cylinder
and cone under internal and external pressures) are investigated. The results
of these solutions can be used to predict failure of the voids of different form
and dimensions in soil.
In the first part of Chap.6, devoted to the ultimate state of structures
at finite strains, the Hoff’s method of infinite elongations at the moment of
fracture is used. A plate and a bar at tension under hydrostatic pressure are
considered. Thick-walled elements (axi-symmetrically stretched plate with a
hole,sphere,cylinderandconeunderinternalandexternalpressure)arestud-
ied in the same way. The reference to other structures is made. The second
part of the chapter is devoted to mixed fracture at unsteady creep. The same
problems from its first part are investigated and the comparison with the
results by the Hoff’s method is made. The ultimate state of shells (a cylinder
and a torus of revolution) under internal pressure as well as different mem-
branesunderhydrostaticloadingisstudied.Thecomparisonwithtestdatais
given. The same is made for a short bar in tension and compression. In con-
clusion the fracture of an anisotropic plate in biaxial tension is investigated.
The results are important not only for similar structures but also for a find-
ing the theoretical ultimate state of a material element (a cube), which are
formulated according to the strength hypotheses. The found independence of
critical maximum strain for brittle materials on the form of a structure and
the stress state type can be formulated as a “law of nature”.
