Experimentation, Numerical Simulation and the Role of Engineering Judgement in the Fracture Mechanics of Concrete and Concrete Structures

7/25/2019 Experimentation, Numerical Simulation and the Role of Engineering Judgement in the Fracture Mechanics of Concrete and Concrete Structures

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( )J.G.M. an Mier, M.R.A. an Vliet Construction and Building Materials 13 1999 3144

course these questions can not be answered positively.First there is the conceptual design, a creative designprocess, and in spite of spectacular progress in com-puter technology, this still remains a human activity.Overlooking the implications of a certain design deci-

sion can be judged in an instant by an experiencedengineer, but takes millions of searches using a com-puter-based approach; not very realistic at present. Infact, such considerations bring us close to questionssuch as, How does the mind work?, which by somescientists is regarded as a computer as well. There iscertainly ground for research in the field of conceptualdesign, as became clear for example at the IASS Con-

ference on the topic in Stuttgart in 1996 1 . However,let us turn to other aspects of the problem, and con-sider the state of affairs when the conceptual design iscompleted and the engineer must check whether the

structure can be built, and what the final dimensions ofall the structural elements would be. With the mainstructural shape being defined, as well as the structuralprinciple, there are more certainties, and the computercan now play a more important role. Essentially it isstill no more than a tool, a number crunchier to assistwith the computations of structures and the manydetails.

At this stage of the design process, the structure isschematised either in one, two or three dimensions.

.Supports boundary conditions and loading cases aredefined, which form the real crux of the problem: theyare always approximations from reality. For example,

hinges are never perfect in practice but are assumed tobe frictionless in mechanics models. Uniform loads canbe defined, but in reality they never are. When hetero-geneous materials like concrete or rock are used, thematerial itself is the most important source of devia-tions from uniformity. The constitutive models repre-sent the material of which the structure was made. Thematerial laws are tuned to laboratory experimentswhere the boundary conditions are supposedly betterknown.

Typical experiments for measuring the mechanicalproperties of a material are the uniaxial tension test,

the uniaxial compression test, different types of multi-axial compression experiments and flexural tests. In atypical diagram of concrete, rock, or other brittle dis-ordered materials like some non-transformable cer-amics and metal matrix composites, a non-linearstressstrain curve is measured up till a certain peakstress is reached, after which the diagram displays aloss of carrying capacity with increasing deformation.This latter part of the curve is called the softeningbranch. During softening, localisation of deformationsoccurs. In tension this leads to a discrete crack whichseparates the specimen into two halves; in compression

a shear band may develop at low confinement such.that brittle failure prevails . Frictional stress transfer is

generally still possible in the shear band, and depend-ing on the confining stress a lower or higher residualstress level is measured. The pre-peak non-linearities

.are caused by quasi- stable microcrack processes, aswas for the first time demonstrated for concrete in tests

by Hsu et al. 2 . .Because the size of the critical crack in tension and

.the shear bands in compression are of the same orderof magnitude as the characteristic specimen dimen-sions, new free boundaries are created which changethe problem completely. As a result, the post-peakfracture process is affected by boundary conditions andsize effects. Thus, certainty about the true fractureprocesses diminishes, and a model can be tuned onlythrough some inverse modelling process.

What is the way out of this paradoxical situation,which can be formulated as follows: when for the final

check, the structural size has been decided, and detailsof the structure have been figured out, it seems thatcertainty about the material models evades. Before thefinal stage the uncertainty about the material modelswas just a single element from the complete set ofuncertainties of the entire design process, and of limitedimportance. The way to proceed seems to bifurcate.One possibility is to dive further into the materials, andtry to model the behaviour to an ever increasing degreeof detail. Microscopic processes are dealt with, whichrequire highly accurate experimental information thatcan be obtained at large costs only. Computations atthe same detailed level require a huge computer capac-

ity, again at increasing cost. The process seems tocorrespond to the match between Achilles and thetortoise: a never ending story, where however, thematch itself is the enjoyable part of it. The secondpossibility is to revert to an engineering approach, andto view the entire design process in the same way in theconceptual design stage. In other words, consider thecomplete system and optimise it, mostly through expe-rience and through trial and error.

Thus, will it ever be possible to develop models forconcrete fracture with a sufficient amount of predictivepower for full-proof structural design? Can everything

be computed, or is engineering judgement just as im-portant as anything else? These questions are a com-mon area of debate between design and research engi-neers.

In this paper we will not try to solve these questions.Rather we will focus on our ability to use numericalmodels as a helpful tool to better understand themechanical behaviour of concrete as well as for devel-oping constitutive equations. Moreover, the numericaltools can be used to engineer new materials. As far asmechanical behaviour is concerned, in particular thefracture stage is of interest. The limit state is also ofinterest to the structural engineer. For instance, therotational capacity of reinforced concrete structures is


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( )J.G.M. an Mier, M.R.A. an Vliet Construction and Building Materials 13 1999 314 5

important as it determines the safety against suddencollapse. This means that the full stressstrain curve ofconcrete must be known, but, because of the depen-dency of this relation to boundary conditions and size,the problem is not easily solved. Moreover, solving the

.problem by means of numerical continuum mechan-ics, brings in a wealth of numerical problems, which

.might be overcome or simply avoided when the physicsof the fracture phenomenon are incorporated in thedescription of the material. Microfracturing and crackgrowth are at the heart of the problem, and cause thecurvilinear shape of the stressstrain curve.

In this paper we will initially assume that everythingcan be computed. More and more detail is brought in.Immediately thereafter we will regard these mattersfrom the other perspective, and will argue thatknowledge of fracture mechanisms is essential to apply

fracture mechanics: an engineering approach wherejudgement and intuition play a major role This seemsparticularly so when we want to engineer new materi-als. Judgement is essential there as well, because weare essentially facing a major design task.

2. Improving the reliability of material parameters

2.1. Numerical simulation: elasticity

Concrete is a composite material. CalculatingYoungs modulus of the composite has been attempted

for several decades. Knowledge of the phase composi-tion of the concrete, as well as of the properties of theseparate phases, may if thought mathematicallysound help to compute the overall stiffness of a

.representative volume element RVE . The definitionof the RVE is important. Normally it is claimed thatwhen the ratio between the smallest element size and

largest material entity for example a pore, air bubble.or aggregate particle is larger than a certain minimum

number, usually 35, the material can be considered asa continuum. The simplest estimates for the compositestiffness are obtained with a series or parallel spring

model of a two-phase model i.e. the bond zone is not.incorporated . According to the series model, the com-posite stiffness of a two-phase material composed of

. .aggregate a and matrix m material is equal to

V V1 a m . , 1E E Ec a m

where V and V are the volume fractions of thea maggregate and matrix phase, and E , E and E arec a mYoungs moduli of the composite material, the aggre-gate and matrix phases. The parallel model reads asfollows,

.E V E V E 2c a a m m

The series and parallel models are considered as thetwo extremes between which Youngs modulus of the

real composite would lie. In the 1960s see the overview .by Newman 3 , many more refined models were devel-

oped such as the Counto and Hirsch models, which areessentially combinations of the series and parallel mod-els. Quite well known also are the HashinShtrikman

upper and lower bounds 4 . With the development ofnumerical tools, the computation of Youngs modulusof the composite can be based on the real materialstructure. A technique which we have used in the

Stevin laboratory 5 , and which was based on develop- .ments in theoretical physics e.g. 6 is the so-called

lattice model. The material is schematised as a regularor random network of elastic, purely brittle beam ele-

ments. Earlier a similar network method was proposedfor estimating the solutions of problems in elasticity by

Hrennikoff 7 . The results presented below can also beobtained with conventional finite element methods, forexample by means of simple triangular plane stresselements. After generating a triangular grid of nodes,

the connectivity with beam elements truss elements.would suffice for the example as well is made. Next the

particle structure of the concrete is described, either bymeans of a probability density function of a distribution

.of particles often represented by circles in a plane, orby means of a digital image of a planar cross-section of

a real concrete 8 . The material structure and latticeare then superimposed on each other, and lattice ele-ments falling in certain areas defined by the aggregatecircles are given the properties of the aggregate,whereas the lattice elements falling in the matrix phaseare given the elastic properties of the matrix. In addi-tion, lattice elements crossing the boundary betweenaggregate and matrix are given interface properties.This latter problem is particularly important for frac-ture simulations, although some effect of the interfacestiffness on the composite Youngs modulus cannot be

. .denied e.g. 9 . The smaller aggregates e.g. 1 mmare usually omitted because a tremendously fine lattice

would be needed to incorporate them in a calculation;Fig. 1 demonstrates the problem. In this figure a com-puter-generated particle structure of concrete is shown,and an overlay was made with a regular triangularlattice of beam elements with a length of 0.5 and 2.0

mm, respectively 10 . For shorter beam lengths, theshape of the aggregates is followed more precisely, butat the cost of a rapidly expanding number of elements,and thus with increasing computational effort. Compar-ison of Fig. 1b and c demonstrates that large circularparticles are transformed to irregularly shaped poly-gons, whereas particles smaller than the beam length

are completely missed. As a consequence the intendedparticle density is reduced. Moreover, the particle den-


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sity was already reduced because the small particleswere omitted as well. In Table 1 and Fig. 2 the in-

tended and effective particle densities after omission.of the small particles , and the density after the lattice

overlay are shown. As can be seen, the reduction is

quite large, in particular for very large numbers ofaggregates. The last example, Fig. 2d, was generated bymeans of a computer programme developed by Stroeven

and Stroeven 11 . The other distributions were de-termined by means of a simple programme written by

Schlangen 8 .Using the particle distributions of Fig. 2, as well as

another set of generated microstructures based on asubsequent omission of aggregates of a certain size 10 , Youngs modulus of the composite was computed.This was done for the case where the aggregate stiff-

ness was larger than the matrix stiffness E 70 GPaa

.and E 25 GPa , as well as for the case where themaggregate stiffness was smaller than the matrix stiffness .E 10 GPa and E 25 GPa . This first case woulda mresemble concrete made with, for instance, river gravelas aggregate material, whereas the second case wouldresemble a lightweight concrete with expanded clayaggregates. The two sets of analyses, i.e. with compos-ites where the aggregates of different sizes are gradu-ally omitted and with composites where the relativearea fraction of aggregate circles of all sizes is varied . denoted as lattice model P , are shown in Fig.k3a,b, together with a comparison with the series and

parallel model, and an approximation following Hashin 12 . For both composites with low and high stiffness

Table 1Effective relative aggregate areas for different values of Pk

P P Pk, intended k,eff k,lattice2 . .area 80 mm grains 1 d after overlay

. 16 mm

0.10 0.06 0.030.40 0.32 0.190.70 0.56 0.341.00 0.80 0.48

aggregates, the computational results with the latticelie between the two extremes defined by the series andparallel models. Hashins model also lies between thesetwo extremes, and the computational outcome comesclose to this generally accepted solution. The relative

aggregate content, which is plotted along the x-axis,stops at almost 60%. Even with an intended aggregate

fraction P 1.0 which would imply only aggre-k,intended.gate particles and no matrix material , the real relative

aggregate fraction does not exceed 60%. The reasonfor this is that the aggregate remains lumped in circu-lar aggregates, and the space between the aggregatesmust be filled with matrix material. If the completeplane is to be filled with circular particles, a welldefined particle distribution, including the very small

.particles all the way down to infinitely small size mustbe included. This would then lead to an exaggerated

small size of the lattice elements in order to include allthese small particles in the analysis. Clearly this is not arealistic option. Quite remarkable is that in experi-ments high particle densities also can not be obtained 13 , simply because not enough matrix is available tofill all the gaps between the aggregates. As a conse-quence, porosity is introduced, which tends to reduce

the measured overall Youngs modulus, 13 . Note thatin parallel and series models it is possible to obtainrelative aggregate fractions of 100%. The two fractionsare assumed to be lumped in one element. When theaggregate fraction is 100%, the composite has becomea continuum consisting of aggregate material only.

In conclusion, it can be stated that the choice for adistribution of circular aggregates is the limiting factorfor obtaining composites with a high relative aggregatefraction, at least when limits are set to the smallestaggregate particle in the distribution. Solving the mat-ter by means of statistical theories would not show theproblem of limiting aggregate densities. In practicalmaterials, including too much fine grains leads to prob-lems in manufacturing the material in the first place. Asimilarity with the numerical simulations is again thatthe computation becomes too lengthy, if not impossible

. . .Fig. 1. Computer-generated particle distribution of concrete a with overlay of a regular triangular lattice b,c . In b the length of the lattice . elements is 0.5 mm, in c the length is 2.0 mm, after Van Mier et al. 10 .


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. . .Fig. 2. Four different particle distributions with variation of the fraction of rounded aggregates: P 0.10 a , 0.40 b , 0.70 c and 1.00k,intended . d , after Van Mier et al. 10 .

when all the small particles are included and a verysmall size of the lattice elements must be selected. Thecomposite Youngs modulus was calculated by meansof a lattice model. However, the same fundamentalproblems are encountered when plane stress elementsare used in finite element analysis.

2.2. Numerical simulation: brittleness and pattern growth

The next step that can be made with the numericallattice model is to try to compute the strength of acomposite. In that case, not only Youngs moduli of the

.Fig. 3. Effect of aggregate content on Youngs overall modulus for the particle composites of Fig. 2. a The case with E 70 GPa and E 25a m . GPa; b the case with E 10 GPa and E 25 GPa, after Van Mier et al. 10 .a m


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different phases must be specified, but thresholds forthe local energy consumption when an element breaksin the material must de defined too. As mentioned, in

1991 we developed a simple lattice model 5 based on developments in theoretical physics 6 . In the model,

the lattice elements are Bernoulli beams, and they areremoved from the mesh as soon as the strength thresh-old is exceeded. In this way purely brittle fracture issimulated, whereas the analysis remains completely .stepwise elastic. Three examples of lattice analyses

2 .are shown in Fig. 4. Square plates 80 mm are usedcontaining a computer-generated particle distribution,similar to those shown in Fig. 2. The P -valuesk,intendedfor the three analyses that are shown here were 0.20,0.50 and 0.80. The distribution of the lattice elementsover the three phases is shown in Table 2. The totalnumber of beams in the lattice was 22 114. Table 2

shows both the absolute number of elements in thethree distinct phases as well as the relative number ofbeams in each phase. The bond strength was 25% ofthe matrix strength, whereas the aggregate strengthwas double the matrix strength. Note that for a latticeanalysis only the relative strength values are of impor-

tance 5 . The matrix and bond zone Youngs moduluswas 25 GPa, whereas for the aggregate beams E 70aGPa was assumed. In Fig. 4a,b the crack patterns at

.peak load F and at F in the descendingpeak peakbranch are shown, respectively. The value of differedslightly for the three analyses, namely 0.25, 0.24

and 0.19 for P 0.20, 0.50 and 0.80, respec-k,intended .tively. The pre-peak crack patterns Fig. 4a show the

beams that were removed, in the post-peak crack pat- .terns Fig. 4b only the remaining elements are plotted

in the deformed state. In Fig. 5 the three loaddefor-mation diagrams are shown.

The behaviour was completely different in the three . .examples. At low 0.20 and middle 0.50 values of

P quite a number of bond beams must be failedk,intendedbefore the peak-load was reached, but after peak aclear localised crack developed. At the lowest P ,k,intendedthe aggregates are rather isolated in the matrix. Table

2 shows that only 8% of the beams are inside theaggregates, whereas the matrix makes up 80% of thenumber of beams. Seldom do particles touch or dis-tances smaller than one lattice element between the

neighbouring bond zones are found. Because of thedifferences in stiffness, stress concentrations will ap-pear in the interfaces between aggregates and matrix,and this is where the first cracks will develop. This canbe seen from the crack patterns in Fig. 4, in particular

for the lowest and middle aggregate content. WhenP is increased to 0.50, the amount of aggregatek,intendedtriples, whereas a spectacular decrease of matrix beamsis found. The number of bond beams increases as well,namely to 30%. The increasing amount of bond beamsleads to a higher percentage of initial bond cracks, butbecause the different bond cracks are separated bystronger matrix elements, the external load still has tobe increased to enforce further crack propagation.Thus, cracking is still stable at this stage. However, inthe third analysis, with P 0.80, the situationk,intendedchanges dramatically. The matrix phase has decreased

to 16% of the total number of beams, whereas the .bond zone has increased markedly again i.e. to 44% .

The result is that continuous patterns of connectedbond zones are present in the concrete structure. Inother words, we have exceeded the percolation thresh-old. If the first cracks appear in such a continuous bondzone, rapid crack propagation cannot be avoided be-cause all the neighbouring bond elements have a lowstrength as well, whereas a local stress concentrationappears where the first beam was removed. In otherwords, a crack is propagating in a homogeneous bondzone where the only deviations are caused by the

positions of stiff particles. The bond zone spans thewidth of the specimen because the percolation thresh-old was exceeded. The local situation determineswhether the crack will propagate or whether moreisolated cracks will develop. Fig. 4a shows that for thehighest aggregate content obviously a situation abovethe percolation threshold has been created. Only threeelements are broken at the peak, all the rest follow inthe descending branch. One of the conclusions that canbe drawn on the basis of these three simulations is thatof controlling the amount of aggregates, as well as thenumber of weak interface elements may lead to an

increased interval of stable crack propagation where atany step the external load must be increased againbefore the next element fractures. The loaddeforma-tion diagram for the low and middle aggregate content

Table 2 . .Absolute and relative number of beams in different phases the total number of lattice elements beams is 22 114

P Absolute number of beams Relative number of beamsk,intended

Aggregate Bond Matrix Aggregate Bond Matrix

0.20 1741 2668 17 705 0.08 0.12 0.800.50 5163 6707 10 244 0.24 0.30 0.46

0.80 8759 9815 3540 0.40 0.44 0.16


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. . Fig. 4. Crack patterns at peak load a and at the end of the simulation b for three different aggregate contents from left to right.P 0.20, 0.50 and 0.80, respectively .k,intended

in Fig. 5 shows that the pre-peak curve is indeed morepronounced in comparison to the high aggregate con-

.tent 0.80 . There is also a marked difference in peakstrength. The strength of the low aggregate specimen .0.20 is governed by the strong ligaments between the .isolated weak bond zones, whereas in the high aggre-

.gate case 0.80 continuous patterns of low-strengthinterface elements control the specimen behaviourcompletely.

Note that after the first matrix beam has been failed .in the low aggregate simulation 0.20 , failure is immi-

nent. In this case, the matrix material has exceeded itspercolation threshold, and a continuous path of matrixmaterial will span the specimens width. The situation isidentical to the case where a high aggregate content is

.present 0.80 , but there as argued before thebond elements have exceeded their percolation thresh-old.

. . .Fig. 5. Loaddeformation diagrams for the three analyses of Fig. 4: a P 0.20, b 0.50, and c 0.80.k,intended


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2.3. The role of experiments: crack detection andloaddeformation response

These simulations create an artificial world, and itremains to be investigated to what extent the computa-

tional results resemble reality. Some features of thefracture process have been observed before. For exam-ple the propagation of the macroscopic crack throughthe specimens cross-section was found in photo-elastic

coating experiments on concrete and mortar plates 14 .In Fig. 6 a result from such a test is shown. In this case,a major crack runs from the left side of the specimen,whereas a second minor crack develops from the otherside. Notches were made in the specimen for testcontrol purposes. The fine detail in the macro-crackcannot be revealed from the photo-elastic coating ex-periments.

For revealing more detail in the fracture processother techniques are needed. The first, and most easyto perform is the impregnation technique. Specimensare fractured until a certain crack width has beenreached, and subsequently the fractured specimen isfilled with fluorescent epoxy. After the epoxy has hard-ened, the specimen is cut open, and fine detail in thecrack patterns can be visualised under ultra-violet light;an example is given in Fig. 7. Two crack patterns are

.shown, i.e. in lytag concrete Fig. 7a and in high- .strength concrete Fig. 7b . The patterns show small

scale overlaps between cracks, similar to the crackoverlap that was observed in Fig. 6. Increasing the

resolution, for example by performing tests whileobserving the fracturing process under an optical mi-croscope reveals more of these bridges. Failure of thebridges occurs when one of the crack branches growsand coalesces in the wake of the second crack. The

same mechanism is found in the numerical simulations,as can be observed from Fig. 4b. The overlap mecha-

.nism, at least when viewed at the macro-scale Fig. 6 ,depends on the actual boundary conditions imposedduring the test. Would the specimen ends be allowed torotate freely, the large scale overlap would not have

developed, but at the scale of the aggregates meso-. level , the mechanism would appear again 15 . The

number of these overlaps, plus their size relative to thespecimen size determines how much load can be car-ried in the tail of the softening diagram.

The size of the crack overlap also determines the

global curvatures in the crack pattern. Obviously, afterthe specimen has been fully fractured this is a compli-cated undulating surface. Projecting the crack patternsfrom different slices on a plane will reveal the bandwidth as shown in Fig. 8. The band width changes fordifferent concretes. As a matter of fact, the larger thesize of stiff and strong aggregate in the mixture, thewider the crack band. For 2-mm cement mortar a verynarrow crack band is found, the next is lytag concretewhich contained 12-mm lytag particles and sand withparticles up to 4 mm. Fig. 7a clearly shows that thecracks in lytag concrete will intersect the lytag particleswhich appear as the large speckled circular areas in

.Fig. 6. Photo-elastic coating experiment on a double-edge-notched concrete plate subjected to uniaxial tension uniform boundary displacement . . . .

In a

c the propagating cracks are visualised, d shows a comparison of the photo-elastic crack path and the crack trajectory after complete . failure of the specimen. In e the loaddeformation curve is shown, after Van Mier 15 .


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.Fig. 7. Crack face bridging in lytag lightweight concrete a , and .crack branching in high strength concrete b , from impregnation

tests of Van Mier 15 .

.the cross-section of the material , but will grow aroundthe larger stiff sand particles. This latter phenomenoncan also be seen in Fig. 7a. The largest crack bandwidth is found for 16-mm normal and high-strengthconcrete. It is generally known that the interface innormal strength concrete is the weakest link in the

material. By adding condensed silica fume to obtainhigh-strength concrete, it is expected that not only thedensity of the concrete is improved, but that also theinterface strength increases. For the high-strength con-

crete which has a compressive strength of approx. 88.MPa of Fig. 7b and Fig. 8d this was obviously not the

case, and cracks were found to grow along the interfaceof the aggregate particles. This explains why the crackband width is almost similar in the normal and high-strength 16-mm concretes.

The crack band width in Fig. 8 should not give theimpression that at each cross-section of the specimen

this width would be found. It is emphasised that thecrack pattern shows the global undulations of thethree-dimensional crack as if we were looking throughthe specimen. Using X-ray techniques, similar observa-

tions can be made. For example in Otsuka et al. 16 and Landis and Nagy 17 results from recent X-ray

imaging experiments are shown, which give the sameimpression as Fig. 8. The X-ray results confirm thefindings of the earlier impregnation experiments.

Other techniques to detect internal cracking in con-crete and rock include acoustic monitoring, for exam-

. ple by means of acoustic emission AE 16,18 or ultrasonic pulse technique 19 . By means of ultrasonic

pulses, reflection and diffraction patterns are mea-

. .Fig. 8. Crack bands for 2-mm mortar a , 16-mm concrete b , lytag . .lightweight concrete c , and high-strength concrete d , from Van

Mier 15 .

sured, which are subsequently converted to informationabout crack patterns. For the ultrasonic pulse tech-nique pre-existing cracks can be measured as well.Using AE, energy bursts from crack propagation bydetecting the vibrations by means of detectors that arefixed at the specimens surface are analysed. The sim-plest method is to count the number of events above acertain threshold. By fixing more transducers to thespecimen a location analysis can be carried out, whereasby means of a moment tensor analysis conclusions can

be drawn about the nature of the event mode I or

mixed modes I and II, etc.; see Ohtsus paper in this.special issue .

There seems to exist a relationship between thenumber of beams removed in a simple lattice model for

.fracture as described above and the number of AEevents recorded during a fracture experiment Kari-

.1haloo, private communication . It is normal procedureto record only the events above a certain threshold.The removal of a beam from the lattice representsbrittle crack propagation. The dissipation of a smallamount of energy occurs, which is represented by the

. area under the local linear stressstrain diagram see.

Fig. 9a . This energy dissipation seems equivalent to anacoustic event. The threshold value in the lattice modelis represented by the size of the beam in the lattice,whereas in the AE analysis, the energy level is specified .e.g. see 16 .

In Fig. 9b the cumulative removal of beams for thethree lattice analyses of Figs. 4 and 5 are shown. Thefirst removals occur after approximately 3 m of defor-mation. After that the number gradually increases untila distinct plateau is reached. Depending on the value

1

The resemblance between AE activity and the stepwise removalof beams was suggested by Prof. Karihaloo.


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.Fig. 9. Energy dissipation from a single beam removal in the lattice model a , and number of elements removed plotted against deformation for .the three analyses of Fig. 4 b .

of P , these kinks occur at different numbers.k,intendedTheir value is governed by the actual fracture process,which was shown to vary widely for the three analyses.After the kink, a complete macroscopic crack pattern ispresent, and the last removals of lattice elements oc-curs when the bridges in the main crack fail. Thenumber of AE events against axial deformation in a

uniaxial tension test has the same shape see Fig. 10 .which has been reproduced from Wissing 20 .

The type of fracture law can be changed in thelattice model, and can perhaps be related to specificAE events. The similarity should be further elucidated

in the future. Extension to compressive failure can bedone by considering the details of the fracture process along the same lines as shown here for tension 15 .

3. Fracture mechanics in structural analysis: the case

of compression

The above discussion is related to what happens

Fig. 10. Comparison of stressdeformation diagram and hit-rate all.

arrivals against deformation for a uniaxial tensile test, from Wissing 20 .

during tensile softening of concrete. As a potentialapplication, the design of new concrete materials ismentioned. Analysing the mechanical behaviour is justa small part of the design of the new composite. In theend, the material must also be manufactured and ap-plied in structures. For structural analysis, consideringthe details of the fracture process is not practical, atleast at present, and we are mainly interested in thevalidity of the macroscopic material laws that are used.This implies that we are interested in tensile stressdeformation diagrams as well as compressive stressdeformation diagrams. Note that for fully three-dimen-

sional codes information about the multiaxial be-haviour is needed, including of course detailedknowledge about the softening behaviour.

At the macroscopic level, the use of the fictitious crack model 21 , or the related crack band model 22

is quite popular. In the fictitious crack model, thetensile diagram is separated in a pre-peak stressstraincurve and a post-peak stresscrack opening diagram.The opening of a tensile crack forms a discontinuity inthe strain field in a tensile bar as shown in Fig. 6.Along the same lines, the compressive diagram must besplit in a pre-peak stressstrain curve and a post-peak

stress

displacement diagram because here also localisation of deformations has been observed 23 . TheRILEM TC 148 SSC Strain Softening of Concrete hasrecently paid attention to this phenomenon in an ex-tensive Round Robin test on fracture under uniaxial

compression 24 . Basically the separation of the com-pressive diagram in a pre- and post-peak curve wasconfirmed for a large range of concretes. The type ofloading platen used in the tests was found to have asignificant effect on stress, but the localisation pheno-menon was always found. In a uniaxial compressivetest, the amount of boundary restraint has a significanteffect on the measured compressive strength. Thestrength is overestimated when rigid steel platens are


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used, but when a teflongrease sandwich is insertedbetween the loading platen and the concrete, a size-in-

dependent compressive strength is measured 24 . Thismakes one wonder about claims about size effect oncompressive strength. The difference between uniaxial

compressive tests on prisms of different slenderness .loaded between low friction teflon and high friction

.steel loading platens is shown in Fig. 11. The questionmust now be posed which diagram should be used instructural analysis of reinforced concrete structures.An example is currently worked out by RILEM TC 148SSC, and concerns the behaviour of an over-reinforcedconcrete beam which fails in the compression zone.Researchers were asked to predict the behaviour of thebeam given the two different softening diagrams for

uniaxial compression i.e. one determined between steelplatens, the other between low friction teflon platens,

.both on prisms with slenderness 2.0 , as well as thetensile softening behaviour of the concrete and the

stressstrain curves for the reinforcing steel 25 . Theresults were discussed during the third InternationalConference on Fracture Mechanics of Concrete Struc-

.tures FraMCoS-3 , and some of the contributions appear in the proceedings of this meeting 26 . A compar-

ison will be published in the near future 27 .The same question, i.e. which diagram should be

used for structural analysis, can be posed for the tensilesoftening diagram also, although due to the impressiveamount of research carried out in the past decades,much is known already. The use of benchmark prob-

lems is encouraged, because in this way the certaintyabout the best input parameters in the numericalmodels used can only improve.

4. Conclusion

In this paper methods for determining the mechani-cal properties of materials are debated. In design,knowledge of the properties of materials is essential.Depending on the stage at which the design is, the

properties should be known to a more or lesser degreeof detail. Determining the material properties of con-crete, in particular the fracture properties is notstraightforward. The result is always affected by sizeand boundary condition effects, whereas the influence

of moisture distributions cannot be neglected either.The latter point has not been discussed in the present

paper, but can be found elsewhere, e.g. Foure 28 .It seems that the certainty about the material models

could be improved by adding an increasing amount ofdetail. In practice it means that where the engineering

models were used traditionally at the macro-level con-.tinuum , now one reverts to more scientific approaches,

for example by stepping down to the meso- or even tothe micro-level. At these levels numerical models are apractical tool to arrive at some solution. Whether thisis the real world or just an artefact of our imagination

must be settled by experiment. The alternative approach is to apply statistical models, e.g. Dyskin 29 ,where the information about the internal materialstructure is dealt with in a rather indirect way. In thatcase however, many aspects found in a direct simula-tion are missed because they are in some sensesmeared over a basic area or volume.

As mentioned, the role of experiment is important. Itis interesting to note that limits reached in generatingthe numerical material structures seem to correspondto limits reached in manufacturing real materials. Theexample given in the paper is the density of particles in

a mixture that can be encircled with a continuousmatrix. Other features of the fracture process from thenumerical simulations are recognised in experiments aswell. They are non-uniform macro-crack growth andcrack face bridging in specimens subjected to uniaxialtension. These specific crack features lead to size-de-pendent and boundary condition-dependent results. Incompression, similar phenomena are observed althoughsome of the details are different because friction playsa larger role. The use of benchmark problems andblind Round Robins may help to improve the accuracyof material properties used at the macroscopic level.

Fig. 11. Effect of specimen slenderness and boundary restraint on the stressstrain diagram in uniaxial compression 24 .


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Thus, a close interaction between experiment and com-putation seems essential for improving the reliability ofthe numerical models, which in the end should hope-fully lead to qualitative better designs. However, let usnot forget the role of engineering judgement as a

crucial human role in the process. This role remainsvague to date, in particular it is ignored in so-calledrational approaches. However, it should get more at-tention in the future.

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Documents

Experimentation, Numerical Simulation and the Role of Engineering Judgement in the Fracture Mechanics of Concrete and Concrete Structures