Advances in strength theories for materials under complex stress state in the 20th Century

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Advances in strength theories for materials under
complex stress state in the 20th Century
Mao-hong Yu
School of Civil Engineering & Mechanics, Xi’an Jiaotong University, Xi’an, 710049, China
mhyu@xjtu.edu.cn; isstad@xjtu.edu.cn
It is 100 years since the well-known Mohr-Coulomb strength theory was established in 1900.
A considerable amount of theoretical and experimental research on strength theory of materi-
als under complex stress state was done in the 20th Century. This review article presents a
survey of the advances in strength theory~yield criteria, failure criterion, etc! of materials~in-
cluding metallic materials, rock, soil, concrete, ice, iron, polymers, energetic material, etc!
under complex stress, discusses the relationship among various criteria, and gives a method of
choosing a reasonable failure criterion for applications in research and engineering. Three se-
ries of strength theories, the unified yield criterion, the unified strength theory, and others are
summarized. This review article contains 1163 references regarding the strength theories. This
review also includes a brief discussion of the computational implementation of the strength
theories and multi-axial fatigue.@DOI: 10.1115/1.1472455#
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1 INTRODUCTION
Strength theory deals with the yield and failure of materi
under a complex stress state. Strength theory is a gen
term. It includes yield criteria and failure criteria, as well
multiaxial fatigue criteria, multiaxial creep conditions, a
material models in computational mechanics and comp
codes. It is an important foundation for research on
strength of materials and structures. Strength theory
widely used in physics, mechanics, material science, e
science, and engineering. It is of great significance in th
retical research and engineering application, and is also
important for the effective utilization of materials. Partic
larly for design purposes, it is important that a reliab
strength prediction be available for various combinations
multiaxial stresses. It is an interdisciplinary field where t
physicist, material scientist, earth scientist, and mechan
and civil engineers interact in a closed loop.
Strength theory is a very unusual and wonderful subj
The objective is very simple, but the problem is very co
plex. It is one of the earliest objectives considered by
onardo da Vinci~1452-1519!, Galileo Galilei ~1564-1642!,
Coulomb~1736-1806!, and Otto Mohr~1835-1918!, but it is
still an open subject. Considerable efforts have been dev
to the formulation of strength theories and to their correlat
with test data, but no single model or criterion has emer
which is fully adequate. Hundreds of models or criteria ha
been proposed. It seems as if an old Chinese said: ‘‘L
hundred flowers bloom and a hundred schools of thou
contend.’’
Timoshenko ~1878-1972! was an outstanding scientis
distinguished engineer, and a great and inspiring profes
Transmitted by Associate Editor F Ziegler
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Timoshenko’s summers of the years from 1903 to 1906 we
spent in Germany where he studied under Foppl, Prandtl,
Klein. After his return from Germany in 1904, he wrote hi
first paper on the subject of ‘‘various strength theories’’
1904@1#. ‘‘Strength theories’’ was also the title of sections i
two of his books@2,3#. Now, ‘‘strength theories’’ is a chapter
of most courses of ‘‘Mechanics of Materials,’’ sometime
referred to as ‘‘Strength of Materials.’’ Moreover, ‘‘yield cri-
teria’’ or ‘‘failure criteria’’ is a chapter of most courses in
Plasticity, Geomechanics, Soil Mechanics, Rock Mechani
and Plasticity of Geomaterials, etc.
This subject, although there are some review articles a
books, is difficult and heavy to survey. Some of the surve
were contributed by Mohr@4#, Westergaard@5#, Schleicher
@6#, Nadai @7,8#, Marin @9#, Gensamer@10#, Meldahl @11#,
Dorn @12#, and Prager@13# in the first half of the 20th cen-
tury. It was also reviewed by Freudental and Geiringer@14#,
Naghdi @15#, Filonenko-Boroditch@16#, Marin @17#, Paul
@18#, Goldenblat and Kopnov@19#, and Taira~creep under
multiaxial stress! @20# in the 1960s. This subject was furthe
reviewed by Tsai and EM Wu~anisotropic material! @21#,
Bell ~experiments! @22#, Krempl @23#, EM Wu ~anisotropic
failure criteria! @24#, Michino and Findley~metals! @25#, Sa-
lencon~soil! @26#, Genievet al ~concrete! @27# in the 1970s;
and by Yu @28,29#, Zyczkowcki @30#, WF Chen~concrete!
@31#, Ward~polymer! @32#, WF Chen and Baladi~soils! @33#,
Hamza~ice! @34#, Shaw@35#, Hosford@36#, Rowlands@37#,
Ikegami~low temperature! @38#, and Desai@39# in the 1980s.
Strength theories were subsequently reviewed by Klaus
@40#, WF Chen@41,42#, Du @43#, Jiang~concrete! @44#, An-
dreev~rock! @45#, Shen~rock, soil! @46#, Kerr ~ice! @47#, Gao
and Brown~Multiaxial fatigue! @48#, You and SB Lee~Mul-
© 2002 American Society of Mechanical Engineers9
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170 Yu: Advances in strength theories Appl Mech Rev vol 55, no 3, May 2002
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tiaxial fatigue! @49#, Sheorey~rock! @50#, WF Chen ~con-
crete! @51#, Yu, Zhao, and Guan~rock, concrete! @52#, Shen
and Yu@53#, and Munx and Fett~ceramics! @54# in the 1990s.
Two books regarding strength theory are published@55,56#.
The advances in strength theories of materials under c
plex stress state in the 20th century will be summarized
the framework of continuum and engineering application
2 STRENGTH THEORIES
BEFORE THE 20th CENTURY
2.1 Early work
Leonardo da Vinci~1452-1519! and Galileo Galilei~1564-
1642! were among the most outstanding scientists of t
period. They may be the earliest researchers of the stre
of materials and structures. Da Vinci and Galileo did tens
tests of wire and stone, as well as bending tests. Da V
believed that the strength of an iron wire would depend s
nificantly on its length. Galilei believed that fracture occu
when a critical stress was reached@2#.
Coulomb ~1736-1806! may be the first researcher in th
maximum shear stress strength theory. No other scientis
the eighteenth century contributed as much as Coulomb
to the science of mechanics of elastic bodies@2#. Coulomb’s
Memoir Essay@57# was read by him to the Academy o
France on March 10 and April 2, 1773, and published
Paris in 1776. The paper began with a discussion of exp
ments which Coulomb made for the purpose of establish
the strength of some kind of sandstone; then, Coulomb g
a theoretical discussion of the bending of beams, the c
pression of a prism, and the stability of retaining walls a
arches.
Coulomb assumed that fracture is due to sliding alon
certain plane, and that it occurs when the component of fo
along this plane becomes larger than the cohesive resist
in shear along the same plane. To bring the theory into be
agreement with experimental results, Coulomb propo
that, not only should cohesive resistance along the sh
plane be considered, but also friction caused by the nor
force acting on the same plane. This was the first descrip
of the famous Mohr-Coulomb Strength Theory.
2.2 Strength theories in the 19th century
There were three strength theories in the 19th century.
The maximum stress theory was the first theory relating
the strength of materials under complex stress. It consid
the maximum or minimum principal stress as the criter
for strength. This criterion was assumed by such scientist
Lamé ~1795-1870! and Rankine~1820-1872!, and was ex-
tended with the well known textbook of Rankine’s,Manual
of Applied Mechanics@58#, the first edition of which ap-
peared in 1858 at Glasgow University,and was publishe
1861, the 21st edition entitledApplied Mechanicsbeing pub-
lished in 1921. Only one principal stresss1 of the 3D
stressess1 ,s2 ,s3 was taken into account.
The second strength theory was the maximum str
theory. Mariotte~1620-1684! made the first statement on th
maximum elongation criterion or maximum strain criterio
@59#. Sometimes, it was called Saint-Venant’s criterion or
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second Strength Theory in the Russian and Chinese lit
ture, and the maximum normal stress criterion was called
First Strength Theory.
Maximum strain theory was generally accepted, prin
pally under the influence of such authorities as the t
French academicians Poncelet~1788-1867! and Saint-Venant
~1797-1886! @60#. In this theory it is assumed that a materi
begins to fail when the maximum strain equals the yie
point strain in simple tension. This theory does not ag
well with most experiments. It was very popular at one tim
but no one uses it today.
In 1864, Tresca presented two notes dealing with the fl
of metals under great pressure to the French Academy@61#.
He assumed that the maximum shear stress at flow is equ
a specific constant. It is called the Tresca yield criterion no
A maximum shear stress criterion was also proposed
Guest@62#. This theory gives better agreement with expe
ment for some ductile materials and is simple to apply. T
theory takes two principal stressess1 and s3 of the 3D
stressess1 , s2 , s3 into account, and the intermediate prin
cipal stresss2 is not taken into account.
Beltrami suggested that, in determining the critical valu
of combined stresses, the amount of strain energy shoul
adopted as the criterion of failure@63#. This theory does not
agree with experiments and has not been used in plast
and engineering. In 1856, Maxwell suggested that the t
strain energy per unit volume can be resolved into two pa
1! the strain energy of uniform tension or compression and!
the strain energy of distortion. Maxwell made the statem
in his letter to William Thomson as follows: ‘‘I have stron
reasons for believing that when~the strain energy of distor
tion! reaches a certain limit then the element will begin
give way.’’ Further on he stated: ‘‘This is the first time that
have put pen to paper on this subject. I have never seen
investigation of this question. Given the mechanical strain
three directions on an element, when will it give way?’’@2#.
At that time, Maxwell already had the theory of yieldin
which we now call the maximum distortion energy theo
But he never came back again to this question, and his id
became known only after publication of Maxwell’s letter
the 1930s. It took researchers considerable time before
finally developed@2# the theory identical with that of Max-
well.
Strength theory was studied by Foppl@64#, Voigt @65#,
Mohr, Guest, and others at the end of the 19th century. C
parisons of strength theories as applied to various de
problems were given in a paper by Marin@9# and a book by
Nadai @66#. A comprehensive bibliography on streng
theory before the 1930s can be found in the article by From
@67#. A discussion of various strength theories with a co
plete bibliography of the subject was given by Ros and Ei
inger @68#.
3 THREE SERIES OF STRENGTH THEORIES
Mohr used the stress circle method@69# in developing his
theory of strength in 1900@70#. Otto Mohr ~1835-1918! was
a very good professor. When thirty-two years old, he w
already a well-known engineer and was invited by the S
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 171
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tgart Polytechnicum Institute~Stuttgart University! to be-
come the professor of engineering mechanics. His lectu
aroused great interest in his students, some of whom w
themselves outstanding, such as Bach and Foppl.
Foppl stated that all the students agreed that Mohr
their finest teacher@2#. Mohr always tried to bring somethin
fresh and interesting to the students’ attention. The reason
his students’ interest in his lectures stemmed from the
that he not only knew the subject thoroughly, but also h
himself done much in the creation of the science which
presented.
Mohr made a more complete study of the strength of m
terials. He considered failure in broad senses; that is, it
be yielding of the material or fracture. Mohr’s criterion ma
be considered as a generalized version of the Tresca crite
@61#. Both criteria were based on the assumption that
maximum shear stress is the only decisive measure of
pending failure. However, while the Tresca criterion assum
that the critical value of the shear stress is a constant, Mo
failure criterion considered the limiting shear stress in
plane to be a function of the normal stress in the same
tion at an element point.
Mohr considered only the largest stress circle. He calle
the principal circle and suggested that such circles should
constructed when experimenting for each stress conditio
which failure occurs. The strength of materials under a co
plex stress state can be determined by the corresponding
iting principal circle.
At that time, most engineers working in stress analy
followed Saint-Venant and used the maximum strain the
as their criterion of failure. A number of tests were ma
with combined stresses with a view to checking Moh
theory@65,71,72#. All these tests were made with brittle ma
terials and the results obtained were not in agreement
Mohr’s theory. Voigt came to the conclusion that the quest
of strength is too complicated, and that it is impossible
devise a single theory for successful application to all kin
of structural materials@2#.
The idea of Mohr’s failure criterion may be tracked ba
to Coulomb~1773! @67#. This criterion is now referred to a
the Mohr-Coulomb strength theory~failure criterion!. In the
special case of metallic materials with the same strengt
tension and in compression, the Mohr-Coulomb stren
theory is reduced to the maximum-shear stress criterion
Tresca@61#.
When Otto Mohr was teaching at the Stuttgart Polyte
nicum, his lectures caused August Foppl to devote mos
his energy to study of the theory of structures. Like Mo
Foppl’s activity in both research and teaching at the Po
technical Institute of Munich was remarkably successful.
was an outstanding professor and knew how to hold stude
interest, although his classes were very large.
At that time, Foppl followed Saint-Venant’s notion an
used the maximum strain theory in deriving formulas
calculating safe dimensions of structures. But at the sa
time he was interested in the various other strength theo
and to clarify the question of which should be used, he c
ducted some interesting experiments. By using a thi
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walled cylinder of high-grade steel, he succeeded in mak
compressive tests of various materials under great hy
static pressures. He found that an isotropic material co
withstand very high pressure in that condition. He design
and constructed a special device for producing compres
of a cubic specimen in two perpendicular directions a
made a series of tests of this kind with cement specimen
is the earliest high-pressure test.
Haigh @73# and Westgaard@5# introduced the limit surface
in a 3D principal stress space. The advantage of such s
lies in its simplicity and visual presentation. It is called th
Haigh-Wesagaard space or stress space. Photographs o
metric models ofsuch surfaces corresponding to vari
yield criteria before 1944 can be found in the papers
Burzynski @74# and Meldahl@11#. The yield surface in the
stress space can be transformed into the strain space@75–
77#. A comparison of various strength theories applied
machine design before the 1930s was given by Marin@78#.
Prandtl was a great scientist. He was in the Academic
the USA, France, and other countries. Strength theory
one of the subjects studied by him. Prandtl himself drew
his theory of the two types of fracture of solids and devis
his model for slip@2#. At that time, Prandtl’s graduate stu
dents worked mainly on strength of materials until Pran
went to the USA in 1941. Von Karman did experiment
work on the strength of stone under confining lateral press
at Goettingen University, and did much work in aerodyna
ics in the USA; Nadai did important work on strength a
plasticity at the Westinghouse Research Lab; Prager
leading the new research in strength and plasticity at Bro
University, and several well-known professors worked
Brown University with Prager@79,80#; Flügge and Timosh-
enko worked at Stanford University.
A lot of strength theories and expressions were presen
after Mohr. The proposed criteria and material models in
20th century are too many, and it is difficult to classify the
Fortunately, a fundamental postulate concerning the y
surfaces was introduced by Drucker@81,82# and Bishop and
Hill @83# with the convexity of yield surface determined
After that the convexity of yield surface was generalized
the strain space by Il’yushin in 1961. Since then the study
strength theory has been developing on a more reliable
oretical basis.
The convex region and its two bounds are most intere
ing. One method we used for representing these theories
use the principal shear stressest13, t12, t23 and the normal
stresss13, s12, s23 acting on the same planes. Streng
theories may be divided into three kinds~Fig. 1 and Table 1!.
Three principal shear stresses and relating normal stre
are
t i j 5
1
2 ~s i2s j !; s i j 5
1
2 ~s i1s j !; i , j 51,2,3
3.1 Single-shear strength theory„SSS theory…
This series of strength theories considers the maximum s
stresst13 and the influence of the normal stresss13 acting on
the same section. It can be written mathematically as
F~t13,s13!5C (1)
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172 Yu: Advances in strength theories Appl Mech Rev vol 55, no 3, May 2002
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Table 1. Summary of three series of strength theories
SSS series
~Single shear stress series
of strength theory!
OSS series
~Octahedral shear stress series
of strength theory!
TSS series
~Twin shear stress series
of strength theories!
Model of element Hexahedron Isoclinal octahedron Dodecahedron or orthogon
octahedron
Shear stress yield criterion SSS yield criterion OSS yield criterion TSS yield criterion
t135C, Tresca, 1864; Guest, 1900 t85C, von Mises 1913;
Eichinger 1926; Nadai 1937
t131t12(or t23)5C, Hill, 1950;
Yu, 1961; Haythornthwaite, 1961
Shear strain yield criterion SSS~Strain! yield criterion
g135C
OSS~Strain! yield criteriong85C TSS ~Strain! yield criterion.
g131g12(or g23)5C
Failure criterion SSS failure criterion
t131bs135C
OSS failure criteriont81bs85C TSS failure criterion
t131t12(or t23)1bs13
1bs12(or s23)5C
Coulomb, 1773 Burzynski, 1928
Mohr, 1882-1900 Drucker-Prager, 1952 Yu-He-Song, 1985
Slip condition SSS Slip condition OSS Slip condition TSS Slip condition
Schmid, 1924 von Mises, 1926 Yu-He, 1983
Cap model SSS Cap model OSS Cap model TSS Cap model
Roscoe1963; Wei,1964 Baladi, Roscoe, Mroz Yu-Li, 1986
Smooth ridge model SSS ridge model OSS ridge model TSS ridge model
Argyris-Gudehus, 1973; Lade-Duncan, 1975 Yu-Liu, 1988
Matsuoka-Nakai,1974
Multi-parameter criterion Ashtonet al ~1965!
Hobbs~1964!
Murrell ~1965!
Franklin ~1971!
Hoek-Brown~1980!
Pramono-Willam~1989!
Bresler-Pister~1958!;
Willam-Warnke~1974!;
Ottosen~1977!;
Hsieh-Ting-Chen~1979!
Podgorski~1985!;
Desai, de Boer~1988!;
Song-Zhao~1994!;
Ehlers~1995!
TSS Multi-parameter criterion
Yu-Liu, 1988 ~1990!
h
y
x-
sion
)
d
According to the shear stress, it may be referred to as
single-shear strength theory~SSS theory!.
3.1.1 Single-shear yield criterion [61]
The expression is
f 5t135C, or f 5s12s35ss (2)
It is the one-parameter criterion of the SSS~single-shear
strength! theory. This yield criterion is also referred to as t
maximum shear stress criterion or the third strength theor
Russian and in Chinese. It is adopted only for one kind
material with the same yield stress both in tension and
compressions t5sc5ss .
The generalization of the Tresca criterion by adding
hydrostatic stress termsm was given by Sandel@30#, Davi-
ing.
r-
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ub-
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the
e
in
of
in
a
genkov @84#, Drucker @85#, Volkov @86#, and Hara@87,88#
and others. The limit surface is a pyramid with regular he
agonal cross section similar to the Tresca’s. The expres
of the extended Tresca criterion is
f 5t131bsm5C (3)
3.1.2 Single-shear strength theory (Mohr-Coulomb 1900
The expression is
F5t131bs135C, or
F5s12as35s t , a5s t /sc (4)
It is a two-parameter criterion of the SSS~single-shear
strength! theory. It is the famous Mohr-Coulomb theory an
is also the most widely used strength theory in engineer
The failure locus of SSS theory on thep plane ~deviatoric
plane! has the inner hexagonal threefold symmetry~lower
bound! as shown in Fig. 1. It is interesting that Shield@89#
was the first to publish the correct form of the Moh
Coulomb limit locus in the deviatoric plane in 1955@18#. It
is also indicated by Shield that after the paper was co
pleted, he learned that the correct yield surface was obta
previously by Professor Prager and Dr Bishop in an unp
lished work @89#. Before Shield, the limit surface of th
Mohr-Coulomb strength theory was always consistent wit
sixfold symmetry hexagonal pyramid failure surface that
intercepted by a Tresca-type hexagonal cylinder.
Single-shear strength theory~Mohr-Coulomb 1900! forms
the lower ~inner! bound for all the possible convex failur
surfaces coincided with the Drucker postulation on the
Fig. 1 Limiting loci of SSS, OSS, and TSS theories
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 173
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viatoric plane in stress space. No admissible limit surfa
may exceed the Mohr-Coulomb limit surface from below,
shown in Fig. 1.
The disadvantage of the Mohr-Coulomb theory is that
intermediate principal stresss2 is not taken into account
Substantial departures from the prediction of the Mo
Coulomb theory were observed by many researchers,eg,
Shibat and Karuhe@91#, Mogi @92,93#, Ko and Scott@94#,
Green and Bishop@95#, Vaid and Campanella@96#, Lade and
Musante@97#, Michelis @98,99# and others.
3.1.3 Multi-parameter Single-Shear criteria
Multi-parameter Single-Shear criteria are nonlinear Mo
Coulomb criteria~Mogi @92#, Salencon@26#, Hoek-Brown
@100#, et al! used in rock mechanics and rock engineering
Some forms are expressed as follows
F5t131ls13
n 50 Murrell @101# (5)
F5S s12s32c D
n
512
s11s3
2t
Ashton et al @102# (6)
F5~s12s3!1Ams12c50 Hoek-Brown @100# (7)
F5F ~12k! s12sc2 1 s12s3sc G
2
1k2m
s1
sc
5k2c
Pramono-Willam @103,104# (8)
in which kP(0,1) is the normalized strength parameter,c
andm are the cohesive and frictional parameters.
3.1.4 Single-shear cap model
It is used in soil mechanics and engineering. It will be d
cussed in the next section.
3.1.5 Application of the SSS theory
Single-shear yield criterion~Tresca yield criterion! has been
widely used for metallic materials and in mechanical en
neering.
Mohr’s theory ~Single-shear strength theory! attracted
great attention from engineers and physicists. ‘‘The Mo
Coulomb failure criterion is currently the most widely us
in soil mechanics’’~Bishop@105#!. ‘‘The Mohr-Coulomb
theory is currently the most widely used for soil in practic
applications owing to its extreme simplicity’’@41,106#. ‘‘In
soil mechanics, the Coulomb criterion is widely used; and
applied mechanics, Mohr’s criterion has been widely us
for concrete Mohr-Coulomb criterion appears to be m
popular.’’ ‘‘Taking into account its extreme simplicity, th
Mohr-Coulomb criterion with tension cutoffs is in man
cases a fair first approximation and therefore suitable
manual calculation. However, the failure mechanism ass
ated with this model is not verified in general by the te
results, and the influence of the intermediate principal str
is not taken into account,’’ as indicated by Chen@31#.
SSS theory is the earliest and simplest series of stre
theory. A considerable amount of research was done in c
nection with it. However, it is still studied up to the prese
~Shield @89#; Paul @18,107#; Harkness@108#; Pankaj-Moin
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ce
as
he
r-
r-
.
is-
gi-
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d
al
in
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st
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for
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@109,110#; Heyman @111#; Schajer@112#!. Multi-parameter
Single-Shear criteria were used in rock mechanics and r
engineering.
3.2 Octahedral-shear strength theory„OSS Theory…
This series of strength theories considers the octahedral s
stresst8 and the influence of the octahedral normal stresss8
acting upon the same section. It can be written mathem
cally as
F~t8 ,s8!5C or t85 f ~s8! (9)
This is a fruitful series in the strength theory. It contain
3.2.1 Octahedral-shear stress yield criterion
(von Mises yield criterion)
It is a one-parameter criterion of the OSS theory
f 5t85C, or J25C, or tm5C (10)
It is the widely used yield criterion for metallic materia
with the same yield stress both in tension and in compr
sion. It is also referred to as the von Mises criterion@113#, or
octahedral shear stresst8 yield criterion by Ros and Eich-
inger @68# as well as Nadai@7,8#. Sometimes, it was referre
to as theJ2 theory~second invariant of deviatoric stress te
sor!, shear strain energy theory~energy of distortion, Max-
well @2#, Huber @114#, Hencky @115#!, equivalent stress cri-
terion ~effect stress or equivalent stressse!, or mean root
square shear stress theory. It was also referred as the m
square shear stresstm averaged over all planes by No
vozhilov @116#, mean square of principal stress deviations
Paul @18#, tri-shear yield criterion by Shen@46#, and the
fourth strength theory in Russian, Chinese, etc. All the
pressions mentioned above are the same, because of
t85
A15
3
tm5
&
3
se5A23 J25
2
3
At122 1t232 1t132
5
1
3
A~s12s2!21~s22s3!21~s32s1!2 (11)
3.2.2 Octahedral-shear failure criterion (Drucker-Prage
criterion etc)
It is the two-parameter criterion of the OSS~Octahedral-
shear strength! theory as follows
F5t81bs85C. (12)
This criterion is an extension of the von Mises criterio
for pressure-dependent materials, and called the Druc
Prager criterion expressed by Drucker and Prager@117# as a
modification of the von Mises yield criterion by adding
hydrostatic stress termsm ~or s8!. The Drucker-Prager cri-
terion was used widely in soil mechanics. The extended
Mises criterion, however, gives a very poor approximation
the real failure conditions for rock, soil, and concrete. It w
indicated by Humpheson-Naylor@118#, Zienkiewicz-Pande
@119#, and WF Chen@31,41,42# et al.
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3.2.3 Multi-parameter octahedral shear failure criterion
The first effective formulation of such a condition in gene
form was given by Burzynski@74#. The general function of a
three-parameter criterion is expressed as follows
F5At8
21B s8
21Cs82150 or
F5t81bs81as8
25C. (13)
The general equation~13! and its alternations, or particu
lar cases, were later proposed, more or less independentl
many authors. The formulations of more than 30 papers w
similar as indicated by Zyczkowski@30#.
OSS theory contains many smooth~ridge! models and
three, four, and five-parameter failure criteria used in c
crete mechanics. Many empirical formulas, typically fitt
with different functions, were proposed around the 1980s
cater to the various engineering materials. Among those w
the ridge models and many multi-parametric criteria as
lows:
F5t81g~u!S C1 1
)
s8
t8
D 50 William et al @120# (14)
F5
I 1I 2
I 3
5C Matsuoka-Nakai@121# (15)
F5
I 1
3
I 3
5C Lade-Duncan@122# (16)
F5
3
2
t8
21
1
3
As85C Chen-Chen@31# (17)
F5
3
2
t8
22
1
6
s8
21
1
3
As85C Chen-Chen@31# (18)
F5t8t1a1s81a2s8
25C1 ~u560°!
F85t8c1b1s81b2s8
25C2 ~u50°!
Willam-Warnke @120# (19)
F5t81at8
21bs85C Ottosen @123# (20)
F5S I 13I 3 227D S I 1paD
m
5C Lade @124# (21)
F5t81a
2t8
21bt81ds15C Hsieh et al @31# (22)
F5t81a~s81b!
n5C Kotsovos @125# (23)
F5asm
2 1bsm1g1~t8 /g~u!!
250
Zienkiewicz-Pande@119# (24)
whereg(u) is a shape function, and various functions we
proposed as follows:
g~u!5
2k
~11K !2~12K !sin 3u
Argyris-Gudehus@127#
(25)
This function was improved by Lin-Bazant in@128# and
Shi-Yang in@129# as follows:
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ere
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to
ere
ol-
re
g~u!5r c
2k~c11c2 cos 3u!
~c31k!1~c32k!cos 3u
Lin-Bazant (26)
g~u!5
~712k!22~12k!sin 3u
9
Yang-Shi (27)
Elliptic function proposed by Willams and Warnke@120#
is
g~u!5
~12K2!~) cosu2sinu!
~12K2!~21cos 2u2) sin 2u!1~122K !2
1
~2K21!A~21cos 2u2) sin 2u!~12K2!15K224K
~12K2!~21cos 2u2) sin 2u!1~122K !2
(28)
Hyperbolic function proposed by Yu-Liu@131# is
g~u!5
2~12K2!cosu1~2K21!A4~12K2!cos2 u15K224r t
4~12K2!cos2 u1~K22!2
(29)
F5at8
21bt81cs11ds851 Chen @31# (30)
F5t8
21c1P~u!t81c2s85C, Podgorski @132# (31)
where
P5cos@1/3 arccos~cos 3u!2b#
F5~at8!
21m@bt8 P~u,l!1cs8#5C
Menetrey-Willam @133# (32)
F5J31cJ22~12h!c
350, Krenk @134# (33)
Some other failure criteria were proposed as follows:
F5~)/& !t82k~123s8 /s ttt!
a2~3s8 /sccc!
b50
Yu-Liu @44# (34)
wherea andb are the shape functions, 0<a<1 and 0<b
<1.
F5t81a~12l!S s81bs812aD
a
1alS s81bs813aD
b
Qu @44# (35)
wherel5(sin32u)
(0.81 3/21u)
F5
sm
12~h/h0!
n Shen @135# (36)
where
h5
1
&
F S t12s12D
2
1S t13s13D
2
1S t23s23D
2G1/2
F5J22a1bI1J3
1/35C, Yin-Li et al @136# (37)
F5t82aS b2s8c2s8D
d
Guo-Wang @137,138# (38)
wherec5ct(cos 3u/2)
1.51cc(sin 3u/2)
1.5,
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 175
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F5at8
1.51bt8 cosu1as85C,
Zhang-Huang@139# (39)
F5at8
21~b1c cosu!t81ds85C, Jiang @44,140# (40)
F5t8t1a1s81b1s8
25C1, ~u50°!
F85t8c1a2s81b2s8
25C2, ~u560°!
Song-Zhao@141,142# (41)
wheret8(u)5t8 cos
2(3u/2)1t8 sin
2(3u/2)
F5t8
21~As81B!@12~12c!~12cos 3u!#5C,
Genev-Kissyuk@27# (42)
F5t81As8A12B cos 3u5C, Gudehus@127# (43)
Interested readers are referred to reviewing literatu
written by WF Chen@31# and Shen-Yu@52#. Some failure
surfaces with cross section of quadratic curve and reg
triangle were derived from hypo-elasticity by Tokuoka@143#.
Two J3-modified Drucker-Prager criteria were proposed
Lee and Ghosh in@144#. Another modified von Mises crite
rion proposed by Raghava-Cadell for polymers was used
viscoplastic analysis by Hu, Schimit, and Francois in@145#.
Other yield criteria joining all the three invariants were pr
posed by Hashiguchi in@146#, Maitra-Majumdar in@147#,
Haddow-Hrudey in@148# et al.
3.2.4 Octahedral shear cap model
Druckeret al was upon the first to suggest that soil might
modeled as an elasto-plastic work-hardening material~see
Section 6.3 of this article!. They proposed that successiv
yield functions might resemble Drucker-Prager cones w
convex end caps. Based on the same idea of using a ca
part of the yield surface, various types of cap models h
been developed at Cambridge University.They are refe
to as the Cam-clay model and the Modified Cam-clay mod
Discussions of various Cambridge models were summar
by Parry@149#, Palmer@31#, and Wood@150#.
Multi-parameter criterion of SSS theory takes three pr
cipal shear stresses and the hydrostatic stress into acc
They are the curved failure surfaces mediated between
failure surface of the SSS theory~Single-shear strength
theory! and the failure surface of the TSS theory~Twin-shear
strength theory! proposed and developed in China from 19
to 1990 as shown in Fig. 1.
According to Eq.~11!, all the failure criteria of OSS serie
strength theory can be expressed in terms of three princ
stresst13, t12, and t23. So, this series of strength theo
may be also referred to as the three-shear strength th
@46#.
3.2.5 Applications of the OSS theory
The octahedral-shear stress yield criterion~von Mises crite-
rion! has been widely used for metallic materia
Octahedral-shear failure criterion~Drucker-Prager criterion!
and the octahedral-shear cap model were used in soil
chanics and geotechnological engineering, and impleme
: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 05/10/2
res
lar
by
for
o-
e
e
ith
p as
ve
red
el.
zed
in-
unt.
the
1
ipal
y
ory
s.
me-
ted
into nonlinear FEM codes. Various multi-parameter octa
dral shear failure criteria were used for concrete.
It is very interesting, as indicted by Zyczkowski@30#, that
various expressions of SSS equations and OSS equatio
general form ~13! or its particular cases were late
repeated—more or less independently—by many auth
@30#. If we utilize Eq. ~11!, many expressions are the sam
3.3 Twin-shear strength theory„TSS theory…
It is clear that there are three principal shear stressest13,
t12, andt23 in a stressed element. However,t13, t12, and
t23 are not independent, there are only two independent c
ponents in three principal shear stresses because the m
mum principal shear stresst13 equals the sum of the othe
two, ie, t135t121t23. So, the idea oftwin-shearwas intro-
duced and developed by Yu and Yuet al in 1961-1990@152–
160#.
This series of strength theories considers the maxim
principal shear stresst13 and intermediate principal shea
stresst12 ~or t23!, and the influence of the normal stress
s13 ands12 ~or s23! acting on the same section, respective
It is referred to as the twin-shear strength theory~TSS
theory!, and can be written mathematically as
F@t13,t12;s13,s12#5C,
when f ~t12,s12!> f ~t23,s23! (44)
F8@t13,t23;s13,s23#5C,
when f ~t12,s12!< f ~t23,s23! (448)
The first criterion in the TSS category was originally po
tulated in 1961, and has since developed into a new serie
strength theory. Among the main stream are the twin-sh
yield criterion ~one-parameter! @151,152#, the generalized
twin-shear strength theory~two-parameter! @155#, the twin-
shear ridge model@131#, the twin-shear multiple-slip condi
tion for crystals@157#, the multi-parameter twin-shear crite
rion @158,159#, and the twin-shear cap model@160#. The
systematical theories and their applications were summar
in a new book@156#.
3.3.1 Twin-shear yield criterion (Yu 1961)
This is a one-parameter criterion of the twin-shear stren
theory. The idea and expressions of twin-shear yield criter
are as follows@152,153#:
F5t131t125s12
1
2~s21s3!5ss ,
when s2<
s11s3
2
(45)
F5t131t235
1
2 ~s11s2!2s35ss ,
when s2>
s11s3
2
(458)
Twin-shear yield criterion is a special case of the Tw
shear strength theory@155#.
The generalization of the Twin-shear yield criterion b
adding a hydrostatic stress term was given by Yu in 19
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@154#. The limit surface is a pyramid with regular hexagon
cross section similar to the yield locus of the twin-shear yi
criterion. The expression is
F5t131t121bsm5C F5t131t231bsm5C (46)
3.3.2 Twin-shear strength theory (Yu-He 1985)
A new and very simple strength theory for geomaterials w
proposed by Yuet al @155#. This is a two-parameter criterio
of the twin-shear strength theory. The idea and mathema
modeling are as follows:
F5t131t121b~s131s12!5C,
when t121bs12>t231bs23 (47)
F5t131t231b~s131s23!5C,
when t121bs12<t231bs23 (478)
It can be expressed in terms of three principal stresse
follows:
F5s12
a
2
~s21s3!5s t , when s2<
s11as3
11a
(48)
F85
1
2
~s11s2!2as35s t , when s2>
s11as3
11a
(488)
The SD effect and the effect of hydrostatic stress are ta
into account in the twin-shear strength theory. The limit s
face of the twin-shear strength theory is a hexagonal pyra
whose cross sections~in thep plane! are symmetric but non
regular hexagons. It is the upper~external! bound of all the
convex limit loci as shown in Fig. 1. No admissible conv
limit surface may exceed the twin-shear limit surface.
3.3.3 Verifications of the twin-shear strength theory
The verifications of the TSS theory were given by Li-X
et al @161# and Ming-Sen-Gu@162# by testing the Laxiwa
granite of a large hydraulic power station in China and ro
like materials under true tri-axial stresses. The twin-sh
strength theory predicted these experimental results. T
conclusion was also given by comparing the experime
data of Launay-Gachon’s tests@163# for concrete and othe
experimental data@164–166#. The experimental results o
Winstone@167# agreed well with the twin-shear yield crite
rion.
3.3.4 Twin-shear multi-parameter criteria
The twin-shear strength theory can also be extended
various multi-parameter criteria for more complex con
tions. The expressions are as follows@158,159#:
F5t131t121b1~s131s12!1A1sm1B1sm
2 5C (49)
F5t131t231b2~s131s23!1A2sm1B2sm
2 5C (498)
whereb,A,B,C are the material parameters.
3.3.5 Twin-shear cap model
Twin-shear cap model was proposed by Yu and Li@160#. It
has been implemented into an elasto-plastic FE program
oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org
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u
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.
3.3.6 Applications of the TSS theory
TSS theory~twin-shear series of strength theory! has pushed
the strength theory study to a new advance by forming
upper~outer! bound for all the possible convex failure su
faces coincided with the Drucker postulation on the dev
toric plane in stress space as shown in Fig. 1.
The twin-shear yield criteria had been used successf
in the plane strain slip line field by Yu-Liu@168#, plane stress
characteristic field by Yan-Bu@169–171#, metal forming by
Zhaoet al @172–176#, and limiting analysis of structures b
Li @177#, Huang-Zeng@178#, JJ Chen@179#, Wang@180#, etc.
TSS theory was implemented into some finite element p
grams by Anet al @181#, by Quint Co@182–184#, etc, and
applied in elasto-plastic analysis and elasto-visco-pla
analysis of structures by Li-Ishii-Nakazato@185#, Luo-Li
@186#, Li and Ishii @187#, and Liuet al @188#.
TSS theory was also applied in the plasticity of geoma
rials by Zhang@189# and Zhu @190#, in wellbore analysis
@191#, in gun barrel design@192#, punching of concrete slab
@193#, and soil liquefaction@194#, etc. Some applications o
the twin-shear strength theory can be seen in@195–199#. The
results of the applications indicate that it could raise the be
ing capacities of engineering structures more than the Mo
Coulomb’s, ie, improves on the conservative Moh
Coulomb’s. So, there is a considerable economical benefi
using the new approach if the strength behavior of mater
is adaptive to the twin shear theory@156,185#.
Three series of strength theories, ie, SSS series, OSS
ries, and TSS series are summarized briefly in Table 1.
4 YIELD CRITERIA FOR METALLIC MATERIALS
Many articles and books, as indicated in Table 1, have
viewed this object. We will supplement the maximum dev
toric stress criterion, or twin shear criterion, and the unifi
yield criterion in Section 5. Some experimental data are su
marized as shownin Table 2.
The experimental results are taken from Guest@62#,
Scoble@200,201#, Smith @202#, Lode @203#, Taylor-Quinney
@204#, Ivey @205#, Paul@18#, Bell @22#, Michno-Findley@25#,
Pisarenko-Lebedev@206#, and others@207–219#. The dis-
crepancies among different experiments and different m
rials are large. Up to now, none of the above yield crite
agreed with the experiments for different materials. After t
comparison of the shear yield strength and tensile yi
strength for thirty materials, Kishkin and Ratner@220# di-
vided the metals into four kinds according to the ratio of t
shear yield strength with tensile yield strengthts /ss as fol-
lows:
a) ts /ss,0.40 ~0.31–0.41, eight materials!. It is a non-
convex result, no yield criterion agreed
b) ts /ss>0.50~0.48–0.53, five materials!. It is agreed with
the single-shear yield criterion~Tresca yield criterion!
c) ts /ss>0.58~0.54–0.62, nine materials!. It is agreed with
the tri-shear yield criterion~von Mises yield criterion!
d) ts /ss>0.68 ~0.67–0.71, eight materials!. It is agreed
with the twin-shear yield criterion.
Four sets of experimental results of Winstone@167# show
that the initial yield surfaces indicated a ratio of shear yie
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Mises
Mises
Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 177
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Table 2. Comparison of three yield criteria with experimental results
Researchers Materials Specimen shearÕtension ty /sy Suitable Criterion
Guest, 1900 steel, 0.474, 0.727
Guest, 1900 steel, brass, etc. tubes 0.474
Hancok, 1906,1908 mild steel solid rods,tube 0.50 to 0.82
Scoble, 1906 mild steel solid rods 0.45 to 0.57 Tresca
Smith, 1909 mild steel solid rods 0.55-0.56 between Tresca and von M
Turner, 1909-1911 steels 0.55 to 0.65 Tresca
Mason, 1909 mild steel tubes 0.5 0.64 Tresca
Scoble,1910 steel 0.376, 0.432, 0.452 no one agreed
Becker, 1916 mild steel tubes no one agreed
Seeley and Putnam steels bars & tubes 0.6 . von Mises
Seigle and Cretin, 1925 mild steel solid bars 0.45 to 0.49 Tresca
Lode, 1926 iron, copper etc. tubes von Mises
Ros and Eichinger, 1926 mild steel tubes von Mises
Taylor and Quinney, 1931 aluminum, steel tubes von Mises
copper, mild steel .von Mises
Morrison, 1940 mild steel tubes Tresca, von Mises
Davis, 1945-1948 copper, tubes von Mises
Osgood, 1947 aluminum alloy tubes von Mises
Cunningham, 1947 magnesium alloy tubes von Mises
Bishop-Hill, 1951 polycrystals tubes 0.54 von Mises
Fikri, Johnson, 1955 mild steel tubes . von Mises
Marin and Hu, 1956 mild steel tubes von Mises
Naghdi, Essenberg, and Koff, 1958 aluminum alloy tubes . von Mises
Hu and Bratt, 1958 aluminum alloy tubes von Mises
Ivey, 1961 aluminum alloy tubes Twin shear
Bertsch-Findley, 1962 aluminum alloy tubes von Mises
Mair-Pugh, 1964 copper tubes Twin shear
Mair-Pugh, 1964 copper tubes von Mises
Chernyaket al, 1965 mild steel tubes von Mises
Miastkowski, 1965 brass von Mises
Rogan, 1969 steel tubes Tresca
Mittal, 1969, 1971 aluminum tubes 0.57 von Mises
Dawson, 1970 polycrystals 0.64 near Twin shear
Phillips et al, 1968- 1972 aluminum tubes at elevated temperature between Tresca and von
Deneshiet al, 1976 aluminum, copper tubes, low temperature 0.6 . von Mises
Pisarenkoet al, 1984 copper, Cr-steel tubes, low temperature von Mises
Winstone, 1984 nickel alloy tubes at elevated temperature 0.7 Twin shear
Ellyin, 1989 titanium tubes 0.66-0.7 Twin shear
Wu-Yeh, 1991 Aluminum tubes 0.58 von Mises
stainless steel 0.66-0.7 Twin shear
Ishikawa, 1997 stainless steel tubes 0.6-0.63 . von Mises
Granlund,Olsson, 1998 structural steel flat cruciform specimens between Tresca and von
l
es
e
ld
in
.
nt.
d
-
e
stress to tensile yield stress of 0.7. He said: ‘‘The ratio of
torsion shear yield stress to tensile yield stress is>0.7, sur-
prisingly high when compared with the values of 0.58 and
0.5 expected from the von Mises and Tresca yield criteria,
respectively. Clearly neither of these criteria can accurately
model the bi-axial yield behavior of Mar-M002 castings.’’
@167#
The Tresca and von Mises yield criteria excepted, Hay-
thornthwaite proposed a new yield criterion@90#. This new
yield criterion is referred to as the maximum reduced stress
~deviatoric stress! Smax yield criterion
f 5Smax51/3~2s12s22s3!52/3sS (50)
The similar idea of maximum deviatoric stress criterion
may be traced back to the deviatoric strain~shape change! by
Schmidt in 1932@30# and Ishlinsky in 1940@222#, and the
linear approximation of the von Mises criterion by Hill in
1950 @18,223#, then first proposed independently from the
idea of maximum deviatoric stress by Haythornthwaite in
1961 @90#. The expression of Hill was
f 5~2s12s22s3!5msS (51)
Comments to this criterion were made by Paul@18# and
by Zyczkowski@30#.
Another new idea was proposed by Yu@152#. It assumed
that yielding begins when the sum of the two larger principa
shear stresses reaches a magnitudeC. It is clear that only
two principal shear stresses in three principal shear stress
t13, t12, andt23 are independent variables, because of th
maximum principal shear stresst13 equals the sum of the
other two. So, it is referred to as the twin-shear stress yie
criterion that is given in Eqs.~44! and ~448!.
The twin-shear yield criterion can also be introduced from
the generalized twin-shear strength theory proposed by Yu
1985@155# ~whena51 in Eq.~46!!. This yield surface is the
upper~outer! bound of all the convex yield surfaces.
All the yield criteria, including the Tresca, von Mises, and
twin-shear yield criterion are single-parameter criterion
Many researchers followed Bridgman@224–228# and as-
sumed that materials are hydrostatic pressure independe
Numerous experiments carried out by Bridgman~1882-
1961! at Harvard proved that the yielding of metals is unaf-
fected by hydrostatic pressure. His experiments include
thirty metals. Most non-metallic materials, however, are hy
drostatic pressure dependent@229–233#.
The yield criteria had been used successfully in the plan
strain slip line field by Hencky@234#, Geiringer @235#,
Prandtl@236–238#, Prager@239,240#, Johnson@241#, Yu et al
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@168#, and others. The uses of yield criterion in plane str
characteristic field and axisymmetric characteristic field a
metal forming can be seen in Kachanov@242#, Yan-Bu@169–
171#, Hill @243#, Haar-von Karman@244#, Sokolovski@245#,
and Thomsen-Yang-Kobayash,@247#, etc. The applications
in damage and yield of ductile media with void nucleati
can be found in Gurson@248,249#, Tvergaard@250–252#,
Gologanu-Leblond-Perrin-Devaux@253# et al. Limiting
analysis and elasto-plastic analysis of structures by using
yield criteria can be seen in Drucker@254,255#, Hodge@256#,
and Saveet al @257#. The implementations of various yiel
criteria to variety finite element programs were summariz
by Brebbia in@258#. However, how to choose a reasonab
yield criterion is an important problem.
It is still a problem to find a unified yield criterion that ca
be applicable to more than one kind of material and estab
the relationship among various yield criteria.
5 UNIFIED YIELD CRITERIA
5.1 Curved general yield criteria
5.1.1 Curved general yield criterion
between SS and OS yield criteria
A curved general yield criterion lying between SS~Single-
shear, Tresca! and OS~Octahedral-shear, von Mises! criteria
was proposed by Hershey in 1954@259#, Davis in 1961
@260,261#, Paul in 1968@18#, Hosford in 1972@262#, Barlat-
Lian in 1989 @263#, and explained by Owen & Peric@264#
et al as follows:
f 15~S12S2!
2k1~S22S3!
2k1~S32S1!
2k52ss
2k
or f 15us22s3um1us32s1um1us12s2um52ss
2k
(52)
This expression is a generalization of Bailey’s flow ru
@260# for combined stress creep by Davis@261# as a yield
surface lies insidethe von Mises yield criterion and outs
the Tresca yield criterion. This kind of yield criteria is som
times called the Bailey-Davis yield criterion.
5.1.2 Curved general yield criteria between OS and
yield criteria
The curvilinear general yield criteria lying between the O
yield criterion ~octahedral shear yield criterion! and the TS
yield criterion ~twin-shear yield criterion! were proposed by
Tan in 1990@265# and Karafillis & Boyca in 1993@266#. The
expression is
f 5S1
2k1S2
2k1S3
2k5
22k12
32k
ss
2k (53)
5.1.3 Curved general yield criteria between SS and
yield criteria
Tan @265# and Karafillis & Boyca@266# obtain a genera
yield criterion lying between the lower bound~SS yield cri-
terion! and the upper bound~TS yield criterion! yield crite-
rion as follows:
f5~12c! f 11c
32k
22k2111
f 2 , cP@0,1# (54)
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S
TS
5.1.4 Drucker criterion
Edelman and Drucker suggested the following criterion@82#
J2
32CdJ3
25F (55)
Dodd-Narusec@267# extended this equation in the followin
expression
~J2
3!m2Cd~J3
2!m5Fm (56)
It is a series of curved yield criteria~m51 or m52! lying
outside the SS~Tresca! yield criterion.
5.1.5 Hosford criterion
Hosford @262# proposed a criterion as follows
@ 12 ~s12s2!
m1 12 ~s12s3!
m#1/m5ss (57)
A series of yield criteria can be given whenm51 to m
5`.
5.1.6 Simplification of anisotropic yield criterion
Hill proposed a new yield criterion as follows@268#
f us22s3um1gus32s1um1hus12s2um1Cu2s12s2
2s3um1u2s22s12s3um1u2s32s12s2um5ss
m
(58)
where m>1, the six parametersf , g, h, a, b, and c are
constants characterizing the anisotropy. For the isotro
case, f 5g5h, a5b5c, it is three-parameter criterion
Dodd and Naruse@267# take f 5g5h51, from which it fol-
lows that
f 5us22s3um1us32s1um1us12s2um1Cu2s12s2
2s3um1u2s22s12s3um1u2s32s12s2umss
m
(588)
A series of curved yield criteria between SSS criterion a
TSS criterion can be given whenm51 to m5`. Similar
yield criteria can also be introduced from the anisotro
yield criteria of Hosford @262# and Barlat et al
@263,269,270,735#.
All the generalized yield criteria mentioned above a
smooth, convex, and curvilinear yield criteria lying betwe
single-shear and twin-shear yield criteria. They are the n
linear unified yield criteria. However, they are not conv
nient to use in the analytic solution of elasto-plastic pro
lems.
5.2 Linear unified yield criterion
5.2.1 Unified yield criterion
A new linear unified yield criterion was introduced from th
unified strength theory by Yu-He@271–274#. The mathemati-
cal modeling is
f 5t131bt125C, when t12>t23 (59)
f 85t131bt235C, when t12<t23 (598)
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 179
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whereb is a coefficient of the effect of the other princip
shear stresses on the yield of materials. The unified y
criterion can be expressed in terms of three principal stre
as follows
f 5s12
1
11b
~bs21s3!5ss , when s2<
1
2
~s21s3!
(60)
f 85
1
11b
~s11bs2!2s35ss , when s2>
1
2
~s21s3!
(608)
It is a linear unified yield criterion. It contains two fam
lies of yield criterion: one is the convex unified yield crit
rion lying between the single-shear and twin-shear yield
teria ~when 0<b<1!. Another is the non-convex yield
criterion lying outside the twin-shear yield criterion~when
b.1! or lying inside the single-shear yield criterion~when
b,0!. So, it can be predicted to most results listed in Ta
2.
This unified yield criterion encompasses the single-sh
~Tresca! yield criterion ~whenb50!, twin-shear yield crite-
rion ~when b51!, and the octahedral-shear yield criterio
~von Mises! as special cases or linear approximationb
51/2). A lot of new linear yield criteria can be also intro
duced@272#. It can be adopted for all the metallic materia
with the same yield strength both in tension and compr
sion. The linear unified yield criterion is also a special ca
of the unified strength theory proposed by Yu in 19
@271,272#, which will be described in Section 7 of this a
ticle.
The varieties of the yield loci of this unified yield crite
rion at thep plane and plane stress are shown in Fig. 2 a
Fig. 3.
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5.2.2 Extension of the unified yield criterion
The relationship between shear yield stressts and tensile
yield stressss of materials can be introduced from the un
fied yield criterion as follows
ts5
11b
21b
ss or at5
ts
ss
5
11b
21b
(61)
It is shown that:~1! The shear strength of ductile mater
als is lower than tensile strength;~2! Yield surfaces are con
vex when 0<b<1 or 1/2<at<2/3; ~3! Yield surfaces are
non-convex whenb,0 andb.1; or the ratio of shear yield
stress to tensile yield stressat,1/2 andat.2/3.
5.2.3 Non-convex yield criterion
Non-convex yield criterion was merely investigated befo
The unified yield criterion can be used to describe the res
of ts /ss,0.5 (b,0) andts /ss.2/3, (b.1). It is a non-
convex result. Recently, Wang and Dixon@275# proposed an
empiric failure criterion in (s2t) combined stress state.
can be fitted in with those experimental results in (s2t)
combined stress state of Guest@62#, and Scoble@200,201#
with ts /ss50.376, 0.432, 0.451, and 0.474. It is easy
match these results by using the unified yield criterion w
b520.4, b520.24, b520.18, andb520.01, respec-
tively.
5.2.4 Applications of the unified yield criterion
The linear unified yield criterion is convenient to use in t
analytical solution of elasto-plastic and other problems. T
unified solutions of simple-supported circular plate we
given by Ma-He@276# and Ma-Yuet al @277,278#. Ma, Yu,
and Iwasakiet al also gave the unified elasto-plastic solutio
of rotating disc and cylinder by using the unified yield crit
rion @279#. The unified solutions of limiting loads of rectan
gular plate and oblique plates were obtained by Zaoet al
@280# and Li, Yu, and Xiao@281#, respectively.
ss
Fig. 2 Varieties of the unified yield criterion inp plane@271,272#
Fig. 3 Varieties of the unified yield criterion in plane stre
@271,272#
013 Terms of Use: http://asme.org/terms
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180 Yu: Advances in strength theories Appl Mech Rev vol 55, no 3, May 2002
Downl
The further studies of limit speeds of variable thickne
discs using the unified yield criterion were given by M
Hao, and Miyamoto in 2001@282#. The plastic limit analyses
of clamped and simple-supported circular plates with resp
to the unified yield criterion were obtained by Guowe
Iwasaki-Miyamotoet al @283#, and Ma, Haoet al @284,285#.
The dynamic plastic behavior of circular plates using
unified yield criterion was studied by Ma, Iwasakiet al
@286#. Qiang and Xuet al @287# gave the unified solutions o
crack tip plastic zone under small scale yielding and the li
loads of rectangular plate, etc, respectively, by using the
fied yield criterion. A series of results can be introduced fro
these studies.
Some research results concerning the yield criterion w
given in @288–318#.
All the yield criteria including the Tresca criterion, vo
Mises criterion, twin-shear criterion, and the unified yie
criterion can be adopted only for those materials with sa
yield stress in tension and in compression. They canno
applied to rock, soil, concrete, ice, iron, ceramics, and th
metallic materials which have the SD effect~strength differ-
ence at tension and compression!. The SD effects of high
strength steels, aluminum alloys, and polymers were
served in 1970s, eg, Chait@319#, Rauch and Leslie@320#,
Drucker @321#, Richmond and Spitzig@322#, and Casey and
Sullivan et al @323#. The SD effect is related to the effect o
the hydrostatic stress.The hydrostatic pressure produce
fects increasing the shearing capacity of the materials.
fects of hydrostatic pressure on mechanical behavior and
formation of materials were summarized recently
Lewandowski and Lowhaphandu@324#. The effects of hy-
drostatic stress and the SD effect of metals, rock, polym
etc were summarized recently by Yu in 1998@156#.
The generalized failure criteria considering the SD eff
and the influence of hydrostatic stress have to be used.
limit loci of the generalized failure criteria considering th
SD effect and the influence of hydrostatic stress are show
Fig. 1. The Mohr-Coulomb strength theory@70# and the Yu’s
twin-shear strength theory@155# are two bounds of all the
convex criteria.
oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org
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6 FAILURE CRITERIA FOR
SPECIFIC MATERIALS
The development of strength theories is closely associ
with that of the experimental technology for testing materi
in complex stress states. A considerable account of tria
stress tests were done in the 20th century. There are
kinds of triaxial tests.
In most laboratories, for triaxial tests, cylindrical rock a
soil specimens are loaded with an axial stresssz5s1 ~or
sz5s3!, and a lateral pressures25s3 ~or s25s1!; both
can be varied independently, but alwayss25s3 ~or s2
5s1!. The first research seems to be due to Foppl@64#, von
Karman@71#, and Boker@72#. Von Karman and Boker were
supervised under Prandtl. Today such tests are done i
rock mechanics and soil mechanics laboratories. This kin
test, unfortunately, is usually called the triaxial test, althou
it involves only very special combinations of triaxial stres
It is better to refer to this test as the confined compress
test, since it is a compression test with a confining late
pressure@18#. Sometimes it is called the untrue triaxial te
or false triaxial test. All the combinations of comple
stresses in a confined compression test lie on a special p
in stress space. So, most triaxial tests are only a plane s
test.
In 1914, Boker retested the type of marble used by v
Karman in a confined pressure test in which the lateral p
sure was the major principal stress. The corresponding
hr’s envelope did not agree with von Karman’s~in von Kar-
man’s tests, the axial pressure exceeded the lateral press!.
This means that the Mohr-Coulomb criterion could not fit t
data adequately in the range of low hydrostatic pressure
though the more general hexagonal pyramid criterion w
not ruled out@18,72#. It is evident that the confined compre
sion test is not capable of proving that the intermediate st
is of no influence on the failure criterion.
Another is seldom a true tri-axial test, in which all thre
principal stresses can be varied independently. Several w
ers have designed specialized equipment for conducting
triaxial test, eg, Shibata-Karube@91#, Mogi @92,93,325–328#,
Launay-Gachon@163#, Desai et al @333#, Hunsche@336#,
Fig. 4 Effect of the intermediate principal stress@355# Fig. 5 Effect of the intermediate principal stress@98,99#
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 181
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Ming et al @162#, Xu-Geng@349#, Michelis @98,99#, Li, Xu,
and Geng@350#, Mier @348#, and Wawersik, Carsonet al
@345#.
A great amount of effort was dedicated to the develo
ment of true-triaxial testing facilities, and the facilities we
then used to test engineering materials. Some represent
efforts were seen on rocks at Tokyo University and others
soil at Cambridge University, Karlsruhe University, Kyo
University, and others, and on concrete in Europe and
United States.
Mogi’s persistent effort revealed that rock strength var
with the intermediate principal stresss2 , which was quite
different from what had been depicted in the conventio
Mohr-Coulomb theory. The study was further extend
@346,353,354# to a understanding that thes2 effect had two
zones: the rock strength kept on increasing, whens2 built up
its magnitude froms3 , until reaching a maximum value
beyond that, the rock strength decreased with the further
crease ofs2 . Xu and Geng also pointed out that varyings2 ,
only, while keeping the other principal stressess1 and s3
unchanged, could lead to rock failure, and this fact co
also be attributable to inducing earthquakes@347#. Michelis
indicated that the effect of intermediate principal stress is
essential behavior of materials@98,99#. Figures 4 and 5 are
taken from Mazanti-Sowers@355# and Michelis@98,99#. The
effect of the intermediate principal stress on rock is evide
On a modified high pressure true tri-axial test facility,
and his colleagues@161,350# tested some granite and show
that thes2 effect is significant. This result is consistent wi
the twin-shear strength theory. Ming and his coworkers@162#
and Lu @358,359# also reached the same conclusion.
Most true tri-axial tests are three compressive stres
Triaxial tension-compression tests for multiaxial stres
were done by Ming, Shen, and Gu@162#, and by Calloch and
Marquis@344#. A true triaxial cell for testing cylindrical rock
specimens was developed by Smartet al @341–343#. This
true triaxial cell can be suited for testing cylindrical roc
specimens.
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ergy
for
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Recently, at the University of Wisconsin, a new true t
axial testing system was designed, calibrated, and succ
fully tested by Haimson and Chang@356#. It is suitable for
testing strong rocks, which emulates Mogi’s original desi
@93# with significant simplifications. Its main feature is ver
high loading capacity in all three orthogonal directions, e
abling the testing to failure of hard crystalline rocks su
jected to large minimum and intermediate principal stres
@346#.
A mathematical proof regarding the twin-shear theory a
the single-shear theory was given by citing the mathemat
concept of convex sets@360,361#. It is shown that the twin
shear strength theory is the exterior~upper! bound and the
single-shear theory is the interior~lower! bound of all the
convex limiting loci on thep plane as shown in Fig. 1.
The true tri-axial tests on concrete bear many similarit
with that on rocks, both in testing facilities and test resu
Many such tests have been reported by researchers in Fra
Japan, Germany, the former Soviet Union, the United Sta
and China.
Through numerous true tri-axial tests on both rock a
concrete, the existence of thes2 effect has now been wel
recognized as characteristic of these materials@90–99, 137–
142, 325–328, 346–373#. Figure 6 shows the effects of th
intermediate principal stress of concrete given by Launa
Gachon in 1972@163#. The effects of the intermediate prin
cipal stress of metals, rock, concrete, etc were summar
by Yu in 1998@156#.
In the United States, an enhancement factor was in
duced in the ACI Standard@362# guiding designs of pre-
stressed concrete pressure vessels and safety shell
nuclear power station as shown in Fig. 7.
This standard and many experimental results allow hig
permissible strength to be used in concrete and in rock un
triaxial compression stress states, and hence lead to hi
economical effectiveness in construction use. More imp
tantly, the impact of the concept is expected to be enorm
to the design of ordinary engineering structures.
The wider application of the enhancement-factor conc
on a global scale is, on one hand, to bring tremendous en
saving and pollution curbing; it calls, on the other hand,
Fig. 6 Effects of the intermediate principal stress of concr
@163#
te
Fig. 7 Enhanceds2 effect in concrete strength@362#
013 Terms of Use: http://asme.org/terms
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182 Yu: Advances in strength theories Appl Mech Rev vol 55, no 3, May 2002
Downl
a theoretical support on which the concept could be ba
The engineering practice in general has a desire to ha
new strength theory, which should be more rational and m
consistent to the experimental data than what can be don
the Mohr-Coulomb single-shear strength theory. Some f
ure criteria~Tresca, von Mises, Mohr-Coulomb, and max
mum tensile stress theory! were reviewed by Shaw@35#.
6.1 Failure criteria for rock
Up to now, more than 20 strength~yield or failure! criteria
for rocks have been developed, but only a few of the crite
are widely used in rock engineering. Failure criteria of roc
were summarized by Jaeger and Cook@363#, Lade@122,364#,
Andreev@45#, and Sheorey@50#. Various researches and a
plications can be found in@363–438#.
The Mohr-Coulomb theory is the most widely applie
one. Some other nonlinear Mohr-Coulomb criteria similar
the Hoek-Brown criterion were summarized in the literatu
of Andrew @45# and Sheorey@50#. The Ashton criterion was
extended by JM Hill and YH Wu@365#. All the Mohr-
Coulomb, Hoek-Brown, and most kinds of empirical ro
failure criteria ~Eqs. ~4!–~7!! only take thes1 and s3 into
account. They may be referred to as the single-shear stre
theory (t135(s12s3)/2). The effects of the intermediat
principal stresss2 were not taken into account in these c
teria ~see Eqs.~3!–~7!!. The general form of these failur
criteria may be expressed as follows:
F5 f 1~s12s3!1 f 2~s11s3!1 f 3~s1!5C (62)
Some single-shear type failure criteria for rock are given
section three~Eqs.~4!–~7!!, the other two failure criteria for
rock are
s12s35sc1as3
b Hobbs @366# (63)
s12s35a~s11s3!
b Franklin @367# (64)
Mogi @92,93,325–328# proposed a combined failure criterio
of octahedral shear stressest8 ands13 for rock as follows:
F5t81A~s11s3!
n, F5t81 f ~s11as21s3! or
F5t131bs131Asm5C, F5s12as31Asm5C (65)
The von Mises-Schleicher-Drucker-Prager criterion w
modified and applied to rock by Aubertin, L Li, Simon, an
Khalfi @368#.
The strength tests for various rocks under the action
complex stresses were conducted by Foppl@64#, Voigt @65#,
von Karman@71#, Boker @72#, Griggs @370#, Jaeger@371#,
Mogi @91,92,325–328#, Michelis @98,99# et al Many experi-
mental investigations were devoted to the studies on the
fect of the intermediate principal stress, eg, Murrell@372#,
Handin et al @375#, Mogi @325–328,385#, Amadei et al
@398,406#, Kim-Lade @401#, Michelis @98,99,409#, Desai
@403–405#, Yin et al @136#, Gao-Tao@357#, Wang, YM Li,
and Yu@166#, Lu @164,165,358,359#, and others. Some octa
hedral shear type criteria~OSS theory! for rocks were pro-
posed which included the failure criterion for natural po
crystalline rock salt by Hunsche@336,408,410#.
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According to Wang and Kemeny@416#, s2 has a strong
effect ons1 at failure even ifs3 equals zero. Their polyaxia
laboratory tests on hollow cylinders suggested a new emp
cal failure criterion in which the intermediate principal stre
was taken into account. Effect of intermediate princip
stress on strength of anisotropic rock mass was investig
by Singh-Goel-Mehrotraet al @422#. The Mohr-Coulomb
theory was modified by replacings3 by the average ofs2
ands3 . A multiaxial stress criterion for short- and long-ter
strength of isotropic rock media was proposed by Aubertin
Li, Simon et al @368,369#.
Vernik and Zoback@427# found that use of the Mohr-
Coulomb criterion in relating borehole breakout dimensio
to the prevailingin situ stress conditions in crystalline rock
did not provide realistic results. They suggested the use
more general criterion that accounts for the effect on stren
of the intermediate principal stress. Recently, Ewy repor
that the Mohr-Coulomb criterion is significantly too conse
vative because it neglects the perceived strengthening e
of the intermediate principal stress@428#.
The twin-shear strength theory was verified by the exp
mental results of XC Li, Xu, and Liuet al @161,350#, Gao-
Tao@357#, and Ming, Sen, and Gu@162#. The comparisons of
the twin-shear strength theory with the experimental res
presented in literature were given by Lu@164,165,358,359#.
The applications of the twin-shear strength theory to ro
were given by Li, Xu, and Liuet al @350#, An, Yu, and X Wu
@181#, and Luo and ZD Li@186# et al This strength theory
has been applied to the stability analyses of the undergro
cave of a large hydraulic power station at the Yellow River
China@161,350,429,430#. It was also used in the research
the stability of the high rock slopes in the permanent sh
lock at three Gorges on the Yangtze River by Yangtze S
ence Research Institute in 1997@432–433#, and the analysis
of ultimate inner pressure of rings@438#. The twin-shear
strength theory was introduced by Sun@436# and Yang@437#
to rock mechanics.
The strength criteria of rock joints were described a
reviewed by Jaeger@439#, Goodman@441#, Zienkiewicz-
Valliappan-King @442#, Barton @439–446#, Ghaboussi-
Wilson-Isenberg@447#, Singh@448,449#, Ge @450#, Shiryaev
et al @451#, Stimpson@452#, Heuze-Barbour@455#, Desai-
Zaman@456#, Lei-Swoboda-Du@459#, and recently by Zhao
@461,463#, WS Chen, Feng, Ge, and Schweiger@464# et al. A
series of conferences on Mechanics of Joint and Fau
Rock ~MJFR! were held@462#.
The systematical researches on rock rheology were g
by Crestescu@407# and Sun@435#.
The threshold associated with the onset of microcrack
was seen as a yield criterion for an elasto- plastic model o
a damage initiation criterion in a state variable model.
multiaxial damage criterion for low porosity rocks was pr
posed by Aubertin and Simon@418#.
A monogragh on Advanced Triaxial Testing of Soil an
Rock was published by the American Society for Testing a
Materials~Donagle, Chaney and Silver, eds@339#!.
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Appl Mech Rev vol 55, no 3, May 2002 Yu: Advances in strength theories 183
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6.2 Failure criteria for concrete
Failure criteria for concrete including high strength concre
light concrete, steel fibre concrete, etc were studied by m
researchers@465–536#. Various criteria for concrete wer
proposed by Bresler and Pister@465,466#, Geniev @27#,
Mills-Zimmerman@471#, Buyukozturk-Liu-Nilson-Slate, and
Tasuji @473#, HC Wu @476#, Willam-Warnke@120#, Ottosen
@485#, Cedolin-Crutzen-Dei Poli@480#, Hsieh, Ting, and WF
Chen@492#, Dafalias@478#, Yang-Dafalias-Herrmann@496#,
Schreyer-Babcock@499,507#, de Boeret al @505#, Faruque
and Chang @508#, Jiang @44,140#, Song-Zhao-Peng
@141,142,516,518#, Voyiadjis and Abu-Lebdeh@514,517#,
Labbane, Saha, and Ting@515#, Menetrey-Willam@133#, JK
Li @523# et al
In general, these criteria are the OSS theory~Octahedral-
Stress Strength theory! as described above~Eqs.~12!–~44!!.
WF Chenet al @31,41#, Zhang@189#, and Jiang@44# made a
general survey of these criteria. A microplane model for
clic triaxial behavior of concrete was proposed by Baz
and Ozbolt@512,513#. Recently, a key paper@51# entitled
‘‘Concrete plasticity: Past, present, and future’’ was p
sented in theProceedings of ISSTAD ’98~International Sym-
posium on Strength Theory: Application, Development a
Prospects for the 21st Century!. The yield criteria of concrete
used in concrete plasticity were summarized by WF Ch
@51#. Great contributions in the research of fracture and f
ure of concrete are due to Bazant@79,129,482,486,503# and
others. A failure criterion for high strength concrete was p
posed by QP Li and Ansari@526,527# at ISSTAD ’98.
Smooth limit surfaces for metals, concrete, and geotechn
materials was proposedby Schreyer@499,507#. A new book
entitled,Concrete Strength Theory and its Applications, was
published recently@504#.
Considerable experimental data regarding the strengt
concrete subjected to multi-axial stresses were given, eg
chartet al @229#, Balmer@230#, Bresler and Pister@465,466#,
Kupfer et al @470#, Launay and Gachon@163#, Kotsovos and
Newman @479#, Tasuji et al @484#, Ottosen@485#, Gerstle
@489–491#, Michelis @98,99#, Song and Zhao
@141,142,516,518#, Wang-Guo @137,138#, Traina-Mansor
@510#, et al.
Lu gave some applications of the twin-shear stren
theory for concrete under true triaxial compressive st
@164,165,358,359#. The solution of the axisymmetrica
punching problem of concrete slab by using the twin-sh
strength theory was obtained by Yan@193#. The application
of the unified strength theory~see next section! to concrete
rectangular plate was given by Zao@533#.
The strength theory of concrete was also applied to
~reinforced concrete! and the nonlinear FEM analysis of R
structures by Nilsson@537#, Villiappan-Doolan @538#,
Zienkiewicz-Owen-Phillips @539#, Argyris-Faust-Szimmat
et al @540#, Buyukozturk @541#, Bathe-Ramaswang@542#,
Chen @31#, Bangash@543#, Jiang@44#, et al The twin-shear
strength theory and the unified strength theory were use
finite element analysis of reinforced concrete beams
plate by Guo-Liang@545#, Wang@547#, and others.
: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 05/10/2
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Damage model for concrete@506,517#, multi-axial fatigue
@504#, bounding surface model@496#, blast and hard impac
damaged concrete@529#, etc were proposed.
6.3 Failure criteria for soils
The behavior of soil under the complex stress state are m
complex. Many studies were devoted to these problems s
the 1960s@549–627#. In classical soil mechanics, soil prob
lems have generally been solved on the basis of an id
elastic soil, where the deformation and stability propert
are defined by a single value of strength and deforma
modules. Sometimes, the Tresca criterion and the von M
criterion were used. More sophisticated solutions of the be
ing capacity problem involving contained plasticity approa
reality more closely by use of the elasto-plastic idealizati
The Mohr-Coulomb failure criterion was the most wide
used in soil mechanics. However, the failure mechanism
sociated with this model is not verified in general by the t
results, and the influence of the intermediate principal str
is not taken into account. An extended von Mises criter
was proposed by Drucker and Prager in 1952@117# and now
is refereed as the Drucker-Prager criterion.
Although it is widely used, the Mohr-Coulomb mode
does not yield agreement with experimental data for m
materials. Furthermore, the great disadvantage of the M
Coulomb model at present is the lack of indication of beh
ior in the direction of the intermediate principal stress, and
gives far too much deformation. Previous researches
Habib @549#, Haythornthwaite~remolded soil! @551,552#,
Broms and Casbarian@555#, Shibata and Karube~clay! @91#,
Bishop and Green@83,91,95,105,556#, Ko and Scott@94#,
Sutherland and Mesdary@563#, Lade and Duncan@122,564#,
Green@561,562#, Gudehus@127,128,588#, Matsuoka, Nakai
et al @121,577,601#, Lade and Musante~remolded clay! @97#,
Vermeer @569,589#, Dafalias-Herrmann~boundary surface!
@581#, Tang ~sand! @571#, Fang @591#, de Boer @505,600#,
Xing-Liu-Zheng ~loess! @603#, Yumlu and Ozbay@417#,
Wang-Ma-Zhou~dynamic characteristics of soil in comple
stress state! @614#, and others have indicated appreciable
fluences of the intermediate principal stress on the beha
involved in the stress-strain relations, pore pressure,
strength characteristics of most materials. It is obvious t
the third stress~the intermediate principal stress! influences
all three principal strains and the volumetric strain.
After many studies, Green@562# came to the following
conclusion in the Roscoe Memorial Symposium held
Cambridge University in 1971: ‘‘Mohr-Coulomb failure cri
terion will tend to underestimate the strength by up to 5°
cohesive angle for the dense sand as the value of the in
mediate principal stress increases. This would be a sig
cant error in many analyses of engineering problems but
resents an extreme case in as much as medium dense
loose sand, and probably most clays show a small increa
Bishop @105,556# also indicated that the failure surfaces
extended Tresca and extended von Mises criteria are cle
impossible for a cohesionless material.
At the same Symposium, Harkness@108# indicated that:
‘‘The great disadvantage of the Mohr-Coulomb criterion
013 Terms of Use: http://asme.org/terms
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184 Yu: Advances in strength theories Appl Mech Rev vol 55, no 3, May 2002
Downl
present is the lack of indication of behavior in the directi
of the intermediate principal stress. Further developmen
Mohr-Coulomb in this direction would be most interesting
Some international symposia were held, the purposes
which were to allow a comparison to be made of vario
mathematical models of mechanical behavior of soils@339#.
The introduction of a spherical end cap to the Druck
Prager criterion was made by Druckeret al @550# to
control the plastic volumetric change or dilation of so
under complex stress state. Since then, a specific Cam
model was suggested by Roscoeet al @553#. The Cam Clay
model and critical state soil mechanics had been develo
by the research group at Cambridge Univers
@150,151,553,559,628,629#. The cap model had been furthe
modified and refined by DiMaggioet al @631,632#, Farque-
Chang@637# et al. Critical state concept gained widespre
recognition as a framework to the understanding of the
havior of soils, eg, Atkinson and Branoby@633#, Atkingson
@634#, Wood@151#, and Ortigao@638#. The critical state con-
cept was applied also to rock by Gerogiannopoulos
Brown @386# and to concrete@31#.
The cyclic behavior of soil under complex stress w
studied by many researchers. A single-surface yield func
with seven-parameter for geomaterials was proposed
Ehlers@626#. The multisurfaces theory was originally intro
duced by Mroz@639# and Iwan@640# and applied to two-
surfaces or three-surfaces model by Krieg@641#, Dafalias-
Popov @478#, Prevost@642,643#, Mroz et al @568,576,635#,
Hashiguchi @147,616#, Shen @644#, Hirai @646#, de Boer
@505#, Simo et al @647#, Zheng @649# et al. A generalized
nonassociative multisurface approach for granular mate
was given by Pan@648#. The concept of the bounding surfac
was proposed by Dafalias and Popov@478# in metal plastic-
ity, and applied to soil plasticity by Mroz, Norris, and Zien
kiewicz @568,576,635#, as well as Dafalias and Herrman
@581# and Borjaet al @606#.
Strength theory is also generalized to act as rigid-pla
and elasto-plastic models in RS~reinforced soils!. Some cri-
teria for RS were proposed, such as Sawicki@651#, Micha-
lowski and Zhao~RS with randomly distributed short fiber!
@613#. A global yield surface considering thes1 ands3 was
given by Sawicki. The summary of yield conditions for R
and its applications in RS structures was presented in S
icki recently @618#.
The joint failure criterion@437# and the Mohr-Coulomb
failure criterion were adopted as the yield criterion of s
and interface in research for dynamic soil structure inter
tion system@595#. A new interface cap model was recent
developed by Lourenco-Rots@650# that is bounded by a
composite yield surface that includes tension, shear and c
pression failure as follows
F5cnnsm
2 1cns1csst8
25s0 (66)
6.4 Failure criteria for iron
Studies of the fracture of iron date back to the work of Co
and Robertson in 1911~thick-walled tubes subjected to inte
nal pressure and compression! @207#, Ros and Eichinger in
oaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org
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