Size effect on structural strength:a review
Z.P.Baz Ïant
Summary The article attempts a broad review of the problem of size
得心应手的近义词effect or scaling of failure,which has recently come to the forefront of attention becau of its importance for concrete and geotechnical engineering,
geomechanics,arctic ice engineering,as well as for designing large load-bearing parts made of advanced ceramics and for aircraft or ships.First,
the main results of Weibull statistical theory of random strength are brie¯y summarized,and
its applicability and limitations described.In this theory as well
as plasticity,elasticity with a
strength limit,and linear elastic fracture mechanics (LEFM),the size effect is a simple power
law,becau no characteristic size or length is prent.Attention is then focud on the deterministic siz
e effect in quasibrittle materials which,becau of the existence of a non-negligible material length characterizing the size of the fracture process zone,reprents the bridging between the simple power-law size effects of plasticity and of LEFM.The energetic
theory of quasibrittle
size effect in the bridging region is explained,and then a host of recent
re®nements,extensions and rami®cations are discusd.Comments on other types of size effect,including that which might be associated with the fractal geometry of fracture,are also made.The historical development of the size-effect theories is outlined,and the recent trends of rearch are emphasized.Key words Scaling,size effect,fracture mechanics,quasibrittle materials,asymptotic methods 1Introduction The size effect
is a problem of scaling,which is central to every physical theory.In ¯uid mechanics rearch,the problem of scaling continuously played a prominent role for over a hundred years.In solid mechanics rearch,though,the attention to scaling had many inter-ruptions and became inten only during the last decade.
Not surprisingly,the modern studies of nonclassical size effect,begun in the 1970's,were
stimulated by the problems of concrete structures,for which there inevitably is a large gap
between the scales of large structures (e.g.dams,
reactor containments,
bridges)
and
of labo-
ratory tests.This gap involves in such structures about one order of magnitude;even in the rare
cas when a full-scale test is carried out,it is impossible to acquire a suf®cient statistical basis on the full scale.
The question of size effect recently became a crucial consideration in the efforts to u advanced ®ber composites and sandwiches for large ship hulls,bulkheads,decks,stacks and masts,as well as for large load-bearing fulage panels.The scaling problems are even greater in geotechnical
engineering,arctic engineering,and geomechanics.In analyzing the safety of an
excavation wall or
a tunnel,
the risk
of a mountain
slide,
the
risk of slip of a fault in the earth crust or the force exerted on an oil platform in the Arctic by a moving mile-size ice ¯oe,the
scale jump from the laboratory spans many orders of magnitude.Received 13April 1999;accepted for publication 6June 1999
Z.P.Baz
Ïant
Walter P.Murphy Professor of Civil Engineering and Materials Science,Northwestern University,Evanston,Illinois 60208,USA
Preparation of the prent review article was supported
by the
Of®ce of Naval Rearch under Grant N00014-91-J-1109to Northwestern University,monitored by Dr.Yapa D.S.Rajapak.Archive of Applied Mechanics 69(1999)703±725ÓSpringer-Verlag 1999
703
In most of mechanical and aerospace engineering,on the other hand,the problem of scaling has been less pressing becau the structures or structural components can usually be tested at full size.It must be recognized,however,that even in that ca the scaling implied by the theory must be correct.Scaling is the most fundamental characteristic of any physical theory.If the scaling properties of a theory are incorrect,the theory itlf is incorrect.
The size effect in solid mechanics is understood as the effect of the characteristic structure size(dimension)D on the nominal strength r N of structure when geometrically similar structures are compared.The nominal stress(or strength,in ca of maximum load)is de®ned as
r N c N P
bD
or
c N P
D2
for two-or three-dimensional similarity,respectively,where P is the load(or load
parameter),b structure thickness and c N arbitrary coef®cient chon for convenience
(normally,c N 1).So r N is not a real stress but a load parameter having the dimension of
stress.The de®nition of D can be he beam depth or half-depth,the beam
span,the diagonal dimension,etc.)becau it does not matter for comparing geometrically
similar structures.
The basic scaling laws in physics are power laws in terms of D,for which no characteristic
size(or length)exists.The classical Weibull[107]theory of statistical size effect caud by
randomness of material strength is of this type.During the1970's it was found that a major
deterministic size effect,overwhelming the statistical size effect,can be caud by stress re-
distributions caud by stable propagation of fracture or damage and the inherent energy
relea.The law of the deterministic size effect provides a way of bridging two different power
laws applicable in two adjacent size ranges.The structure size at which this bridging transition
occurs reprents a characteristic size.
The material for which this new kind of size effect was identi®ed®rst,and studied in the
greatest depth and with the largest experimental effort by far,is concrete.In general,a size
effect that bridges the small-scale power law for nonbrittle(plastic,ductile)behavior and the
large-scale power law for brittle behavior signals the prence of a certain nonnegligible
characteristic length of the material.This length,which reprents the quintesntial property
of quasibrittle materials,characterizes the typical size of material inhomogeneities or the
fracture process zone(FPZ).Aside from concrete,other quasibrittle materials include rocks,
cement mortars,ice(especially a ice),consolidated snow,tough®ber composites and
particulate composites,toughened ceramics,®ber-reinforced concretes,dental cements,bone
and cartilage,biological shells,stiff clays,cemented sands,grouted soils,coal,paper,wood,
wood particle board,various refractories and®lled elastomers as well as some special tough
metal alloys.Keen interest in the size effect and scaling is now emerging for various`high-tech'
applications of the materials.
Quasibrittle behavior can be attained by creating or enhancing material inhomogeneities.
治干眼症的妙招Such behavior is desirable becau it endows the structure made from a material incapable of
plastic yielding with a signi®cant energy absorption capability.Long ago,civil engineers
subconsciously but cleverly engineered concrete structures to achieve and enhance quasibrittle
characteristics.Most modern`high-tech'materials achieve quasibrittle characteristics in much
the same way±by means of inclusions,embedded reinforcement,and intentional micro-
挑拨cracking(as in transformation toughening of ceramics,analogous to shrinkage microcracking
of concrete).In effect,they emulate concrete.
In materials science,an inver size effect spanning veral orders of magnitude must be
tackled in passing from normal laboratory tests of material strength to microelectronic com-
ponents and micromechanisms.A material that follows linear elastic fracture mechanics
(LEFM)on the scale of laboratory specimens of sizes from1to10cm may exhibit quasibrittle
or even ductile(plastic)failure on the scale of0.1or100microns.
The purpo of this article is to prent a brief review of the basic results and their
history.For an in-depth review with a thousand of literature references,the recent article
[16]may be consulted.A full exposition of most of the material reviewed here is found in
the recent book[28].The problem of scale bridging in the micromechanics of materials,
review.
704
2
History of size effect up to Weibull
Speculations about the size effect can be traced back to Leonardo da Vinci(1500's)[112].He
obrved that``among cords of equal thickness the longest is the least strong,''and propod
that``a cord is so much stronger F F F as it is shorter,''implying inver proportionality.A
century later,Galileo Galilei(1638)[58],the inventor of the concept of stress,argued that
Leonardo's size effect cannot be true.He further discusd the effect of the size of an animal on
the shape of its bones,obrving that bulkiness of bones is the weakness of the giants.
A major idea was spawned by Mariotte(1686)[76].Bad on his extensive experiments,he
obrved that``a long rope and a short one always support the same weight unless that in a long
rope there may happen to be some faulty place in which it will break sooner than in a shorter'',
705 and propod the principle of``the inequality of matter who absolute resistance is less in one
place than another.''In other words,the larger the structure,the greater is the probability of
encountering in it an element of low strength.This is the basic idea of the statistical theory of
size effect.
Despite no lack of attention,not much progress was achieved for two and half centuries,
until the remarkable work of Grif®th(1921)[60],the founder of fracture mechanics.He showed
experimentally that the nominal strength of glass®bers was raid from42,300psi to
491,000psi when the diameter decread 0.00013in.,and concluded that
``the weakness of is due to the prence of discontinuities or¯The
effective strength of technical materials could be incread10or20times at least if the¯aws
could be eliminated.''In Grif®th's view,however,the¯aws or cracks at the moment of failure
were still only microscopic;their random distribution controlled the macroscopic strength of
the material but did not invalidate the concept of strength.Thus,Grif®th discovered the
physical basis of Mariotte's statistical idea but not a new kind of size effect.
The statistical theory of size effect began to emerge after Peirce(1926)[86],formulated the
weakest-link model for a chain and introduced the extreme value statistics which was origi-
nated by Tippett(1925)[101],Fischer and Tippett(1928)[52],and FreÂchet(1927)[51],and
re®ned by von Mis(1936)[102]and others(e also[50,56,57,97]).The capstone of the
statistical theory was laid by Weibull(1939)[107](also[108±110]).On a heuristic and ex-
perimental basis,he concluded that the tail distribution of low strength values with an ex-
tremely small probability could not be adequately reprented by any of the previously known
distributions.He introduced what came to be known as the Weibull distribution,which gives
the probability of a small material element as a power law of the strength difference from a
®nite or zero threshold.[56,97])later offered a theoretical justi®cation by means of
a statistical distribution of microscopic¯aws or microcracks.Re®nement of applications to
metals and ceramics(fatigue embrittlement,cleavage toughness of steels at low and brittle-
ductile transition temperatures,evaluation of scatter of fracture toughness data)has continued
until [32,35,50,71]).
Most subquent studies of the statistical theory of size effect dealt basically with re®nements
and applications of Weibull's theory to fatigue embrittled metals and to [69,70]).
Applications to concrete,where the size effect was of the greatest concern,have been studied in
[36,37,78,79,82,115,116,117]and elwhere.
Until about1985,most mechanicians paid almost no attention to the possibility of a de-
terministic size effect.Whenever a size effect was detected in tests,it was automatically as-
sumed to be statistical,and thus its study was suppod to belong to statisticians rather than mechanicians.The reason probably was that no size effect is exhibited by the classical con-
tinuum mechanics in which the failure criterion is written in terms of stress and strains:
elasticity with strength limit,plasticity and viscoplasticity,as well as fracture mechanics of
bodies containing only microscopic cracks or¯aws[10].The size effect was not even men-
tioned by S.P.Timoshenko in1953in his monumental History of the Strength of Materials.
The attitude,however,changed drastically in the1980's.In conquence of the well-funded
rearch in concrete structures for nuclear power plants,theories exhibiting a deterministic size
effect have been developed.We will discuss it later.
3
Power scaling and the ca of no size effect
It is proper to explain®rst the simple scaling applicable to all physical systems that involve no
characteristic length.Let us consider geometrically similar systems,for example the beams
shown in Fig.1a,and ek to deduce the respon he maximum stress or the maximum
de¯ection)as a function of the characteristic size (dimension)D of the structure;Y Y 0f D where u is the chon unit of measurement (e.g.1m,1mm).We imagine three structure sizes 1,D
,and
D H
(Fig.1a).If we take size 1as the reference sizes.The respons for sizes D and D H are Y f D and Y H f D H .However,since there is no characteristic length,we can also take size D as the reference size.Conquently,the equation f D H f D f D
H D
1
must hold [10,16](for ¯uid mechanics,e [4,96]).This is a functional equation for the
unknown scaling law f D .It has one and only one solution,namely the power law:f D D c 1
s
X 2
where s const.and c 1is a constant which is always implied as a unit of length measurement.Note that
c 1cancels out of Eq.(1)when the power function (2)is substituted.
On the other hand,when for instance f D log D a c 1 ,Eq.(1)is not satis®ed and the unit
of measurement,c 1,does not cancel out.So,the logarithmic scaling could be possible only if
the system possd a characteristic length related to c 1.The power scaling must apply for every physical theory in which there is no characteristic length.In solid mechanics
such failure theories include elasticity with a strength limit,elastoplasticity,viscoplasticity as well as LEFM (for which the FPZ is assumed shrunken into a point).To determine exponent s ,the failure
criterion
of the material must be taken into account.For
抹茶泡芙elasticity with a strength limit (strength theory),or plasticity (or elasto-plasticity)with a yield
天堂英语surface expresd in terms of stress or strains,or both,one ®nds that s 0when respon Y
reprents the
stress or strain (for example the maximum stress,or the stress at certain
homologous points,or the nominal stress at failure)[10].Thus,if there is no characteristic dimension,all geometrically similar structures of different sizes must fail at the same nominal stress.By convention,this came to be known as the ca of no size effect.
In LEFM,on the other hand,s À1a 2,provided that geometrically similar structures with geometrically similar cracks or notches are considered.This may be generally demonstrated with the help of Rice's J-integral [10].If log r N is plotted versus log D ,the power law is a straight line,Fig.1.For plasticity,or elasticity with a strength limit,the exponent of the power law he slope of this line
is zero.For LEFM,the slope is À1a 2.A recently emerged `hot'subject is the quasibrittle
materials and structures,for which the size effect bridges the two power
laws.Fig.1.a Geometrically similar structures
of different sizes b power scaling laws c size
effect law for quasibrittle failures bridging
the power law of plasticity (horizontal asymptote)and the power law of LEFM
(inclined asymptote)706
4Weibull statistical
size
effect
The classical theory of size effect has been statistical.Three-dimensional continuous general-ization of the weakest link model for the failure of a chain of links of random strength,Fig.2
left,leads to the distribution P f r N 1Àexp À V c r x d V x &'
Y 3
which reprents
the failure probability of a structure that fails as soon as a macroscopic
fracture initiates from a microcrack (or some ¯aw)somewhere in the structure;r is the stress tensor ®eld just before failure,x the coordinate vector,V the volume of structure and c
r a
function giving the spatial concentration of failure probability of the material (=V À1r Âfailure
probability of material reprentative volume V r )
[56],c r %
i P 1 r i
V 0
where r i are the principal stress (i 1Y 2Y
3)and P 1 r the failure probability (cumulative)of
the smallest possible test specimen of volume V 0(or reprentative volume of the material)
海鲜店subjected to uniaxial tensile stress r ,P 1 r r Àr u s 0(
)m
Y 4
[107],where m Y s 0Y
r 1are material constants (m denotes the Weibull modulus,usually between
日记300字大全初中5and 50,s 0is a scale parameter,r u the strength threshold,which may
usually be taken as zero)
and V 0is the
reference volume
understood
as the volume of specimens on which c r was
measured.For specimens under uniform uniaxial stress (and r u 0),(3)and (4)lead to the
following simple expressions for the mean and the coef®cient of variation of nominal strength:r N s 0C 1 m À1 V 0V 1m Y x C 1 2m À1 C 2 1 m À1 À1 !12
Y 5
where C is the gamma function.Since x depends only on
m ,it is often ud for determining m from the obrved
statistical scatter of strength of identical test specimens.The expression for r N includes
the effect of volume V which depends on size D .In general,for structures with
nonuniform multidimensional stress,the size effect of Weibull theory (for r r %0)is of the
type:Fig.2.a Chain with many links
of
random strength b failure proba-bility of a small element c struc-ture with many microcracks of
different probabilities to become critical 707
r N G D Àn d m Y 6 where n d 1Y 2or 3for uni-,two-or three-dimensional similarity.In view of (5),the value r W r N V a V 0 1a m for a uniformly stresd specimen can be adopted as a size-independent stress measure.Taking this viewpoint,Beremin [32],propod taking into account the nonuniform stress in a large crack-tip plastic zone by the so-called Weibull stress:r W i r m I i V i
0231m Y 7
where V i i 1Y 2Y F F F Y N W are elements of the plastic
zone
having maximum principal stress
r I i .The sum
in (5)was replaced by an integral in [95],e also [71].Equation (7),however,considers only the crack-tip plastic zone who size which is almost independent of D .Con-quently,Eq.(7)is applicable only if the crack at the moment of failure is not yet macroscopic,still being negligible compared
to structural
dimensions.As far as quasibrittle structures are concerned,applications of the classical Weibull theory
face a number of rious objections:1.The fact that in (6)the size effect is a power law implies the abnce of any characteristic
length.But
this cannot
be
true if
the material contains
sizable inhomogeneities.
2.The energy relea due to stress redistributions caud by macroscopic FPZ,or stable crack
growth before P max
,gives ri to a deterministic size effect which is ignored.Thus the Weibull theory is valid only if the structure fails as soon as a microscopic crack becomes macroscopic.3.Every structure is mathematically equivalent to a uniaxially stresd bar (or chain,Fig.2),
which means that no information on the structural geometry and failure mechanism is taken into account.4.The size-effect differences between two-and three-dimensional similarities (n d
2or 3)are predicted much too large.5.Many tests of quasibrittle materials (e.g.diagonal shear failure of reinforced concrete
beams)show a much stronger size effect than predicted by Weibull theory,[28]and the review in [11]).6.The classical theory neglects
the
spatial correlations
of material
failure probabilities
of
neighboring elements caud by nonlocal properties of damage evolution (while general-izations bad on some phenomenological load-sharing hypothes have been divorced from mechanics).7.When Eq.(5)is ®t to the test
data on statistical scatter for specimens of one size (V =趴蹄
const.),and
when Eq.(6)is ®t to the mean test data on the effect of size (of unnotched plain
concrete specimens),the optimum values of Weibull exponent m are very different,namely m 12and
m 24,respectively.If the theory were applicable,the value would have to coincide.In view
of the limitations,among concrete structures Weibull theory appears applicable to some extremely thick plain (unreinforced)he ¯exure of an arch dam acting as a horizontal beam (but not for vertical bending or arch dams nor gravity dams,becau large vertical compressive stress
cau long cracks to grow stably before the maximum load).Most
other plain concrete structures are not thick enough to prevent the deterministic size effect
from dominating.Steel or ®ber reinforcement prevents it as well.5Quasibrittle size
effect bridging plasticity and LEFM,and its history
Quasibrittle materials are tho that obey on a small scale the theory of plasticity (or strength
theory),characterized by material strength or yield limit r 0,and on a large scale
the LEFM,
characterized by
fracture energy G f .While plasticity alone,as well as LEFM alone,posss no
characteristic length,the combination of both,which must be considered for the bridging of
plasticity and LEFM,does.Combination of r 0and G f yields Irwin's (1958)characteristic length
(material length) 0 EG f
r 20Y 8 708
