Mesoscopic analysis of concrete under excessively high strain rate compression and implications on...

at SciVerse ScienceDirect

International Journal of Impact Engineering 46 (2012) 41e55

Contents lists available

International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

Mesoscopic analysis of concrete under excessively high strain rate compressionand implications on interpretation of test data

Zhenhuan Song, Yong Lu*

Institute for Infrastructure and Environment, School of Engineering, The University of Edinburgh, William Rankine Building, The King’s Buildings, Edinburgh EH9 3JL, UK

a r t i c l e i n f o

Article history:Received 3 June 2011Received in revised form5 January 2012Accepted 24 January 2012Available online 11 February 2012

Keywords:High strain rateConcreteMesoscale modelStress waveDynamic strength

* Corresponding author. Tel.: þ44 (0)131 6519052;E-mail address: [email protected] (Y. Lu).

0734-743X/$ e see front matter � 2012 Elsevier Ltd.doi:10.1016/j.ijimpeng.2012.01.010

a b s t r a c t

The strain rate effect on the behaviour of brittle materials like concrete has been a classical topic ofinterest in the shock and impact engineering community. For concrete under high strain ratecompression, a dynamic increase factor (DIF) is commonly used to account for the nominal dynamicstrength enhancement for engineering applications. The cause of the experimentally observed DIF onstandard concrete specimens has been a subject of securitization in recent years. This paper presents aninvestigation on the dynamic behaviour of concrete specimens under high strain rate compression withthe aid of mesoscale numerical simulation. Beyond a further observation on the so-called lateral inertiaconfinement effect, special attention is paid to the transient shock wave effect and the propagation ofmaterial failure when a specimen is loaded with a strain rate exceeding a theoretical limit for a givenspecimen size, i.e., in the “excessive” strain rate regime as referred to in this paper. Based on thesimulation, it is argued that the validity of many existing test data on the nominal compression DIF forconcrete, especially those in the very high strain regime, is rather questionable. The correlation betweenthe externally measured (inferred) strength-strain data and the actual material dynamic response withinthe specimen is examined. The influence of the material heterogeneity on the DIF is also discussed withquantification.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The bulk compressive behaviour of concrete-like materials isknown to be strain rate dependent, as commonly observed fromdynamic experiments using typically an SHPB (Split HopkinsonPressure Bar) apparatus [1e4]. Theoretical and numerical studiestend to indicate that such strain rate dependence in compressioncan be largely attributed to the inertial effect [5,6]. Recent studiesby the authors using a newly developed mesoscale numericalmodel with random aggregate shapes and sizes provide furtherevidence favouring this argument [7].

One important consideration in evaluating and interpreting thedynamic response of a specimen under high strain rate compres-sion is how stress and strain distributewithin the sample specimen.As a general guide, reasonable stress uniformity should be achievedbefore the specimen reaches failure so that the externallymeasured(or inferred) strength and deformation data at failure may beconsidered as representative of the bulk material behaviour. Inconventional SHPB testing of metallic materials, a commonapproach for achieving uniform stress state is to restrict the sample

fax: þ44 (0)131 6506554.

All rights reserved.

length within a specific limit for a target strain rate range, so thata uniform stress state can be built up in the course of stress wavereflections [8,9]. The diameter of the specimen is governed bya proper aspect ratio (length to diameter). To minimize the influ-ence of radial and longitudinal inertia as well as friction effects, anoptimum aspect ratio for the specimen is recommended [8]. Fora cylinder metallic specimen, ASM Handbook recommends theratio to be 0.5 w 1.

For metallic materials, it is not difficult to satisfy the stressuniformity requirement in a conventional SHPB test since metalsamples can be made sufficiently small. However, this is not thecase for non-metallic, non-homogeneous materials such asconcrete and rocks, for which large size specimens are required tocater to the heterogeneity, and consequently the need of using largediameter pressure bars becomes inevitable.

One issue that needs to be consideredwhen a large diameter baris used is the wave dispersion effect, which may influence theaccuracy of the test due to the change of pulse shape in propagation.Zhao and Gary [10] reported that strong wave dispersion wasobserved in large diameter pressure bars, and consequentlycorrection for thewave dispersion effect is necessary for small strainmeasurement. Since then, a number of investigations have beenundertaken to develop appropriate correction techniques [11e14].

mailto:[email protected]

www.sciencedirect.com/science/journal/0734743X

http://www.elsevier.com/locate/ijimpeng

http://dx.doi.org/10.1016/j.ijimpeng.2012.01.010



250

300

350

400

450

in ra

te (1

/s)

Z. Song, Y. Lu / International Journal of Impact Engineering 46 (2012) 41e5542

Whereas the aspect ratio requirement is easier to satisfy in largediameter SHPB tests and the dispersion effect may be rectified toa certain extent, one can face a dilemmawhen it comes to choosinga suitable specimen size for testing concrete-like materials in thehigh strain rate regime. On one hand, a reasonable representationof the bulk composite behaviour requires a specimen size to be atleast a few (typically 4w 5) times of the largest grain (aggregate) inthe composite. On the other hand, the need to accommodate testsinto high strain rate regime, for instance above 100 s�1, dictates thatthe specimen length should be kept well within a few centimetres.The commonly adopted practice of using scaled concrete, typicallymortar, is an obvious attempt to balance between the above twoconflicting requirements. However, to better preserve the charac-teristics of the concrete material the choice of larger size specimensbecomes inevitable, and in such cases it is hardly possible to satisfythe size limit for the higher end of strain rates of interest. In fact,a variety of specimen sizes have been used both in experimentaltests and numerical SHPB simulations, ranging from 10 mmmortarcubes to as large as 300 mm concrete cylinders [2,3,6,15,16].The employment of larger specimens without observing therequirement of stress uniformity could lead to incorrect interpre-tation of the strain rate effect. Unfortunately such an importantfactor has not always been examined rigorously in the existingstudies on the dynamic properties of concrete-like materials withlarge heterogeneity.

Dioh [17] argued that the presence of large plastic wave front inthe specimen during a test would render the classical SHPB testmethod erroneous. Meng and Li [18] evaluated qualitatively theaxial and radial stress uniformity using an FEmodel. However, littleeffort has been paid to provide a rigorous assessment regarding theeffect of stress non-uniformity on the behaviour of the test spec-imen and how the actual response within the specimen wouldcorrelate with the externally measured data. As a matter of fact,such correlation could provide vital guidance for the correctinterpretation of the measured data from SHPB type of tests,especially for concrete-like specimens where the specimen sizelimit or strain rate limit may not be strictly observed when testingin the high strain rate regime.

In this paper, we will aim to address the following two mainquestions:

1) What is the stress and strain state of a concrete specimen if thestress uniformity is not satisfied, and how does the materialactually behave within the specimen in such cases?

2) How do the externally measured strength and strain datacorrelate with the actual dynamic behaviour of the material?

In way of the above investigation, the general inertia effect onthe dynamic behaviour of concrete will also be examined.Furthermore, the employment of a mesoscale model facilitates anevaluation of the contribution of the material heterogeneitytowards the bulk dynamic behaviour of the material, in addition tothe general inertia effect.

0

50

100

150

200

0 10 20 30 40 50 60 70 80Length of concrete specimen (mm)

Lim

iting

stra

Fig. 1. Relationship between specimen length and strain rate limit for 30-MPaconcrete.

2. Background and basic model considerations

2.1. Strain rate limit in SHPB tests

Strictly speaking, two basic assumptions should be satisfied inorder for an SHPB test to be valid, namely, a) one dimensional elasticwave propagation is ensured within the pressure bars and thespecimen, and b) stress state within the specimen is uniaxial anduniform. To meet the second requirement, several reflections of thestress wave are needed for a uniform stress state to build-up within

the specimen. Thus, the time required for the stress to reach equi-librium may be written as a multiple of wave travel time, i.e.,

t>nlsc0

(1)

where ls is the length of specimen, c0 is the soundwave speed in thespecimen. The integer n is recommended to be 3 w 4 based onexperimental observations [8,19,20].

Accordingly, an upper limit of strain rate that can be accom-modated in a specimen satisfying the above requirement will be:

_3¼ 3c

t<

3cc0nls

(2)

where 3c denotes the strain at the maximum strength.This equation indicates that for a given specimen length,

a limiting strain rate exists for a reliable SHPB test. Conversely, ifone intends to test the material strength up to a specific strain ratelevel, the length of the specimen should be smaller than a criticalvalue. Take grade 30MPa concrete for example, assuming a strain atmaximum strength of 0.002, mass density of 2300 kg/m3, andYoung’s modulus of 20 GPa, a standard specimen of 50 mm inlength can only accommodate a strain rate limit of approximately50 s�1. For this class of concrete, the relationship between thespecimen length and the strain rate limit can be depicted by a graphshown in Fig. 1.

Examining the experimental data available in the literature forconcrete-like materials under dynamic compression, for examplethose presented in Bischoff and Perry [15], Ross et al. [2], Grote et al.[3], numerous data points towards the high strain rate end appearto fall beyond the upper strain rate limits for the respective speci-mens. For example, In Ross et al. [2] concrete specimens of length51 mmwere used; theoretically this would restrict the valid strainrate range to be below 40 w 50 s�1. However, the actual testsextended to a strain rate as high as 1000 s�1. Bischoff and Perry [15]acknowledged that in the higher strain rate regime, “strain ratemeasurements are sometimes quite erratic”.

A special specimen configuration worth mentioning here is thatemployed in an impact experiment by Riedel et al. [21] for strainrates up to 105w106 s�1. The specimen was assembled by multiplesliced blocks of concrete, such that stress gauges could be attachedto the interface of the adjacent blocks, which were then gluedtogether with epoxy. Such an assemblage enabled the measure-ment of the propagating shock wave within the specimen at highstrain rates, which would not have been possible using

Z. Song, Y. Lu / International Journal of Impact Engineering 46 (2012) 41e55 43

a conventional piece of specimen. However, because of the need ofphysical division within the specimen in the loading direction, thistechnique is still subject to the same difficulty in withholding theheterogeneity features when large aggregates are involved.

2.2. Overview of the basic mesoscale numerical model

Under high strain rate loading, drastic variation of stress andstrain occurs as the dynamic stress wave builds up in a specimen. Torealistically represent the travelling stress wave effect, an explicitinclusion of the mesoscale heterogeneity becomes crucial.Although mesoscale modelling of concrete has been a subject ofresearch interest in static loading literature [e.g. [22e24]], analysisof the dynamic behaviour of concrete poses different challenges onthe material descriptions and numerical solutions, and explorationin this area has been a relatively recent development. A few studieshave been published in recent years on the use of mesoscalemodelsfor the analysis of concrete response under high strain ratecompression, with simplified two phase models [25e27] assem-bled from regular FE elements, or models involving idealisedspherical aggregate shapes [16].

The present study adopts a more realistic mesoscale modelincorporating random polygon-shaped coarse aggregatesembedded in the mortar matrix, while the interfacial transitionzone (ITZ) is represented by a thin layer of solid elementssurrounding the individual aggregates. The basic mesoscale modeland an extensive investigation regarding the general modelperformances have been discussed extensively in Tu and Lu [28]. Abrief overview of the model is given in what follows.

Fig. 2 shows a sample 2D mesoscale structure and the finiteelement mesh. Different material properties are used for the threeindividual phases. The mortar matrix and the equivalent ITZ aremodelled using a damage-plasticity model, herein the ConcreteDamage Model or K&C model [29]. This material model hasa comprehensive capability in describing the behaviour ofconcrete-like materials in a wide range of stress and pressureconditions [30]. For a typical 30-MPa grade concrete, the mortarelements are assumed to have 35 MPa peak strength, while for theequivalent ITZ elements, a reduced strength of around 75% of themortar strength is adopted [28], which is approximately 27 MPa.The aggregates are also modelled as nonlinear plastic material,however with a higher strength of 150 MPa representing typicalcrushed granite [31].

The basic 2D mesoscale model is validated with representativequasi-static compression tests. Using a concrete cube undercompression as a benchmark condition, a model is set-up witha side length of 50 mm and coarse aggregates in a range of1.6e8 mm (this is numerically equivalent to standard cube of150 mm and coarse aggregates of 4.75e25 mm). The nominal meshgrid is around 0.3e0.5 mm. The simulation results under a planestress condition are reproduced in Fig. 3 for two loading end friction

Fig. 2. Configuration of th

assumptions, namely a) friction free (lower bound condition), andb) infinitive friction (upper bound condition). The computedcompressive stress-strain curves agree favourably with typicalexperimental observations [32] as depicted in Fig. 3(a). It is notedthat the stress here is evaluated by dividing the total (nodal) forceon the loading face by the cross-section area, while the strain iscalculated as the total axial displacement divided by the initiallength of the specimen, which conform with the engineeringmeasures used in physical experiments. The damage contoursshown in Fig. 3(b) also resemble closely typical experimentalfailure patterns under low and high piston frictions, respectively.Inclined cracks occur under low friction loading, whereas a double-cone failure mode appears when a high frictional constraint isinvolved.

With regard to the effectiveness of modelling the ITZ with anequivalent layer of solid elements, a convergence analysis is alsoconducted inwhich the ITZ layer thickness, along with the nominalgrid size, is varied in a range of 2 w 0.5 mm. Fig. 4 shows thecomparison of the stress histories. It can be seen that the limitedvariation of the equivalent ITZ thickness only affects the descendingbranch, while the stress increase phase is almost identical. Theoverall response tends to converge when the ITZ thickness isreduced to 0.5 mm. This thickness is adopted in the subsequentmodelling calculations.

2.3. A pseudo 3D mesoscale model configuration

Among other theories, an overwhelming view about thedynamic strength increase, at least for compression, is directedtowards the confining effect arising from the lateral inertia whenthe material tends to expand under high rate compression[6,7,15]. As far as numerical simulation using a finite elementmodel is concerned, a good representation of the lateral inertiaconfinement thus needs to be ensured, and this necessities anappropriate incorporation of the 3D effect. Comparison between2D and 3D homogeneous models in simulating high straincompression [7] reveals some marked difference in the develop-ment of the confined core and the final damage patterns.However, generation of a realistic 3D mesoscale model isextremely complicated and computationally demanding, for thisreason 3D mesoscale model has been employed in the past onlywith considerable simplifications [e.g., [23,26]], while 2D meso-scale model is still commonly adopted. It should be particularlypointed out that, whereas for a homogeneous specimen underaxial loading the 3D effect may readily be represented using a 2Daxis-symmetrical model, such an idealisation is not suitable ina mesoscale model. This is because with a 2D axis-symmetricmodel the aggregates would effectively resemble continuouscircular rings, which consequently would introduce falseconfining stress when the model tends to expand in the radialdirection, even without lateral inertia.

e mesoscale model.

0

5

10

15

20

25

30

35

40

0 0.001 0.002 0.003 0.004 0.005 0.006

Axial strain

Axi

al s

tress

(MP

a)without friction

with upper bound friction

0

10

20

30

40

50

250 500 750 1000

Stre

ss (M

Pa)

Axial deformation (µm)

Low friction

high friction

100 mm cubes

60

Nominal stress-strain curves (Left = computed for 30MPa concrete; right = experimental for ~40 MPa concrete - after van Vliet [35])

Low friction High friction Low friction High friction

Two representative failure modes (left = computed; right = experimental - after van Vliet [35])

a

b

Fig. 3. Mesoscale model analysis vs. experiment under static compression.


To tackle all these difficulties, a pseudo 3D configuration isproposed by the authors [7] in attempt to satisfy the need ofreproducing realistically the 3D inertial effects while retaining therelative simplicity of a 2D mesoscale description.

The pseudo 3D mesoscale model is constructed by sandwichinga slice of the 2D mesoscale model, with a small nominal thickness,into a pair of homogenous half-3D specimen of similar bulkconcrete properties to form a complete 3D configuration. Fig. 5(a)depicts a pseudo 3Dmesoscale model for a cylindrical specimen. Toavoid unwanted interference from the attached half-cylinders onthe mesoscale model within the primary 2D plane, fromwhich the

0

5

10

15

20

25

30

35

0.00 0.10 0.20 0.30Time (ms)

Stre

ss (M

Pa)

ITZ=2mm

ITZ=1mm

ITZ=0.5mm

Fig. 4. ITZ element size convergence analysis in 2D mesoscale model.

mesoscale behaviour is observed, the interface is coupled only inthe normal direction while relative movement in the tangentialdirection is left free, as schematically illustrated in Fig. 5(a). Thisensures that, other than the inertia stress in the out-of-planedirection, the response within the mesoscale plane is only gov-erned by the 2D mesoscale model itself. The loading and responseof the mesoscale part can thus be evaluated separately.

The effectiveness of such a pseudo 3D configuration in repro-ducing a 3D stress condition for the (2D) mesoscale model isverified by first comparing the general development of damagefrom such a model with an intact 3D homogenous model, andsubsequently an evaluation of the sensitivity of the response in themesoscale part to the thickness of the mesoscale layer. Fig. 5(b)shows a comparison of the damage within the mesoscale plane inthe pseudo 3D model and that in the 3D homogeneous model. It isnoted that in this and the subsequent dynamic analyses, loading isalways applied horizontally from the left end of the specimens,unless otherwise specified. It can be observed that the overalldamage patterns agree well between the two models. Damage isinitiated from the loading side and propagates towards the supportend. As in the 3D homogeneous specimen, the central zone withinthe mesoscale plane exhibits considerably less damage than theouter region, indicating a similar “confining” environment. Asensitivity analysis (not shown) also confirmed that the responsewithin the mesoscale plane is not sensitive to the variation of thethickness of the meso-layer.

It should be particularly mentioned that in all the dynamic anal-ysis hereinafter, the dynamic increase factor (DIF) is not incorporatedat the constituentmaterial descriptions. Thismeans that thematerialitself is considered as strain rate insensitive such that any dynamicbulk strengthenhancement in themodelwill beentirelyattributed tothe dynamic andmeso structural effect within the sample specimen.

Fig. 5. Pseudo 3D mesoscale model for a cylinder and comparison of damage patterns with 3D homogeneous model.



3. Model set-up for high rate compression simulation

3.1. Dimensions and fictitious strips for evaluating wavepropagation effects

A cylinder specimen of 50 mm (diameter) � 50 mm (length), ascommonly used in SHPB tests of concrete, is considered. Themaximum aggregate has a nominal size of 8 mm, which is aboutone sixth of the sample dimension. A pseudo 3D mesoscale model,as shown in Fig. 5(a), is developed to model such a specimen. Meshconvergence trials yielded a final FEmeshwith a gird size of around0.5 mm for the sandwiched mesoscale slice. Totally around 11,700elements are used in the mesoscale slice (with a single elementthickness), while around 30,000 elements are used to model thehomogeneous half-cylinder. A symmetrical condition is imposed onthe front face of the mesoscale slice.

Examination of the responses is focused within the mesoscaleplane of the model. To facilitate the evaluation of the stress-straindistribution while the stress wave propagates and reflectsbetween the two end faces, the mesoscale plane is fictitiouslydivided into five equal strips along the loading (axial) direction,with a width of 10 mm for each strip, as depicted in Fig. 6.

The nominal strain rate in each strip is evaluated from theaverage nodal velocities over the two dividing lines bounding thestrip. The strip-wise axial stress and strain are calculated as theaverage axial stress and strain among all the meso-model elementswithin the strip. With these data, a correlation between the ach-ieved stress-strain behaviour in different strips of the mseo-modeland their global “engineering” counterparts, which are evaluatedusing stresses and velocities measured on the loading (incident)and support (transmission) ends, can be examined in detail.

3.2. Simplified loading scheme resembling SHPB tests

There are two obvious options in the computational model set-up for simulating a high rate compression test with an SHPBapparatus. One is to include the incident and transmitter pressure

Fig. 6. Specimen being divided into 5 strips for evaluation of stress and straindistributions.

bars directly in the model and apply actions on the input end of theincident bar, as adopted in some previous studies [6,33]. The otheris to apply actions directly on the specimen via prescribed timevarying boundary conditions at the two end faces [7,21,25,26]. Theformer approach would simulate more realistically the loadingenvironment as in a physical SHPB test, but computational costwould increase; andmoreover, defining the exact loading conditionthat is actually applied on the test sample would become lessstraightforward. Using the second approach is computationallymore efficient, and when the boundary histories are carefullyspecified it is possible to capture the main loading and responsecharacteristics as in an actual SHPB environment. For these reasons,and considering that an “exact” reproduction of an SHPB testcondition is not of critical importance for the current investigation,the second option is considered as rational and this approach isadopted in the present study.

As a matter of fact, if actual boundary (usually velocity) historiesas measured from an SHPB test are specified in the numericalmodel, the result would be equivalent to inclusion of the pressurebars. Herein only an idealized velocity history is imposed on theloading face, while the end face is simplified as fully restrained inthe axial direction. Analogous to the typical measured velocityhistories at the incident barespecimen interface, the idealisedvelocity history (Fig. 7a) consists of a gradual rise phase, followedby a constant velocity vmax.

For verification on the sensitivity of the simulation results to thesimplification in the velocity boundary condition, a comparativeanalysis was carried out with different velocity boundary settings,including one that was adapted from the measured velocityhistories in an actual SHPB test, as presented in Meng and Li [34].Thus, three velocity boundary scenarios are considered, as shownin Fig. 7, namely, a) idealized incident velocity only (with thetransmitter end fixed), b) idealized incident velocity with“measured” transmitter velocity, and c) “measured” incident and“measured” transmitter velocities. The maximum incident velocityis about 7.5 m/s and the resulting nominal strain rate is around150 s�1 in the present 50-mm specimen.

Representative results from the analysis under the above threevelocity boundary scenarios are compared in Fig. 8. The achievedoverall forces on the loading (incident) face (F1) and transmitterface (F2), shown in Fig. 8(a), are similar among the three loadingschemes. Small fluctuations in F1 of scheme “c” are apparently dueto the fluctuations in the corresponding loading velocity. Damagesat the peak strength of F2 are shown in Fig. 8(b). In scheme “a” and“b”, the development of damage agrees well with each other, whilein scheme “c” elements close to the transmitter boundary areshown to have slightly less damage.

Further examination into the detailed stress and strainresponses in each strip, Fig. 7(c)e(d), indicates that with the ide-alised loading (scheme “a”) some degree of over-representation ofthe stress tends to occur near the fixed end due to the exclusion oftransmission effect. However, the magnitude of stress and strain inthe first few strips resembles well the actual condition, and thecharacteristics of the propagating stress wave are reasonablyretained in the simplified loading scheme.

4. Numerical investigation of dynamic compression: generalresults and DIF

Before concentrating on the phenomena associated with“excessive” high strain rate compression, a comprehensive para-metric study was conducted using both the pseudo 3D mesoscalemodel and 3D homogeneous-only model, and for a wide spectrumof strain rates. The purpose was to further verify the adequacy ofthe current modelling framework, including the material models

0

2

4

6

8

10

0.00 0.01 0.02 0.03 0.04 0.05

Time (ms)

Velo

city

(m/s

)V0

0

2

4

6

8

10

0.00 0.01 0.02 0.03 0.04 0.05

Time (ms)

Velo

city

(m/s

)

V0V5

0

2

4

6

8

10

0.00 0.01 0.02 0.03 0.04 0.05

Time (ms)

Velo

city

(m/s

) V0V5

v0 = idealised; v5 = 0 v0 = idealised; v5 = measured v0 & v5 = measureda b c

Fig. 7. Three velocity boundary schemes.


used, in reproducing the dynamic effect within the sample spec-imen and the associated inertia contribution to the bulk dynamicstrength increase, without the involvement of a DIF at the consti-tutive material property level. By comparing between the pseudo

0

20

40

60

80

0.00 0.01 0.02 0.03 0.04 0.05

Time (ms)

Boun

dary

forc

e (k

N) F1

F2

0

20

40

60

80

0.00 0.01 0.02

Tim

Boun

dary

forc

e (k

N)

Incident (front) and transm

Damage

0

20

40

60

80

0 0.01 0.02 0.03 0.04 0.05Time (ms)

Stre

ss (M

Pa)

Strip1Strip2Strip3Strip4Strip5

0

20

40

60

80

0 0.01 0.02Time

Stre

ss (M

Pa)

Strip-wise st

0

0.002

0.004

0.006

0.008

0.01

0 0.01 0.02 0.03 0.04 0.05

Time (ms)

Stra

in


0

0.002

0.004

0.006

0.008

0.01

0 0.01 0.02

Tim

Stra

in


Strip-wise s

a

b

c

d

Fig. 8. Comparison of responses corresponding to the three

3D and homogeneous model results, an observation on thecontribution of the heterogeneity, particularly the aggregates, onthe DIF can also be established. The detailed response within thespecimen for representative strain rates will be scrutinized in

0.03 0.04 0.05

e (ms)

F1F2

0

20

40

60

80

0.00 0.01 0.02 0.03 0.04 0.05

Time (ms)

Boun

dary

forc

e (k

N) F1

F2

itter (back) force histories

contours

0.03 0.04 0.05 (ms)


0

20

40

60

80

0 0.01 0.02 0.03 0.04 0.05Time (ms)

Stre

ss (M

Pa)


ress histories

0.03 0.04 0.05

e (ms)

0

0.002

0.004

0.006

0.008

0.01

0 0.01 0.02 0.03 0.04 0.05

Time (ms)

Stra

in


train histories

different velocity boundary conditions shown in Fig. 7.


associationwith the discussion of the wave effect later in Section 5.In what follows a summary of the general results is provided topave the way for the subsequent discussion.

Fig. 9 shows the computed DIF and its variation with the strainrate for the 50 mm cylinder specimen using the pseudo 3D meso-scale model and a homogeneous model, respectively. Fora comparison, the curve generated using directly a 2D mesoscalemodel (plane stress) is also included. The DIF from the numericaltests is calculated as the ratio between the bulk dynamic strengthunder a given strain rate and the strength under a quasi-staticloading. The bulk dynamic strength is obtained from the stress onthe loading (incident) face, calculated as the total nodal force in theaxial direction divided by the corresponding area. This effectivelyresembles the “one-wave” approach in standard SHPB tests as willbe discussed later in Section 6. Averaging the stress between thetwo faces, which resembles “three-wave” approach, could beproblematic for higher strain rate regime. This will be discussed indetail in Section 6 as well.

Shown in the figure are also two empirical curves for compar-ison, namely,

a) CEB-FIP DIF model [35],

DIF ¼( �

_3=_3s�1:026as for _3� 30 s�1

gs�_3=_3s

�1=3 for _3>30 s�1(3)

where _3s ¼ 30 � 10�6 s�1 (static strain rate), logg ¼ 6.156as�2,as ¼ 1/(5 þ 9fcs/fc0), fcs is static compressive strength, andfc0 ¼ 10 MPa.

b) A semi-empirical model [36], which may be expressed by thefollowing regression formulae:

DIF ¼�

1þ 0:15_30:2 þ 0:0013_31:1 Compression1þ 1:505_30:295 Tension

(4)

Two immediate observations may bemade from Fig. 9, i) the DIFresults from the numerical experiments in all cases generally followthe trends of the empirical and semi-empirical predictions, despitethe fact that no embedded DIF is involved at the material consti-tutive model level. This tends to confirm the well-known argu-mentation as mentioned in Section 2.3 that the bulk DIF from

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0.1 1 10 100 1000 10000

Strain rate (1/s)

DIF

CEB FIP 1990 [35]3D homogeneous modelPseudo 3D meso-modelSemi-empirical [36]2D meso-model (plane stress)

Fig. 9. DIF curves generated using numerical experiment with pseudo 3D mesoscaleand homogeneous models (based on F1 at the incident end).

testing concrete sample specimens is attributable mainly to thedynamic inertial confinement effect, even if the constituent mate-rial properties are not strain rate sensitive. As can be expected, theDIF results from the 2D mesoscale model without the involvementof the out-of-plane inertia exhibits a relatively lower DIF value thanthe “3D”models. ii) The heterogeneity tends to play an increasinglysignificant role as the strain rate increases, especially towards thehigh strain rate end. Similar general trend has been observed insome recent numerical studies involving SHPB apparatus [16,31,37].Referring to the damage patterns shown in Fig. 10, it can be clearlyobserved that the presence of the much stronger aggregates altersthe fracture paths and induces more distributed damage. The effectof the aggregates in influencing the dynamic behaviour of thespecimen will be discussed further in Section 7.

5. Numerical investigation of dynamic compression: stresswave effect in the “excessive” high strain rate regime

5.1. Stress development path and distribution

Three strain rate levels are chosen to characterise the responseof the sample specimens in the high strain rate regime, namely50 s�1 to represent the strain rate limit for the 50� 50mm concretespecimen, and 100 s�1 and 200 s�1 to represent two levels of“excessive” strain rate conditions for such a specimen dimension.Fig. 10 shows the development of axial stress in the five strips forthe above three strain rates, respectively, along with the corre-sponding damage patterns.

Under the strain rate of 50 s�1, all strips appear to reach themaximum strength more or less at the same time. This suggeststhat a relatively uniform stress state can indeed be achieved whenthe applied strain rate is generally within the theoretical upperlimit. In fact from the stress histories it can be observed that twocomplete reverberations have occurred before the maximumstrength is reached. The damage patterns also demonstrate a rela-tively uniform distribution of damage within the specimen whenthe maximum strength is attained (stage III in the figure).

As the strain rate increases to 100 s�1, which is about twice asmuch the limiting strain rate, individual strips tend to attainmaximum strength at different times. Only the front part (first twostrips on the incident side) appears to have experienced tworeverberations of the stress wave. The non-uniformity of stress anddamage along the length of the specimen can also be clearlyobserved from the damage contours; when the front part (Strips 1and 2) reaches the maximum stress and fails (stage II in the figure),much of the remaining region is still intact. Failure appears todevelop in the specimen in a strip by strip manner.

Under a further increased strain rate of 200 s�1, the stress non-uniformity is further intensified. The stress histories and thedamage contours indicate that individual strips enter themaximum strength (and failure) consecutively as the first peakstress propagates from the loading face through to the end face.

The dynamic axial strength in individual strips apparentlyincreases with the increase of the strain rate, which is consistentwith the increase of the bulk DIF shown in Fig. 9. At the strain ratelevel of 50 s�1, the maximum strengths achieved in different stripsare effectively identical. The maximum strengths achieved inindividual strips under the strain rate of 100 s�1 are also almost thesame. For the 200 s�1 case, the maximum strength tends to showa noticeable decrease towards the middle of the specimen, withstrip 3 showing a maximum strength about 10% lower than that atboth ends. This is deemed to be attributable to a relatively lowerstrain rate attained in the middle portion of the specimen ascompared to the end regions, as will be elaborated further in thesection that follows.

Nominal strain rate = 50s-1

0

20

40

60

80

100

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time (ms)

Stre

ss (M

Pa)


I

II III

Nominal strain rate = 100s-1

0

20

40

60

80

100

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time (ms)

Stre

ss (M

Pa)


I

II

III

0

20

40

60

80

100

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Time (ms)

Stre

ss (M

Pa)


I

II III

I

II

III

I

II

III

I

II

III Nominal strain rate = 200s-1

a

b

c

Fig. 10. Stress time histories in individual strips.


5.2. Strain development path and distribution

Fig. 11 shows the strain development history under a globalstrain rate of 50 and 200 s�1 respectively. The nominal strainwithineach strip can be calculated either as the average of axial strain inall elements within the strip or from average nodal velocities on the

two bounding lines. The results (not shown) are found to agreewith each other. The latter is shown in the following strain historyand stress-strain curve.

For the lower strain rate (50 s�1) case, the strain in differentstrips tends to be comparable after the initial wave build-up stage(Fig. 11(a)), similar to the stress development shown in Fig. 10(a)

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0 0.02 0.04 0.06 0.08Time (ms)

Stra

in


dashed lines indicate similar rates when strips attain max. strength

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Time (ms)

Stra

in


dashed lines indicate varying strain rates when strips attain max. strength

Global strain rate = 50s -1

Global strain rate = 200s-1

a

b

Fig. 11. Strain time histories in individual strips.

0102030405060708090

100

0 0.001 0.002 0.003 0.004 0.005 0.006

Strain

Stre

ss (M

Pa)


Nominal rate = 50 s-1

0

10

20

30

40

50

60

70

80

90

100

0 0.001 0.002 0.003 0.004 0.005 0.006

Strain

Stre

ss (M

Pa)

Strip1

Strip2

Strip3

Strip4

Strip5


a

b

Fig. 12. Stress vs. strain curves in individual strips.


before failure occurs (up to about 0.05 ms). When failure starts tooccur, the strain rates (slope of the strain time history curves) inindividual strips are almost identical, indicating that the globalstrain rate as derived from the measurements on the two endfaces is representative of the actual strain rate inside thespecimen.

In the case with a strain rate of 200 s�1, similar to the stressdevelopment shown in Fig. 10c), individual strips attain the strainat peak strength (around 0.002) in a consecutive manner as thestress wave propagates from the front (incident) end to the supportend. Inspection of the slope (i.e. strain rate) when the failure occurs,as marked with the dashed lines, reveals noticeable variationamong different strips, and it tends to decrease towards the middleand then increase again as the wave approaches the support end.This appears to explain the variation in the maximum strengthattained in different strips, as described in Section 5.1.

5.3. Stress vs. strain relationships

With the stress and strain results as described in Sections5.1 and 5.2, the stressestrain relationships can be plotted foreach individual strip. Fig. 12 shows the strip-wise stress-straincurves under two global strain rates of 50 s�1 and 200 s�1,respectively.

The stress-strain curves are observed to be consistent amongdifferent strips, even for the 200 s�1 case. The peak strengths indifferent strips in the 50 s�1 case are almost identical, although thestrains at the attainment of the peak strength do not coincideexactly. As discussed in Section 5.2, in the 200 s�1 case the peakstrength tends to slightly decrease towards the middle portion ofthe specimen. Nevertheless, the overall stress-strain characteristicsare still similar among individual strips. As can be observedcomparing the stress-strain curves across the two different strainrate scenarios, the Young’s modulus is almost unaffected by thedifferent strain rates, and stands around 30 GPa in both strain ratescenarios. Correspondingly, the strain at the peak strengthincreases almost proportionately as the peak strength increaseswith the strain rate. This will be discussed further in Section 6 inassociation with the discussion on the relevant experimental data.

6. Discussion on the interpretation of global stress-strainresults from SHPB tests

As in typical SHPB tests, the global (sample-wide) nominalstress and strain responses in the mesoscale numerical model canbe evaluated by means of the classical 1-wave, 2-wave, or 3-wavemethod [9]. In this section, we shall examine the results evalu-ated using these approaches, which would represent what wouldbe obtained from a classical SHPB experiment. By examining theseresults against those actually taking place within the samplespecimen, as represented by the “strip-wise” results described inSection 5, it will be possible to understand the soundness andvalidity of the SHPB measurements in the high strain rate regime.

0

10

20

30

40

50

60

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

Strain

Stre

ss (M

Pa)

Based on F1

Based on F2

Based on (F1+F2)/2


0

10

20

30

40

50

60

70

80

90

100

0 0.001 0.002 0.003 0.004 0.005 0.006

Strain

Stre

ss (M

Pa)

Based on F1

Based on F2

Based on (F1+F2)/2


a

b

Fig. 13. Stress-strain curves inferred from the three different wave approaches.


The 1-wave analysis, or “back stress” analysis, is derived basedon the stress equilibrium assumption in the sample, such that theforce on the loading (incident) end, F1, equals the force on thetransmitter end, F2, where

F1 ¼ AbEð 3i þ 3rÞ; F2 ¼ AbE 3t (5)

Thus the stress is calculated as

s ¼ F2=As ¼ AbE 3t=As (6)

where Ab and As are the cross-section area of the pressure bar andthe specimen, respectively; E is the Young’s modulus of the pres-sure bar. Subscripts i, r and t indicate the incident, reflected andtransmitted pulse, respectively. The global engineering strain rateevaluated as:

_3¼ ðv1 � v2Þ=ls ¼ 2c 3r=ls (7)

where v1 and v2 are the velocities at the incident and transmitterends, respectively, c is the wave speed and ls is the length ofspecimen. Thus, in this approach the strain is evaluated using onlythe reflected wave, whereas the stress is evaluated using only thetransmitted wave.

The 2-wave method assumes that a momentum balance isachieved after the incident wave strikes the incident baresampleinterface, and hence

s ¼ F1=As ¼ AEð 3i þ 3rÞ=As (8)

Finally the 3-wave analysis averages the measurements associ-ated with both the incident and transmitter ends, thus,

s ¼ ðF1 þ F2Þ=2As ¼ ðAE=2AsÞð 3i þ 3r þ 3tÞ (9)

_3¼ ðv1 � v2Þ=ls ¼ ðc=lsÞð 3i � 3r � 3tÞ (10)

and

3¼ ðc=lsÞZ

ð 3i � 3r � 3tÞdt (11)

Fig. 13 illustrates the global (engineering) stress-strain curvesobtained using the above-mentioned 1-wave, 2-wave and 3-waveanalysis, respectively, with the strip-wise results shown inthinner lines at the background.

For the 50 s�1 case, the global stressestrain relationship curvesusing the three different wave methods may be considered asconsistent in terms of the peak stress and strain, and theyreasonably resemble the strip-wise results.

For the “excessive strain rate” case of 200 s�1, the discrepancyamong the stress-strain curves using different global approachesand their deviation from the “true” stress-strain results withinindividual strips becomes severe. More specifically, the 1-wave and2-wave approaches, although reasonably capturing the peakstrength, tends to produce a strain response which is essentiallyuncorrelated with the actual strain within the specimen. Conse-quently, the Young’s modulus is considerably overestimated, whilethe peak strain is significantly underestimated. Because of thesignificant delay in the stress response at the transmitter end, thereis virtually a zero-stress phase in the stress vs. nominal straincurves derived based on F2, and as such a direct averaging betweenthe curves from F1 and F2, as shown in the graph for an indicativepurpose, would bemeaningless. Removing the time lag in the stressresponse between the two end faces would seem to be a necessarystep if the responses at the two ends are to be averaged. It should benoted, however, the current practice of removing the time lag isbasically meant for reinstating the stress histories at the specimen

end faces, and they do not warrantee a removal of the time lag inresponses between the two end faces. Moreover, even with a care-ful removal of the time lag between F1 and F2, it would still beimpossible to recover the “true” strain experienced within thespecimen from the strain measurements in the incident andtransmitter bars, and consequently overestimation of the elasticmodulus would occur in a similar way as those derived using F1 orF2 separately.

At this juncture, it is useful to call upon some comparableexperiment studies on concrete using SHPB tests. In the experi-mental study by Ross et al. [2], where specimens of 51mm in lengthwere tested, the uniform stress assumption was adopted andconsequently the global (engineering) strain, stress and strain ratewere calculated following the 1-wave approach. The study by Groteet al. [3] used exactly the same calculation conventions in pro-cessing their experimental results for mortar specimens of lengtharound 10 mm. Li and Meng [6] adopted the 3-wave approach inanalyzing their numerical SHPB results.

Fig. 14 shows two sets of the compressive stress-strain curvesbased on the above-mentioned two experimental studies. Althoughthe overall shapes of the stress-strain curves are similar, there areconsiderable discrepancies in the reported strains and the Young’smodulus. While the results from Ross et al. [2] show no apparentincrease in the Young’s modulus as the strain rate increases(actually an abnormal decrease is seen on the highest strain rate337 s�1 case), results fromGrote et al. [3] exhibit persistent increasein the Young’s modulus, along with an increase in the peak strain. Itshould be noted that the range of the tested strain rates in bothexperiments exceeded the respective strain rate limit (whichwould

0

20

40

60

80

100

120

140

160

0 0.005 0.01 0.015 0.02 0.025 0.03strain

Stre

ss (M

Pa)

strain rate=92.8/sstrain rate=237/sstrain rate=145.8/sstrain rate=268.8/sstrain rate=337.3/s

After Ross et al. [2]

0

20

40

60

80

100

120

140

160

0 0.005 0.01 0.015 0.02 0.025 0.03strain

Stre

ss (M

Pa)

strain rate=290/sstrain rate=620/sstrain rate=1050/sstrain rate=1500/s

After Grote et al. [3]

a

b

Fig. 14. Representative experimental stress-strain curves.


be around 50 s�1 for the case of Ross et al. [2] and 300 s�1 for thecase of Grote et al. [3]).It is not clear as to what exactly caused suchdiscrepancies in the two sets of experimental results, where thesame data processing method (1-wave method) was adopted.However, on the basis of the discussion in association with thenumerical results presented in Fig. 13, it may be reasonable tospeculate that the strain results produced from the above experi-mental studies were not strictly representative of the true strainswithin the test specimen. The “engineering strains” derived fromthe measurements taken at the incident and transmitter ends, i.e.,through measurements in the incident and transmitter bars,requires careful interpretation before an inference may be madetowards the establishment of the local strains. In this respect, highfidelity numerical simulations in the framework of the presentinvestigation could play a constructive role in providing thenecessary correlations.

7. Mesoscopic evaluation of DIF under compression

While the lateral inertial confinement tends to be a dominantfactor causing the bulk dynamic strength increase of concrete-likematerials under compression, the presence of mesoscopic hetero-geneity also plays a noticeable role, as demonstrated in Section 4. Itis noted that similar characteristic trend regarding the influence ofthe aggregates on the dynamic compressive strength has beenobserved in some other recent numerical studies using mesoscalemodels [e.g. [33,37]]; however, quantification is deemed to be an

issue in those studies due to the use of simply a 2D or an axis-symmetric mesoscale model, which does not represent well theactual 3D inertial effect, as discussed in Section 2.3. In this section,we shall examine howeach phase in themeso-structure of concretebehaves and contributes in the dynamic strength enhancementbased on the current pseudo 3D mesoscale modelling results.

For this purpose, the average axial stresses developed in eachindividual phase are computed for the two representative strainrates of 50 and 200 s�1, respectively. Strip1 and Strip 3 are chosenfor an examination within individual strips. For simplicity, the ITZand mortar are combined as the “mortar” phase to compare withthe stresses in the aggregates. The results using a comparablehomogeneous model of 30-MPa concrete, which does not involvethe heterogeneity effect, are also provided as a reference for thecomparison.

Figs. 15 and 16 show the comparison of the development of axialstresses in the mesoscale and homogeneous models under the twostrain rates, respectively. It can be immediately observed that theaggregate phase develops a markedly higher axial stress than themortar phase, and the difference broadens as the strain rateincreases. Comparison with the homogeneous model shows thatthe stress in the mortar phase in the mesoscale model developsalmost the same axial stress as in the homogeneous model for thesame strip and under the same strain rate. This further indicatesthat the extra stress developed in the aggregates in the mesoscalemodel represents a net gain in the dynamic strength, and thuscontributes to the global DIF.

Fig. 17 shows the stresses in the mortar and aggregate phases atthree representative strain rates, including a lower rate of 10 s�1. Ascan be seen, under a low strain rate, the maximum stress in mortarand aggregates are almost identical, which indicates an essentiallyhomogeneous stress state despite the heterogeneity, as is generallythe case with quasi-static compression. When the strain rateincreases to 50 s�1, the maximum stress in the aggregates is higherthan the stress in the mortar by about 20e25%, and the differencefurther increases to about 50% under the strain rate of 200 s�1. Sucha phenomenon is clearly attributable to the stress wave propaga-tion and the associated progressive failure when the stress waveintensity is high enough, as in the cases under “excessive” highstrain rates. In such cases, no re-balance or redistribution ofstresses among different ingredients is possible, and consequentlymuch higher stresses could develop in the stronger and more rigidaggregate phase. It can be envisaged that such phenomenonwouldintensify as the strain rate further increases.

Noting that in a typical concrete specimen, coarse aggregatesgenerallymakes up about 40% in the total volume, thus a significantincrease in the stress in the aggregates would immediately mani-fest as an appreciable increase in the overall dynamic strength, andhence in the DIF. It should be noted that in the strain rate regimecurrently investigated, the aggregates generally remain in theelastic stage, thus the stress development in the aggregates israther independent from the lateral inertia confinement condition.

As indicated earlier in Sections 2.3 and 4, it tends to beincreasingly accepted that the lateral inertia confinement, coupledwith the pressure dependency of the constitutive material behav-iour, is a key mechanism causing the experimentally observeddynamic increase of the bulk compressive strength of concrete, andhence the DIF. Numerical studies using refined FE models withhomogenised material properties, without involving any strain ratesensitivity in the constitutivemodel, have been able to demonstratethat the experimental DIF is overwhelmingly attributable to theinertia confining effect [6]. While the general trend is deemed to bewell-established, at this juncture it should be pointed out that thenumerical evidences associated with the magnitude of the inertiaeffects as derived from a homogeneous model depends directly

0

10

20

30

40

50

60

70

0 0.02 0.04 0.06 0.08Time (ms)

Stre

ss (M

Pa)

AggregatesMortar

0

10

20

30

40

50

60

70

0 0.02 0.04 0.06 0.08Time(ms)

Stre

ss (M

Pa)

a Strip 1 of mesoscale model

0

10

20

30

40

50

60

70

0 0.02 0.04 0.06 0.08Time (ms)

Stre

ss (M

Pa)

AggregatesMortar

0

10

20

30

40

50

60

70

0 0.02 0.04 0.06 0.08Time (ms)

Stre

ss (M

Pa)

c Strip 3 of mesoscale model

b Strip 1 of homogeneous model

d Strip 3 of homogeneous model

Fig. 15. Axial stresses in mortar and aggregates, and comparison with stress in the homogeneous model, strain rate ¼ 50 s�1.


upon the soundness as how the lateral inertia are represented,which in turn depends on the dynamic Poisson’s ratio. However,such a crucial parameter is not entirely understood under dynamicloading, and in the analysis it is by default assumed to be indifferentfrom the static loading.

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04 0.05

Time (ms)

Stre

ss (M

Pa)

AggregatesMortar

a Strip 1 of mesoscale model

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04 0.05Time (ms)

Stre

ss (M

Pa)

AggregatesMortar

c Strip 3 of mesoscale model

Fig. 16. Axial stresses in mortar and aggregates, and comparison w

The contribution of the mesoscale heterogeneity in the DIF inthe high strain rate regime, discussed above and observed also insome other studies [33,37], suggests that there is at least anothermechanism which can play a sensible role in the DIF of concrete-like materials, in addition to the lateral inertia. Moreover, such

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04

Time (ms)

Stre

ss (M

Pa)

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04 0.05

Time (ms)

Stre

ss (M

Pa)

b Strip 1 of homogeneous model

d Strip 3 of homogeneous model

ith stress in the homogeneous model, strain rate ¼ 200 s�1.

0

30

60

90

120

150

180

10 50 200strain rate(/s)

Ax

ial S

tre

ss

(M

Pa

)

mortar aggregates

Fig. 17. Axial peak stress in mortar and aggregates.


a contribution is rather independent of the lateral inertial effect,therefore even in a (somewhat unlikely) scenario where littlelateral inertia exists due to a small dynamic Poisson’s ratio, theheterogeneity effect would still prevail and hence give rise toa certain amount of the bulk DIF. A securitization of this subject isbeyond the scope of the present discussion and will be treated ina separate paper.

8. Conclusions

A mesoscopic investigation into the development and distri-bution of stresses in concrete material specimens under high strainrate compression has been presented. Particular attention has beenpaid to the stress wave effect and the resulting progressive damagein the specimens, especially in the strain rate regime that exceedsthe strain rate limit governing a stress uniformity condition. Thecontribution of the large heterogeneity in concrete-likematerials tothe dynamic increase factor (DIF) of the bulk strength is highlightedwith quantification. More specifically, the following conclusionsmay be drawn:

1) For strain rates up to the theoretical strain rate limit, in thepresent case about 50 s�1 for the 50 mm sample, a reasonableuniformity of stress can generally bemaintained. Consequently,the nominal dynamic strength and strain quantities as obtainedfrom the end face responses (measured from the pressure barsin the case of SHPB tests) appear to represent well the actualbulk dynamic behaviour of the concrete.

2) With increase of the strain rate beyond the theoretical limit, thestress distribution tend to become increasingly non-uniform asthe strain rate increases, and this results in progressive failurein a layer by layer fashion as the stress wave propagates fromthe front (incident) face towards the end (transmission) face.Consequently, the nominal strength and strain responsederived using any of the three different wave methods can notproduce consistent results.

3) Relatively speaking, averaging of the responses between thetwo ends using the 3-wave method tends to produce betterrepresentation of the internal (strip-wise) stress-strain curve.However, because of the significant time lag in the responsebetween the two end faces, such an averaging treatment is notphysically meaningful. On the other hand, the 1-wave or 3-wave analysis produces a strength result which is similar to theback and front layer of the sample, as can be expected; but theytend to considerably overestimate the dynamic strain in the

material. In lieu of a general guide, a dedicated numericalanalysis is recommended to provide necessary correlation forthe interpretation and corrections of any test data that exceedthe respective strain rate limits.

4) The phenomenon of propagating damage under high rateloading enables the mobilisation of the strength of aggregatesin the dynamic resistance of concrete, as evidenced by theincreasingly higher stress in the aggregates than the mortarmatrix as the strain rate increases. This explains the mecha-nism through which large heterogeneity, herein the muchstronger aggregates, affects the dynamic strength of concreteunder compression. It is particularly noteworthy that sucha mechanism is relatively independent from the lateral inertiaconfinement conditions, and therefore is less subject to anyuncertainty in the constitutive model parameters governingthe inertia effect.

Further research in this direction will aim to incorporate a true3Dmesoscale model and develop specific guides for the correlationand the interpretation of the high strain rate test results.

References

[1] Gary G. Essais à grande vitesse sur béton. Problèmes spéci-fiques, Rapportspécifique du GRECO. Paris: GRECO Publisher; 1990 (in French).

[2] Ross CA, Tedesco JW, Kuennen ST. Effects of strain rate on concrete strength.ACI Mate J 1990;92:75e81.

[3] Grote DL, Park SW, Zhou M. Dynamic behavior of concrete at high strain ratesand pressures: I. Experimental characterization. Int J Impact Eng 2001;25:869e86.

[4] Zhang M, Wu HJ, Li QM, Huang FL. Further investigation on the dynamiccompressive strength enhancement of concrete-like materials based on splitHopkinson pressure bar tests, part I: experiments. Int J Impact Eng 2009;36:1327e34.

[5] Brace WF, Jones AH. Comparison of uniaxial deformation in shock and staticloading of three rocks. J Geophys Res 1971;13:4913e21.

[6] Li QM, Meng H. About the dynamic strength enhancement of concrete likematerials in a split Hopkinson pressure bar test. Int J Solids Struct 2003;40:343e60.

[7] Lu Y, Song Z, Tu Z. Analysis of dynamic response of concrete using a mesoscalemodel incorporating 3D effects. Int J Prot Struct 2010;1:197e217.

[8] Davies EDH, Hunter SC. The dynamic compression testing of solids by themethod of the split Hopkinson pressure bar. J Mech Phys Solids 1963;11:155e79.

[9] ASM handbook: mechanical testing and evaluation, vol. 8. Metals Park, OH:ASM International; 2000.

[10] Zhao H, Gary G. A three dimensional analytical solution of the longitudinalwave propagation in an infinite linear viscoelastic cylindrical bar: applicationto experimental techniques. J Mech Phys Solids 1995;43:1335e48.

[11] Zhao H, Gary G. On the use of SHPB techniques to determine the dynamicbehavior of materials in the range of small strains. Int J Solids Struct 1996;33:3363e75.

[12] Zhao H, Gary G. A new method for the separation of waves: application to theSHPB technique for an unlimited duration of measurement. J Mech PhysSolids 1997;45:1185e202.

[13] Gong JC, Malvern LE, Jenkins DA. Dispersion investigation in the split Hop-kinson pressure bar. J Eng Mater Technol 1990;112:309e14.

[14] Bacon C. Separation of waves propagating in an elastic or viscoelastic Hop-kinson pressure bar with three dimensional effects. Int J Impact Eng 1990;22:55e69.

[15] Bischoff PH, Perry SH. Compression behaviour of concrete at high strain-rates.Mater Struct 1991;24:425e50.

[16] Zhou XQ, Hao H. Modelling of compressive behaviour of concrete-likematerials at high strain rate. Int J Solids Struct 2008;45:4648e61.

[17] Dioh NN, Ivankovic A, Leevers PS, Williams JG. Stress wave propagation effectsin split Hopkinson pressure bar tests. P Roy Soc Lond A Mat 1995;449:187e204.

[18] Meng H, Li QM. Correlation between the accuracy of a SHPB test and the stressuniformity based on numerical experiments. Int J Impact Eng 2003;28:537e55.

[19] Follansbee PS. The Hopkinson pressure bar. In: Kuhn H, Medlin D, editors.ASM handbook. Mechanical testing and evaluation, vol. 8. ASM International;1985. p. 198e203.

[20] Pankow M, Attard C, Waas AM. Specimen size and shape effect in split Hop-kinson pressure bar testing. J Strain Anal Eng 2009;44:689e98.

[21] Riedel W, Kawai N, Kondo K. Numerical assessment for impact strengthmeasurements in concrete materials. Int J Impact Eng 2009;36:283e93.


[22] Eckardt, S, Hafner, S, Konke, C. Simulation of the fracture behaviour ofconcrete using continuum damage models at the mesoscale, The ECCOMAS2004 Congress, Jyväskylä, Finland.

[23] Wriggers P, Moftah SO. Mesoscale models for concrete: homogenisation anddamage behaviour. Finite Elem Anal Des 2006;42:623e36.

[24] Wang ZM, Kwan AKH, Chan HC. Mesoscopic study of concrete I: generation ofrandom aggregate structure and finite element mesh. Comput Struct 1999;70:533e44.

[25] Donze FV, Magnier S-A, Daudeville L, Mariotti C, Davenne L. Numerical studyof compressive behavior of concrete at high strain rates. J Eng Mech-ASCE1999;125:1154e63.

[26] Riedel W, Wicklein M, Thoma K. Shock properties of conventional and highstrength concrete: experimental and mesomechanical analysis. Int J ImpactEng 2008;35:155e71.

[27] Dupray F, Malecot Y, Daudeville L, Buzaud E. A mesoscopic model for thebehaviour of concrete under high confinement. Int J Numer Anal Meth Geo-mech 2009;33:1407e23.

[28] Tu ZG, Lu Y. Mesoscale modeling of concrete for static and dynamic responseanalysis part 1: model development and implement. Struct Eng Mech 2011;37:197e213.

[29] Malvar LJ, Crawford JE, Wesevich JW. A plasticity concrete material model forDyna3D. Int J Impact Eng 1997;19:847e73.

[30] Tu ZG, Lu Y. Evaluation of typical concrete material models used inhydrocodes for high dynamic response simulations. Int J Impact Eng 2009;36. 132-126.

[31] Mogi K. Experimental rock mechanics. 1st ed. London: Taylor & Francis; 2006.[32] Van, Vliet MRA, van Mier JGM. Experimental investigation of concrete fracture

under uniaxial compression. Mech Cohes-Frict Mat 1996;1:115e27.[33] Lu, Y, Song, ZH, Tu, ZG. Numerical simulation study of the strain rate effect on

concrete in compression considering material heterogeneity, The 9th Inter-national Conference of DYMAT Association, Sept. 2009, Brussels, Belgium.

[34] Meng H, Li QM. An SHPB set-up with reduced time-shift and pressure barlength. Int J Impact Eng 2003;28:677e96.

[35] Comite Euro-International du Beton. Redwood books. CEB-FIP model code1990. Wiltshire, UK: Trowbridge; 1993.

[36] Lu Y, Xu K. Modelling of concrete materials under blast loading. Int J SolidsStruct 2004;41:131e43.

[37] Hao YF, Hao H. Numerical evaluation of the influence of aggregates onconcrete compressive strength at high strain rate. Int J Prot Struct 2011;2(2):177e206.

Mesoscopic analysis of concrete under excessively high strain rate compression and implications on...

Documents

Transcript of Mesoscopic analysis of concrete under excessively high strain rate compression and implications on...