three-dimensional finite-element simulation of the dynamic brazilian tests on concrete cylinders

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 48:963–994

Three-dimensional �nite-element simulation of thedynamic Brazilian tests on concrete cylinders

Gonzalo Ruiz 1, Michael Ortiz2;∗;† and Anna Pandol� 31Departamento de Ciencia de Materiales; Universidad Polit�ecnica de Madrid; 28040 Madrid; Spain

2 Graduate Aeronautical Laboratories; California Institute of Technology; Pasadena; CA 91125; U.S.A.3Dipartimento di Ingegneria Strutturale; Politecnico di Milano; 20133 Milano; Italy

SUMMARY

We investigate the feasibility of using cohesive theories of fracture, in conjunction with the direct simulationof fracture and fragmentation, in order to describe processes of tensile damage and compressive crushing inconcrete specimens subjected to dynamic loading. We account explicitly for microcracking, the developmentof macroscopic cracks and inertia, and the e�ective dynamic behaviour of the material is predicted as anoutcome of the calculations. The cohesive properties of the material are assumed to be rate-independent andare therefore determined by static properties such as the static tensile strength. The ability of model to predictthe dynamic behaviour of concrete may be traced to the fact that cohesive theories endow the material withan intrinsic time scale. The particular con�guration contemplated in this study is the Brazilian cylinder testperformed in a Hopkinson bar. Our simulations capture closely the experimentally observed rate sensitivity ofthe dynamic strength of concrete in the form of a nearly linear increase in dynamic strength with strain rate.More generally, our simulations give accurate transmitted loads over a range of strain rates, which attests tothe �delity of the model where rate e�ects are concerned. The model also predicts key features of the fracturepattern such as the primary lens-shaped cracks parallel to the load plane, as well as the secondary profusecracking near the supports. The primary cracks are predicted to be nucleated at the centre of the circularbases of the cylinder and to subsequently propagate towards the interior, in accordance with experimentalobservations. The primary and secondary cracks are responsible for two peaks in the load history, also inkeeping with experiment. The results of the simulations also exhibit a size e�ect. These results validate thetheory as it bears on mixed-mode fracture and fragmentation processes in concrete over a range of strainrates. Copyright ? 2000 John Wiley & Sons, Ltd.

KEY WORDS: concrete; fracture; cohesive elements; dynamic strength; mixed mode fracture; size e�ect;strain rate e�ect

∗ Correspondence to: Michael Ortiz, Graduate Aeronautical Laboratories, California Institute of Technology, FirestoneFlight Sciences Laboratory, Pasadena, CA 91125, U.S.A.

† E-Mail: [email protected]

Contract=grant sponsor: Direcci�on General de Ensenanza Superior, Ministerio de Educaci�on y Cultura, SpainContract=grant sponsor: Army Research O�ce; contract=grant number: DAAH04-96-1-0056

Received 29 January 1999Copyright ? 2000 John Wiley & Sons, Ltd. Revised 15 September 1999

964 G. RUIZ, M. ORTIZ AND A. PANDOLFI

1. INTRODUCTION

The dynamic behaviour of brittle materials, including glasses, ceramics, rocks and concrete, sub-jected to high strain rates often involves the development of complex fracture and fragmentationpatterns. Cracks may nucleate from pre-existing aws or defects which populate the material inlarge numbers, or at stress concentrations induced by heterogeneities present in the material onthe microscale. Once nucleated, fracture may proceed in a distributed fashion, as microcracking,or in the form of a few dominant macrostructural cracks. The paths followed by such cracks areoften tortuous and undergo frequent branching, specially under dynamic conditions. Cracks maylink up with free surfaces or with each other to form fragments. In concrete, the microstructurallength scale is commensurate with the aggregate size and fracture processes often occur on a scalecomparable to the geometrical dimensions of the structure. In addition to the energy required tofracture the material, other sources of dissipation often operate simultaneously, including plasticworking, viscosity and heat conduction. Finally, microinertia often plays a signi�cant role in shap-ing the e�ective macroscopic behaviour of solids at high rates of deformation (e.g. References[1; 2]).Several avenues have been traditionally followed in order to model this complex behaviour. One

avenue is to attempt to bury all dissipative mechanisms into the constitutive relations. However, inpractice the e�ective behaviour of systems with complex microsctructures can only be characterizedanalytically, if at all, by recourse to sweeping simplifying assumptions. One particularly vexingdi�culty inherent to constitutive descriptions concerns their inability to endow brittle materialswith a well-de�ned fracture energy, i.e. a material-characteristic measure of energy dissipationper unit area of crack surface. Thus, the conventional thermodynamic route to the formulation ofgeneral constitutive relations regards dissipation as an extensive quantity possessing a well-de�neddensity per unit volume. Attempts to build fracture into such formulations by means of a softeningstress–strain law leads to pathologies such as a spurious scaling of the e�ective fracture energywith the discretization size. The formulation of appropriate functional forms of the energy of asolid which allow for both bulk and fracture-like behaviour is a challenging mathematical problemwhich is presently the subject of active research (e.g. References [3; 4]).Fracture mechanics speci�cally addresses the issue of whether a body under load will remain

intact or new free surface will form. However, the classical theory of fracture mechanics is pred-icated around the assumption of a pre-existing dominant crack. Under these and other restrictiveassumptions, fracture mechanics successfully predicts when a crack will grow, in what direc-tion and how fast, and how the results of laboratory experiments can be scaled up to structuraldimensions. However, the reliance on a pre-existing dominant crack entirely forgoes the issue ofnucleation, which is for the most part foreign to classical fracture mechanics. Situations, such as thisarise in fragmentation or crushing, involving many intersecting cracks also fall outside the purviewof classical fracture mechanics. Additionally, the conditions for crack growth are typically expressedin terms of parameters characterizing the amplitude of autonomous near-tip �elds. The applicabilityof these criteria is therefore contingent upon the existence and establishment of such near-tip �elds.This in turn severely restricts the scope of the theory, e.g. by requiring that the plastic or processzone be small relative to any limiting geometrical dimension of the solid such as the ligament size,conditions which are rarely realized in materials such as concrete. In addition, the structure of theautonomous near-tip �elds depends sensitively on the constitutive behaviour of the material, whichinextricably ties the fracture criteria to the constitution of the material. In dynamic fracture, the frac-ture criterion is additionally responsible for accounting for the microinertia which accompanies the

Copyright ? 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 48:963–994

THREE-DIMENSIONAL FINITE-ELEMENT SIMULATION 965

motion of the crack tip. Another complicating factor is non-proportional and cyclic loading, suchas arises in high-cycle fatigue, which renders important near-tip parameters, such as the J -integral,inapplicable.An alternative approach to fracture is based on the use of cohesive models to describe processes

of separation leading to the formation of new free surface [5–30]. Cohesive theories of fractureaddress many of the same issues contemplated by classical fracture mechanics while e�ectivelyovercoming most of the di�culties alluded to above. Indeed, cohesive models furnish a completetheory of fracture, and permit the incorporation into the material description of bona �de frac-ture parameters such as a fracture energy and a spall strength. By focusing speci�cally on theseparation process, a sharp distinction is drawn in cohesive theories between fracture, which isdescribed by recourse to cohesive laws, and bulk material behaviour, which is described throughan independent set of constitutive relations. The use of cohesive models is therefore not limitedby any consideration of material behaviour, �nite kinematics, non-proportional loading, dynamics,or the geometry of the specimen.Another appealing aspect of cohesive laws as models of fracture is that they �t naturally within

the conventional framework of �nite element analysis. However, one important numerical require-ment pertaining to the use of cohesive theories is that the cohesive zones must be adequatelyresolved by the discretization in order to obtain mesh-size-independent results [23]. In materialsfor which the characteristic cohesive length is small, this may necessitate substantial mesh re�ne-ment near crack tips. In materials such as concrete, by contrast, the characteristic cohesive lengthis comparatively large, typically of the order of 10 cm, and the mesh-size requirements are oftenless severe.The feasibility of using cohesive theories of fracture for the direct simulation of fragmentation

processes in brittle materials subjected to impact was established by Camacho and Ortiz [23].In this approach, individual cracks are tracked as they nucleate, propagate, branch and possiblylink up to form fragments. Clearly, it is incumbent upon the mesh to provide a rich enough setof possible fracture paths, an issue which may be addressed within the framework of adaptivemeshing. The ensuing granular ow of the comminuted material is also simulated explicitly. Inthis manner, the daunting task of developing constitutive relations which account for crushing andfragmentation is avoided altogether.The �delity of cohesive elements in applications involving dynamic fracture of ductile materials

has recently been investigated by Pandol� et al. [30], who have simulated the fragmentation ofexpanding aluminium rings. The numerical simulations have been found to be highly predictiveof a number of observed features, including: the number of dominant and arrested necks; thefragmentation patterns; the dependence of the number of fragments and the fracture strain on theexpansion speed; and the distribution of fragment sizes at �xed expansion speed. Pandol� et al. [31]have additionally shown that simulations of dynamic crack propagation based on cohesive modelsare highly predictive of a number of observed features, including: the crack growth initiation time;the trajectory of the propagating crack tip; and the formation of shear lips near the lateral surfaces.Repetto et al. [32] have applied cohesive models to the simulation of failure waves in glass rodssubjected to impact. Their calculations correctly capture the development and propagation of asharp failure wave, its propagation speed and the bursting of the comminuted material followingthe passage of the failure wave.The simulations of dynamic fracture of concrete available in the literature commonly model the

tensile strength as an increasing function of strain rate [33–43]. In addition, the fracture energyis often presumed to be constant and independent of strain rate [33; 44]. These rate-dependency



laws are necessarily empirical and endeavor to model the e�ective or macroscopic behaviour ofconcrete under dynamic loading.In this paper we investigate the feasibility of using cohesive theories of fracture, in conjunction

with the direct simulation of fracture and fragmentation, in order to describe processes of tensiledamage and compressive crushing in concrete specimens subjected to dynamic loading. We accountexplicitly for microcracking, the development of macroscopic cracks and inertia, and the e�ectivedynamic behaviour of the material is predicted as an outcome of the calculations. Indeed, thecohesive properties of the material are assumed to be rate-independent, and are therefore determinedby static properties such as the static tensile strength. The ability of model to predict key aspectsof the dynamic behaviour of concrete, such as the strain-rate sensitivity of strength may be tracedto the fact that cohesive theories, in addition to building a characteristic length into the materialdescription, they endow the material with an intrinsic time scale as well [23]. This intrinsic timescale permits the material to discriminate between slow and fast loading rates and ultimately allowsfor the accurate prediction of the dynamic strength of the material as a function of strain rate andother rate e�ects.The particular con�guration contemplated in this study is the Brazilian cylinder test performed in

a Hopkinson bar. Dynamic Brazilian tests performed using a split-Hopkinson pressure bar (SHPB)have been proposed as a convenient means of determining the tensile strength of cohesive materials[41; 37; 36; 45–47]. These tests have yielded a wealth of information on the dynamic behaviourof concrete. For instance, it is well-established that the e�ective tensile strength of the materialincreases with strain rate [41; 37; 36; 33]. By contrast, the fracture energy is ostensible insensitiveto strain rate [33; 44]. Detailed measurements of the near-tip stress �elds attendant to crackspropagating in mode I and mixed mode [35] have shown that the length of the cohesive orcrack-bridging zone at a crack tip decreases with increasing strain rate.Dynamic Brazilian tests therefore furnish a convenient yet exacting validation test for cohesive

theories of fracture applied to concrete. Our simulations give accurate transmitted loads over arange of strain rates, which attests to the �delity of the model where rate e�ects are concerned.The model also predicts key features of the fracture pattern such as the primary lens-shaped cracksparallel to the load plane, as well as the secondary profuse cracking near the support. The primarycracks are predicted to be nucleated at the centre of the circular bases of the cylinder and tosubsequently propagate towards the interior, in accordance with experimental observations. Theprimary and secondary cracks are responsible for two peaks in the load history, also in keepingwith experiment. The results of the simulations also exhibit a size e�ect, i.e. a dependence of thee�ective behaviour on the size of the specimen [48]. These results validate the theory as it bearson mixed-mode fracture and fragmentation processes in concrete over a range of strain rates. Oursimulations also provide useful insights into the interpretability of the dynamic Brazilian test, andthe sensitivity of the measurement to details of the experimental design, issues which have beenaddressed extensively in the experimental literature [49–52].The organization of the paper is as follows. A brief outline of the particular form of the cohesive

models and their �nite-element implementation employed in calculations is given in Section 2. InSection 3 we begin by compiling the main parameters which de�ne the simulations, includingthe specimen geometry, material parameters, load and boundary conditions, and the details of themesh design. The section closes with a detailed comparison of the results of the calculationswith corresponding experimental observations, including load and energy histories, crack patternsand estimated crack velocities. In Section 4 we proceed to explore the e�ect of variations inselected parameters of the model, including strain rate, specimen size, the width of the bearing



Figure 1. Cohesive surface traversing a 3D body. Figure 2. Loading–unloading rule from linearly de-creasing loading envelop expressed in terms of ane�ective opening displacement � and traction t.

strips, specimen size and the ratio � of modes II to I toughness. Finally, a summary and someconcluding remarks are collected in Section 5.

2. FINITE-ELEMENT MODEL

By way of a general framework, we start by considering a deformable body occupying an initialcon�guration B0⊂R3. The boundary @B0 of the body is partitioned into a displacement boundary@B0 ;1 and a traction boundary @B0 ;2. The body undergoes a motion described by a deformationmapping e :B0× [0; T ]→R3, where [0; T ] is the duration of the motion, under the action of bodyforces �0b and prescribed boundary tractions �t applied over @B0 ;2. The attendant deformationgradients are denoted F and the �rst Piola–Kirchho� stress tensor P (cf. e.g. Reference [53]).In addition, the solid contains a collection of cohesive cracks. The locus of these cracks on theundeformed con�guration is denoted S0 (Figure 1).Under these conditions, the weak form of linear momentum balance, or virtual work expression,

takes the following form:

∫B0[�0(b− �e) · W− P · ∇0W] dV0 −

∫S0t · <W= dS0 +

∫@B0 ; 2

�t · W dS0 = 0 (1)

where a superposed dot denotes the material time derivative, ∇0 is the material gradient, W isan arbitrary virtual displacement satisfying homogeneous boundary conditions on @B0 ;1; t are thecohesive traction over S0, and < · = denotes the jump across an oriented surface.From (1) is evident that the presence of a cohesive surface results in the addition of a new

term to the virtual work expression. In order to complete the de�nition of the problem, a set ofconstitutive relations for the cohesive tractions t must be provided, in addition to the conventional



constitutive relations describing the bulk behaviour of the material. To this end, we postulate theexistence of a free energy density per unit undeformed area over S0 of the general form

�=�(T; �; q; n) (2)

where

T= <e= (3)

are the opening displacements over the cohesive surface, � is the local temperature, q is somesuitable collection of internal variables describing the current state of decohesion of the surface, andn is the unit normal to the cohesive surface in the deformed con�guration. The explicit dependenceof � on n is required to allow for di�erences in cohesive behaviour for opening and sliding. Byrecourse to the Coleman and Noll method (e.g. References [54; 55]) it is possible to show thatthe cohesive law takes the form

t=@�@T (4)

The potential structure of the cohesive law is a consequence of the �rst and second laws ofthermodynamics. The evolution of the internal variables q is governed by a set of kinetic relationsof the general form

q= f(T; �; q) (5)

A more general class of free energies which allows for surface anisotropy and �nite openingdisplacements have been considered by Ortiz and Pandol� [28].Following Camacho and Ortiz [23] and others [27; 28], we consider a simple class of mixed-

mode cohesive laws accounting for tension–shear coupling obtained by the introduction of ane�ective opening displacement:

�=√�2�2S + �2n (6)

where

�n = T · n (7)

is the normal opening displacement and

�S = |TS|= |T− �nn| (8)

is the magnitude of the sliding displacement. The parameter � assigns di�erent weights to thesliding and normal opening displacements.Assuming that the free energy potential � depends on T only through the e�ective opening

displacement �, i.e.

�=�(�; �; q) (9)



the cohesive law (4) reduces to

t=t�(�2TS + �nn) (10)

where

t=@�@�(�; �; q) (11)

is a scalar e�ective traction, which expression follows from (6) and (10):

t=√�−2|tS|2 + t2n (12)

where tS and tn are the shear and the normal traction, respectively. From this relation, we observethat � de�nes the ratio between the shear and the normal critical tractions. In brittle materials, thisratio may be estimated by imposing lateral con�nement on specimens subjected to high-strain-rateaxial compression [56; 57]. In addition, the parameter � roughly de�nes the ratio of KIIc to KIc ofthe material.Upon closure, the cohesive surfaces are subjected to the contact unilateral constraint, including

friction. We regard contact and friction as independent phenomena to be modelled outside thecohesive law. Friction may signi�cantly increase the sliding resistance in closed cohesive sur-faces. In particular, the presence of friction may result in a steady—or even increasing—frictionalresistance while the normal cohesive strength simultaneously weakens.Figure 2 depicts a particular type of irreversible cohesive laws envisioned here; �c is the tensile

strength and �c the critical opening displacement. Irreversibility manifests itself upon unloading.Therefore, an appropriate choice of internal variable is the maximum attained e�ective openingdisplacement �max. Loading is then characterized by the conditions: �= �max and �¿0. Conversely,we shall say that the cohesive surface undergoes unloading when it does not undergo loading. Weassume the existence of a loading envelop de�ning a relation between t and � under conditions ofloading. A simple and convenient relation is furnished by the linearly decreasing envelop shownin Figure 2. Following Camacho and Ortiz [23] we shall assume unloading to the origin, Figure 2,giving

t=tmax�max

� if � ¡ �max or � ¡ 0 (13)

For the present model, the kinetic relations (5) reduce to a straightforward computation of �max.It is a well-known fact [5–8; 58] that cohesive theories introduce a well-de�ned length scale

into the material description and, in consequence, are sensitive to the size of the specimen. Thecharacteristic length of the material is

‘c =EGcf2ts

(14)

where Gc is the fracture energy and fts the static tensile strength. Camacho and Ortiz [23] havenoted that in conjunction with inertia cohesive models introduce a characteristic time as well.This characteristic or intrinsic time is

tc =�c�c2fts

(15)



Figure 3. Geometry of cohesive element. The surfaces S− and S+ coincide in the referencecon�guration of the solid.

where � is the mass density and c the longitudinal wave speed. Owing to this intrinsic time scale,the material behaves di�erently when subjected to fast and slow loading rates. This sensitivityto the rate of loading confers to the cohesive models the ability to reproduce subtle features ofthe dynamic behaviour of brittle solids, such as crack-growth initiation times and propagationspeeds [31]; and the dependence of the pattern of fracture and fragmentation on strain rate [30].In addition, the calculations presented subsequently demonstrate the ability of cohesive theories toaccount for the dynamic strength of brittle solids, i.e. the dependence of the dynamic strength onstrain rate.Another appealing aspect of cohesive laws as models of fracture is that they �t naturally within

the conventional framework of �nite-element analysis. We follow Camacho and Ortiz [23] andadaptively create new surfaces as required by the cohesive model by duplicating nodes alongpreviously coherent element boundaries. The introduction of cohesive surfaces may result in drasticchanges in the topology of the model [29]. The nodes are subsequently released in accordancewith a tension–shear cohesive law. The particular class of cohesive elements used in calculationshas been developed by Ortiz and Pandol� [28] and consists of two six-node triangles endowedwith quadratic displacement interpolation (Figure 3).Inserting the displacement interpolation into the virtual work expression (1) leads to a system

of semi-discrete equations of motion of the form

M �x + f int(x)= fext(t) (16)

where x is the array of nodal co-ordinates, M is the mass matrix, fext is the external force array,and f int is the internal force array. In calculations we use the second-order accurate central di�er-ence algorithm to discretize (16) in time [59–61]. Despite the fact that the time step is boundedby stability [60], explicit integration is particularly attractive in three-dimensional calculations,where implicit schemes lead to system matrices which often exceed the available in-core storagecapacity. Yet another advantage of explicit algorithms is that they are ideally suited for concurrentcomputing [62].



3. APPLICATION OF COHESIVE MODEL TO THE PREDICTION OFTHE DYNAMIC BEHAVIOUR OF CONCRETE

We proceed to simulate the dynamic behaviour of concrete using the cohesive model just described.The particular con�guration contemplated in the study is the Brazilian cylinder test performed in aHopkinson bar. The simulations give accurate transmitted loads over a range of strain rates, whichattests to the �delity of the model where rate e�ects are concerned. The model also predicts keyfeatures of the fracture pattern such as the primary lens-shaped cracks parallel to the load plane,as well as the secondary profuse cracking near the support. The primary cracks are predicted to benucleated at the centre of the circular bases of the cylinder and to subsequently propagate towardsthe interior, in accordance with experimental observations. The primary and secondary cracks areresponsible for two peaks in the load history, also in keeping with experiment.These results validate the theory as it bears on mixed-mode fracture and fragmentation processes

in concrete over a range of strain rates. It is particularly noteworthy that neither the bulk materialbehaviour, which remains essentially elastic throughout the calculations reported here, nor thecohesive law are themselves rate dependent. As noted earlier, cohesive laws, in conjunction withinertia, endow the material with a characteristic or intrinsic time scale, an attribute which ultimatelyaccounts for the ability of cohesive theories to capture rate e�ects accurately.

3.1. Experimental set-up

Dynamic Brazilian tests performed using a split-Hopkinson pressure bar (SHPB) have been pro-posed as a convenient means of determining the tensile strength of cohesive materials[41; 37; 36; 45–47]. The SHPB consists of an incident bar and a transmitter bar, with a shortspecimen placed between them. In addition, a striker bar impacts the incident bar and producesa longitudinal compressive pulse which propagates toward the specimen (Figure 4(a)). The spec-imen is typically in the form of a cylinder—although other geometries can be used, includingcubes and prisms—and it is placed transversely to the bars, i.e. such that the axis of the specimenis perpendicular to the axis of the bars. The specimen is connected to the bars by two bearingstrips (Figure 4(b)). The amplitude and duration of the pulse depends on the material properties,velocity and length of the striker. The pulse is partially re ected back into the incident bar, andpartially transmitted through the specimen. As is well known [63], under quasistatic conditions thediametral loading of a cylindrical specimen generates tension perpendicular to the loading planewhich eventually causes the specimen to split. This quasistatic solution is often taken as a basisfor interpreting the dynamic experiments as well [51].The bars are designed to remain elastic throughout the test, and by virtue of their slenderness,

one-dimensional stress-wave theory applies to the bars to a good approximation. The strain cor-responding to the incident, re ected and transmitted pulses may then be measured using gaugesglued to the surface of the bars, and these signals, recorded by means of an oscilloscope anda data acquisition system, may be used to calculate the corresponding stress pulses (Figure5). In addition, the dynamic splitting tensile stress, ftd, may be inferred from the followingequation:

ftd =2Pmax�WD

(17)

where Pmax is the maximum load transmitted through the cylinder and W and D are the width anddiameter of the cylinder, respectively. Pmax is in turn calculated from the maximum transmitted



Figure 4. Experimental set-up (a), and details of the specimen (b).

Figure 5. Typical stress pulses (adapted from Reference [41]). Figure 6. Simpli�ed inci-dent load pulse used in

the analysis.

stress, �max, with the result

Pmax = �R2�max (18)

where R is the SHPB radius. The strain rate follows as

�=ftdE�

(19)

where E is the elastic modulus of the specimen material and � is the time delay between the startand the peak of the transmitted pulse.



Table I. Concrete parameters.

Density (in Reference [40]) 2405 kg=m3

Elastic modulus 37:9GPaAcoustic velocity 3620m=s

Static compressive strengthCylinder 152× 305mm 48:3MPaCylinder 51× 51mm 57:1MPa

Static splitting tensile strengthCylinder 51× 51mm 3:86MPa

Static direct tensile strengthCylinder 51× 51mm, square notch 4:53MPa

Fracture energy 66:2N=mCharacteristic length 122mm

3.2. Specimen geometry and material parameters

We speci�cally aim to simulate the experiments reported by Hughes et al. [37], Tedesco et al.[41] and Ross et al. [36]. The specimens in these experiments were concrete cylinders of 50:8mmin diameter and height obtained by coring from a concrete block. The maximum aggregate sizewas 8:5mm, or 1

6 of the specimen diameter. The corresponding material parameters are listed inTable I. All these parameters were measured through independent tests with the exception of thefracture energy, which we have estimated following the recommendations set forth in the ModelCode [64].It is important to bear in mind the uncertainties which are inevitably inherent to the material

properties listed in Table I. Thus, for instance, the measured compressive strength depends onboth the shape and size of the specimen, ranging from 48.3MPa, corresponding to the standardspecimen [65], to 57.1MPa, when the 51×51mm cylinder is used. This variation is in agreementwith the expected size e�ect in concrete, namely, an increasing strength with decreasing specimensize and increasing sti�ness of the loading device. In the calculations of interest here, however,the response of the specimen is dominated by tensile cracking and the compressive strength of thematerial is seldom reached. Under these conditions, the bulk behaviour may simply be modelledas elastic to a good approximation (see, e.g., Reference [48; Section7:1:6]). As a slight—and fairlyinconsequential—improvement, we assume that the material obeys J2-plasticity upon the attainmentof its compressive strength. Details of the speci�c implementation of J2-plasticity employed incalculations may be found elsewhere [66].The tensile strength is highly dependent on the testing technique and conventions adopted for

its measurement. The static direct tensile strength of 4.53MPa listed in Table I was measuredby Hughes et al. [37], Tedesco et al. [41] and Ross et al. [36] using cylinders similar to thoseemployed in the dynamic splitting tests. The specimens included a circumferential square notch3:2mm deep around their mid-section. In addition, the same authors performed static Brazilian testson specimens of the same geometry and measured a tensile strength of 3:86MPa. However, it hasbeen shown by Rocco [52] that the static Brazilian test yields estimates of the tensile strength whichare strongly dependent on the shape and size of the specimen, and on the boundary conditions,



and are, therefore, unreliable. In view of these uncertainties, we have employed the tensile strengthof 4:53MPa measured by direct tensile testing on notched specimens.The material is assumed to obey the linear irreversible cohesive law shown in Figure 2 with the

measured tensile strength taken as the cohesive strength of the material. According to this law, thematerial remains coherent up to the attainment of its tensile strength. This criterion is veri�ed atall interfaces between tetrahedral elements throughout the calculations. When the criterion is met,a cohesive element is inserted at the interface, which subsequently opens in accordance with thelinearly decreasing cohesive law. This approach is particularly well suited to explicit dynamics asit does not a�ect the critical time step.There is experimental evidence [67–69] which suggests that the intrinsic fracture toughness of

concrete, i.e. the critical stress intensity factor required to advance a semi-in�nite crack within itsplane in the absence of kinking, is much larger in pure mode II that in pure mode I owing to theinterlocking of aggregate particles. This in turn suggests adopting a large value of the couplingparameter � in (6), since, as remarked earlier, � gives the ratio of modes II to I fracture toughness.Based on a suggestion by Gustafsson and Hillerborg [70] on the relative strengths of concrete intension and shear, in calculations we take �=10. Under the assumptions just stated, a semi-in�nitecrack subjected to mixed-mode loading will tend to kink at an angle roughly corresponding to themaximum circumferential stress in its K-�eld. Indeed, the maximum circumferential stress criterionis known to lead to accurate predictions of crack paths in concrete [71; 72].It should be carefully noted that, under the conditions envisioned in this study, and more gener-

ally when dealing with the fracture of concrete, conventional concepts from linear-elastic fracturemechanics must be applied with caution or not at all. Thus, from (14) and the properties listedin Table I one computes a characteristic length for concrete of 122mm, which is larger thanthe diameter of the specimen, or 50:8mm. The conditions of the analysis correspond to a ‘fullyyielded’ situation and linear-elastic fracture mechanics, which fundamentally rests on the small-scale yielding condition, does not apply. Cohesive theories of fracture, by way of contrast, are notso constrained, and can be applied to situations such as envisioned here in which the size of theprocess zone is comparable to the ligament length or any other limiting geometrical dimension.Further discussions of this and related issues may be found in References [73; 20; 48].The simulations of dynamic fracture of concrete available in the literature commonly model the

tensile strength as an increasing function of strain rate [33–43]. In addition, the fracture energyis often presumed to be constant and independent of strain rate [33; 44]. These rate-dependencylaws are necessarily empirical and endeavor to model the e�ective or macroscopic behaviour ofconcrete under dynamic loading. By way of contrast, we account explicitly for microcracking, thedevelopment of macroscopic cracks and inertia, and the e�ective dynamic behaviour of the materialis predicted as an outcome of the calculations. In particular, we stress again that the cohesive law,despite being rate-insensitive, endows the material with an intrinsic time scale which discriminatesbetween slow and fast loading rates and ultimately allows for the accurate prediction of the dynamicstrength of the material as a function of strain rate and other rate e�ects.

3.3. Load and boundary conditions

We consider several loading cases covering a range of load levels and strain rates and correspond-ing to the tests reported by Hughes et al. [37], Tedesco et al. [41] and Ross et al. [36]. In theactual tests, the width of the steel bearing strips connecting the specimen to the bars equalled 1

8of the diameter of the cylinder. The load applied to the specimen—and the load transmitted by



Table II. Parameters for the incident load pulses.

Rise time Duration Stress level Related velocityLoad case no. tr (�s) td (�s) �i (MPa) v (m=s)

1 in Reference [41]—1 66 100 60.2 1.52 in Reference [41]—2 72 100 72.8 1.81 in Reference [37]—3 80 100 79.4 2.02 in Reference [37]—4 85 100 122.5 3.13 in Reference [37]—5 41 100 184.5 4.73 in Reference [41]—6 48 100 264.3 6.7

it—may be readily obtained from the strain recorded by the gages glued to the bars. The loadingpulse can be idealized as a linearly rising ramp followed by a plateau (Figure 6).In calculations it is convenient to simply prescribe displacements at the contact between the

striker bar and the specimen. To this end, the prescribed velocity at the contact may be estimatedas [74]

v=�i�c

(20)

where � and c are the one-dimensional density and wave speed of the incident bar, respectively, and�i is the incident stress. Equation (20) simply amounts to requiring that the motion of the specimenmatches the incident wave pro�le. The rise time, duration, and stress and velocity levels for theloading cases under consideration are collected in Table II. By way of contrast, the contact betweenthe specimen and the transmitter bar is assumed to be rigid, leading to constrained displacementsover the width of the transmitter bearing strip. This simpli�cation is warranted by the observationthat the transmitted pulses measured in the tests were very weak, roughly one order of magnitudelower than the incident pulses.

3.4. Computational mesh

Figure 7 shows the mesh used in the calculations. All the surfaces and the interior of the specimenare meshed automatically by the advancing front method [75] using 10-node quadratic tetrahedra.It should be carefully noted that the midplane—or load plane—of the specimen, which is a likelycrack path is not built into the mesh. Instead, the load plane is crossed by tetrahedra, (Figure7(b)), which forces some degree of roughness on any macroscopic crack contained within it. Post-mortem examination of actual concrete specimens tends to reveal a similar degree of roughnessin the form of protruding aggregate particles. The computational mesh comprises 8378 nodes and5669 tetrahedra, (Figure 7(a)), and is designed so as to be �ne and nearly uniform on and in thevicinity of the load plane, with a size commensurate with half the maximum aggregate size, andto gradually coarsen away from the load plane up to the maximum aggregate size.It is important to note that the mesh size roughly ranges from 1

15 to130 of the characteristic co-

hesive length (14) of the material and may, therefore, be expected to yield objective and mesh-sizeinsensitive results [23]. In order to verify this point, we have compared test solutions obtained fromthree di�erent meshes: the computational mesh just described and two increasingly �ner meshes.The results of the three tests are collected in Figure 8. The maximum transmitted load obtained



Figure 7. External appearance of the mesh (a), split specimen by the load plane (b) andby the middle circular cross-section (c).

Figure 8. A comparison of numerical results corresponding to three meshes of varying degrees of re-�nement, illustrating the sensitivity—or lack of it thereof—of the various phases of the solution to meshsize. (a) Histories of transmitted load and expended cohesive energy, and (b) asymptotic behaviour of the

cohesive energy consumption for the same mesh sizes.



Figure 9. Comparison of experimental and numerical rate-sensitivity curves and dynamic strength of concrete.

from the intermediate and �ne meshes di�ers from that obtained from the coarser computationalmesh by −2 per cent and +0:6 per cent, respectively, which we take to be acceptable error. Inaddition, in all cases the cohesive energy consumption curves di�er negligibly up to the peak load,which provides an indication of the good accuracy of the coarse solution.It is interesting to note that the post-peak solution, which is characterized by extensive micro-

cracking and macroscopic crack propagation and is therefore inherently unstable, is highly sensitiveto small changes in the parameters of the model, including the details of the mesh design. Thus,because of the vast number of possible paths which the system can choose from, correspondingto variations in local microcracking and fragmentation patterns, the solution inevitably exhibits amarked random or stochastic character in the unstable region following the peak load. Indeed,short-term and local uctuations in the solution exhibit broad variation even between meshes pos-sessing the same element-size distribution. Under these highly stochastic conditions, convergenceof the solution can only reasonably be understood in terms of aggregate or average macroscopicquantities such as energy, dissipation, microcrack densities, and other similar features. For instance,in calculations the accumulated cohesive energy expenditure remains within a narrow band andconverges to a mesh-independent asymptote independent of mesh size (Figure 8(b)), which atteststo the requisite convergence in the mean.

3.5. Simulation results

Selected results of the calculations and comparisons with experimental data are shown inFigures 9–11. The main features of these results, as regards dynamic strength load histories, crackpatterns, and crack velocity are next discussed in turn.

3.5.1. Dynamic strength and rate sensitivity. In keeping with the experimental procedure, weidentify the dynamic strength of the material with the peak transmitted load. Our computation ofthe strain rate is based on the time interval between the start and the peak of the transmittedload wave. In order to avoid spurious e�ects due to the bearing strip-cylinder contact, we haveestimated the experimental starting time of the load wave as the intersect of the tangent to thelinear ramp-up with the time axis.



Figure 10. Comparison of experimental and numerical transmitted load and energy histories for loadingcase 2. (b) Experimental crack-pattern sequence [41].

Figure 9 compares the predicted and observed ratio of static to dynamic strengths for all loadingcases under consideration, and the dependence of the dynamic strength on strain rate. In addition,the curve inset in Figure 9(a) represents a linear �t to the experimentally observed strain-ratedependence of the dynamic strength. As may be seen from Figure 9(a), the calculations capturewell the overall rate sensitivity of the material, which takes the form of a steady rise in dynamicstrength with increasing strain rate. It is interesting to note that the simulations corresponding toloading cases 1–3 yield comparable results (Figure 9(a)), which may be due to the compensatinge�ect of a simulataneous increase in rise time and impact velocity in the load pulse (cf. Table II).The accuracy in the calculation of the peak transmitted load and, by extension, of the dynamicstrength, Equation (17), is equally satisfactory, with the exception of loading case 1 (Figure9(b)). Interestingly, the match between the numerical and experimental maximum transmitted loadsimproves at high strain rate, with the error decreasing below 10 per cent for load cases 5 and 6.This trend may be attributed to the fact that the e�ective opening displacements of the cracks atthe peak load tend to be larger at low strain rates, with the result that the computed peak load issensitively dependent on the shape of the cohesive law. Contrarily, at high strain rates the openingdisplacements are comparatively smaller and the peak load correlates more closely with the tensilestrength. This suggests that the �delity of the predicted peak loads at low strain rates may beimproved by a more realistic choice of cohesive law.Again, it bears emphasis that these features of the dynamic behaviour of concrete are predicted

by—and not built into—the theory. Indeed, the sole strength parameter of the theory is the staticstrength of concrete. In addition, by itself, the cohesive law of the material is rate-independent. Thegood qualitative and quantitative agreement between theory and experiment just reported suggeststhat microinertia associated with microcracking and dynamic fracture is the dominant mechanismunderlying the e�ective rate sensitivity of concrete.It is evident in Figure 9 that the experimental scatter is substantial, specially where load cases 1

and 6 are concerned. Thus, while the loading conditions corresponding to load case 1 are ostensiblysimilar to those corresponding to load case 2 (cf. Table II), the strain rate and peak load reportedfor the former are considerably lower that those reported for the latter. In particular, the dynamic



Figure 11. Development of fracture patterns on the surface and in the interior of the specimen. Displace-ments are magni�ed by a factor of 100 to aid visualization. The damage �eld represents the fraction of

expended fracture energy to total fracture energy per unit surface.

strength reported for load case 1 is below the static strength, which constitutes a clear anomaly.In addition, the strain rate reported for load case 6 is abnormally low in comparison to that forload case 5, which corresponds to a �i roughly 50 per cent lower than that of load case 6, tr beingnearly identical in both cases.These anomalies, and the overall scatter of the experimental data, attest to the di�culties inherent

to the testing of highly heterogeneous materials such as concrete. For instance, nominally identicalloading conditions may result in sharply di�ering local conditions at the supports, leading tonoticeable di�erences in the overall response of the specimens. These experimental uncertainties,as well as the uncertainties in the computed solution alluded to above, must be borne in mind inmaking meaningful comparisons between the numerical predictions and experimental data.



Figure 11. Continued.

3.5.2. Load and energy history curves. A comparison between computed and experimental loadhistories for loading case 2 is shown in Figure 10(a). The two histories match nearly exactly duringthe early stages of loading. The discrepancy in the peak loads, of the order of 17 per cent, and inthe slope of the ramp portion of the record are well within the experimental scatter. Indeed, locale�ects near the bearing strips are notoriously di�cult to control experimentally, which detractsconsiderably from the reproducibility of the ramp slope. In view of the good match between theexperimental and computed strain rates for loading case 5, we may reasonably surmise that theexperimental load history is likewise better captured in that case. Regrettably, that history was notreported in the original publications.It is interesting to note that the time required for elastic pressure waves to travel through the

diameter of the cylinder is of the order of 14 �s. In view of the smoothness of the rising partof the transmitted load history, it would appear that the number of reverberations which take



Figure 11. Continued.

place during the ramp-up time of the loading pulse is su�cient to establish a rather uniform stateof deformation within the specimen. It follows as a corollary that the waviness of the softeningpart of the load history curve is due to fracture processes rather than wave e�ects. Furthermore,the absence of signi�cant wave e�ects justi�es the use of the test data for purposes of inferringconstitutive properties of the material.The horizontal dash line inset in Figure 10(a) represents the maximum load attained during the

static Brazilian test and is shown for reference. As may be seen from the �gure, the dynamic peakload is of the order of 2.71 times the static maximum load, which attests to the importance of thedynamic e�ects under the conditions of the test.Figure 10(a) also depicts the consumption of cohesive energy and the kinetic energy as a

function of time for the same loading case. The various stages of cracking observed experimentally



are also shown in Figure 10(b) for comparison. As is evident from Figure 10(a), the kineticenergy initially builds up to the initiation time of the main crack contained within the mid-plane of the specimen. The kinetic energy subsequently increases slowly during the growth of themain crack. The peak load is attained when the main crack extends clear through the specimen,Figures 10(a) and 10(b), point 7. It therefore follows that the dynamic growth of the main crack isstable in the sense of requiring an increasing applied load to proceed. By way of contrast, followingthe peak load, a sharp upturn in the kinetic energy occurs while the applied load simultaneouslydecreases. This phase of the energy-history record is accompanied by a marked increase in theexpenditure of cohesive energy (Figure 10(a)). These features are indicative of profuse and unstablemicrocracking—hence the high cohesive energy expenditure—resulting in widespread dynamicfragmentation—hence the loss of load-carrying capacity and increase in kinetic energy. Eventually,the specimen loses all load-bearing capacity, the fragmentation process arrests and the expendedcohesive energy saturates asymptotically to a constant value (Figure 8(b)).The horizontal dash line shown for reference in Figure 10(a) represents the fracture energy

expended in the formation of a single planar crack cutting through the mid-plane of the specimen.As may be observed, the actual fracture energy expended is greatly in excess of that value, whichis indicative of a far more complex and intricate crack pattern. It is also interesting to note thatno energy is dissipated through plastic work at any time during the calculations. Hence, cohesivefracture accounts for the totality of energy dissipation.

3.5.3. Crack pattern. The predicted sequence of crack patterns also follows closely the experi-mental patterns observed by means of a high-speed camera (Figure 10(b)) [41]. Figure 11 depictsa sequence of snapshots of the deformed specimens at intervals of 10 �s, showing the distributionof cracks. It should be carefully noted that, in this plot, displacements have been magni�ed bya factor of 100 in order to aid visualization. It may also be recalled that the peak load occursroughly at 70 �s, corresponding to snapshot (Figure 11(f)). Also shown in the �gures are levelcontours of damage, de�ned as the fraction of expended fracture energy to total fracture energy perunit surface, or critical energy release rate. Thus, a damage density of zero denotes an uncrackedsurface, whereas a damage density of one is indicative of a fully cracked or free surface. It bearsemphasis that this damage �eld is de�ned on any internal surface of the body, i.e. it represents adensity per unit area—as opposed to a density per unit volume. In Figure 11 we have chosen torepresent the extent of damage on the mid-plane, or load plane, of the specimen.Remarkably, both the experimental observations and the numerical solution clearly exhibit a

main crack on the mid-plane of the specimen which initiates near the centre of the cylinder andsubsequently propagates towards the bearing strips, eventually causing the specimen to split intotwo main fragments (cf. Figure 10(b)). The observed initiation time is roughly 30 �s, which isin fair agreement with the results of the calculations (Figures 11(b) and 11(c)). Furthermore,the simulation also captures some early localized cracking in the loading area (Figures 10(b)and 11(b)). Evidently, the numerical simulation provides a wealth of information regarding thethree-dimensional character of the evolving fracture patterns which is inaccessible to experimentaldiagnostics, which are for most part limited to the observation of the surface of the specimen. Inthis regard it is interesting to note that, as expected, the initiation and growth of the main crackis far from being uniform through the width of the specimen. Indeed, our simulations suggest thatlenticular cracks initiate from the surface of the specimen, i.e. the ends of the cylinder (Figures11(c) and (d)) and subsequently propagate inward within the mid-plane with increasing load.Eventually, the surface cracks coalesce and form a single through-crack (Figures 11(d) –11(f)).



The development of the main crack on the mid-plane of the specimen is by no means a cleanprocess. Thus, the main crack is in constant competition with incipient cracks initiating on adjacentparallel planes, e.g. Figure 11(d). However, these competing cracks are quickly shut out by theunloading set forth by the dominant crack and their growth is stunted. A similar process of ‘trialcracks’ has been observed by Xu and Needleman to accompany the dynamic growth of cracks [17].The softening regime is likewise far from monotonic and often exhibits multiple peaks, which is

indicative of a halting—rather than smooth—crack propagation. Thus, as noted earlier, the �rst peakload coincides with the formation of a well-developed through-crack. This event generates reliefwaves which temporarily halt the splitting process. The specimen subsequently reloads, inducingfurther crack growth and a new peak load is attained (Figures 10(a) and 12).Figures 11(e)–11(f) further reveal the development of profuse cracking at the impact point on the

surface of the specimen, in keeping with observation (Figure 10). As noted earlier, this generalizedmicrocracking, which often leads to fragmentation, occasions a sharp upturn in energy dissipation.The ability to account for such complex fracture patterns with relative ease is a remarkable featureof cohesive theories of fracture.

3.5.4. Crack velocity. Tedesco et al. [41] also reported crack speeds estimated from high-speedcamera pictures. These rough crack-speed estimates range from 10 to 50 per cent of the Rayleighwave speed, which equals 3620m=s for the concrete used in the tests. An identical procedureapplied to loading case 5 returns a crack speed of 1320m=s, or 36 per cent the Rayleigh wavevelocity, which is within the experimental range.

4. PARAMETRIC STUDY

The preceding comparisons with experiment may be regarded as validation of the cohesive theoryof fracture adopted in the calculations as a model of the dynamic behaviour of concrete. Inparticular, the predictions of the theory have been found to be in good agreement with severalsalient features of the experimental record, including the crack pattern and the transmitted loadhistories measured by Tedesco et al. [41] in a split Hopkinson-bar Brazilian test con�guration,and the attendant e�ective rate sensitivity of the dynamic strength of concrete. Now that the�delity of the model has been established, we proceed to apply the model predictively with aview to exploring the e�ect of variations in selected parameters of the model, including strainrate, specimen size, the width of the bearing strips, the ratio � of modes II to I toughness andspecimen geometry. This parametric study reveals useful insights into the interpretability of thedynamic Brazilian test and the role of material parameters in shaping the dynamic behaviour ofconcrete.We de�ne a base case with respect to which we introduce systematic variations in parameters

of interest. The specimen geometry in the base case is as previously considered. However, inorder to speed up calculations, we con�ne the analysis to a thin slice of the cylinder of thicknessD=12, and constrain the normal displacement on the lateral surfaces of the slice. Thus, whereasthe calculations are carried out in three dimensions, the conditions of the analysis approximate aplane-strain constraint such as may be expected to be in force at the centre of the cylinder. Thematerial properties remain as listed in Table I. The loading pulse at the end of the incident bar isidealized as a linearly rising ramp in velocity of a duration of ti=50 �s, followed by a constantplateau at 2m=s maintained up to a time td = 100 �s (Figure 6). The boundary conditions remain



as described in Section 3.3. The mesh design conforms to the pattern described in Section 3.4.However, owing to the restriction of the analysis to a single slice of the cylinder the computationalmesh is conveniently reduced to 1231 nodes and 630 tetrahedra only.Finally, throughout this section results are presented in dimensionless form in order to facilitate

comparisons between di�erent cases. To this end, we de�ned the normalized transmitted load P as

P∗=2P

�WDfts(21)

With this convention, P∗ gives the ratio ftd=fts at peak load, and P∗=1 when the transmitted loadis equal to the theoretical maximum static load. We choose to normalize the cumulative cohesiveenergy Wcoh by the fracture energy required to cleave the cylinder along its load plane into twoequal fragments with the result:

W ∗coh =

WcohWDGc

(22)

Finally, we de�ned the normalized time as

t∗=ttc

‘cD

(23)

where tc and ‘c are the characteristic time and length of the material, respectively (Equations (15)and (14)). The factor ‘c=D is introduced in (23) in order to bring into coincidence the initialstages of the P∗–t∗ curves for specimens of di�erent sizes.

4.1. In uence of impact velocity

In order to gain additional insight into the role of impact velocity in the dynamic Brazilian test, weproceed to apply incident pulses of amplitudes v=2; 4; 10; 20; 40 and 100m=s. All the remainingparameters of the model, including the ramp-up time tr and the duration time td, are held constant asin the previously de�ned base case. It should be carefully noted that the highest impact velocitiesconsidered here are well beyond the range of impact velocities in Tedesco et al. experiments(cf. Table II).Figure 12 collects the computed load-history curves plotted in terms of normalized variables.

As may be seen from the �gure, the �rst load peak increases monotonically at low impact speeds,or, alternatively, at low strain rates, and marks the maximum load transmitted by the specimen. Inparticular, the �rst load peak is consistently well above the second load peak, and the secondaryload peaks occur during the softening part of the load-history curve. By contrast, the �rst loadpeak increases less rapidly at high impact speeds and is of comparable magnitude to—and insome cases exceeded by—the second load peak. As a consequence, for high impact speeds theascending part of the load-history curve tends to exhibit multiple peaks. Our calculations showthat incident velocities v=10–100m=s cause considerable local crushing, in the form of extensivemicrocracking and fragmentation, near the bearing strips which impart the loads to the specimen.This observation suggests that the increased fracture and fragmentation near the bearing stripsaccounts for the change in the peak load structure at high impact speeds.The transition from monotonic load histories to load histories with multiple peaks prior to the

maximum load is also noteworthy. Indeed, our calculations reveal the existence of a thresholdimpact velocity, or, equivalently, a threshold nominal strain rate, beyond which a su�ciently



Figure 12. Normalized load-history curves for di�erent prescribed velocities.

uniform state of deformation is not attained prior to the onset of cracking. The existence of thisthreshold has been noted by Wu and Gorham [76]. The presence of multiple peaks in the risingpart of the load-history curve is a telltale sign of strong wave e�ects and e�ectively disquali�esthe dynamic Brazilian test as a constitutive test at high impact speeds.

4.2. Size e�ect

Because of its large characteristic length, which in the material under study here equals 122mm,the fundamental postulates of linear-elastic fracture mechanics, most notably the requirement ofsmall-scale yielding, are often violated in practice. When the limiting geometrical dimensions arecomparable to the characteristic length of the material, the structure exhibits transitional behaviourstraddling strength of materials and conventional fracture mechanics. This transitional behaviouris often summarily referred to a ‘size e�ect’. One pernicious consequence of this size e�ect isthat conventional fracture tests, which are designed to meet all the requirements of linear-elasticfracture mechanics, require inordinately large specimens and prohibitively expensive experimentalfacilities [77; 78].In order to investigate the role of specimen size in the dynamic Brazilian test, we have performed

calculations for three cylinder diameters: D=50:8, 100 and 150mm, or D=0:4‘c; 0:8‘c and1:2‘c. The smaller diameter coincides with that which has been considered previously, whereasthe remaining diameters bear a ratio of 2 and 3 to the smaller diameter, respectively. In orderto facilitate comparisons, the width of the bearing strips and the prescribed impact velocity arescaled in direct proportion to the diameter. The prescribed velocities are chosen as 2; 4 and 6m=s.In consequence of this scaling, the nominal strain rate and contact pressure are independent of thespecimen size. The minimum mesh size is kept constant in all cases so as to provide an equalresolution of the cohesive length of the material. The resulting meshes are shown in the inset of



Figure 13. Normalized load-history curves for three di�erent cylinder diameters, D=50:8; 100 and 150mm.

Figure 13, which also shows the normalized load and cohesive-energy histories resulting from thecalculations.By virtue of the time normalization, the initial rising part of the load history is ostensibly

identical in all cases. As may be seen from Figure 13, the peak load increases with decreasingspecimen size, which is consistent with the well-known size e�ect in fully yielded specimens, i.e.‘smaller is stronger’ [48]. Our results suggest that the peak or failure load becomes insensitive tothe size of the specimen beyond a diameter of the order of one characteristic length, in agreementwith the �ndings of Ba�zant and Planas [48].The cohesive energy consumption scales almost exactly with the size of the specimen, as may

be seen from Figure 13. The increase in speci�c fracture energy with specimen size is indicativeof a higher extent of localized and distributed microcracking, as has been pointed out for statictests by Guinea [49], Guinea et al. [50] and Planas et al. [79].The contribution of microcracking to the fracture energy may be estimated as follows. For small

specimens, the zone of distributed microcracking may be expected to engulf the entire specimen,and the fracture energy consumed by microcracking may be estimated as

W ∗coh ≈ GcW

�D2

41l

(24)

The normalized fracture energy then follows as

WcohWDGc

≈ 1 + �4Dl

(25)

where l is the fragment size, which may be taken to be commensurate with the aggregate size,and term 1 in (25) accounts for the fracture energy required to split the specimen along the loadplane, which equals WDGc. For large specimens, the zone of distributed microcracking may beexpected to be concentrated around the bearing strips. Using Boussinesq’s solution for a



line load on an elastic half-space, the depth of the microcracking zone may be estimated as

d ≈ Pmax2W�c

(26)

and the fracture energy then follows in the form

WcohWDGc

≈ 1 + �4

(Pmax2W�c

)2 1Dl

(27)

The maximum applied load may in turn be estimated from (20) as

Pmax ≈ �iA= �cvA (28)

where A is the cross-sectional area of the incident bar. The cross-over between the fracture energies(25) and (27) occurs for a diameter

Dc ≈ Pmax2W�c

≈ �cvA2W�c

(29)

It follows from these estimates that, provided that the impact velocity v and the specimen width Ware scaled in proportion to the diameter D, the fracture energy decreases for D/Dc as D−1, andthe e�ect of microcracking becomes increasingly localized. Under these conditions, the extendedfracture energy tends asymptotically to the nominal value WDGc for large D. For small specimens,D.Dc, microcracking extends to the entire specimen and the fracture energy increases linearlywith D.For the calculations shown in Figure 13 one has Dc ≈ 360mm. Thus, the specimen diameters

may be expected to be below the transitional regime in which the e�ect of microcracking saturates.This accounts for the variation in fracture energy between the three specimens, and the substantialexcess fracture energy over that which is strictly required to cleave the specimens (Figure 13). Itcan hardly be overemphasized the relative ease with which the e�ects are accounted for within acohesive fracture framework.

4.3. In uence of the width of the bearing strips

As already noted, local e�ects near the bearing strips can have a marked in uence on the overallresponse of the specimen and are notoriously di�cult to control experimentally. In order to gagethe sensitivity of the test results to these e�ects, we have carried out calculations for two bearingstrips of widths b=D=8 and D=12. The �rst bearing-strip width is as in the previously de�nedbase case, whereas the second smaller bearing strip is reduced in size by a factor of 2

3 .Figure 14 shows the resulting load-history curves for the two bearing-strip sizes. Also shown

inset is the slight local modi�cation to the mesh required to accommodate the narrower strip. Asmay be seen from the �gure, the maximum load diminishes with the size of the bearing strip,an e�ect reported by Rocco et al. [80] and by Rocco [52] for the static test. For the geometriesconsidered in the calculations, the reduction in peak load is of the order of 16 per cent. Thisresult strongly suggests that the early stages of cracking are governed by the stress P=(Wb) underthe bearing strips. It should be carefully noted that a direct application of the static formula (17)yields two di�erent values of the dynamic strength of the material, which cautions against theindiscriminate use of such formulae. The cohesive energy consumption is nearly identical in bothcases (Figure 14).



Figure 14. Normalized load-history curves for two di�erent bearing-strip widths b=D=8 and D=12.

4.4. E�ect of �

As noted earlier, the parameter � determines the ratio of modes II to I fracture toughness. FollowingReference [70], in calculations we have adopted the value �=10 in order to account for thetoughening e�ect of the aggregate interlocking mechanism under mode II loading. It should becarefully noted that for a large (small) value of � the critical condition for the initiation ofa cohesive crack is ostensively independent of the magnitude of the shear (normal) tractions,and the ensuing sliding (opening) displacements are constrained to remain close to zero at alltimes. In the limit of � → ∞ (� → 0), the initiation criterion becomes completely independentof the shear (normal) traction and the sliding (normal) displacements are constrained to vanishidentically.In order to ascertain the sensitivity of the results to the choice of �, we have compared the

cases �=0:1; 1; 10, and 100. Figure 15 shows the results of the simulations in the form of load-history and fracture energy curves, and deformed meshes and damage patterns at peak load. Itis concluded from these �gures that, as expected, the choice of � has a marked in uence onthe predictions of the model. Thus, for the small �=0:1, the specimen fails along slip lines andtransmits a small peak load (Figure 15). The deformed mesh and the damage pattern at the peakload exhibit profuse distributed shear cracking. The cohesive energy consumption is very small upto the peak load, for large sliding displacements in the cracks are necessary to reach the e�ectivecritical opening displacement. The value �=1 gives the same weight to normal and shear stressesfor crack initiation and opening, and thus represents a transition between predominantly shearcracking and predominantly normal opening, e.g. due to the aggregate interlocking constraint.Figure 15 shows that, for �=1, the specimen undergoes considerable local crushing near theincident bearing strip. In addition, a through-crack develops in which some degree of sliding isevident. Both mechanisms dissipate cohesive energy before the maximum load is attained. Theextent of distributed cracking is smaller than in the case of �=0:1. The results for �=10 and100 di�er only slightly, which demonstrates that a choice of �=10 approximates closely the caseof perfect aggregate interlocking. The load-history and the fracture energy curves are identicalin both cases up to the peak transmitted load, while the deformed meshes and damage patterns



Figure 15. Normalized load-history curves and deformed meshes for �=0:1; 1; 10 and 100, representingdi�erent ratios of modes II to I toughness.

reveal the predominance of a single through-crack devoid of any appreciable sliding. For �=10,some degree of local crushing remains in the loading area which is almost entirely suppressed for�=100. These results demonstrate the ability of the cohesive model to reproduce di�erent typesof mixed-mode fracture behaviours.

4.5. In uence of the geometry of the specimen

Finally, we investigate the e�ect and shape of the specimen. To this end, we consider an additionalspecimen of square cross-section and dimensions such that the cross-section through the load planeis identical to that of the base specimen. The mesh design near the load plane is likewise identicalin both cases. The details of the specimen geometry and mesh design are shown in Figure 16.The resulting load and fracture energy histories are also shown in the �gure. It follows from theseresults that the shape of the specimen has a modest e�ect on the peak load. The higher peakload corresponds to the square cross-section and is within 10 per cent of the peak load for thecircular cross-section. The post-peak response of the square specimen is accompanied by increaseddistributed cracking near the bearing strips, which results in a higher fracture energy consumption.



Figure 16. Normalized load-history curves for two specimen shapes of identicalcross-section through the load plane.

5. SUMMARY AND CONCLUSIONS

We have investigated the feasibility of using cohesive theories of fracture, in conjunction with thedirect simulation of fracture and fragmentation, in order to describe processes of tensile damageand compressive crushing in concrete specimens subjected to dynamic loading. The particularcon�guration contemplated in this study is the Brazilian cylinder test performed in a Hopkinsonbar [41; 37; 36], which furnishes a demanding validation test of the theory. Our approach accountsexplicitly for microcracking, the development of macroscopic cracks and inertia. The e�ectivedynamic behaviour of the material is predicted as an outcome of the calculations. In particular,our simulations capture closely the experimentally observed rate sensitivity of the dynamic strengthof concrete, i.e. the nearly linear increase in dynamic strength with strain rate [41; 37; 36; 33].More generally, our simulations give accurate transmitted loads over a range of strain rates, whichattests to the �delity of the model where rate e�ects are concerned. The model also predicts keyfeatures of the fracture pattern such as the primary lens-shaped cracks parallel to the load plane,as well as the secondary profuse cracking near the support. These results validate the theory asit bears on mixed-mode fracture and fragmentation processes in concrete over a range of strainrates.We have assumed that the cohesive properties of the material are rate independent and therefore

determined by static properties such as the static tensile strength. However, we have noted thatcohesive theories, in addition to building a characteristic length into the material description, endowthe material with an intrinsic time scale as well [23]. This intrinsic time scale accounts for theability of model to predict key aspects of the dynamic behaviour of concrete, such as the strain-rate sensitivity of strength. Our results suggest, therefore, that most of the strain rate sensitivityof concrete is attributable to the microinertia attendant to dynamic microcracking and fracture.We have additionally carried out a parametric study with a view to ascertaining the sensitivity

of the dynamic Brazilian test to details of the experimental set-up such as the size and geometryof the specimen and the size of the bearing strips. The results of the simulations also exhibit the



expected size e�ect, i.e. smaller specimens are stronger [48]. Our results suggest that the peak orfailure load becomes insensitive to the size of the specimen beyond a diameter of the order ofone characteristic length, in agreement with the �ndings of Ba�zant and Planas [48]. We have alsofound that the cohesive energy expenditure is considerably larger in the dynamic test than in thestatic case, which re ects the dissipation due to microcracking. An additional insight provided bythe simulations is that the load-history curve exhibits multiple peaks within its rising part at highstrain rates. Yet another �nding is that the peak load is sensitively dependent on the size of thebearing strips through which the loading is imparted to the specimen.These observations, taken together, should caution experimentalists against a simplistic inter-

pretation of the dynamic Brazilian test, specially as regards the inference of intrinsic materialproperties. Similar warnings have been voiced by Rodr��guez et al. [45] and Johnstone and Ruiz[46] in the context of SHPB tests on ceramics.

ACKNOWLEDGEMENTS

Gonzalo Ruiz gratefully acknowledges the �nancial support for his stay at the California Institute of Technol-ogy provided by the Direcci�on General de Ensenanza Superior, Ministerio de Educaci�on y Cultura, Spain.Anna Pandol� and Michael Ortiz are grateful for support from the Department of Energy through Caltech’sASCI Center of Excellence for Simulating Dynamic Response of Materials. Michael Ortiz also wishes togratefully acknowledge the support of the Army Research O�ce through grant DAAH04-96-1-0056. We areindebted to Dr. Ra�ul A. Radovitzky for his assistance in the development of the meshes used throughout thisresearch.

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three-dimensional finite-element simulation of the dynamic brazilian tests on concrete cylinders

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