Research Article Effect of End Friction on the Dynamic

Research ArticleEffect of End Friction on the Dynamic Compressive MechanicalBehavior of Concrete under Medium and Low Strain Rates

Liu Jin12 Chengshun Xu1 Yaqiang Han1 and Xiuli Du1

1Key Laboratory of Urban Security and Disaster Engineering Ministry of Education Beijing University of TechnologyBeijing 100124 China2Department of Civil Engineering Tsinghua University Beijing 100084 China

Correspondence should be addressed to Chengshun Xu xuchengshunbjuteducn

Received 8 September 2015 Revised 25 October 2015 Accepted 26 October 2015

Academic Editor Nerio Tullini

Copyright copy 2016 Liu Jin et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The objective of the study is to examine the quantitative influence of end friction on the dynamic mechanical behavior of concreteunder medium and low strain rates Considering the concrete heterogeneity a mesoscale mechanical model was established tostudy the confinement effect of end friction in which the concrete was assumed to be composed of aggregates mortar matrixand the interfacial transition zones between them The friction behavior was utilized to describe the interaction between theconcrete specimens and loading apparatus The dynamic axial compressive mechanical behavior of concrete subjected to differentmedium and low strain rates and friction coefficients was simulated Furthermore the confinement mechanism of end frictionon the compressive dynamic increase factor (DIF) of concrete was studied The simulation results indicate that with the increaseof end friction coefficient the uniaxial compressive strength of concrete first increases and then becomes stable the end frictionconfinement changes the local stress state and damage distribution of concrete and it thus contributes to the increase in compressivestrength of concrete the friction contribution factor presents a descending tendency with increasing the strain rate and decreasesobviously when the end friction coefficient increases

1 Introduction

Strain rate effects on the tensile and compressive strengthsof concrete-like materials play an important role in theconstruction of material constitutive models which in turnexert a great deal of influences on the numerical simulationsof concrete structures subjected to intense dynamic loadings[1]The dynamic increase factor [DIF] defined by the ratio ofthe dynamic strength to the quasistatic strength in uniaxialcompression and tension has been widely accepted as animportant parameter to measure the strain-rate effect on thestrength of concrete-like materials [2] A great number oftests have been carried out to find the dependence of DIF onstrain-rate by using various test methods for example drop-hammer techniques servo-hydraulic loading rigs Split Hop-kinson pressure bar [SHPB] and explosive devices [2ndash4]Theobtained test data demonstrates the strain rate sensitivity ofconcrete dynamic strengths and indicates that the tested DIF

increases with strain rate However the test data presentslarge discrepancy which may come from the followingsources (1) different testing techniques (2) specimen sizeeffect (3) materials differences for example concrete qualityaggregate grade curing andmoisture condition and age and(4) dynamic and boundary effects as well as specimen lateralinertial effect

Recentlymany efforts have been conducted to explore themechanism of apparent dynamic strength enhancement ofconcrete such as in the publications [1 2] It is widely agreednow that the DIF obtained from dynamic test is affected bylateral inertia confinement [4 5] The friction at the interfaceof concrete specimen and loading measurement in the testesmay also constrain the lateral deformation of the specimenand changes the internal stress status and thus it might affectthe test results as well The strain rate effect described by thetensile and compressive DIFs should purely reflect the strain-rate-dependence of material itself

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 6309073 20 pageshttpdxdoiorg10115520166309073

2 Shock and Vibration

As known the end friction cannot be completely avoidedwhen conducting dynamic compressive tests The quantita-tive influence of end friction confinement should be deductedin the compressive DIF and it has been an interesting topicTo discover the influence of end friction on the macroscopicmechanical behavior of concrete including the dynamicfailure patterns and the nominal stress-strain curves as well asthe compressive DIF of concrete subjected to dynamic load-ings numerous efforts have been conducted For instanceusing the numerical methods Li and Meng [2] discussed theend friction confinement effect and found that for specimenswith 119871119863 = 05 [119871 means length of specimen and 119863 meansthe diameter of the specimen] the influence of the frictioncoefficient 120583 on DIF is insignificant when 120583 is less than01 but becomes considerable when it is larger than 02 Intheir simulations the strain rate employed ranges from 27times 101 sminus1 to 39 times 102 sminus1 and the friction coefficient utilizedranges from 0 to 07 Mu et al [6] modeled the dynamicmechanical properties of concrete subjected to the strain rateranging from 44 times 101 sminus1 to 147 times 102 sminus1 They found thatthe compressive strain-rate effect of concrete-like materialsis a pseudo property and the main mechanisms that initiatelateral confinement include the material lateral inertia andthe interface friction between the loading apparatus and theloaded specimen In addition in the works of Kim et al [7]Li et al [40] and Hao et al [4] the interface friction betweenthe loading apparatus and the loaded specimen was alsosimulated and examined They concluded that the frictioneffect is another factor in the so-called strain-rate effectThesesimulation efforts contribute to the understandings aboutthe dynamic compressive strength enhancement of concreteHowever these efforts are all focused on high strain ratesand the effect of end friction on the failure of concrete undermedium and low strain rates is rarely involved

The objective of the present study is therefore to examinethe quantitative influence of the confinement due to end fric-tion on the mechanical behavior of concrete under mediumand low strain rates Experimental results show that concreteis a heterogeneous brittle material that fractures throughthe formation growth and coalescence of microcracks [8]Moreover the failure behavior of concrete is closely relatedto the meso-microstructure of concrete and the hetero-geneities should therefore be considered [9ndash13] In this workconcrete was assumed to be a composite material consistingof mortarmatrix coarse aggregates and interfacial transitionzone (ITZ) between the former two phases Including thesecomponents in both experimental and numerical investiga-tions is essential because the mechanical properties of thesematerials are distinct In the computational model the dam-aged plasticity model implemented in a strain-rate sensitiveframework was utilized for the description of the dynamicmechanical behavior of the mortar matrix and the ITZ andelastic model was employed to characterize the mechanicalproperties of the aggregateThe friction behavior was utilizedto describe the interaction between the concrete specimensand loading apparatus By using the material model togetherwith the commercial software ABAQUS the reliability ofthe present approach was verified through the comparison

with the available test data Subsequently the dynamic axialcompressive mechanical behavior of concrete subjected todifferent medium and low strain rates was simulated underdifferent friction coefficients Furthermore the confinementmechanism of end friction on the compressive dynamicincrease factor (DIF) of concrete was studied and discussed

The paper is decomposed as follows The next sectiondescribes the rate-dependent constitutive framework forconcrete Section 3 presents the mesoscale computationalmodel for the simulation of the end friction confinementeffect and illustrates the verification of the present approachIn Section 4 the quantitative influences of the end frictionconfinement on the failure patterns the nominal stress-straincurves and the compressive DIF of concrete are exploredAnd finally the concluding remarks are given in Section 5

2 The Constitutive Framework forConcrete Mesocomponents

21 Concrete Damaged Plasticity Model The behavior ofconcrete is complex due to an array ofmorphological featuresaswell as deformation and failuremechanisms inherent in theconcrete microstructure The complicating factors includeheterogeneous constitution randomphase distribution arbi-trary microcrack patterns and internal granular frictionThecomplexity of the meso- or microscopic processes rendersexplicit resolution of the influences on a highly challengingtask

As known plasticity continuum damage mechanicsfracture mechanics elastic-damage and combined plasticityand damage mechanics are the common theories that havebeen widely used for the description of concrete mechanicalbehavior However the plasticity theory failed to address thedegradation of the material stiffness due to microcrackingthe fracture mechanics method was hard to reflect the nucle-ation of numerousmicrocracks and the growth of themicroc-racks to macrocracks and the continuum damage mechanicswith elasticity also could not describe the irreversible defor-mations and the inelastic volumetric expansion in compres-sion [13] Since both the stiffness degradation behavior andirreversible plastic deformations are contributing to the non-linear behavior of concrete a reasonable constitutive modelshould characterize the two distinct mechanisms In recentyears many coupled plasticity-damage models were pro-posed to describe the mechanical behavior of concrete suchas the work in the references [13 15ndash18] Among these effortsthe concrete damaged plasticity model was first proposed byLubliner et al [15] for monotonic loading and it was devel-oped later by Lee and Fenves [16] to consider the dynamic andcyclic loadings The model is a continuum plasticity basedmodel for concrete and it assumes that themain failuremech-anisms of the concrete are cracking in tension and crushingin compression The model which has been included in theABAQUS package has been widely used for the description ofthe static and dynamic mechanical behaviors of concrete-like materials for example in the publications [11 12 19 20]A brief presentation of the damaged plasticity model fromABAQUS is presented below

Shock and Vibration 3

Crack closure

ft

fc

E0(1 minus dc)(1 minus dt)

E0(1 minus dc)

E0(1 minus dt)

120576

120590c

120590t

E0

E0

Figure 1 Unaxial loading path of concrete plasticity damagemodel

211 Isotropic Damage Model The stress-strain relation forthe general three-dimensional multiaxial condition is givenby the scalar damage elasticity equation

120590 = (1 minus 119889)119863el0 (120576 minus 120576

pl) (1)

in which119863el0is the initial elasticity matrix 119889means the dam-

age variable and 120576pl is the plastic strain tensor Tensile damageand compressive damage are characterized independently bytwo hardening parameters 120576pl

119905and 120576pl119888(subscripts ldquo119905rdquo and ldquo119888rdquo

stand for tension and compression resp) which are referredto as equivalent plastic strains in tension and compressionThe evolution equations of the hardening variables 120576pl

119905and

120576pl119888 are conveniently formulated by considering uniaxial

loading conditions first and then extended to multiaxial con-ditionsThemodel assumes that the uniaxial tensile and com-pressive responses of concrete are characterized by damagedplasticity as shown in Figure 1 It is assumed that the stress-strain response follows a linear elastic relationship under uni-axial tension until the value of the failure stress119891

119905 is reached

The failure stress corresponds to the onset of microcrackingin the concrete material Beyond the failure stress the for-mation of microcracks is often represented macroscopicallywith a softening stress-strain response which induces strainlocalization in the concrete structure The uniaxial stress-strain curves can be converted into stress versus plastic straincurves of the form as

120590119905= 120590119905(120576

pl119905 120576

pl119905 120579 119891119894) (2a)

120590119888= 120590119888(120576

pl119888 120576

pl119888 120579 119891119894) (2b)

where 120576pl119905and 120576

pl119888are the equivalent plastic strain rates 120579 is the

temperature and 119891119894(119894 = 1 2 ) are other predefined field

variablesAs shown in Figure 1 when the specimen is unloaded

from any point on the strain softening branch of the stress-strain curves the unloading response is weakened the elasticstiffness of the material appears to be degraded Herein

the degradation of the elastic stiffness is characterized by twodamage variables tensile damage factor 119889

119905and compressive

damage factor 119889119888 and the stress-strain relationships under

tension and compression are respectively

120590119905= (1 minus 119889

119905) 1198640(120576119905minus 120576

pl119905) (3a)

120590119888= (1 minus 119889

119888) 1198640(120576119888minus 120576

pl119888) (3b)

This representation although somewhat simplified cap-tures the main features of concrete behavior Sometimesin order to avoid unreasonable mesh sensitive results thetensile post-failure behavior is given in terms of a fractureenergy cracking criterion by specifying a stress-displacementcurve instead of a stress-strain curve Under multiaxialloading conditions the evolution equation of the hardeningparameters for uniaxial loading should be extended

According to Lee and Fenvesrsquos work [16] tensile and com-pressive equivalent plastic strains have the following forms

120576pl119905= 119903 () 120576

plmax (4a)

120576pl119888= minus (1 minus 119903 ()) 120576

plmin (4b)

in which 120576plmax and 120576

plmin are the maximum and minimum

plastic strain tensors respectively 119903() means a multiaxialstress weight factor and is defined as

119903 () =

sum3

119894=1⟨119894⟩

sum3

119894=1

10038161003816100381610038161003816119894

10038161003816100381610038161003816

0 le 119903 () le 1 (5)

in which 119894are principal stress components (119894 = 1 2 for 2D

problems and 119894 = 1 2 3 for 3D problems) and the Macauleybracket ldquo⟨sdot⟩rdquo is defined by ⟨119909⟩ = 05 times (|119909| + 119909)

212 Plasticity Yield Surface The model makes use of theyield function of Lubliner et al [15] with the modificationsproposed by Lee and Fenves [16] to account for differentevolution of strength under tension and compression Theevolution of the yield surface is controlled by the two harden-ing variables 120576pl

119905and 120576pl119888 In terms of effective stress the yield

function 119865 is defined as

119865 = 119865 (120590 120576pl)

=1

1 minus 120572[119902 minus 3120572119901 + 120573 (120576

pl) ⟨120590max⟩ minus 120574 ⟨minus120590max⟩]

minus 120590119888(120576

pl119888) = 0

(6)

in which 119901 = minus(13)120590 119868 is the effective hydrostatic pressurestress 119868 is a stress invariant 119902 = radic(32)119878 119878means theMisesequivalent effective stress and 119878 is the effective stress deviatordefined as 119878 = 119901119868 + 120590 120590max is the maximum principaleffective stress Typical yield surface is depicted in Figure 2 forplane stress conditions


Uniaxial compression

Uniaxial tension

Biaxial tension

Biaxial compression 120590c

120590t

(120590cc 120590cc)

1205901

1205902

Figure 2 Yield surface for plane stress conditions

1205901

1205902

1205903

Kc = 1

Kc = 23

Figure 3 Yield surfaces in the deviatory plane [119870119888

= 23

corresponds to Rankine formulation and119870119888= 1 corresponds to the

Drucker-Prager criterion]

Moreover parameters of 120572 120573(120576pl) and 120574 respectively canbe calculated by

120572 =120590119888119888120590119888minus 1

2120590119888119888120590119888minus 1

(7a)

120573 (120576pl) =

120590119888(120576

pl119888)

120590119905(120576

pl119905)

(1 minus 120572) minus (1 + 120572) (7b)

120574 =3 (1 minus 119870

119888)

2119870119888minus 1

(7c)

in which 120590119888119888120590119888means the ratio of initial equivalent-biaxial

compressive yield stress to initial uniaxial compressive yieldstress and its default value of the ratio is 116 according to[19 20] 119870

119888is the ratio of the tensile to the compressive

meridian and defines the shape of the yield surface in thedeviatory plane (see Figure 3)

213 Plastic Flow Rule The nonassociated potential plasticflow is employed in the present damaged plasticity modelThe flow potential function 119866(120590) used for this model is

the nonassociatedDrucker-Prager hyperbolic function and isdefined according to

119866 (120590) = radic(120578120590119905tan120595)2 + 1199022 minus 119901 tan120595 (8)

in which 120595 is the dilation angle measured in the 119901-119902 planeat high confining pressure 120590

119905is the uniaxial tensile stress at

failure 120578 is the eccentricity that gives the rate at which theplastic potential function approximates the asymptote (theflow potential tends to a straight line as the eccentricity tendsto zero) and its default value is equal to 01Thenonassociatedflow rule means that the plastic strain vector is normal to theplastic potential function that differs from the yield surface

The plastic strain rate 120576pl and the effective plastic strain

rate 120576pl can be derived as follows

120576pl=

120597119866

120597120590(9a)

120576pl= ℎ (120590 120576

pl) sdot 120576

pl (9b)

where is the plastic multiplierBecause the selected plastic flow is nonassociated to

obtain an acceptable rate of convergence the unsymmetricmatrix storage and solution scheme is used For more detailsabout the constitutive model refer to Lubliner et al [15] Leeand Fenves [16] and Du et al [11]

22 Dynamic Increase Factor (DIF) for Concrete The avail-able experimental and numerical observations have demon-strated that concrete and other cementitiousmaterials are ratesensitive [3 21 22] The strain rate effect has been studiedon different mechanical characteristics of concrete Youngrsquosmodulus Poissonrsquos ratio energy-absorption capacity andaxial strain atmaximumstrength are rate-sensitive quantitiesbut at amuch lower intensity than the compressive and tensilestrengths [3 23] Similar to the work [4 11] only the strengthenhancement was considered in the present study

So far many empirical and theoretical relations have beendeveloped to quantitatively describe the strain rate effect onconcrete properties In particular the code by the ComiteEuro-international du Beton (CEB) recommends variousvalues of the DIF for the tensile and compressive strengthsunder different strain rates based on the available test resultsThe formula of the compressive DIF recommended by CEB[3] has the following form

CDIF =119891cd119891cs

= (120576119889

120576cs)

1026120572

for 120576119889le 30 sminus1 (10a)

CDIF =119891cd119891cs

= 120574 ( 120576119889)13 for 120576

119889gt 30 sminus1 (10b)

where 119891cd is the dynamic compressive strength at the strainrate of 120576

119889(in the range of 30 times 10minus6ndash1000 sminus1) 120576cs = 30 times

10minus6 sminus1 (static strain rate) log 120574 = 6156120572 minus 049 120572 = (5 +

3119891cu4)minus1 119891cs is the static compressive strength and 119891cu is

the static cube compressive strength [in MPa] The DIF for


the compressive strength in (10a) and (10b) was oftenadopted for example in the publications [10 11]

The CEB also gives empirical formulae to estimate theDIF for concrete tensile strength However comparisonbetween the available experimental results in the literaturewith the CEB empirical equations indicates that the CEBtensile DIF substantially underestimates the tensile DIF ofconcrete [24 25] In Malvar and Rossrsquos [24] work an empiri-cal formula was proposed and the formula for the tensile DIFcan be written as

TDIF =119891td119891ts

= (120576119889

120576ts)

120575

for 120576119889le 1 sminus1 (11a)

TDIF =119891td119891ts

= 120573(120576119889

120576ts)

13

for 120576119889gt 1 sminus1 (11b)

where 119891td and 119891ts are respectively the dynamic uniaxial ten-sile strength at the strain rate 120576

119889and the quasistatic uniaxial

tensile strength 120576ts = 10minus6 sminus1 (static strain rate) log120573 = 6120575minus2

120575 = 1(1 + 81198911015840

1198881198911015840

1198880) 11989110158401198880= 10MPa and 119891

1015840

119888is the quasistatic

uniaxial compressive strength of concrete (in MPa)

3 Mesoscale ComputationalModel for Concrete

Theoretical experimental and numerical studies indicatethat the failure pattern crack mode and even the velocity ofthe crack propagation are influenced by loading rate [11 26ndash28] The mechanical behaviors of concrete including themacroscopic mechanical properties and the failure patternsare associated with the micro-mesostructure of concreteDue to the complexity of concrete the failure behavior can bestudied numerically In the numerical simulations concreteheterogeneity should be accounted for and thus it requiresthe adoption of a mesoscale mechanical model in which theheterogeneity can be explicitly described

At themesoscale concretewas often assumed to be amul-tiphase composite composed of aggregate mortar matrix theinterfacial transition zones [ITZs] voids and so forth A lot ofwork [9 10 19 29] has demonstrated that the ITZs have a dra-matic influence on the failure pattern and the macromechan-ical properties of concrete However the mechanical proper-ties especially the dynamic mechanical behavior of the ITZsare not well understood yet [13 19 30ndash32] Moreover thereal typical thickness of ITZ is about 10ndash50120583m [9 10] whichlimits the minimum size of the finite element mesh In mostsimulation efforts the ITZ thickness is set as 02ndash08mm[9 10] 05mm [8] 04mm [19] and 1mm [12] for the sake ofsimplicity

31 Mesostructure and Mechanical Model of Concrete Con-crete exhibits heterogeneity in material and mechanicalproperties onmany length scales [29] At the mesoscale con-sidering the effect of heterogeneities of concrete the concretein the present work was regarded as a three-phase compositecomposed of coarse aggregate particles mortar matrix andthe ITZs between the former two phases Random aggregatemodels have long been demonstrated to be an effective

mesomechanical approach to simulating the failure processand the deformation of concrete and they have been widelyutilized in fracturemechanics of concrete for example in [11ndash13 33] Based on the concrete random aggregate model inwhich the shape size and distribution of the coarse aggregateparticles closely resemble real concrete in the statistical senselots of efforts have been carried out to investigate the macro-scopic mechanical behavior and failure mechanism of con-crete [10ndash12 29] In this work the concrete random aggregatemodel was therefore also employed

Due to great computational expense and memorydemand of mesoscale simulations it is in general notpossible to model the full structure at the mesoscale with thecomputational power available nowadays Herein the presentstudy only two-dimensional simulations were thus con-ducted Similar to the work [9ndash12 33] the aggregate particlesare modeled as circles with random spatial distributionenfolded by a thin interface layer of ITZ Similar to those in[11 12] the ldquotake-and-placerdquo method was utilized to generatethe coarse aggregate particles The fine ones and othercomponents (eg air voids) were assumed to be mixed upwith the cement paste establishing the mortar matrix phaseThis matrix phase can thus be regarded as a homogeneouscomponent at the mesoscale The material properties of thepaste phase are thus strongly dependent on the type of cementused amount of pores water to cement ratio and otheradditives present in the concrete which can be determinedby microscale simulation efforts Furthermore the Fullercurve widely utilized for example in [11 12] was selected todetermine the distribution of aggregate with different sizesemployed in the concrete specimen The aggregate particleswere then placed into the matrix one by one at randomlydetermined positions and in such a way that no overlappingwith particles already placed Finally at the mesoscale aggre-gate particles (typically sand or gravel) and the surroundingITZ layers behave as inclusions in the matrix phase of mortarmatrix

Figure 4 presents a typical schematic diagram of concretemesostructure generated by employing the above approachThe size of the heterogeneous concrete specimen is 150mm times

150mm Note that aggregate particles in real concrete arenot perfectly circular However the efforts in [10 11] haveindicated that the effect of aggregate shape on macroscopicstress-strain curve of concrete seems to be very smallSimilar to that in [11 34] the concrete sample contains 6medium stones with a diameter of 30mm and 56 small stoneswith a diameter of 12mm The concrete specimens involveabout 469 of aggregate per volume In Figure 4 the greenareas mean the aggregate phase the red thin-layer zonessurrounding the aggregate particles are the ITZs and the greyzones denote the mortar matrix Herein the interfacial zonesare viewed as uniform thickness shells placed concentricallyaround each aggregate Similar to the work [34] the thicknessof the ITZ was set as 1mm

To describe the influence of end friction confinement onthe dynamic compressive mechanical properties of concretetwo rigid blocks (see the two blue blocks in Figure 4) wereplaced respectively at the top and the bottom of the con-crete specimen The interactions between the heterogeneous


y

x

AggregateITZ

Mortar matrixRigid block


Unit mm End

fric

tion

25 25150

2525

150

Figure 4 Mesoscale mechanical model of concrete under dynamic uniaxial compression considering the influence of end frictionconfinement The specimen has 22346 elements [For interpretation of the references to color in this figure legend the reader is referredto the web version of this paper]

concrete specimen and the two rigid blocks that is the twored-line regions are implemented by the classic coulombfriction behavior The friction coefficients considered in thefollowing simulations are 00 (completely smooth) 01 0203 06 08 and infin (ie the top and bottom surfaces ofthe specimen were fixed in the horizontal direction) fora quantitative assessment of the end friction confinementeffect As known the DIF difference obtained from thesimulation results with different friction coefficients canthen be attributed purely to the influence of end frictionconfinement effects

Themeshing of these concrete specimens were conductedwith four node solid elements with a linear strain field Tohave a better understanding of concrete fracture the averagemesh size was l mm which is consistent with that in the workfrom Hentz et al [23] and Du et al [11 12]

32 Mechanical Parameters for the Three MesocomponentsIn Zhou and Haorsquos [11 13] and Snozzi et alrsquos [35] workthe uniaxial tensile and compressive mechanical behaviorof concrete at high strain rate were studied From theirsimulation observations it can be noticed that no damagegenerates in the aggregate phaseMoreover in Pedersen et alrsquos[8] simulation of the dynamic tensile behavior of concretethe aggregate grains were assumed to follow a linear elasticmaterial behaviorThis is because the tensile and compressivestrengths of the aggregate are much larger than those of mor-tar matrix and the ITZs Similar to the work [8 11] the aggre-gate was assumed as elastic in the following simulationsThistreatment should work for the concrete subjected to mediumand low strain rates as discussed in the present study

To the mortar matrix and the ITZs the continuumdamage mechanics theories mentioned previously wereemployed to describe their mechanical properties In contin-uum mechanics the constitutive model is often expressed in

terms of stress-strain relations When the material exhibitsstrain-softening behavior leading to the strain localizationbehavior this formulation leads to a strongmesh dependencyof the finite element results in which the energy dissipateddecreases upon mesh refinement [11 19] Similar to the workin [10 11 19] to avoid or release unreasonable mesh sensitiveresults the tensile postfailure behavior was given in terms ofa fracture energy cracking criterion by specifying a stress-displacement curve instead of a stress-strain curve In addi-tion unlike discrete crack models [36 37] the continuumdamaged plasticity model based on continuum mechanicsuses damage factors such asDAMAGEC for compression andDAMAGET for tension to describe the fracture behavior andit is not capable of providing accurate crack geometry such ascrack width

The main mechanical parameters including the elasticmodulus Poissonrsquos ratio and uniaxial tensilecompressivestrengths of the three mesoconstituents used in this study forthe following numerical analysis were listed in Table 1

It is to be noted that the DIFs given in Section 22 are builtbased on the test results on concrete material The strain rateeffects for the aggregate phase mortar matrix and the ITZare essentially distinct as illustrated in [38 39] NeverthelessGrote et alrsquos [21] experimental observations indicate that thedynamic mechanical behavior of mortar matrix is closelysimilar to that of concrete In addition the ITZ layer can betreated as a kind of mortar matrix with a high(er) porosity[34] and thus its dynamic mechanical behavior can also beregarded as similar to those of mortar matrix and concreteIn summary it is assumed that the ITZ and the mortarmatrix have the same DIFs mentioned previously The treat-ments are consistent with the ones in [9 10] The strainrate effect for both the mortar matrix and the ITZs ontheir tensilecompressive strengths was incorporated intothe plasticity damaged model which was represented by


Table 1 Mechanical parameters of the concrete mesocomponentsused in the study

Mortar matrix ITZ AggregateMass density 120588 [kgm3] 275 times 103 275 times 103 288 times 103

Elastic modulus [GPa] 30lowast

25lowast

70lowast

Poissonrsquos ratio 02lowast

022lowast

016lowast

Dilation angle 120595 [∘] 30 30Eccentricity 120578 01 01Shape factor 119870

1198880667 0667

Stress ratio 120590119888119888120590119888

116 116Tensile yield stress 120590

119905[MPa] 143

lowast12lowast

Compressive yield stress 1205901198880

[MPa] 143lowast

12lowast

Ultimate compressive yieldstress 120590cu [MPa] 143 12

Note data with ldquolowastrdquo are quoted from Du et al [12]

a dynamic increase factor (ie the DIF) with respect to strainrate

33 BoundaryLoading Conditions and FE Simulations Toexamine the effects of end friction confinement on themacromechanical behavior of concrete subjected to dynamiccompressive loadings the failure behavior of the concretespecimen was explored The corresponding loading andboundary conditions for the interaction system were illus-trated as follows (1) constant vertical velocity V was appliedon the top surface of the rigid block and the left and rightsides of the specimen are free of stress (2) the movementat the bottom rigid block was fixed at both the vertical andhorizontal directions (3) friction behavior between the twoblocks and the concrete sample was set The nominal strainrate 120576 is defined as the applied velocity divided by the heightof the specimen that is 120576 = Vℎ where ℎ is the height of thespecimen

In the following simulations ABAQUS increased auto-matically the size of step time going from smaller one to themaximumauthorized defined by the user For all the concretespecimens the numerical simulations were performed in aplane strain condition

34 Verification of the Mesoscale Approach In reality theavailable tests on loading rate effect of concrete are not veryrich and the test data is also not complete Comparativelyspeaking Dilger et alrsquos [14] test data is the most completeone and it is often used to verify the reliability of the analysismethod or model such as in [26]

To illustrate the reliability and accuracy of the presentmesoscale simulation method Dilger et alrsquos [14] test on thedynamic uniaxial compressive behavior of concrete was alsoselected for the comparison in this section In Dilger et alrsquos[14] experiment the macroscopic mechanical behavior ofconcrete under different nominal strain rates (ie the strainrates of 333 times 10minus5 sminus1 333 times 10minus3 sminus1 and 02 sminus1) was

explored and the complete compressive stress-strain curvesof concrete were obtained

In the previouswork [33] a series of dynamic compressivetest of concrete under different strain rates were conductedbased on the present mesoscale mechanical method includ-ing 120576 = 333 times 10minus5 sminus1 1 times 10minus4 sminus1 333 times 10minus3 sminus1 1 times 10minus2 sminus15 times 10minus2 sminus1 02 sminus1 1 sminus1 100 sminus1 and 200 sminus1 It is to benoted that in the simulations the concrete specimens werefixed at one end and loaded longitudinally through an appliedconstant velocity at the other end The detailed mechanicalparameters and analysis method can refer to [33]

Figure 5 presents the failure patterns of the concretespecimen subjected to different nominal strain rates Onecan notice that with the increase of strain rate the numberof cracks (ie the red damaged region) increases obviouslyand the crack branching behavior occurs Therefore withthe increase of strain rate the fracture of concrete requiresmore energy demand to reach failure which should be themain reason for the improvement of the dynamic strength ofconcrete

Figure 6(a) reports the comparison between the sim-ulations results (solid curves) and the experimental data(symbols) in terms of nominal stress (applied forces dividedby the specimen cross sectional area) versus nominal strain(the displacement generated at the applied-load end dividedby the length of the specimen) curves and the agreementis very good It should be noted herein that as mentionedpreviously the relatedmechanical and physical parameters ofthe concrete mixture used in the test are not full especiallythe mechanical parameters of the ITZ Accordingly in thepreviouswork [33] themechanical parameters (especially theelasticmodulus and the strength) of the ITZwere determinedby the inversion method on the basis of only two strainrates (333 times 10minus5 sminus1 and 333 times 10minus3 sminus1) The simulationof the third available strain (ie 02 sminus1) can be regarded asa validation of the ability of the model to predict concretestrength outside the range of strain rates used for calibrationThis treatment is similar to that in [26]

Furthermore the compressive DIFs were evaluatedaccording to the simulation results and the comparison of thepresent simulation results and some available experimentalobservations is plotted in Figure 6(b) It can be observedthat the present simulation results agree very well with thecollective experimental data This indicates that the presentanalysis method can well simulate the dynamic compressivebehavior of concrete The validated mesoscale mechanicalapproach would be employed in the following simulations tostudy the effect of end friction confinement on the dynamiccompressive behavior of concrete

4 Dynamic Compressive FailureBehavior of Concrete

On the basis of the mesoscale mechanical model describedin Section 3 the simulations of the dynamic axial com-pression of the concrete specimen with various end fric-tion coefficients were implemented The loading velocitiesadopted on the upper rigid block in the simulations are


10Damage

120576 = 333 times 10minus5 120576 = 1 times 10minus4 120576 = 333 times 10minus3 120576 = 1 times 10minus2 120576 = 5 times 10minus2

120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200

Figure 5 Compressive failure patterns of the concrete specimen under different strain rates ranging from 1 times 10minus5 sminus1 to 200 sminus1

Compression test(Dilger et al 1984) and present model

Nom

inal

com

pres

sive

stre

ss (M

Pa)

0002 00030001 0004 0005 00060Compressive strain

0

10

20

30

40

120576 = 333 times 10minus5 sminus1


120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF

Test data Present method

1E minus 6 1E minus 4

Strain rate (sminus1)

(b)

Figure 6 (a) Comparison of the present simulation results and Dilger et alrsquos [14] test data and (b) compressive DIFs for the simulation resultsand some available test data


Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3

Mesh size 15mmMesh size 10mmMesh size 05mm

End friction coefficient 120583 = 06

1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)

Figure 7 (a) The simulation results of three concrete specimens with the same distribution of aggregates and different mesh sizes and(b) damage distribution of the concrete specimens with different mesh sizes under the same strain rate and friction coefficients at theirpeak stresses

V = 15 times 10minus6ms 15 times 10minus4ms 15 times 10minus3ms 15 times

10minus25ms 15 times 10minus2ms 15 times 10minus15ms and 15 times 10minus1msin an ascending sequence and in every loading cycle the con-stant velocity was utilized The corresponding macroscopicnominal strain rates are 120576 = 1 times 10minus5 sminus1 (quasistatic) 1 times10minus3 sminus1 1 times 10minus2 sminus1 1 times 10minus15 sminus1 1 times 10minus1 sminus1 1 times 10minus05 sminus1and 1 times 100 sminus1 respectively Moreover as mentioned previ-ously the seven end friction coefficients 120583 considered in thefollowing simulations are 00 (completely smooth) 01 0203 06 08 and infin (ie the top and bottom surfaces of thespecimenwere fixed in the horizontal direction) respectively

To check the mesh sensitivity in the simulations threedifferentmesh sizeswere utilized that is 05mm 10mm and

15mm Figure 7(a) presents the nominal compressive stress-strain curves of the concrete specimen subjected to a nominalstrain rate of 01 sminus1 with the same end friction coefficientand the three different mesh sizes It can be found clearlyfrom Figure 7(a) that the mesh sizes (ie 05mm 10mmand 15mm) have less influence on the stress-strain curvesThe three curves are also overlapped Figure 7(b) shows thedamage distribution of the three concrete specimens men-tioned above with the same distribution of aggregates Onecannotice that the failure patterns of the three specimenswithdifferent mesh sizes are closely similar to each other Conse-quently the mesh size of 15mm was used in the followingsimulations


= 1 times 10minus1 sminus1 = 1 times 100 sminus1= 1 times 10minus3 sminus1 = 1 times 10minus2 sminus1= 1 times 10minus5 sminus1

Incr

easin

g fr

ictio

n co

effici

ent

Increasing strain rate

Equivalent plastic strain times10minus300 56

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576

Figure 8 Equivalent plastic strain distribution within the concrete specimen under different strain rates and end friction coefficients at theirpeak compressive stresses

41 Effects of Strain Rate and End Friction Confinement onFailure Patterns of Concrete Figure 8 presents the equivalentplastic strain distribution within the concrete specimensunder five groups of different strain rates ( 120576 = 1 times 10minus5 sminus11 times 10minus3 sminus1 1 times 10minus2 sminus1 1 times 10minus1 sminus1 and 1 times 100 sminus1) and fourgroups of different friction coefficients (120583 = 0 03 08 andinfin) at their peak compressive stresses (ie the compressivestrength) One can note from Figures 8(a)ndash8(d) that for thespecimen with the same friction coefficient with the increase

of nominal strain rate the equivalent plastic strain withinconcrete increases obviously In addition as the frictioncoefficient 120583 increases from 0 to infin the equivalent plasticstrain within concrete at reaching the peak stress varies

For the specimenwith the friction coefficient of 0 and 03the obtained plastic strain distributions within the concretespecimens are obviously different For the one with the fric-tion coefficient of 03 and 08 the obtained plastic strain dis-tributions are slightly different And for that with the friction


Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Maximum principal stress (MPa)


120576

120576

120576

120576

Figure 9 Maximum principal stress distribution within the concrete specimen under different strain rates and end friction coefficients attheir peak compressive stresses

coefficient of 08 andinfin the obtained plastic strain distribu-tions are almost totally consistent That is to say when thefriction coefficient becomes larger than 08 the effect of endfriction confinement on the failure patterns of concrete canbe ignored

The maximum principal stress distributions within theconcrete specimens under the five groups of different strainrates and the four different friction coefficients at the peakcompressive stresses are presented in Figure 9 It can be

noted intuitively from Figure 9 that the changes of nominalstrain rate and friction coefficient have a significant influenceon the maximum principal stress distribution of the speci-menWith the increase of end friction coefficient the value ofmaximum principal stress gets large and the distribution areadraws close to central section from the top and bottom of theconcrete specimen respectively at same loading velocity

The damage (ie the cracks) distributions within theconcrete specimens corresponding to Figures 8 and 9 are


Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576

Figure 10 Damage distribution within the concrete specimens under different strain rates and friction coefficients at their peak stresses

presented in Figure 10 for different loading velocities anddifferent end friction coefficients In Figure 10 differentcolors represent different damage indexes The red colordenotes the most serious damage while the blue color meansthat no damage generates The white rounded areas whichare regarded as aggregates in these models have no damage

It can be noticed clearly from Figure 10 that the damagedareas increase with the growth of nominal strain rate Whenthe nominal strain rate is relatively low(er) (eg 1 times 10minus5 sminus1)the damaged area of the specimens is small when reaching

their peak stress Accordingly the damage regions (iecracks) are narrow and short and the number of cracks islimited As the nominal strain rate rises from 1times 10minus5 sminus1 to 1times100 sminus1 the damaged area and the number of cracks increasegradually When the nominal strain rate is relatively high(er)(eg 1 times 100 sminus1) a lot of areas are damaged in different levelsThe cracks widen and link together forming a crack networksimultaneously

Therefore it can be concluded that with the increase ofloading velocity the nominal strain rate increases gradually


Table 2 Dynamic compressive strength of concrete under different end friction coefficients

120583Dynamic compressive strength [MPa]

1 times 10minus5 sminus1 1 times 10minus3 sminus1 1 times 10minus2 sminus1 1 times 10minus15 sminus1 1 times 10minus1 sminus1 1 times 10minus05 sminus1 1 times 100 sminus1

0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749

and the number of cracks or damaged zones increases whenthe apparent compressive strength of concrete is reachedTheenergy consumption of the loading process increases corre-spondinglyThis should be the major reason for the improve-ment of dynamic compressive strength of concrete materials

It can be found obviously from the comparison amongFigures 10(a)ndash10(d) that the damage states of concrete spec-imens are diverse at their peak stresses for different endfriction conditions

Compared with the concrete specimen without end fric-tion confinement (ie 120583 = 0) the deformation at the end ofthe specimen considering the effect of end friction (ie120583 gt 0)was obviously hindered In addition with the increase of endfriction coefficient the obstruction degree of deformationraises For the specimen with the two ends completely fixed(ie 120583 = infin) there is no horizontal deformation at the endsmaking the confinement effect very strong

From Figure 10 one can note that a larger end frictioncoefficient leads to a lower damage level in the top and bottomof the specimen with a constant nominal strain rate Forlarger end friction coefficient the damaged areas draw closeto the central section from the top and bottom of the concretespecimenThese phenomena should generate due to the effectof the end friction confinement which changes the uniaxialloading state in the local regions of concrete These can alsobe illustrated from the simulation results in Figures 8 and 9

42 Effect of End Friction Confinement on Dynamic Compres-sive Strength The macroscopic nominal compressive stress-strain curves of the concrete specimen under different strainrates and end friction coefficients are plotted and compared inFigure 11 Moreover the corresponding peak stresses that isthe nominal dynamic compressive strengths of concrete arereported in Table 2 Obviously both the stress-strain curvesof concrete in Figure 11 and the axial dynamic compressivestrengths in Table 2 demonstrate the effects of strain rates andend friction coefficients

It can be noted obviously from Figure 11 and Table 2 thatthe dynamic uniaxial compressive strength of concrete isimproved due to the end friction confinement effect Further-more under the same strain rates the ascending curves of theconcrete sample with various friction coefficients are almostoverlapped indicating that the elastic modulus of concreteis almost not influenced by the end friction confinementeffect The obtained compressive strengths and the softening

curves are distinct significantly because the stresses andstrains status as well as the fracture paths within the concretespecimen for different friction coefficients vary

For various strain rates the quantitative relationshipsbetween the obtained dynamic compressive strengths of theconcrete specimen and end friction coefficient are presentedin Figures 12(a) and 12(b) It can be seen clearly fromFigure 12 that the concrete dynamic compressive strengthincreases with the increase of end friction coefficient Whenthe end friction coefficient 120583 grows from 0 to 03 the uniaxialcompressive strength increases rapidly while the uniaxialdynamic compressive strength increases slightly and thengets to be stable when the end friction coefficient exceeds 03This is to say when the end friction coefficient exceeds 03 theend confinement effect on the dynamic compressivemechan-ical behavior of concrete can be ignored This can also beconcluded from the analysis resultsmentioned previously forexample the damage factor distribution of concrete pre-sented in Figure 10

Furthermore the results about the effect of end frictionconfinement on concrete dynamic compressive strength atmedium and low strain rates are similar to those at high strainrate For more information about the end confinement effectfor high strain rate refer to Li et al [40]

Figure 13 presents the damage distribution within theconcrete specimens at their corresponding peak stressesunder different end friction confinement conditions at thesame loading velocities Figure 13(a) shows the results for theloading condition of a relative lower strain rate that is 120576 = 1 times10minus2 sminus1 One can note that for ldquopurerdquo uniaxial compressivebehavior of concrete (ie the end friction 120583 between theconcrete specimen and the rigid block is zero 120583 = 0) thenumber of cracks generated or the damage degree withinthe specimen is very limited when reaching the compressivestrength of the concrete sample With the increase of endfriction coefficient for example 120583 = 01 and 02 the totalnumber of cracks ascends gradually but the number of cracksnear the top and bottom regions of the specimen descendsWhen the end friction coefficient becomes larger than 03(eg 120583 = 03 06 08 and infin) the number of cracks turnsto be stable but the width of the cracks broadens as can beseen in Figure 13(a)

Figure 13(b) shows the simulation results for the loadingcondition of a relative higher strain rate that is 120576 = 1 times10minus05 sminus1 For ldquopurerdquo uniaxial compression (ie the end


7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading

Dynamic loading Dynamic loading= 1 times 10minus1 sminus1

= 1 times 10minus3 sminus1



times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

3 6 90Strain 120576 times10minus3

3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading

Friction coefficient

= 1 times 100 sminus1

120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576

Figure 11 Dynamic compressive stress-strain curves of concrete under different strain rates and end friction coefficients

Com

pres

sive s

treng

th (M

Pa)

Strain rate1 times 10minus3 sminus1

1 times 10minus2 sminus1 1 times 10minus5 sminus11 times 10minus15 sminus1

02 04 06 08 1000Friction coefficient 120583

6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)

Strain rate1 times 10minus1 sminus11 times 100 sminus1

1 times 10minus05 sminus1

6

8

10

12

14

16

18

20


(b)

Figure 12 The relationship between the obtained dynamic compressive strengths of concrete and end friction coefficient

friction coefficient 120583 = 0) the cracks generated within thespecimen exhibit a network distribution With the increaseof the end friction coefficient (eg 120583 = 01 and 02) thedamage distribution concentrates on the central section from

the top and bottom of the specimen When the end frictioncoefficient becomes larger than 03 (eg 120583 = 03 06 08 andinfin) the damage distributions within the specimen becomestable For the five end friction coefficients that is 120583 = 03


120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)

Figure 13 Damage distribution within concretes at their corresponding peak stresses (a) for 120576 = 1 times 10minus2 sminus1 and (b) for 120576 = 1 times 10

minus05 sminus1

06 08 and infin the damage distribution and the damagedegree within the specimen are almost totally the same

According to the above analysis on the simulation resultsit can be concluded that at the medium and low strainrate levels (1 times 10

minus5 sminus1 sim 1 times 100 sminus1) the dynamic axial

compressive strength of concrete increases first (0 le 120583 le 03)and then keeps stable (120583 gt 03) In addition the end frictionconfinement effect changes the local stress distribution withconcrete and the damage pattern

43 Effect of End Friction Confinement on Compressive DIFsLi et al [40] explored the effect of end friction confinementon the dynamic increase factor (DIF) of concrete subjectedto high strain rate ranging from 27 sminus1 to 390 sminus1 and Haoet al [4] explored the one ranging from 10 sminus1 to 590 sminus1Herein the present work concentrates on the one subjected tomedium and low strain rate ranging from 10minus5 sminus1 to 100 sminus1

Seven groups of curves which describe the relationshipof compressive DIFs and nominal strain rate under different


Takeda (1959)Takeda and Tachikawa (1962)Hatano and Tsutsumi (1960)Kvirikadze (1977)

08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF

minus5 minus4 minus3 minus2 minus1 0minus6log( )

Present model (120583 = 0)

120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)

Figure 14 The relationship between the obtained compressive DIFs and strain rate (a) comparison of the present simulation results and theavailable test data and (b) compressive DIFs of concrete under different end friction coefficients

end friction coefficients are obtained as shown in Figure 14It can be found obviously from Figure 14(a) that the presentsimulation results for 120583 = 0 coincide well with the availableexperimental observations The feasibility of the presentmesomechanical method is therefore demonstrated againFigure 14(b) plots the obtained compressive DIFs of concretepossessing various end friction coefficients For the nominalstrain rate 120576 le 1times10

minus3 sminus1 the dynamic compressive strengthof concrete improves slightly While for the nominal strainrate 120576 ge 1 times 10

minus2 sminus1 the dynamic compressive strengthincreases rapidly Exactly for this the effect of end frictionconfinement on the compressive DIFs of concrete with thenominal strain rate of 120576 ge 1 times 10

minus2 sminus1 draws much attentionTo examine the quantitative effect of end friction con-

finement a new factor 120573120583 120576 contribution of end friction

confinement in the improvement of concrete dynamic axialcompressive strength is given in the form of the followingequation

120573120583 120576=

DIF120583 120576minus DIF

0 120576

DIF120583 120576minus 1

(12)

where the subscript 120583 is the end friction coefficient 120576 meansthe nominal strain rate DIF

120583 120576is the dynamic increase factor

for the end friction coefficient 120583 and the nominal strain rate120576 DIF0 120576is the dynamic increase factor when the end friction

coefficient 120583 is equal to zero (ie without considering theeffect of end friction confinement) and the nominal strainrate is 120576

Figure 15(a) shows the quantitative relationships betweenthe friction contribution factor 120573 and end friction coefficient120583 under different nominal strain rates 120576 It can be seen fromFigure 15(a) that when the end friction coefficient 120583 growsfrom01 to 03 the friction contribution factor120573 increases sig-nificantly However as the end friction coefficient grows from03 to infin the friction contribution factor increases slightlyand then gets to be stable In addition one can see fromFigure 15(a) that with the increase of strain rate the frictioncontribution factor decreases especially when the frictioncoefficient exceeds 03

Figure 15(b) plots the relationship between friction con-tribution factor 120573 and nominal strain rate (described bylog( 120576)) under different end friction confinement conditionsWith the increase of nominal strain rate the friction contri-bution factor presents a descending trendMoreover with theincrease of end friction coefficient the descending trend getsto be much more obvious When the strain rate is relativelylower (ie for 120576 = 10minus2 sminus1) the larger the end frictioncoefficient is the larger the friction contribution factor is

For further research considering the extreme situation ofend confinement that is 120583 = infin the ascending parts of theaxial dynamic compressive stress-strain relationship curvesof the concrete specimens are obtained as shown in Figure 16for the relatively low strain rate 120576 = 1 times 10minus2 sminus1 and the rela-tively high strain rate 120576 = 1 times 100 sminus1 In Figure 16 119864

1denotes

the energy consumptionwhen the concrete specimenwithoutconsidering the effect of end friction confinement (ie 120583 = 0)


Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2

1 times 10minus11 times 10minus15

1 times 10minus05

1 times 100

120573120583 = (DIF120583 minus DIF0 )(DIF120583 minus 1)120576

120576

120576

120576

(a)

minus20 minus15 minus10 minus05 00


log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)

Figure 15 Effects of friction coefficient 120583 and strain rate 120576 on the friction contribution factor 120573 (a) 120573 and 120583 and (b) 120573 and 120576


(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)

(1) Quasistatic loading

(2) (3) Dynamic loading= 0

= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)


(4) (5) Dynamic loading

(5)

(1)

(4)

Friction coefficient(5) 120583 (4) 120583(1) 120583 = 0

= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)

Figure 16 The ascending part of concrete dynamic uniaxial compressive stress-strain curves (a) for a lower strain rate of 1 times 10minus2 sminus1 and(b) for a higher strain rate of 1 times 10minus0 sminus1


reaches its quasistatic nominal compressive strength (thestrain rate 120576 = 1 times 10minus5 sminus1) and its magnitude is equal tothe area occupied by the ascending curve and the horizontalaxial both 119864

2and 119864

4denote the energy consumption for

the strength improvement due to material strain rate effect1198643and 119864

5denote the energy consumption for the strength

improvement due to end friction confinement It can be seenclearly from Figure 16 that

1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)

That is to say when the nominal strain rate is relativelylow(er) (eg 120576 = 1 times 10minus2 sminus1) the contribution of endfriction confinement effect to the enhancement of dynamiccompressive strength is relatively high(er)While the nominalstrain rate is relatively high(er) (eg 120576 = 1 times 100 sminus1) thecontribution of end friction confinement effect to dynamicstrength improvement is relatively low(er) The simulationresults are consistent with those in the work [4]

5 Conclusions

Considering the heterogeneities two-dimensional mesoscalemechanicalmodels of concretewere established to investigatethe quantitative influence of end friction confinement onthe dynamic compressive behavior of concrete specimensunder medium and low strain rates The proposed mesoscaleanalysis method was firstly verified by comparing the presentsimulation results with the available test data In the simula-tions seven groups of friction coefficients namely 00 0102 03 06 08 and infin are taken into account The effectsof end friction confinement on the dynamic failure patternsthe nominal compressive stress-strain relations and the com-pressive DIFs were explored subsequently One can find thatthe friction at the interface of concrete specimen and loadingmeasurement constrains the lateral deformation of the spec-imen and changes the internal stress status and thus it hasa significant influence on the tested DIF Furthermore thefriction contribution factor was proposed and it was used todescribe the quantitatively influence of end friction confine-ment According to the simulations the following conclu-sions can be obtained

(1) At the medium and low strain rate level (1 times

10minus5 sminus1 sim 1times10

0 sminus1) the uniaxial dynamic compres-sive strength increases first (0 le 120583 le 03) and thenkeeps stable (120583 gt 03)

(2) End friction confinement effect changes the uniaxialforce state of local concrete and the failure patternsMeanwhile the concrete uniaxial dynamic compres-sive strength is improved due to the end frictionconfinement effect

(3) For same end friction confinement the friction con-tribution factor decreases with the increase of nomi-nal strain rate (ranging from 1times 10minus2 sminus1 to 1times 100 sminus1)And with the increase of end friction coefficient thedescending trend gets to be obvious

It is to be noted that further studies are still necessary fora fully understanding about the strain-rate effect on the ulti-mate uniaxial compressive strength of concrete and concrete-like materials subjected to dynamic loadings Both tests andsimulations could offermore information to separate pseudo-strain-rate effects from genuine strain-rate effects in dynamicmaterial properties tests which is of significant importancefor material model identification

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the National Key Basic Researchand Development Program of China (no 2011CB013600)and the Science Fund for Creative Research Groups ofthe National Natural Science Foundation of China (no51421005) The support is gratefully acknowledged

References

[1] H Xu and H M Wen ldquoSemi-empirical equations for thedynamic strength enhancement of concrete-like materialsrdquoInternational Journal of Impact Engineering vol 60 pp 76ndash812013

[2] Q M Li and H Meng ldquoAbout the dynamic strength enhance-ment of concrete-like materials in a split Hopkinson pressurebar testrdquo International Journal of Solids and Structures vol 40no 2 pp 343ndash360 2003

[3] P H Bischoff and S H Perry ldquoCompressive behaviour ofconcrete at high strain ratesrdquo Materials and Structures vol 24no 6 pp 425ndash450 1991

[4] Y Hao H Hao and Z X Li ldquoInfluence of end frictionconfinement on impact tests of concrete material at high strainraterdquo International Journal of Impact Engineering vol 60 pp82ndash106 2013

[5] M J Forrestal T W Wright andW Chen ldquoThe effect of radialinertia on brittle samples during the split Hopkinson pressurebar testrdquo International Journal of Impact Engineering vol 34 no3 pp 405ndash411 2007

[6] Z C Mu A N Dancygier W Zhang and D Z YankelevskyldquoRevisiting the dynamic compressive behavior of concrete-likematerialsrdquo International Journal of Impact Engineering vol 49pp 91ndash102 2012

[7] D J Kim K Sirijaroonchai S E El-Tawil and A NaamanldquoNumerical simulation of the split Hopkinson pressure bartest technique for concrete under compressionrdquo InternationalJournal of Impact Engineering vol 37 no 2 pp 141ndash149 2010

[8] R R Pedersen A Simone and L J Sluys ldquoMesoscopicmodeling and simulation of the dynamic tensile behavior ofconcreterdquo Cement and Concrete Research vol 50 pp 74ndash872013

[9] X Q Zhou and H Hao ldquoMesoscale modelling of concretetensile failure mechanism at high strain ratesrdquo Computers andStructures vol 86 no 21-22 pp 2013ndash2026 2008

[10] XQ Zhou andHHao ldquoModelling of compressive behaviour ofconcrete-likematerials at high strain raterdquo International Journalof Solids and Structures vol 45 no 17 pp 4648ndash4661 2008


[11] X L Du L Jin and G W Ma ldquoNumerical simulation ofdynamic tensile-failure of concrete at meso-scalerdquo InternationalJournal of Impact Engineering vol 66 pp 5ndash17 2014

[12] X L Du L Jin and R B Zhang ldquoModeling the cracking ofcover concrete due to non-uniform corrosion of reinforcementrdquoCorrosion Science vol 89 pp 189ndash202 2014

[13] S-M Kim and R K Abu Al-Rub ldquoMeso-scale computationalmodeling of the plastic-damage response of cementitious com-positesrdquo Cement and Concrete Research vol 41 no 3 pp 339ndash358 2011

[14] WHDilger R Koch and R Kowalczyk ldquoDuctility of plain andconfined concrete under different strain ratesrdquoACI Journal vol81 no 1 pp 73ndash81 1984

[15] J Lubliner J Oliver S Oller and E Onate ldquoA plastic-damagemodel for concreterdquo International Journal of Solids and Struc-tures vol 25 no 3 pp 299ndash326 1989

[16] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[17] P Grassl and M Jirasek ldquoDamage-plastic model for concretefailurerdquo International Journal of Solids and Structures vol 43no 22-23 pp 7166ndash7196 2006

[18] P Badel V Godard and J-B Leblond ldquoApplication of someanisotropic damage model to the prediction of the failureof some complex industrial concrete structurerdquo InternationalJournal of Solids and Structures vol 44 no 18-19 pp 5848ndash58742007

[19] Y Huang Z Yang W Ren G Liu and C Zhang ldquo3D meso-scale fracture modelling and validation of concrete based onin-situ X-ray computed tomography images using damageplasticity modelrdquo International Journal of Solids and Structuresvol 67-68 pp 340ndash352 2015

[20] A S Genikomsou and M A Polak ldquoFinite element analysisof punching shear of concrete slabs using damaged plasticitymodel in ABAQUSrdquo Engineering Structures vol 98 pp 38ndash482015

[21] D L Grote S W Park and M Zhou ldquoDynamic behavior ofconcrete at high strain rates and pressures I Experimentalcharacterizationrdquo International Journal of Impact Engineeringvol 25 no 9 pp 869ndash886 2001

[22] SW Park Q Xia andM Zhou ldquoDynamic behavior of concreteat high strain rates and pressures II Numerical simulationrdquoInternational Journal of Impact Engineering vol 25 no 9 pp887ndash910 2001

[23] S Hentz F V Donze and L Daudeville ldquoDiscrete elementmodelling of concrete submitted to dynamic loading at highstrain ratesrdquo Computers amp Structures vol 82 no 29-30 pp2509ndash2524 2004

[24] L J Malvar and C A Ross ldquoReview of strain rate effects forconcrete in tensionrdquo ACI Materials Journal vol 95 no 6 pp735ndash739 1998

[25] X Q Zhou V A Kuznetsov H Hao and JWaschl ldquoNumericalprediction of concrete slab response to blast loadingrdquo Interna-tional Journal of Impact Engineering vol 35 no 10 pp 1186ndash1200 2008

[26] G Cusatis ldquoStrain-rate effects on concrete behaviorrdquo Interna-tional Journal of Impact Engineering vol 38 no 4 pp 162ndash1702011

[27] J Ozbolt A Sharma and H-W Reinhardt ldquoDynamic fractureof concretemdashcompact tension specimenrdquo International Journalof Solids and Structures vol 48 no 10 pp 1534ndash1543 2011

[28] S Pyo and S El-Tawil ldquoCrack velocity-dependent dynamictensile behavior of concreterdquo International Journal of ImpactEngineering vol 55 pp 63ndash70 2013

[29] N Tregger D Corr L Graham-Brady and S Shah ldquoModelingthe effect of mesoscale randomness on concrete fracturerdquoProbabilistic Engineering Mechanics vol 21 no 3 pp 217ndash2252006

[30] W Riedel M Wicklein and K Thoma ldquoShock properties ofconventional and high strength concrete experimental andmesomechanical analysisrdquo International Journal of Impact Engi-neering vol 35 no 3 pp 155ndash171 2008

[31] F Fakhari Tehrani J Absi F Allou and C Petit ldquoHeteroge-neous numerical modeling of asphalt concrete through use ofa biphasic approach porous matrixinclusionsrdquo ComputationalMaterials Science vol 69 pp 186ndash196 2013

[32] F Bernard and S Kamali-Bernard ldquoNumerical study of ITZcontribution on mechanical behavior and diffusivity of mor-tarsrdquo Computational Materials Science vol 102 pp 250ndash2572015

[33] L Jin and X-L Du ldquoEffects of loading rate and its suddenchange on concrete compressive failurerdquo Journal of Vibrationand Shock vol 33 no 19 pp 187ndash193 2014

[34] X L Du L Jin and G W Ma ldquoA meso-scale analysis methodfor the simulation of nonlinear damage and failure behavior ofreinforced concrete membersrdquo International Journal of DamageMechanics vol 22 no 6 pp 878ndash904 2013

[35] L Snozzi A Caballero and J F Molinari ldquoInfluence of themeso-structure in dynamic fracture simulation of concreteunder tensile loadingrdquo Cement and Concrete Research vol 41no 11 pp 1130ndash1142 2011

[36] H-KMan and J GM vanMier ldquoDamage distribution and sizeeffect in numerical concrete from lattice analysesrdquo Cement andConcrete Composites vol 33 no 9 pp 867ndash880 2011

[37] W Y Ren Z J Yang R Sharma C H Zhang and P J WithersldquoTwo-dimensional X-ray CT image based meso-scale fracturemodelling of concreterdquoEngineering FractureMechanics vol 133pp 24ndash39 2015

[38] Y Hao H Hao and X H Zhang ldquoNumerical analysis ofconcrete material properties at high strain rate under directtensionrdquo International Journal of Impact Engineering vol 39 no1 pp 51ndash62 2012

[39] Y Hao and H Hao ldquoNumerical investigation of the dynamiccompressive behaviour of rock materials at high strain raterdquoRock Mechanics and Rock Engineering vol 46 no 2 pp 373ndash388 2013

[40] Q M Li Y B Lu and H Meng ldquoFurther investigation on thedynamic compressive strength enhancement of concrete-likematerials based on split Hopkinson pressure bar tests Part IInumerical simulationsrdquo International Journal of Impact Engi-neering vol 36 no 12 pp 1335ndash1345 2009

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



As known the end friction cannot be completely avoidedwhen conducting dynamic compressive tests The quantita-tive influence of end friction confinement should be deductedin the compressive DIF and it has been an interesting topicTo discover the influence of end friction on the macroscopicmechanical behavior of concrete including the dynamicfailure patterns and the nominal stress-strain curves as well asthe compressive DIF of concrete subjected to dynamic load-ings numerous efforts have been conducted For instanceusing the numerical methods Li and Meng [2] discussed theend friction confinement effect and found that for specimenswith 119871119863 = 05 [119871 means length of specimen and 119863 meansthe diameter of the specimen] the influence of the frictioncoefficient 120583 on DIF is insignificant when 120583 is less than01 but becomes considerable when it is larger than 02 Intheir simulations the strain rate employed ranges from 27times 101 sminus1 to 39 times 102 sminus1 and the friction coefficient utilizedranges from 0 to 07 Mu et al [6] modeled the dynamicmechanical properties of concrete subjected to the strain rateranging from 44 times 101 sminus1 to 147 times 102 sminus1 They found thatthe compressive strain-rate effect of concrete-like materialsis a pseudo property and the main mechanisms that initiatelateral confinement include the material lateral inertia andthe interface friction between the loading apparatus and theloaded specimen In addition in the works of Kim et al [7]Li et al [40] and Hao et al [4] the interface friction betweenthe loading apparatus and the loaded specimen was alsosimulated and examined They concluded that the frictioneffect is another factor in the so-called strain-rate effectThesesimulation efforts contribute to the understandings aboutthe dynamic compressive strength enhancement of concreteHowever these efforts are all focused on high strain ratesand the effect of end friction on the failure of concrete undermedium and low strain rates is rarely involved

The objective of the present study is therefore to examinethe quantitative influence of the confinement due to end fric-tion on the mechanical behavior of concrete under mediumand low strain rates Experimental results show that concreteis a heterogeneous brittle material that fractures throughthe formation growth and coalescence of microcracks [8]Moreover the failure behavior of concrete is closely relatedto the meso-microstructure of concrete and the hetero-geneities should therefore be considered [9ndash13] In this workconcrete was assumed to be a composite material consistingof mortarmatrix coarse aggregates and interfacial transitionzone (ITZ) between the former two phases Including thesecomponents in both experimental and numerical investiga-tions is essential because the mechanical properties of thesematerials are distinct In the computational model the dam-aged plasticity model implemented in a strain-rate sensitiveframework was utilized for the description of the dynamicmechanical behavior of the mortar matrix and the ITZ andelastic model was employed to characterize the mechanicalproperties of the aggregateThe friction behavior was utilizedto describe the interaction between the concrete specimensand loading apparatus By using the material model togetherwith the commercial software ABAQUS the reliability ofthe present approach was verified through the comparison

with the available test data Subsequently the dynamic axialcompressive mechanical behavior of concrete subjected todifferent medium and low strain rates was simulated underdifferent friction coefficients Furthermore the confinementmechanism of end friction on the compressive dynamicincrease factor (DIF) of concrete was studied and discussed

The paper is decomposed as follows The next sectiondescribes the rate-dependent constitutive framework forconcrete Section 3 presents the mesoscale computationalmodel for the simulation of the end friction confinementeffect and illustrates the verification of the present approachIn Section 4 the quantitative influences of the end frictionconfinement on the failure patterns the nominal stress-straincurves and the compressive DIF of concrete are exploredAnd finally the concluding remarks are given in Section 5

2 The Constitutive Framework forConcrete Mesocomponents

21 Concrete Damaged Plasticity Model The behavior ofconcrete is complex due to an array ofmorphological featuresaswell as deformation and failuremechanisms inherent in theconcrete microstructure The complicating factors includeheterogeneous constitution randomphase distribution arbi-trary microcrack patterns and internal granular frictionThecomplexity of the meso- or microscopic processes rendersexplicit resolution of the influences on a highly challengingtask

As known plasticity continuum damage mechanicsfracture mechanics elastic-damage and combined plasticityand damage mechanics are the common theories that havebeen widely used for the description of concrete mechanicalbehavior However the plasticity theory failed to address thedegradation of the material stiffness due to microcrackingthe fracture mechanics method was hard to reflect the nucle-ation of numerousmicrocracks and the growth of themicroc-racks to macrocracks and the continuum damage mechanicswith elasticity also could not describe the irreversible defor-mations and the inelastic volumetric expansion in compres-sion [13] Since both the stiffness degradation behavior andirreversible plastic deformations are contributing to the non-linear behavior of concrete a reasonable constitutive modelshould characterize the two distinct mechanisms In recentyears many coupled plasticity-damage models were pro-posed to describe the mechanical behavior of concrete suchas the work in the references [13 15ndash18] Among these effortsthe concrete damaged plasticity model was first proposed byLubliner et al [15] for monotonic loading and it was devel-oped later by Lee and Fenves [16] to consider the dynamic andcyclic loadings The model is a continuum plasticity basedmodel for concrete and it assumes that themain failuremech-anisms of the concrete are cracking in tension and crushingin compression The model which has been included in theABAQUS package has been widely used for the description ofthe static and dynamic mechanical behaviors of concrete-like materials for example in the publications [11 12 19 20]A brief presentation of the damaged plasticity model fromABAQUS is presented below


Crack closure

ft

fc


E0(1 minus dc)

E0(1 minus dt)

120576

120590c

120590t

E0

E0



120590 = (1 minus 119889)119863el0 (120576 minus 120576

pl) (1)





119905and



119905 is reached


120590119905= 120590119905(120576

pl119905 120576

pl119905 120579 119891119894) (2a)

120590119888= 120590119888(120576

pl119888 120576

pl119888 120579 119891119894) (2b)

where 120576pl119905and 120576









120590119905= (1 minus 119889

119905) 1198640(120576119905minus 120576

pl119905) (3a)

120590119888= (1 minus 119889

119888) 1198640(120576119888minus 120576

pl119888) (3b)



120576pl119905= 119903 () 120576

plmax (4a)

120576pl119888= minus (1 minus 119903 ()) 120576

plmin (4b)




119903 () =

sum3

119894=1⟨119894⟩

sum3

119894=1

10038161003816100381610038161003816119894

10038161003816100381610038161003816

0 le 119903 () le 1 (5)






119865 = 119865 (120590 120576pl)

=1

1 minus 120572[119902 minus 3120572119901 + 120573 (120576


minus 120590119888(120576

pl119888) = 0

(6)




Uniaxial tension

Biaxial tension


120590t

(120590cc 120590cc)

1205901

1205902


1205901

1205902

1205903

Kc = 1

Kc = 23


= 23




120572 =120590119888119888120590119888minus 1

2120590119888119888120590119888minus 1

(7a)

120573 (120576pl) =

120590119888(120576

pl119888)

120590119905(120576

pl119905)

(1 minus 120572) minus (1 + 120572) (7b)

120574 =3 (1 minus 119870

119888)

2119870119888minus 1

(7c)













120576pl=

120597119866

120597120590(9a)

120576pl= ℎ (120590 120576

pl) sdot 120576

pl (9b)





CDIF =119891cd119891cs

= (120576119889

120576cs)

1026120572

for 120576119889le 30 sminus1 (10a)

CDIF =119891cd119891cs

= 120574 ( 120576119889)13 for 120576

119889gt 30 sminus1 (10b)









TDIF =119891td119891ts

= (120576119889

120576ts)

120575

for 120576119889le 1 sminus1 (11a)

TDIF =119891td119891ts

= 120573(120576119889

120576ts)

13

for 120576119889gt 1 sminus1 (11b)




120575 = 1(1 + 81198911015840

1198881198911015840

1198880) 11989110158401198880= 10MPa and 119891

1015840













y

x

AggregateITZ



Unit mm End

fric

tion

25 25150

2525

150













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Crack closure

ft

fc


E0(1 minus dc)

E0(1 minus dt)

120576

120590c

120590t

E0

E0



120590 = (1 minus 119889)119863el0 (120576 minus 120576

pl) (1)





119905and



119905 is reached


120590119905= 120590119905(120576

pl119905 120576

pl119905 120579 119891119894) (2a)

120590119888= 120590119888(120576

pl119888 120576

pl119888 120579 119891119894) (2b)

where 120576pl119905and 120576









120590119905= (1 minus 119889

119905) 1198640(120576119905minus 120576

pl119905) (3a)

120590119888= (1 minus 119889

119888) 1198640(120576119888minus 120576

pl119888) (3b)



120576pl119905= 119903 () 120576

plmax (4a)

120576pl119888= minus (1 minus 119903 ()) 120576

plmin (4b)




119903 () =

sum3

119894=1⟨119894⟩

sum3

119894=1

10038161003816100381610038161003816119894

10038161003816100381610038161003816

0 le 119903 () le 1 (5)






119865 = 119865 (120590 120576pl)

=1

1 minus 120572[119902 minus 3120572119901 + 120573 (120576


minus 120590119888(120576

pl119888) = 0

(6)




Uniaxial tension

Biaxial tension


120590t

(120590cc 120590cc)

1205901

1205902


1205901

1205902

1205903

Kc = 1

Kc = 23


= 23




120572 =120590119888119888120590119888minus 1

2120590119888119888120590119888minus 1

(7a)

120573 (120576pl) =

120590119888(120576

pl119888)

120590119905(120576

pl119905)

(1 minus 120572) minus (1 + 120572) (7b)

120574 =3 (1 minus 119870

119888)

2119870119888minus 1

(7c)













120576pl=

120597119866

120597120590(9a)

120576pl= ℎ (120590 120576

pl) sdot 120576

pl (9b)





CDIF =119891cd119891cs

= (120576119889

120576cs)

1026120572

for 120576119889le 30 sminus1 (10a)

CDIF =119891cd119891cs

= 120574 ( 120576119889)13 for 120576

119889gt 30 sminus1 (10b)









TDIF =119891td119891ts

= (120576119889

120576ts)

120575

for 120576119889le 1 sminus1 (11a)

TDIF =119891td119891ts

= 120573(120576119889

120576ts)

13

for 120576119889gt 1 sminus1 (11b)




120575 = 1(1 + 81198911015840

1198881198911015840

1198880) 11989110158401198880= 10MPa and 119891

1015840













y

x

AggregateITZ



Unit mm End

fric

tion

25 25150

2525

150













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Uniaxial tension

Biaxial tension


120590t

(120590cc 120590cc)

1205901

1205902


1205901

1205902

1205903

Kc = 1

Kc = 23


= 23




120572 =120590119888119888120590119888minus 1

2120590119888119888120590119888minus 1

(7a)

120573 (120576pl) =

120590119888(120576

pl119888)

120590119905(120576

pl119905)

(1 minus 120572) minus (1 + 120572) (7b)

120574 =3 (1 minus 119870

119888)

2119870119888minus 1

(7c)













120576pl=

120597119866

120597120590(9a)

120576pl= ℎ (120590 120576

pl) sdot 120576

pl (9b)





CDIF =119891cd119891cs

= (120576119889

120576cs)

1026120572

for 120576119889le 30 sminus1 (10a)

CDIF =119891cd119891cs

= 120574 ( 120576119889)13 for 120576

119889gt 30 sminus1 (10b)









TDIF =119891td119891ts

= (120576119889

120576ts)

120575

for 120576119889le 1 sminus1 (11a)

TDIF =119891td119891ts

= 120573(120576119889

120576ts)

13

for 120576119889gt 1 sminus1 (11b)




120575 = 1(1 + 81198911015840

1198881198911015840

1198880) 11989110158401198880= 10MPa and 119891

1015840













y

x

AggregateITZ



Unit mm End

fric

tion

25 25150

2525

150













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












TDIF =119891td119891ts

= (120576119889

120576ts)

120575

for 120576119889le 1 sminus1 (11a)

TDIF =119891td119891ts

= 120573(120576119889

120576ts)

13

for 120576119889gt 1 sminus1 (11b)




120575 = 1(1 + 81198911015840

1198881198911015840

1198880) 11989110158401198880= 10MPa and 119891

1015840













y

x

AggregateITZ



Unit mm End

fric

tion

25 25150

2525

150













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y

x

AggregateITZ



Unit mm End

fric

tion

25 25150

2525

150













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













25lowast

70lowast


022lowast

016lowast


1198880667 0667

Stress ratio 120590119888119888120590119888


119905[MPa] 143

lowast12lowast


[MPa] 143lowast

12lowast
















10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10Damage


120576 = 2 times 10minus1 120576 = 1 120576 = 100 120576 = 200



Nom

inal

com

pres

sive

stre

ss (M

Pa)


0

10

20

30

40



120576 = 02 sminus1

(a)

001 1 100

09

12

15

18

21

24

Com

pres

sive D

IF




(b)



Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Dynamic loading

0

2

4

6

8

10

12

14

Com

pres

sive

stre

ss (M

Pa)


times10minus3



1 432 5 6 70Strain 120576

(a)


Damage factor0 1

(b)








Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Incr

easin

g fr

ictio

n co

effici

ent



(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

120576

120576

120576

120576

120576






Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Incr

easin

g fr

ictio

n co

effici

ent


000 49

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin



120576

120576

120576

120576







Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Incr

easin

g fr

ictio

n co

effici

ent


0 1

(a) 120583 = 0

(b) 120583 = 03

(c) 120583 = 08

(d) 120583 = infin

Damage factor


120576

120576

120576

120576










0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0 793 806 858 919 1042 1197 153901 829 838 906 977 1113 1297 161202 846 857 926 1007 1137 1349 168203 849 864 939 1016 1167 1355 169606 857 877 955 1036 1175 1382 173608 860 881 962 1042 1185 1391 1745infin 875 892 980 1067 1212 1406 1749













7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










7252015

9

Quasistatic loading

8

95

85

752515

Dynamic loading

20

Dynamic loading

20

10

9

8

25 30 35302520

11

10

9

8

Dynamic loading





times10minus3

0

3

6

9

12

15

18C

ompr

essiv

e st

ress

(MPa

)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576

times10minus3


3 6 90Strain 120576



120576

120576

120576

120576

120576

120576

Figure 11 Continued


Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Dynamic loading



120583 = 01

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = 0

120583 = infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3

3 6 90Strain 120576

120576


Com

pres

sive s

treng

th (M

Pa)




6

7

8

9

10

11

12

(a)

Com

pres

sive s

treng

th (M

Pa)



6

8

10

12

14

16

18

20


(b)





120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

0 1

Damage factor

(a)

0 1

Damage factor

120583 = 01 120583 = 02 120583 = 03

120583 = 06 120583 = 08

120583 = 00

120583 = infin

(b)


minus05 sminus1









08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











08

10

12

14

16

18

20

22C

ompr

essiv

e D

IF



120576

(a)



Concrete


120583 = 01

120583 = 0

120583 = 02

120583 = 03

120583 = 06

120583 = 08

120583 = infin

08

10

12

14

16

18

20

22

Com

pres

sive

DIF

120576

(b)








120573120583 120576=


0 120576

DIF120583 120576minus 1

(12)








1denotes



Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Strain rate

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573


1 times 10minus2


1 times 10minus05

1 times 100


120576

120576

120576

(a)



log( 120576)

000

005

010

015

020

025

030

035

Fric

tion

cont

ribut

ion

fact

or 120573

120583 = 01

120583 = 02120583 = 03

120583 = 06

120583 = 08

120583 = infin


120576

120576

120576

(b)



(3)(2)

(3) 120583 (2) 120583(1) 120583

(1)



= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576



E1

E2

E3

120576

120576

(a)



(5)

(1)

(4)


= 0

= infin

0

3

6

9

12

15

18

Com

pres

sive

stre

ss (M

Pa)

times10minus3620 4

Strain 120576


E4

E5

E1


120576

120576

(b)




2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2and 119864





1198643

1198642+ 1198643

gt1198645

1198644+ 1198645

(13)


5 Conclusions










Acknowledgments


References












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Effect of End Friction on the Dynamic