Further investigation on the dynamic compressive strength enhancement of concrete-like materials bad on split Hopkinson pressure bar tests.Part I:Experiments
M.Zhang a ,H.J.Wu a ,Q.M.Li a ,b ,*,F.L.Huang a
a State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology,Beijing 100081,PR China
b
School of Mechanical,Aerospace and Civil Engineering,The University of Manchester,PO Box 88,Manchester M601QD,UK
a r t i c l e i n f o
Article history:
Received 1April 2008Accepted 17April 2009
Available online 23May 2009Keywords:
Concrete-like materials
Split Hopkinson pressure bar Dynamic increasing factor Compressive strength Experimental study
a b s t r a c t
Effects of the inertia-induced radial confinement on the dynamic increa factor (DIF)of a mortar specimen are investigated in split Hopkinson pressure bar (SHPB)tests.It is shown that axial strain acceleration is unavoidable in SHPB tests on brittle samples at high strain-rates although it can be reduced by the application of a wave shaper.By introducing proper measures of the strain-rate and axial strain acceleration,their correlations are established.In order to demonstrate the influence of inertia-induced confinement on the dynamic compressive strength of concrete-like materials,tubular mortar specimens are ud to reduce the inertia-induced radial confinement in SHPB tests.It is shown that the DIF measured by SHPB tests on tubular specimens is lower than the DIF measured by SHPB tests on solid specimens.This paper offers experimental support for a previous publication [Li QM,Meng H.About the dynamic strength enhancement of concrete-like materials in a split Hopkinson pressure bar test.Int J Solids Struct 2003;40:343–360.],which claimed that inertia-induced radial confinement makes a large contribution to the dynamic compressive strength enhancement of
concrete-like materials when the strain-rate is greater than a critical transition strain-rate between 101and 102s À1.It is concluded that DIF formulae for concrete-like materials measured by split Hopkinson pressure bar tests need to be corrected if they are going to be ud as the unconfined uniaxial compressive strength in the design and numerical modelling of structures made from concrete-like materials to resist impact and blast loads.
Ó2009Elvier Ltd.All rights rerved.
1.Introduction
The dynamic increa factor (DIF),defined as the ratio of the dynamic strength to the quasi-static strength in uniaxial compression,has been widely accepted as an important parameter to measure the strain-rate effect on the compressive strength of concrete-like materials.Numerous experiments have been per-formed using various experimental techniques to characterize the DIF of concrete-like materials at strain-rates from 100to 103s À1.A critical review was given by Bischoff and Perry [1]to compare DIFs of concrete and mortar specimens from experiments con-ducted between 1910and 1990.It clearly demonstrated the increa of DIF with strain-rate for concrete and mortar specimens although great discrepancies were obrved due to differences in material and dimensions of the specimen and the method ud in testing and measurement.
Among all available testing methods for dynamic compressive strength of concrete-like materials,the split Hopkinson pressure bar (SHPB),propod originally by Kolsky [2],has been ud widely to measure the DIF of concrete-like materials at strain-rates between 101and 103s À1since the 1980s.Bad on studies of the applications of the SHPB to the dynamic behaviours of some metals and polymers,Davies and Hunter [3]propod an optimum
dimension for the SHPB L =D ¼ð1=2Þffiffiffiffiffiffiffi
3n s p where n s is Poisson’s ratio and L and D are the length and the diameter of the specimen,respectively,in order to reduce the effects of inertia and friction on the measured dynamic stress in the specimen.Such optimal dimension is also adopted in SHPB tests for concrete-like materials (ar,concrete,geo-material,ceramic,etc.)[4].However,rearches have suggested that inertia effect cannot in general be cancelled by adjusting specimen geometry [5–8],which indicates that the inertia effect needs to be checked carefully at high strain-rates,especially in SHPB tests for brittle materials where large diameter specimens are ud.Bad on SHPB tests,DIFs of concrete-like materials have been studied extensively to obtain many empirical formulae,reprented by CEB formula
*Corresponding author.
E-mail address:qingming.li@manchester.ac.uk (Q.M.
Li).
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International Journal of Impact Engineering
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International Journal of Impact Engineering 36(2009)1327–1334
[9]for concrete.It is interesting to note that all experimental data on the variation of DIF with strain-rate clearly demonstrate the existence of a critical transition strain-rate,beyond which the dependence of DIF on strain-rate becomes significant.This transi-tion strain-rate varies in different testing 30s À1in CEB formulae [9],63.1s À1in publications [10–13]and 266s À1in Grote et al.[4].
甩炮Bad on a numerical study,Li and Meng [14]demonstrated that the significant increa of DIF with strain-rate beyond the transi-tion strain-rate is mainly due to the inertia-induced radial confinement effects.The dependence of the compressive strength of concrete-like materials on radial confinement had already been shown in confined quasi-static tests and strength models (e.g.Drucker–Prager model)and confined dynamic tests and analys (e.g.[14–18]).It was pointed out by Li and Meng [14]that the misinterpretation of the confinement enhancement as the strain-rate enhancement in a S
HPB test leads to non-conrvative design or analysis of a concrete structure against impact or blast loading.It should be noted that a similar point of view has been indicated in other publications (e.g.[1,6,7,19–21],Field et al.,2004).Forrestal et al.[22]recently suggested a method to investigate the effect of the axial strain acceleration on the additional axial stress and radial confinement in a brittle cylindrical sample,which further supports the previous findings on SHPB tests of concrete-like materials in Li and Meng [14].The radial stress induced by axial strain acceleration in an elastic cylinder is given by [22]:
s r ¼s 0r ¼Àn ð3À2n Þ8ð1Àn Þ
h r 2
Àb 2i r €30z ðt Þ(1)
where b is the radius of the cylindrical specimen,n and r are the Poisson’s ratio and the density of the specimen material,respec-tively;€30z ðt Þis the axial strain acceleration in the specimen.Eq.(1)
shows that the radial stress is influenced by two he radius of the specimen and the axial strain acceleration in the specimen.The maximum radial stress occurs at the centre of the cross-ct
ion of the cylinder and reduces to zero on the outer surface of the cylinder according to a parabolic function.Unfortu-nately,the effect of the radial confinement on the measurement of
DIF of concrete-like materials in dynamic compressive tests has been largely ignored by the urs of the DIF data and formulae,which were derived mainly from SHPB tests.Many recent publi-cations still employ DIF data and formulae as the dynamic compressive strength in uniaxial stress state to define the strain-rate effect on the compressive strength of concrete-like materials in corresponding constitutive models [23–26].Similar recommenda-tions also appeared in recent concrete model (K&C model)in LS-DYNA Version 971.The applications of the misinterpreted DIF in the design and numerical simulation may cau significant increa of the predicted impact or blast resistance of structures made from concrete-like materials,and thus,lead to dangerous non-conr-vative design or asssment of the structures against impact and blast loads.
In the prent paper,experimental evidences bad on SHPB tests on solid and tubular mortar specimens are given to demon-strate the correlations between the reprentative axial strain-rate and the axial strain acceleration in a SHPB test on concrete-like material.A tubular specimen is introduced in order to reduce the inertia-induced radial confinement,and thus,demonstrate the influence of the inertial-induced radial confinement on DIF in SHPB tests,which further confirms the fi
ndings in Li and Meng [14].Experiments are described in Section 2,which is followed by data analys in Section 3and discussion and conclusions in Sections 4and 5.
2.Descriptions of experiment 2.1.SHPB tup
Series of experiments of solid and tubular specimens with different diameters (37mm,50mm and 74mm)were tested at various strain-rates from 50to 400s À1on a SHPB system (Fig.1).A gas gun was ud to launch the striker bar.The velocity of the striker bar,which is controlled by gas pressure,is measured by two parallel light gates and an electronic time counter.The signals from the strain gauges on the incident and transmitted pressure bars are amplified and then recorded by a transient recorder.Wave shaper is ud in order to reduce the stress non-equilibrium in specimen.2.2.SHPB specimen
The SHPB specimens are made of mortar which is a mixture of cement,water and medium fine sand.The mass ratio of the three materials is 533:302:1600.Typical solid and tubular specimens are shown in Fig.2.Dimensions of the specimen are shown by the specimen a specimen with code LT-aa-bb-cc-dd contains following information,(i)if T ¼S ,it is a solid cylinder with outer diameter ‘‘aa’’,length ‘‘bb’’,inner diameter ‘‘cc’’¼‘‘00’’and testing
Fig.1.The schematic diagram of a SHPB tup.
Fig.2.Typical solid and tubular SHPB specimens.(a)Solid SHPB specimen and (b)Tubular SHPB specimen.
M.Zhang et al./International Journal of Impact Engineering 36(2009)1327–1334
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number ‘‘dd’’;(ii)if T ¼H ,it is a tubular cylinder with outer diameter ‘‘aa’’,length ‘‘bb’’,inner diameter ‘‘cc’’and testing number ‘‘dd’’.‘‘L’’is a letter ud for internal reference.The quasi-static mechanical properties of the mortar are given in Table 1.
Typical signals obtained from strain gauges on incident and transmitted bars are shown in Fig.3.A thin rubber ring is ud as the wave shaper.A wave shaper has two functions:(i)the achievement of stress equilibrium and (ii)the achievement of nearly constant strain-rate (arly zero axial strain accelera-tion)in the SHPB specimen.The first requirement is satisfied by increasing the ri time of the input pul through a wave shaper.But the cond requirement is not satisfied in the prent SHPB tests although the u of a wave shaper generally reduces the
axial strain acceleration in the mortar specimen in SHPB test.The design of a proper wave shaper to meet the nearly constant strain-rate requirement is not always straightforward becau it strongly depends on good matching between the dynamic prop-erties of the wave shaper material and the tested material as well as their geometrical dimensions [27].It will be shown in Section 3that the axial
strain acceleration increas with strain-rate in the prent SHPB tests.Typical incident stress puls obtained at different impact velocities in the prent SHPB tests are shown in Fig.4.The duration of the incident stress pul is nearly a constant determined by the length of the striker.However,the amplitude of the incident stress pul increas linearly with the impact velocity of the striker,as shown in Fig.5.The incident stress pul can be expresd by Eqs.(2),(3a)and (3b)using data-fitting
安卓是什么意思s ¼s i $
11þe Àð
t À120
Þ$ 1À
11þe Àð
t À274
Þ
(2)
where t (in m s)is time.s i is the amplitude of the incident stress,which increas linearly with the impact velocity of the striker,n i (in m/s),i.e.
s i ¼20:6v i À7:9for 37mm SHPB
(3a)
and
s i ¼9:2v i À16:0for 74mm SHPB :
(3b)
Eqs.(2),(3a)and (3b)are only suitable for the prent SHPB system and will be ud in the companion numerical analysis in Li et al.[28].
The recovered specimens after SHPB tests are shown in Fig.6.Spalling fragments and the longitudinal cracks are obrved in the recovered specimens.The spalling fragments are almost axis
ym-metrically formed from the outer surface toward the centre of the specimen.An intact core is left if the specimen is not
totally
Table 1
50100150200250
300
350
i n c i d e n t s t r e s s p u l s e (M P a )
s)
Fig.4.Typical incident stress puls for different impact velocities.
15
25
5
10
20
30
50
100
厂房转让150
200
250
300
350
i n c i d e n t s t r e s s a m p l i t u d e (M P a )
impact velocity(m/s)
Fig.5.Variations of the incident stress amplitude and the impact velocity of the striker for 37mm and 74mm SHPBs.
200
400
600
800
1000
-1.0
-0.5
0.0
0.5
1.0
v o l t a g e (V )
s)
Fig.3.Typical strain gauge signals obtained from incident and transmitted pressure
bars when wave shaper is applied.
M.Zhang et al./International Journal of Impact Engineering 36(2009)1327–1334
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smashed,which implies that the central part of the specimen can resist large compressive loads.
The voltage signals measured in each SHPB test are procesd according to the 3-wave formulae [29]to obtain the variations of the engineering stress,strain and strain-rate with time as well as the engineering stress–strain relation.The signals show that stress equilibrium in the SHPB specimen in each SHPB test is approxi-mately satisfied.Values of the dynamic longitudinal compressive strength at various strain-rates are obtained.The will be further discusd in Section 3.
3.Data analysis
3.1.Axial strain acceleration
The instantaneous average strain-rate in the specimen is obtained bad on one-dimensional elastic stress wave theory
_3
s ðt Þ¼v 1ðt ÞÀv 2ðt Þl 0¼c
l 0
ð3i À3r À3t Þ:
(4)
The variations of strain-rate with time for solid specimens tested on a 74mm diameter SHPB at different impact velocities are shown in Fig.7.The figure shows that strain-rate during the effective loading period cannot be treated as a constant,especially when the impact velocity is incread to achieve high strain-rate.The gradient of the strain-rate curve in Fig.he axial strain
acceleration,increas with impact velocity.Since the DIF measured in each SHPB test is associated with a reprentative strain-rate,it is necessary to give a clear definition of such repre-ntative strain-rate ud in SHPB tests.Usually,such reprenta-tive strain-rate is defined as the mean value of the strain-rate over the loading period [4].Since mortar is a type of brittle material and most of the loading period is in the elastic deformation stage,the mean strain-rate during the loading period in a
SHPB test is less relevant to the compressive failure of the specimen than the strain-rate at the failure point.Therefore,the strain-rate at the failure he end of the strain-rate curve in Fig.7,is ud as the reprentative strain-rate in a SHPB test in the prent paper.However,linear correlations between the strain-rate at the failure point and the mean strain-rate are revealed for all SHPB tests in the prent Fig.8for 74mm diameter SHPB tests on solid specimens.
The axial strain acceleration is defined by
€3¼
d _3
d t
(5)
Although the axial strain acceleration is not a constant during the loading period,its variation with time near the failure point is very small,and therefore,the reprentative axial strain acceleration in the following discussion is determined as the mean value of the axial strain acceleration over the time duration of 10m s before the failure point on the strain-rate curve.It is found that the reprentati
ve axial strain acceleration increas with
the
Fig.6.Recovered specimens after SHPB test.
50100150200250
300350s t r a i n r a t e (s -1)
s)
Fig.7.Variations of strain-rate with time for solid specimens tested on 74mm SHPB at different impact velocities.
20
40
60
80
100
白菜烧粉条120
140
50
100150200250300
350F a i l u r e s t r a i n r a t e (s -1)
Mean strain rate(s -1)
Fig.8.Correlation between the failure strain-rate and the mean strain-rate for 74mm SHPB tests on solid specimens.
M.Zhang et al./International Journal of Impact Engineering 36(2009)1327–1334
1330
02
4
6
8
10
s t r a i n a c c e l e r a t i o n (x 106 s -2)
strain rate(s -1
蒜蓉龙虾
)
Fig.9.Correlations between the axial strain acceleration and strain-rate in solid and tubular SHPB specimens with 37mm outer diameter.
50
蘑菇的功效100
150
江畔独步寻花的意思200
250
300泊字怎么读
1
2
3
4
s t r a i n a c c e l e r a t i o n (x 106 s -2)
strain rate(s -1 )
Fig.10.Correlation between the axial strain acceleration and strain-rate in solid and tubular specimens with 50mm outer diameter.
2
4
6
8
10
s t r a i n a c c e l e r a t i o n (x 106 s -2)
strain rate(s -1)
Fig.11.Correlations between the axial strain acceleration and strain-rate in solid and tubular specimens with 74mm outer diameter.
2
4
6
8
10
s t r a i n a c c e l e r a t i o n (*106 s -2)
strain rate(s -1)
Fig.12.Correlations between the axial strain acceleration and strain-rate in solid specimens with 37mm and 74mm outer diameters.
1.01.2
1.4
1.6
1.8
2.0
2.2
D I F
D I F
strain rate(s -1)
strain rate(s
-1)
1.0
1.21.41.6
1.8
2.0
a
b
Fig.13.Variation of DIF with strain-rate for 37mm solid and tubular specimens.(a)Strain-rate with logarithm scale,and (b)strain-rate with linear scale.
M.Zhang et al./International Journal of Impact Engineering 36(2009)1327–13341331
