Dimensional Analysis Used For Concrete Damaged Under Impact Loading
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Dimensional Analysis Used For Concrete Damaged Under ImpactLoadingGuoping Jiang, Weijun TaoEarthquake Engineering Research Test Center Of GuangzhouUniversity,Guangzhou510405,ChinaAbstractThe dynamic response is analyzed. The numerical simulation method has been used to projectile penetrating concrete for many years. However, most previous research work has concentrated on the cuboids concrete model. Very few studies have been conducted to investigate the penetration process of complex concrete model. In this paper, numerical simulations of laboratory tests are conducted to show the penetration process of wedge concrete model. A double scalar damage model based on the concept of continuum damage mechanics is applied to modeling the failure of concrete. In the numerical model of wedge concrete, the wedges with different angles are modeled. By the numerical simulation, the relationship of projectile penetrating into the wedges concrete with different angles are demonstrated.Keywords: concrete; relationship; steel fiberIntroductionReliable numerical simulation is important for accurately calculating, predicting structural response under the impact loads, as well as carrying out protection measures for disaster research , rapid assessment and decision-making extremely . On the other hand, based on numerical simulation compared to the test of impact with the damaging effects can also save a lot of money. The meaning is obvious.(Tang, 2004 and 1992; Jacques and Cete, 2004; Bonneau et al.,1996; Richard and Cheyrezy, 1995).Theoretical analysis and numerical calculation of the building structures with impact resistant character depend on the parameters and results of the experiment. The dimensions of the plototype is shrunken to the experimental model . The minification of the dimension and strength is not accordant between the plototype and the experimental model .Usually, the relativestrength of the experimental model is greater than the plototype. Thus, the research on convergent-divergent ratio is needed when designing the experimental model with impact resistant character. The relationship between convergent-divergent of physical dimension and deformation, stress, pressure, wave speed, time, mass point speed is researched based on impact dimensional analysis. The lamination crack of concrete is also obtained. The dimensions of the plototype is shrunken to the experimental model in many investigations(Hao and Tarasov, Zhou and Hao,2008)..The dimensional analysis is becoming more and more important in the investigations(Li et al., 2004,Cai et al.,2010)..1.Numerical simulation1.1 Material modelThe concrete subjected to large strains, high strain rates and high pressures can be described by the Johnson-Holmquist-Concrete material model(Bonneau et al., 1996,Cotsovos et al., 2007;Grote et al.,2001). The equivalent strength is expressed as a function of pressure, strain rate and damage. The pressure is expressed as a function of the volumetric strain and includes the effect of permanent crushing. The damage is accumulated as a function of the plastic volumetric strain, the equivalent strain and pressure.Fig1 Numerical calculation model 1Fig2 numerical calculation model 2 Hydrostatic-pressure p is the function ofμ.Three response domains(linear elastic zone, intermediate zone, close-grained zone) are included in the relationship function ofμ-p.Three zones included in compressed domain:⎪⎪⎩⎪⎪⎨⎧-≤++---+≤=grainedclose P P K K K ermediateP elastic linear p p K P LOCK crush crushlock crush crush crush elastic 33221max int )(μμμμμμμμμμIn tensil e domain ：{}⎪⎩⎪⎨⎧-+-≤=grained close FK K F ermediate K elastic linear p p K P elastic crushelastic μμμ11)1(intWhere crush lock crushF μμμμ--=max ，elastic K is elasticbulk modulus ，321,,K K K are constants ，crush Pis critical pressure andcrush μ is volumedeformation when the voids of the concrete become clogging ，1/0-=ρρμgrain lock ，grainρ is crystal density ，0ρis initial density ，max μis the max volume deformation beforeunloading ，Tis tensile-strength[Ma etal.,2006,Schule et al.,2006,Xiao et al.,2010].The ball has been simplified to a lumped mass on the flyer and 1/4 model adopted because of the symmetry in the numerical simulation. The calculation time is shrinked.1.2 Finite modelQuarter of the geometrical models are adopted in the model because of the symmetrical character (Fig1，Fig2).Two conditions are numerical simulated in the present paper. SOLIDE164 elements are adopted in all the parts. The dimensions between the two states are 1:0.7 for the wedges concrete. The specific dime nsionsfor model 1 are list in table 1.The similarresearcher is carried out for the wedges concrete. The concrete subjected to large strains, high strain rates and high pressures can be described by the Johnson-Holmquist-Concrete material model.2.Dimensional AnalysisThe main dimensions are listed in Table 1. The tunnel diameter can be expressed as function below:(,,,,,,,;;,,,,,)p p p p p p c t t t t F f d L E Y V E Y c ρψνϕρνμ= （1）The tunnel depth can be expressed as function below:(,,,,,,,;;,,,,,)p p p p p p c t t t t G g d L E Y V E Y c ρψνϕρνμ= （2）Table1 The main dimensionalsThe avalanche diameter can be expressed as function below:(,,,,,,,;;,,,,,,)p p p p p p c t t t t t H h d L E Y V E Y c d ρψνϕρνμ= （3）Dimensional analysis is used to analyze the equations (1),(2),(3), The dimensions are magnified to M 、L 、T double.The following equations can be obtained from equations (1),(2),(3)322323(,,,,,,,;;,,,,,)p p p p p p c t t t t M M M L M M M M LLF f d L L L E Y V E Y c L LT LT T L LT L LT Tρψνϕρνμ= （4）322323(,,,,,,,;;,,,,,)p p pp p p c t t t t M M M L M M M M L LG g d L L L E Y V E Y c L LT LT T L LT L LT Tρψνϕρνμ= （5）322323(,,,,,,,;;,,,,,,)p p pp p p c t t t t t M M M L M M M M LLH h d L L L E Y V E Y c d L L LT LT T L LT L LT Tρψνϕρνμ= （6）It is supposed that 1p Ld =；1c L V T =；31p M L ρ= because of the L M T ’s randomcharacter :1pL d =；c pV T d =；31p PM d ρ=Then the following equations can be obtained ：222211111(,,,,;;,,,,,)p t p pp p t t t pcp c p p c p c p p p c cL cF d f E Y E Y d V d V d V d V d d V V ρψνϕνμρρ= （7）222211111(,,,,;;,,,,,)p t p pp p t t t pcp c p p c p c p p p c cL cG d g E Y E Y d V d V d V d V d d V V ρψνϕνμρρ= (8)11111(,,,,;;,,,,,,)p t t p pp p t t t pcp c p p c p c p p p c c pL dc Hd h E Y E Y d V d V d V d V d d V V d ρψνϕνμρρ= (9)The six front combination independent variables of functions f 、g 、h are related to the structure and movement of steel striker .The following variables are related to concrete target.Separation of variables method is done in the equations .12222211111(,)(,,)()(,,,,)p p p pp p t t t t pcp c p c p c p p p c cL c F d f f E Y f f E Y d V d V d V d V d d V V ψνϕνμρ= (10)1211111(,)(,,)()(,,,,)p p p pp p t t t t pc p c p c p c p p p c cL cG d g g E Y g g E Y d V d V d V d V d d V V ψνϕνμρ= (11)123222211111(,)(,,)()(,,,,)()p t p p pp p t t t t pcp c p c p c p p p c c pL d cH d h h E Y h h E Y g d V d V d V d V d d V V d ψνϕνμρ= (12)Where 1(,)p pL f d ψ is the function affected bytheshapeofsteelstriker.2211(,,)p pp pc p c pf E Y V d V d ν is the functionaffected by the material character of steel striker. 2()f ϕ is the function affected by thedrift angle of steel striker.22111(,,,,)t t t t c p c p p p c cc f E Y Vd V d d V V νμρ is thefunction affected by the material character oftarget ，21tc pY V d ，21t c pE V d are stictions.1p p cd V μρ，c cV are the function affected bythe mediums.3()t pd g d is the function affectedby the shape of target.The same method can be used to the othermodel:1222211111(,)(,,)()(,,,,)p p p p p p t t t t c pp c p c p c p p p c c L c F d f f E Y f f E Y V d d V d V d V d d V V ψνϕνμρ'''''''''''''=''''''''''' （13）12222211111(,)(,,)()(,,,,)p p p p p p t t t t c pp c p c p c p p p c c L c G d g g E Y g f E Y V d d V d V d V d d V V ψνϕνμρ'''''''''''''=''''''''''''' （14）123222211111(,)(,,)()(,,,,)()p t p p p p p t t t t c p pp c p c p c p p p c c L d c H d h h E Y h h E Y g V d d d V d V d V d d V V ψνϕνμρ''''''''''''''='''''''''''''' （15） If the materials of between the striker and the target are the same, the following can be obtained:1221(,)()()(,)p ppp p p L f d d f F F f d L f d ψϕϕψ=''''''(16)1221(,)()()(,)p ppp p p L g d d g G G g d L g d ψϕϕψ=''''''(17)132231(,)()()()()(,)p tppp p p t pp L d h h d d d h H H h d L d h h d d ψϕϕψ=''''''''（18） lateral effect, therefore long specimenscompared with the diameter must be avoided.A second point after striking, a stresswave propagate in the flyer and the specimens. When it arrives the head face of the flyer, asparsity reflection stress wave produced. To avoid the reflection wave catch up with thespecimens, the length of flyer must be longcompared with the diameter.3 3 Calculation models and ResultsThe models 1 on the two conditions are calculated to testify the dimension analysis. Thedimensions on the two state and the calculation results (Table1) are shown .The Model 2 is also calculated and the same results are obtained.(a) state 1: Projectile: diameter: 1.5cm, height:4.3cm,Length: 40cm, width 40cm, high-3.9cm, 8.1cm,Crater diameter: 8.2cm Depth 2.8cm crater Spall diameter: 7.8cm.(b) state 2:Projectile: diameter 1.05cm, height3.1cm,Length: 28cm, width 28cm, high-2.73cm, 5.67cm,Crater diameter: 5.6cm crater depth of 1.98cm,Spall diameter: 5.5cmFrom which it can be seen that with the similarity ratio 0.7, the same effect of projectile caused in the both models.4 Conclusions1 The relationship between convergent-divergent of physical dimension and deformation ,pressure, wave speed, time, mass point speed is researched based on impact dimensional analysis. The lamination crack of concrete is also obtained on projectile penetrating wedge concrete .2 It can be seen from results that the avalanche diameter is not obey the dimensional analysis ,the other two( crater depth and Spall diameter)obey the dimensional analysis. It is because that the shatter’s crack coming into being ,growth and chain connection lead to the lamination crack which has a complex process.AcknowledgementsThis work was supported by the National Natural Science Foundation of China through grant number 11102100 and the Scientific Research Special Foundation for Provincial University of Education Department of Fujian Province of China through grant number JK2011056.REFERENCES[1]Bonneau O, Poulin C, Dugat J. ―Reactive powder concrete: from theory to practice‖,Concret. Int., 18 (4):47–49(1996). [2]Cai JC, Yu B, Zou M, MQ . Fractal characterization of spontaneous co-current imbibition in porous media. 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