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eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Research Article Volume 1 Issue 3

Numerical simulation of the reinforced concrete barrier destruction at impact

PA Radchenko, SP Batuev, AV Radchenko

Tomsk State University of Architecture and Building, Russia

Correspondence: A V Radchenko, Tomsk State University of Architecture and Building, 2 Solyanaya Sq, Tomsk 634003, Russia

Received: July 19, 2017 | Published: October 16, 2017

Citation: Radchenko PA, Batuev SP, Radchenko AV. Numerical simulation of the reinforced concrete barrier destruction at impact. Aeron Aero Open Access J. 2017;1(3):103-106. DOI: 10.15406/aaoaj.2017.01.00013

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Abstract

The given paper considers penetration of ogival projectile made from titanium and filled with the explosive substance into massive reinforced concrete barrier. Numerical simulation is three-dimensional conducted by means of EFES program software developed by the authors. Mathematical models take into account plastic properties of materials, and fracture. Projectile penetration has been considered as a surface normal, and under angle with the initial velocities of 300 and 700 m/s. Fracture patterns of barrier and explosive substance have been obtained with regards to the interaction conditions.

Introduction

Recent authors’ works describe results of simulation of Boeing 747 interaction with protective shell1 in that case contact interaction of airplane body with the shell has been replaced by impulse. Such approach is justified when initial stage of interaction is described, taking that concrete structure contacts quite thin aluminum body (from 3 to 8 mm thick). In this case one does not need to calculate contact boundaries thus simplifying significantly the objective and enabling obtaining the results in short terms. When projectile impacts concrete barrier it is crucial to consider contact interaction, deformation and fracture of both barrier material and projectile material. Reinforced concrete structures used for nuclear power stations protection are generally designed with quite complex scheme of reinforcement. Modeling of projectiles penetration into such structures can be held by means of two approaches depending on the reinforcement design, size of reinforcing elements and the assigned objectives. The first approach suggests clear separation of reinforcing elements and therefore one need to calculate multiple contact boundaries thus complicating the given objective solution. The second approach is based on separation of locations in reinforced concrete structure which are adjacent to areas of reinforcement placement where material is described by mixture model. In this case there is only one contact boundary which is between the projectile and the barrier. The first approach can find application when it is necessary to obtain information on the influence of boundaries on shock-wave processes, to evaluate the adhesion properties of contact surfaces. However when assessment of integral parameters is required, second approach is appropriate to be applied. The given work contains calculations using second approach.

Mathematical model

The set of equations describing non-stationary adiabatic motions of compressible medium in general coordinates (i=1,2,3) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiikaiaadMgacqGH9aqpcaaIXaGaaiilaiaaykW7caaIYaGa aiilaiaaykW7caaIZaGaaiykaaaa@4091@ , includes the following equations:2

  1. Continuity equation

ρ t +ρ i υ i =0, MathType@MTEF@5@5@+= feaagKart1ev2aqaM5fvLHfij5gC1rhimfMBNvxyNvgaCzMCHn2ECb xyYrxAHXgiCjhAVbsF7XfCHjhDPfgBGqxFRWLCO9gi7ThxUfMySfgi 991EP1xF7Thx1b3CPX2BUbsFETxA91xpWWcatCvAUfeBSjuyZL2yd9 gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wD YLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbb f9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as 0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaq aaaOqaaKqbacbaaaaaaaaapeWaaSaaa8aabaWdbiabgkGi2kabeg8a YbWdaeaapeGaeyOaIyRaamiDaaaacqGHRaWkcqaHbpGCcqGHhis0pa WaaSbaaKqbGeaapeGaamyAaaqcfa4daeqaa8qacqaHfpqDpaWaaWba aeqajqwba+Faa8qacaWGPbaaaKqbakabg2da9iaaicdacaGGSaaaaa@6EA7@                         (1)

  1.  Motion equation

ρ a k = i σ ik + F k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyWdiNaamyya8aadaahaaqabKqbGeaapeGaam4AaaaajuaG cqGH9aqpcqGHhis0paWaaSbaaKqbGeaapeGaamyAaaqcfa4daeqaa8 qacqaHdpWCpaWaaWbaaeqajuaibaWdbiaabMgacaqGRbaaaKqbakab gUcaRiaadAeapaWaaWbaaeqajuaibaWdbiaadUgaaaaaaa@4768@                      (2)

Where

a k = υ k t + υ i i υ k , MathType@MTEF@5@5@+= feaagKart1ev2aqaM5hvLHfij5gC1rhimfMBNvxyNvga7Txy951ER1 xF9WLzYf2y7XfCHjhDPfgBGS3ECvhCZLgBV5gi951ER1xF9ThxWfMC 0LwySbcD9TYE7XvDWnxAS9MBG0Nx7LwF9T3EC5wyIXwyG03x7LwF9T 3ECvhCZLgBV5gi951ER1xFSaWexLMBbXgBcf2CPn2qVrwzqf2zLnha ruavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3b qee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8F4rqqrFfpeea0xe9 Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8 frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaG qaaaaaaaaaWdbiaadggapaWaaWbaaeqajuaibaWdbiaadUgaaaqcfa Oaeyypa0ZaaSaaa8aabaWdbiabgkGi2kabew8a19aadaahaaqabKqb GeaapeGaam4AaaaaaKqba+aabaWdbiabgkGi2kaadshaaaGaey4kaS IaeqyXdu3damaaCaaabeqcfasaa8qacaWGPbaaaKqbakabgEGir=aa daWgaaqcfasaa8qacaWGPbaajuaGpaqabaWdbiabew8a19aadaahaa qabKqbGeaapeGaam4AaaaajuaGcaGGSaaaaa@81CB@

i σ ik = σ i ik Γ im k σ im + Γ im m σ ik , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaey4bIe9damaaBaaabaWdbiaadMgaa8aabeaapeGaeq4Wdm3d amaaCaaabeqcfasaa8qacaWGPbGaam4AaaaajuaGcqGH9aqpcqaHdp WCpaWaa0baaKqbGeaapeGaamyAaaWdaeaapeGaamyAaiaadUgaaaqc faOaeu4KdC0damaaBaaajuaibaWdbiaadMgacaWGTbaajuaGpaqaba WdbiaadUgacqaHdpWCpaWaaWbaaeqajuaibaWdbiaadMgacaWGTbaa aKqbakabgUcaRiabfo5ah9aadaqhaaqcfasaa8qacaWGPbGaamyBaa WdaeaapeGaamyBaaaajuaGcqaHdpWCpaWaaWbaaeqajuaibaWdbiaa dMgacaWGRbaaaKqbakaacYcaaaa@5A35@

  1. Energy equation

dE dt = 1 ρ σ ij e ij MathType@MTEF@5@5@+= feaagKart1ev2aqaMnfvLHfij5gC1rhimfMBNvxyNvgaCzMCHn2EKv uF7rgD91dxMjxyJTxm9ThxYH2BG03E7X1CPDwBHbsFEThxY1giPPwF 9T3EL13x7XLCTbsAQ1xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefq vATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2Daebb nrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0= OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0x fr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaa aaaaaaa8qadaWcaaWdaeaapeGaamizaiaadweaa8aabaWdbiaadsga caWG0baaaiabg2da9maalaaapaqaa8qacaaIXaaapaqaa8qacqaHbp GCaaGaeq4Wdm3damaaCaaabeqcfasaa8qacaqGPbGaaeOAaaaajuaG caWGLbWdamaaBaaajuaibaWdbiaabMgacaqGQbaajuaGpaqabaaaaa@6750@                             (3)

Where F k MathType@MTEF@5@5@+= feaagKart1ev2aqaM1bvLHfij5gC1rhimfMBNvxyNvga7TNr951ER1 xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wy UbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8 YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=h GuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaae GaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGgbWd amaaCaaabeqcfasaa8qacaWGRbaaaaaa@43C3@  is components of mass force vector; Γ ij k MathType@MTEF@5@5@+= feaagKart1ev2aqaMHcvLHfij5gC1rhimfMBNvxyNvgaCDuyT1wyG8 1ECjxBGKMA951ER1hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwB Lnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtub sr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0x bba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0= vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaa aaaapeGaeu4KdC0damaaDaaajuaibaWdbiaabMgacaqGQbaapaqaa8 qacaWGRbaaaaaa@4A11@  - Christoffel symbols; σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqaMHcvLHfij5gC1rhimfMBNvxyNvga7ThxZL2zTf gi951ECjxBGKMA91hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwB Lnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtub sr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0x bba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0= vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaa aaaapeGaeq4Wdm3damaaCaaabeqcfasaa8qacaqGPbGaaeOAaaaaaa a@49B9@  - contravariant components of symmetric stress tensor; E- specific internal energy; e ij MathType@MTEF@5@5@+= feaagKart1ev2aqaMjcvLHfij5gC1rhimfMBNvxyNvga7Txz991ECj xBGKMA91hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2D Gi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHb GeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8Wq FfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpW qaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGa amyza8aadaWgaaqcfasaa8qacaqGPbGaaeOAaaqcfa4daeqaaaaa@474F@  - components of symmetric strain velocity tensor: ρ MathType@MTEF@5@5@+= feaagKart1ev2aqaMHbvLHfij5gC1rhimfMBNvxyNvgaCjhAVbWexL MBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wBH5garmWu 51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8 yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj 0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGaca GaaeqabaWaaeaaeaaakeaajuaGqaaaaaaaaaWdbiabeg8aYbaa@4102@  - density of medium; υ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqaMjcvLHfij5gC1rhimfMBNvxyNvgaCjwyYThx1b 3CPX2BU1hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2D Gi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHb GeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8Wq FfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpW qaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGa fqyXduNbaebaaaa@4575@  - velocity vector; a k MathType@MTEF@5@5@+= feaagKart1ev2aqaM1bvLHfij5gC1rhimfMBNvxyNvga7Txy951ER1 xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wy UbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8 YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=h GuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaae GaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGHbWd amaaCaaabeqcfasaa8qacaWGRbaaaaaa@43F9@  - components of acceleration vector;

e ij = 1 2 ( i υ j + j υ i ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyza8aadaWgaaqcfasaa8qacaqGPbGaaeOAaaqcfa4daeqa a8qacqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaaaada qadaWdaeaapeGaey4bIe9damaaBaaajuaibaWdbiaadMgaaKqba+aa beaacaaMc8+dbiabew8a19aadaWgaaqcfasaa8qacaWGQbaajuaGpa qabaWdbiabgUcaRiabgEGir=aadaWgaaqcfasaa8qacaWGQbaajuaG paqabaWdbiabew8a19aadaWgaaqcfasaa8qacaWGPbaajuaGpaqaba aapeGaayjkaiaawMcaaiaac6caaaa@50C9@                 (4)

Behavior of materials under study both metal and concrete has been described by elasto-plastic model.

Stress tensor is presented as a sum of deviatoric S ki MathType@MTEF@5@5@+= feaagKart1ev2aqaMjcvLHfij5gC1rhimfMBNvxyNvga7T3u951ECj xBGSwA91hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2D Gi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHb GeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8Wq FfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpW qaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGa am4ua8aadaahaaqabKqbGeaapeGaae4AaiaabMgaaaaaaa@4690@  and spherical part P :

σ ij =P g ij + S ij , MathType@MTEF@5@5@+= feaagKart1ev2aqaMLevLHfij5gC1rhimfMBNvxyNvga7ThxZL2zTf gi951E7LwF7PwF91xpTa1E7DwFET3EP13EQ1xF9TYE7nvFET3EP13E Q1xF9XcatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi 1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGe aGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf eaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqa aeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGaeq 4Wdm3damaaCaaabeqcfasaa8qacaWGPbGaamOAaaaajuaGcqGH9aqp cqGHsislcaWGqbGaam4za8aadaahaaqabKqbGeaapeGaamyAaiaadQ gaaaqcfaOaey4kaSIaam4ua8aadaahaaqabKqbGeaapeGaamyAaiaa dQgaaaqcfaOaaiilaaaa@64B0@                      (5)

where g ij MathType@MTEF@5@5@+= feaagKart1ev2aqaMLevLHfij5gC1rhimfMBNvxyNvga7ThxZL2zTf gi951E7LwF7PwF91xpTa1E7DwFET3EP13EQ1xF9TYE7nvFET3EP13E Q1xF9XcatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi 1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGe aGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFf eaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqa aeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGaam 4za8aadaahaaqabKqbGeaapeGaamyAaiaadQgaaaaaaa@5779@  - metric tensor. Pressure inside the materials has been calculated using Mie-Gruneisen equation as a function of specific internal energy E and density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqaMHbvLHfij5gC1rhimfMBNvxyNvgaCjhAVbWexL MBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wBH5garmWu 51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8 yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj 0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGaca GaaeqabaWaaeaaeaaakeaajuaGqaaaaaaaaaWdbiabeg8aYbaa@4102@ :

P= n=1 3 K n ( V V 0 1 ) n [ 1 K 0 ( V V 0 1 )/2 ]+ K 0 ρE, MathType@MTEF@5@5@+= feaagKart1ev2aqaMzlvLHfij5gC1rhimfMBNvxyNvgaq1dxZvxBCX wATLgDZ91EU1tm951EZ03E7T3s991EU1xF9T3ECXwzMrhkCzMCHn2E w13E7TNv991EW0xF91smGWLCPDgA0LsFETNB91hxSvMz0TvmTS3E7T uFFThm91hxSvMz0HcxMjxyJTNv9T3E7zvFFThm91xFTediCjxANHgD P03l7jtFCTdiCjxANHgDDTYE7TuFFThm91hxYH2BGuelamXvP5wqSX MqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2B Sbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9 vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9 q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabe qaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGqbGaeyypa0ZaaabC a8aabaWdbiaadUeapaWaaSbaaKqbGeaapeGaamOBaaqcfa4daeqaaa qcfasaa8qacaWGUbGaeyypa0JaaGymaaWdaeaapeGaaG4maaqcfaOa eyyeIuoadaqadaWdaeaapeWaaSaaa8aabaWdbiaadAfaa8aabaWdbi aadAfapaWaaSbaaKqbGeaapeGaaGimaaqcfa4daeqaaaaapeGaeyOe I0IaaGymaaGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaamOBaa aajuaGdaWadaWdaeaapeGaaGymaiabgkHiTiaadUeapaWaaSbaaKqb GeaapeGaaGimaaqcfa4daeqaa8qadaqadaWdaeaapeWaaSaaa8aaba WdbiaadAfaa8aabaWdbiaadAfapaWaaSbaaKqbGeaapeGaaGimaaqc fa4daeqaaaaapeGaeyOeI0IaaGymaaGaayjkaiaawMcaaiaac+caca aIYaWdaiaaysW7a8qacaGLBbGaayzxaaGaey4kaSIaam4sa8aadaWg aaqcfasaa8qacaaIWaaajuaGpaqabaWdbiabeg8aYjaadweacaGGSa aaaa@A8D3@         (6)

where K 0 , K 1 , K 2 , K 3 MathType@MTEF@5@5@+= feaagKart1ev2aqaMXdvLHfij5gC1rhimfMBNvxyNvga7T3s991EW0 xFSS3El13x7ftF9XYE7bYs991EY0xFSS3El13x7ntF9bWexLMBbXgB cf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wBH5garmWu51MyVX garuWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0x bbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0d Xdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqa baWaaeaaeaaakeaajuaGqaaaaaaaaaWdbiaadUeapaWaaSbaaKqbGe aapeGaaGimaaqcfa4daeqaa8qacaGGSaGaam4sa8aadaWgaaqcfasa a8qacaaIXaaajuaGpaqabaWdbiaacYcacaWGlbWdamaaBaaajuaiba WdbiaaikdaaKqba+aabeaapeGaaiilaiaadUeapaWaaSbaaKqbGeaa peGaaG4maaqcfa4daeqaaaaa@5A0D@  - material constants, V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqaM1bvLHfij5gC1rhimfMBNvxyNvga7TNv991EW0 xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wy UbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8 YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=h GuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaae GaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGwbWd amaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@440F@  - initial specific volume, V- current specific volume.
Suppose that the principle of minimum work of true stresses on the increments of plastic deformations is true for the medium, than the connection of component of strain velocity tensor and stress deviator is as follows:

2G( g im g jk e mk 1 3 g mk e mk g ij )= D S ij Dt +λ S ij  , ( λ0 ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGOmaiaadEeadaqadaWdaeaapeGaam4za8aadaahaaqabKqb GeaapeGaamyAaiaad2gaaaqcfaOaam4za8aadaahaaqabKqbGeaape GaamOAaiaadUgaaaqcfaOaamyza8aadaWgaaqcfasaa8qacaWGTbGa am4Aaaqcfa4daeqaa8qacqGHsisldaWcaaWdaeaapeGaaGymaaWdae aapeGaaG4maaaacaWGNbWdamaaCaaabeqcfasaa8qacaWGTbGaam4A aaaajuaGcaWGLbWdamaaBaaajuaibaWdbiaad2gacaWGRbaajuaGpa qabaWdbiaadEgapaWaaWbaaeqajuaibaWdbiaadMgacaWGQbaaaaqc faOaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacaWGebGaam4ua8 aadaahaaqabKqbGeaapeGaamyAaiaadQgaaaaajuaGpaqaa8qacaWG ebGaamiDaaaacqGHRaWkcqaH7oaBcaWGtbWdamaaCaaabeqcfasaa8 qacaWGPbGaamOAaaaajuaGcaqGGaGaaeilaiaabccadaqadaWdaeaa peGaeq4UdWMaaGPaVlabgwMiZkaaykW7caqGWaaacaGLOaGaayzkaa Gaaiilaaaa@6CD6@               (7)

In this case time derivatives of stress tensor are accepted by Jaumann definition:

D S ij Dt = d S ij dt g im ω mk S kj g jm ω mk S ik , MathType@MTEF@5@5@+= feaagKart1ev2aqaMXkvLHfij5gC1rhimfMBNvxyNvgaCzMCHn2EeT 3Et1Nx7TxA9TNA91xF9Thr01xpCzMCHn2EKT3Et1Nx7TxA9TNA91xF 9Thz01xl7T3z951E7LwF71wF913E7X1BTv2zHbsFFT3ET13ER1xF9T 3Et1Nx7T3A9TNA91xFTS3EN1Nx7TNA9TxB91xF7ThxV1wzNfgi991E 71wF7TwF913E7nvFET3EP13ER1xF9XcatCvAUfeBSjuyZL2yd9gzLb vyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwz YbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9 pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaO qaaKqbacbaaaaaaaaapeWaaSaaa8aabaWdbiaadseacaWGtbWdamaa Caaabeqcfasaa8qacaWGPbGaamOAaaaaaKqba+aabaWdbiaadseaca WG0baaaiabg2da9maalaaapaqaa8qacaWGKbGaam4ua8aadaahaaqa bKqbGeaapeGaamyAaiaadQgaaaaajuaGpaqaa8qacaWGKbGaamiDaa aacqGHsislcaWGNbWdamaaCaaabeqcfasaa8qacaWGPbGaamyBaaaa juaGcqaHjpWDpaWaaSbaaKqbGeaapeGaamyBaiaadUgaaKqba+aabe aapeGaam4ua8aadaahaaqabKqbGeaapeGaam4AaiaadQgaaaqcfaOa eyOeI0Iaam4za8aadaahaaqabKqbGeaapeGaamOAaiaad2gaaaqcfa OaeqyYdC3damaaBaaajuaibaWdbiaad2gacaWGRbaajuaGpaqabaWd biaadofapaWaaWbaaeqajuaibaWdbiaadMgacaWGRbaaaKqbakaacY caaaa@A9F2@

where ω ij = 1 2 ( i υ j j υ i ) MathType@MTEF@5@5@+= feaagKart1ev2aqaMnivLHfij5gC1rhimfMBNvxyNvga7ThxV1wzNf gi991ECjxBGKMA91xpCzMCHn2EX03EY0hxSvMz0HYE7XLBHjgBHbsF FTxA913E7XvDWnxAS9MBG03x7PwF91YE7XLBHjgBHbsFFTNA913E7X vDWnxAS9MBG03x7LwF9bcxYL2zOrxkamXvP5wqSXMqHnxAJn0BKvgu HDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALj hiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXd bba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0= yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGc baqcfaieaaaaaaaaa8qacqaHjpWDpaWaaSbaaKqbGeaapeGaaeyAai aabQgaaKqba+aabeaapeGaeyypa0ZaaSaaa8aabaWdbiaaigdaa8aa baWdbiaaikdaaaWaaeWaa8aabaWdbiabgEGir=aadaWgaaqcfasaa8 qacaWGPbaajuaGpaqabaWdbiabew8a19aadaWgaaqcfasaa8qacaWG QbaajuaGpaqabaWdbiabgkHiTiabgEGir=aadaWgaaqcfasaa8qaca WGQbaajuaGpaqabaWdbiabew8a19aadaWgaaqcfasaa8qacaWGPbaa juaGpaqabaaapeGaayjkaiaawMcaaaaa@834C@ , G- shear modulus.

Consider that material behaves in elastic manner ( λ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqaM5bvLHfij5gC1rhimfMBNvxyNvgaCXwyTjgzHb spWaWexLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wB H5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie YlNi=xH8yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8 FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaaba qaciGacaGaaeqabaWaaeaaeaaakeaajugibabaaaaaaaaapeGaeq4U dWMaeyypa0JaaGimaaaa@4462@ ), in case when Von Mises criterion is followed:

S ij S ij 2 3 σ d 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4ua8aadaahaaqabKqbGeaapeGaaeyAaiaabQgaaaqcfaOa am4ua8aadaWgaaqcfasaa8qacaqGPbGaaeOAaaqcfa4daeqaaiabgs MiJkaaykW7peWaaSaaa8aabaWdbiaaikdaa8aabaWdbiaaiodaaaGa eq4Wdm3damaaDaaajuaibaWdbiaadsgaa8aabaWdbiaaikdaaaqcfa Oaaiilaaaa@484E@                          (8)

and it behaves in plastic manner ( λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aqaM5bvLHfij5gC1rhimfMBNvxyNvgaCXwyTjgzHb IpWaWexLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wB H5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaie YlNi=xH8yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8 FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaaba qaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaaaaaaWdbiabeU7a Sjabg6da+iaaicdaaaa@4464@ ), when the criterion is not followed. Here σ d MathType@MTEF@5@5@+= feaagKart1ev2aqaMncvLHfij5gC1rhimfMBNvxyNvga7ThxZL2zTf gi991EK1xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz 3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYf gasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXd d9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9 adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qa cqaHdpWCpaWaaSbaaKqbGeaapeGaamizaaqcfa4daeqaaaaa@4797@  - dynamic tensile yield stress that can in the general case be the function of deformations velocity, pressure and temperature. The dependency of yield stress on the pressure has been considered for concrete:3

σ d =7.7+ 11.398P 13.9+0.82P MathType@MTEF@5@5@+= feaagKart1ev2aqaMjevLHfij5gC1rhimfMBNvxyNvga7ThxZL2zTf gi991EK1xF9CJl3ScxMjxyJTxmX4Ym5Gdu9TxmZ4soRaJl4idu9bWe xLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wzZbItLDhis9wBH5garm Wu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlNi=x H8yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYR qj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaWaaeaaeaaakeaajuaGqaaaaaaaaaWdbiabeo8aZ9aada Wgaaqcfasaa8qacaWGKbaajuaGpaqabaWdbiabg2da9iaaiEdacaGG UaGaaG4naiabgUcaRmaalaaapaqaa8qacaaIXaGaaGymaiaac6caca aIZaGaaGyoaiaaiIdacaWGqbaapaqaa8qacaaIXaGaaG4maiaac6ca caaI5aGaey4kaSIaaGimaiaac6cacaaI4aGaaGOmaiaadcfaaaaaaa@617B@           (9)

In case the condition (8) is violated, we apply the procedure of correction of stresses considering the material plasticity for calculation of the component of stress deviator. Components of S ij MathType@MTEF@5@5@+= feaagKart1ev2aqaMjcvLHfij5gC1rhimfMBNvxyNvga7T3u951ECj xBGKMA91hatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2D Gi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHb GeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8Wq FfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpW qaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGa am4ua8aadaahaaqabKqbGeaapeGaaeyAaiaabQgaaaaaaa@468E@  are multiplied by normalizing factor, that equals to description of medium behavior in plastic zone as proved by equations of Prandtl-Reuss.

Limiting value of plastic strain intensity is accepted as a local criterion of shear fracture in metals:

e u = 2 3 3 T 2 T 1 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqaMXevLHfij5gC1rhimfMBNvxyNvga7Txz991E11 xF9WLzYf2y7X1CXjhD7jtF9T3m9X1CXjhD7nZE7rvFFTNm91xlu91E X0Nx7jtF9XcatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov 2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNC HbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8 WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaape Gaamyza8aadaWgaaqcfasaa8qacaWG1baajuaGpaqabaWdbiabg2da 9maalaaapaqaa8qadaGcaaWdaeaapeGaaGOmaaqabaaapaqaa8qaca aIZaaaamaakaaapaqaa8qacaaIZaGaamiva8aadaWgaaqcfasaa8qa caaIYaaajuaGpaqabaWdbiabgkHiTiaadsfapaWaa0baaKqbGeaape GaaGymaaWdaeaapeGaaGOmaaaaaKqbagqaaiaacYcaaaa@6315@                  (10)

Where T1,T2 -first and second invariants of strain tensors.

To describe concrete fracture we use Hoffman criterion.4 The criterion considers the differences in ultimate tensile and compressive strength:

C 1 ( σ 22 σ 33 ) 2 + C 2 ( σ 33 σ 11 ) 2 + C 3 ( σ 11 σ 22 ) 2 + + C 4 σ 11 + C 5 σ 22 + C 6 σ 33 + C 7 σ 12 2 + C 8 σ 23 2 + C 9 σ 31 2 1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaieaa aaaaaaa8qacaWGdbWdamaaBaaajuaibaWdbiaaigdaaKqba+aabeaa peWaaeWaa8aabaWdbiabeo8aZ9aadaWgaaqcfasaa8qacaaIYaGaaG Omaaqcfa4daeqaa8qacqGHsislcqaHdpWCpaWaaSbaaKqbGeaapeGa aG4maiaaiodaaKqba+aabeaaa8qacaGLOaGaayzkaaWdamaaCaaabe qcfasaa8qacaaIYaaaaKqbakabgUcaRiaadoeapaWaaSbaaKqbGeaa peGaaGOmaaqcfa4daeqaa8qadaqadaWdaeaapeGaeq4Wdm3damaaBa aajuaibaWdbiaaiodacaaIZaaajuaGpaqabaWdbiabgkHiTiabeo8a Z9aadaWgaaqcfasaa8qacaaIXaGaaGymaaqcfa4daeqaaaWdbiaawI cacaGLPaaapaWaaWbaaeqajuaibaWdbiaaikdaaaqcfaOaey4kaSIa am4qa8aadaWgaaqcfasaa8qacaaIZaaajuaGpaqabaWdbmaabmaapa qaa8qacqaHdpWCpaWaaSbaaKqbGeaapeGaaGymaiaaigdaaKqba+aa beaapeGaeyOeI0Iaeq4Wdm3damaaBaaajuaibaWdbiaaikdacaaIYa aajuaGpaqabaaapeGaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGa aGOmaaaajuaGcqGHRaWkaOqaaKqbakabgUcaRiaadoeapaWaaSbaaK qbGeaapeGaaGinaaqcfa4daeqaa8qacqaHdpWCpaWaaSbaaKqbGeaa peGaaGymaiaaigdaaKqba+aabeaapeGaey4kaSIaam4qa8aadaWgaa qcfasaa8qacaaI1aaajuaGpaqabaWdbiabeo8aZ9aadaWgaaqcfasa a8qacaaIYaGaaGOmaaqcfa4daeqaa8qacqGHRaWkcaWGdbWdamaaBa aajuaibaWdbiaaiAdaaKqba+aabeaapeGaeq4Wdm3damaaBaaajuai baWdbiaaiodacaaIZaaajuaGpaqabaWdbiabgUcaRiaadoeapaWaaS baaKqbGeaapeGaaG4naaqcfa4daeqaa8qacqaHdpWCpaWaa0baaKqb GeaapeGaaGymaiaaikdaa8aabaWdbiaaikdaaaqcfaOaey4kaSIaam 4qa8aadaWgaaqcfasaa8qacaaI4aaajuaGpaqabaWdbiabeo8aZ9aa daqhaaqcfasaa8qacaaIYaGaaG4maaWdaeaapeGaaGOmaaaajuaGcq GHRaWkcaWGdbWdamaaBaaajuaibaWdbiaaiMdaaKqba+aabeaapeGa eq4Wdm3damaaDaaajuaibaWdbiaaiodacaaIXaaapaqaa8qacaaIYa aaaKqbakabgwMiZkaaigdacaGGSaaaaaa@9D36@                     (11)

where C i MathType@MTEF@5@5@+= feaagKart1ev2aqaM1bvLHfij5gC1rhimfMBNvxyNvga7T3q991EP1 xFamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wy UbqedmvETj2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8 YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=h GuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaae GaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGdbWd amaaBaaajuaibaWdbiaadMgaaKqba+aabeaaaaa@4456@  is defined from the following formulas:

C 1 =[ ( Y t Y c ) 1 + ( Z t Z c ) 1 ( X t X c ) 1 ] 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaaeaacaWGdbWdamaaBaaajuaibaWdbiaaigdaaKqba+aa beaapeGaeyypa0JaaGPaVlaaykW7caGGBbGaaiikaiaadMfapaWaaS baaKqbGeaapeGaamiDaaqcfa4daeqaa8qacaWGzbWdamaaBaaajuai baWdbiaadogaaKqba+aabeaapeGaaiyka8aadaahaaqabKqbGeaape GaeyOeI0IaaGymaaaajuaGcqGHRaWkcaGGOaGaamOwa8aadaWgaaqc fasaa8qacaWG0baajuaGpaqabaWdbiaadQfapaWaaSbaaKqbGeaape Gaam4yaaqcfa4daeqaa8qacaGGPaWdamaaCaaabeqcfasaa8qacqGH sislcaaIXaaaaKqbakabgkHiTiaacIcacaWGybWdamaaBaaajuaiba WdbiaadshaaKqba+aabeaapeGaamiwa8aadaWgaaqcfasaa8qacaWG JbaajuaGpaqabaWdbiaacMcapaWaaWbaaeqajuaibaWdbiabgkHiTi aaigdaaaqcfaOaaiyxaaqaaiaaikdaaaGaai4oaaaa@5FE2@
C 2 =[ ( X t X c ) 1 + ( Z t Z c ) 1 ( Y t Y c ) 1 ] 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSWaaeaacaWGdbWdamaaBaaajuaibaWdbiaaikdaaKqba+aa beaapeGaeyypa0JaaGPaVlaaykW7caaMc8Uaai4waiaacIcacaWGyb WdamaaBaaajuaibaWdbiaadshaaKqba+aabeaapeGaamiwa8aadaWg aaqcfasaa8qacaWGJbaajuaGpaqabaWdbiaacMcapaWaaWbaaeqaju aibaWdbiabgkHiTiaaigdaaaqcfaOaey4kaSIaaiikaiaadQfapaWa aSbaaKqbGeaapeGaamiDaaqcfa4daeqaa8qacaWGAbWdamaaBaaaju aibaWdbiaadogaaKqba+aabeaapeGaaiyka8aadaahaaqabKqbGeaa peGaeyOeI0IaaGymaaaajuaGcqGHsislcaaMc8UaaiikaiaadMfapa WaaSbaaKqbGeaapeGaamiDaaqcfa4daeqaa8qacaWGzbWdamaaBaaa juaibaWdbiaadogaaKqba+aabeaapeGaaiyka8aadaahaaqabKqbGe aapeGaeyOeI0IaaGymaaaajuaGcaGGDbaabaGaaGOmaaaacaGG7aaa aa@62FA@
C 3 = [ ( X t X c ) 1 + ( Y t Y c ) 1 ( Z t Z c ) 1 ] 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qa8aadaWgaaqcfasaa8qacaaIZaaajuaGpaqabaWdbiab g2da9iaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaSGaaeaacaGGBb GaaiikaiaadIfapaWaaSbaaKqbGeaapeGaamiDaaqcfa4daeqaa8qa caWGybWdamaaBaaajuaibaWdbiaadogaaKqba+aabeaapeGaaiyka8 aadaahaaqabKqbGeaapeGaeyOeI0IaaGymaaaajuaGcqGHRaWkcaGG OaGaamywa8aadaWgaaqcfasaa8qacaWG0baajuaGpaqabaWdbiaadM fapaWaaSbaaKqbGeaapeGaam4yaaqcfa4daeqaa8qacaGGPaWdamaa Caaabeqcfasaa8qacqGHsislcaaIXaaaaKqbakabgkHiTiaacIcaca WGAbWdamaaBaaajuaibaWdbiaadshaaKqba+aabeaapeGaamOwa8aa daWgaaqcfasaa8qacaWGJbaajuaGpaqabaWdbiaacMcapaWaaWbaae qajuaibaWdbiabgkHiTiaaigdaaaqcfaOaaiyxaaqaaiaaikdaaaGa ai4oaaaa@6485@
C 4 =( X t 1 X c 1 ); MathType@MTEF@5@5@+= feaagKart1ev2aqaM5dvLHfij5gC1rhimfMBNvxyNvga7T3q991E00 xF9GYE7HvFFThD91Nx71sm91YE7HvFFT3y91Nx71sm9LYoamXvP5wq SXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj 2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYl h9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pg c9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaa beqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaaju aibaWdbiaaisdaaKqba+aabeaapeGaeyypa0JaaiikaiaadIfapaWa aSbaaKqbGeaapeGaamiDaaqcfa4daeqaamaaCaaabeqcfasaa8qacq GHsislcaaIXaaaaKqbakabgkHiTiaadIfapaWaaSbaaKqbGeaapeGa am4yaaqcfa4daeqaamaaCaaabeqcfasaa8qacqGHsislcaaIXaaaaK qbakaacMcacaGG7aaaaa@5F7D@
C 5 =( Y t 1 Y c 1 ); MathType@MTEF@5@5@+= feaagKart1ev2aqaM5dvLHfij5gC1rhimfMBNvxyNvga7T3q991E10 xF9GYE7LvFFThD91Nx71sm91YE7LvFFT3y91Nx71sm9LYoamXvP5wq SXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj 2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYl h9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pg c9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaa beqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaaju aibaWdbiaaiwdaaKqba+aabeaapeGaeyypa0JaaiikaiaadMfapaWa aSbaaKqbGeaapeGaamiDaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacq GHsislcaaIXaaaaKqbakabgkHiTiaadMfapaWaaSbaaKqbGeaapeGa am4yaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacqGHsislcaaIXaaaaK qbakaacMcacaGG7aaaaa@5F83@
C 6 =( Z t 1 Z c 1 ); MathType@MTEF@5@5@+= feaagKart1ev2aqaM5dvLHfij5gC1rhimfMBNvxyNvga7T3q991E20 xF9GYE7PvFFThD91Nx71sm91YE7PvFFT3y91Nx71sm9LYoamXvP5wq SXMqHnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj 2BSbqefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYl h9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pg c9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaa beqaamaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaaju aibaWdbiaaiAdaaKqba+aabeaapeGaeyypa0JaaiikaiaadQfapaWa aSbaaKqbGeaapeGaamiDaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacq GHsislcaaIXaaaaKqbakabgkHiTiaadQfapaWaaSbaaKqbGeaapeGa am4yaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacqGHsislcaaIXaaaaK qbakaacMcacaGG7aaaaa@5F89@
C 7 = S yz 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aqaMjdvLHfij5gC1rhimfMBNvxyNvga7T3q991E30 xF9S3Et13x7XLCTbsE61xFETxlY03oamXvP5wqSXMqHnxAJn0BKvgu HDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALj hiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXd bba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0= yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGc baqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaajuaibaWdbiaaiEdaaK qba+aabeaapeGaeyypa0Jaam4ua8aadaWgaaqcfasaa8qacaqG5bGa aeOEaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacqGHsislcaaIYaaaaK qbakaacUdaaaa@5469@
C 8 = S zx 2 ; MathType@MTEF@5@5@+= feaagKart1ev2aqaMjdvLHfij5gC1rhimfMBNvxyNvga7T3q991E40 xF9S3Et13x7XLCTbIE41xFETxlY03oamXvP5wqSXMqHnxAJn0BKvgu HDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALj hiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXd bba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0= yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGc baqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaajuaibaWdbiaaiIdaaK qba+aabeaapeGaeyypa0Jaam4ua8aadaWgaaqcfasaa8qacaqG6bGa aeiEaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacqGHsislcaaIYaaaaK qbakaacUdaaaa@5469@
C 9 = S xy 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqaMjdvLHfij5gC1rhimfMBNvxyNvga7T3q991E50 xF9S3Et13x7XLCTbcE51xFETxlY0NlamXvP5wqSXMqHnxAJn0BKvgu HDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALj hiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXd bba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0= yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGc baqcfaieaaaaaaaaa8qacaWGdbWdamaaBaaajuaibaWdbiaaiMdaaK qba+aabeaapeGaeyypa0Jaam4ua8aadaWgaaqcfasaa8qacaqG4bGa aeyEaaWdaeqaaKqbaoaaCaaajuaibeqaa8qacqGHsislcaaIYaaaaK qbakaac6caaaa@544F@

Where X t , X c , Y t , Y c , Z t , Z c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiwa8aadaWgaaqcfasaa8qacaWG0baajuaGpaqabaGaaiil a8qacaWGybWdamaaBaaajuaibaWdbiaadogaaKqba+aabeaacaGGSa WdbiaadMfapaWaaSbaaKqbGeaapeGaamiDaaqcfa4daeqaaiaacYca peGaamywa8aadaWgaaqcfasaa8qacaWGJbaajuaGpaqabaGaaiila8 qacaWGAbWdamaaBaaajuaibaWdbiaadshaaKqba+aabeaacaGGSaWd biaadQfapaWaaSbaaKqbGeaapeGaam4yaaqcfa4daeqaaaaa@4B72@ - ultimate strength along axes X,Y,Z under tension and compression respectively, while S xy , S yz , S zx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4ua8aadaWgaaqcfasaa8qacaqG4bGaaeyEaaqcfa4daeqa aiaacYcacaaMc8+dbiaadofapaWaaSbaaKqbGeaapeGaaeyEaiaabQ haaKqba+aabeaacaGGSaGaaGPaV=qacaWGtbWdamaaBaaajuaibaWd biaabQhacaqG4baajuaGpaqabaaaaa@46C1@ -ultimate shear strength along the corresponding axes. In case of isotropic material X t = Y t = Z t = R t MathType@MTEF@5@5@+= feaagKart1ev2aqaMndvLHfij5gC1rhimfMBNvxyNvga7HvFFThD91 ZEz13x7rxF9SNw991E01xp7jvFFThD9bWexLMBbXgBcf2CPn2qVrwz qf2zLnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPv MCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8F4rqqrFf peea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq =JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaa keaajuaGqaaaaaaaaaWdbiaadIfapaWaaSbaaKqbGeaapeGaamiDaa qcfa4daeqaa8qacqGH9aqpcaWGzbWdamaaBaaajuaibaWdbiaadsha aKqba+aabeaapeGaeyypa0JaamOwa8aadaWgaaqcfasaa8qacaWG0b aajuaGpaqabaWdbiabg2da9iaadkfapaWaaSbaaKqbGeaapeGaamiD aaqcfa4daeqaaaaa@599B@ , X c = Y c = Z c = R c MathType@MTEF@5@5@+= feaagKart1ev2aqaMndvLHfij5gC1rhimfMBNvxyNvga7HvFFT3y91 ZEz13x7nwF9SNw991EJ1xp7jvFFT3y9bWexLMBbXgBcf2CPn2qVrwz qf2zLnharuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPv MCG4uz3bqee0evGueE0jxyaibaieYlNi=xH8yiVC0xbbL8F4rqqrFf peea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq =JbbG8A8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaa keaajuaGqaaaaaaaaaWdbiaadIfapaWaaSbaaKqbGeaapeGaam4yaa qcfa4daeqaa8qacqGH9aqpcaWGzbWdamaaBaaajuaibaWdbiaadoga aKqba+aabeaapeGaeyypa0JaamOwa8aadaWgaaqcfasaa8qacaWGJb aajuaGpaqabaWdbiabg2da9iaadkfapaWaaSbaaKqbGeaapeGaam4y aaqcfa4daeqaaaaa@5913@ , S xy = S yz = S zx = R s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeaeaaaaaa aaa8qacaWGtbqcfa4damaaBaaajeaibaqcLbmapeGaaeiEaiaabMha aSWdaeqaaKqzGeWdbiabg2da9iaadofajuaGpaWaaSbaaKqaGeaaju gWa8qacaqG5bGaaeOEaaWcpaqabaqcLbsapeGaeyypa0Jaam4uaKqb a+aadaWgaaqcbasaaKqzadWdbiaabQhacaqG4baal8aabeaajugib8 qacqGH9aqpcaWGsbqcfa4damaaBaaajeaibaqcLbmapeGaam4CaaWc paqabaaaaa@4EC8@ .

 It is suggested that concrete fracture in conditions of intensive dynamic loads occurs the following way 1:

  1. In case strength criterion (11) is violated under conditions of compression ( e kk 0) MathType@MTEF@5@5@+= feaagKart1ev2aqaMLcvLHfij5gC1rhimfMBNvxyNvgaOS3EL13x7X LCTbYAR1xFCXwzGatkamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvA Tv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2Daebbnr fifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=Oq Ffea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr =xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaa aaaaa8qacaGGOaGaamyza8aadaWgaaqcfasaa8qacaqGRbGaae4Aaa qcfa4daeqaa8qacqGHKjYOcaaIWaGaaiykaaaa@4D02@ , than further material behavior is described by hydrodynamic model, in that case concrete can be destructed when satisfying the criterion considering the ultimate value of plastic deformations intensity (10);
  2. In case criterion (11) is violated under conditions of tension ( e kk >0) MathType@MTEF@5@5@+= feaagKart1ev2aqaMLcvLHfij5gC1rhimfMBNvxyNvgaOSxzFT3ECj xB7T2A91xF9bIpGatkamXvP5wqSXMqHnxAJn0BKvguHDwzZbqefqvA Tv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhiov2Daebbnr fifHhDYfgasaacH8YjY=vipgYlh9vqqj=hEeeu0xXdbba9frFj0=Oq Ffea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr =xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaa aaaaa8qacaGGOaGaamyza8aadaWgaaqcfasaa8qacaqGRbGaae4Aaa qcfa4daeqaa8qacqGH+aGpcaaIWaGaaiykaaaa@4C5E@ , than material is considered to be completely destructed and components of stress tensor are supposed to be equal to zero.

Experiments have shown that dynamic loads lead to increasing strength properties of concrete.5 Moreover the dependency of ultimate tensile and compressive strength differs. Connection of static and dynamic ultimate strength is expressed by means of dynamic response factor:

K d = R d / R s MathType@MTEF@5@5@+= feaagKart1ev2aqaMDcvLHfij5gC1rhimfMBNvxyNvga7T0xK1hi9a YEs9fz99YEs91C9bWexLMBbXgBcf2CPn2qVrwzqf2zLnharuavP1wz ZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGu eE0jxyaibaieYlNi=xH8yiVC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vq aqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fv e9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGqaaaaaaa aaWdbiaadUeapaWaaSbaaKqbGeaapeGaamizaaqcfa4daeqaa8qacq GH9aqpcaWGsbWdamaaBaaajuaibaWdbiaadsgaaKqba+aabeaapeGa ai4laiaadkfapaWaaSbaaKqbGeaapeGaam4Caaqcfa4daeqaaaaa@4EC9@

where R d MathType@MTEF@5@5@+= feaagKart1ev2aqaMLbvLHfij5gC1rhimfMBNvxyNvga7j1xK1hatC vAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXa tLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8Ff YJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVc Lq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaci aacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGaamOua8aadaWg aaqcfasaa8qacaWGKbaajuaGpaqabaaaaa@4276@  - dynamic strength, R s MathType@MTEF@5@5@+= feaagKart1ev2aqaMLbvLHfij5gC1rhimfMBNvxyNvga7j1xZ1hatC vAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXa tLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8Ff YJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVc Lq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaci aacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaaaapeGaamOua8aadaWg aaqcfasaa8qacaWGZbaajuaGpaqabaaaaa@4294@  - static strength.

Based on the experimental data dynamic response factor has been obtained for concrete under compression (12) and tension (13). Corresponding curves are given in Figure 1, where curve 1 describes dependency of dynamic response factor under tension on deformations velocity, curve 2 describes dependency under compression.

K dt =0.00158333 e 5 +0.0252855 e 4 +0.15255 e 3 +0.47898 e 2 +1.01959e+2.36037. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaieaa aaaaaaa8qacaWGlbWdamaaBaaajuaibaWdbiaabsgacaqG0baajuaG paqabaWdbiabg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIXaGaaG ynaiaaiIdacaaIZaGaaG4maiaaiodacaWGLbWdamaaCaaabeqcfasa a8qacaaI1aaaaKqbakabgUcaRiaaicdacaGGUaGaaGimaiaaikdaca aI1aGaaGOmaiaaiIdacaaI1aGaaGynaiaadwgapaWaaWbaaeqajuai baWdbiaaisdaaaaakeaajuaGcaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVladasHHRaWkcGaGuGimaiacasHGUaGaiaifigdacGaGuGynaiac asbIYaGaiaifiwdacGaGuGynaiacasXGLbWdamacasbhaaqajaifju aibGaGu8qacGaGuG4maaaajuaGcWaGuy4kaSIaiaificdacGaGuiOl aiacasbI0aGaiaifiEdacGaGuGioaiacasbI5aGaiaifiIdacGaGum yza8aadGaGuWbaaeqcasrcfauaiaifpeGaiaifikdaaaqcfaOamaif gUcaRiacasbIXaGaiaifc6cacGaGuGimaiacasbIXaGaiaifiMdacG aGuGynaiacasbI5aGaiaifdwgacWaGuy4kaSIaiaifikdacGaGuiOl aiacasbIZaGaiaifiAdacGaGuGimaiacasbIZaGaiaifiEdacGaGui Olaaaaaa@9145@ (12)

K dc =0.000832308 e 5 +0.0110547 e 4 + +0.0447734 e 3 +0.0475887 e 2 +0.0184316e+1.20895. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaieaa aaaaaaa8qacaWGlbWdamaaBaaajuaibaWdbiaabsgacaqGJbaajuaG paqabaWdbiabg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIWaGaaG ioaiaaiodacaaIYaGaaG4maiaaicdacaaI4aGaamyza8aadaahaaqa bKqbGeaapeGaaGynaaaajuaGcqGHRaWkcaaIWaGaaiOlaiaaicdaca aIXaGaaGymaiaaicdacaaI1aGaaGinaiaaiEdacaWGLbWdamaaCaaa beqcfasaa8qacaaI0aaaaKqbakabgUcaRaGcbaqcfaOaey4kaSIaaG imaiaac6cacaaIWaGaaGinaiaaisdacaaI3aGaaG4naiaaiodacaaI 0aGaamyza8aadaahaaqabKqbGeaapeGaaG4maaaajuaGcqGHRaWkca aIWaGaaiOlaiaaicdacaaI0aGaaG4naiaaiwdacaaI4aGaaGioaiaa iEdacaWGLbWdamaaCaaabeqcfasaa8qacaaIYaaaaKqbakabgUcaRi aaicdacaGGUaGaaGimaiaaigdacaaI4aGaaGinaiaaiodacaaIXaGa aGOnaiaadwgacqGHRaWkcaaIXaGaaiOlaiaaikdacaaIWaGaaGioai aaiMdacaaI1aGaaiOlaaaaaa@751B@ (13)

Figure 1 Dependency of ultimate strength of concrete on deformation velocity: 1 - under tension, 2 -under compression.

Ultimate shear strength of concrete is defined from the values of ultimate compressive and tensile strength:3

R s =0.55 R c R t MathType@MTEF@5@5@+= feaagKart1ev2aqaMTdvLHfij5gC1rhimfMBNvxyNvga7TNu991EZ1 xF9aJl1udxZfNC0T3E7jvFFT3y913E7jvFFThD91xFamXvP5wqSXMq HnxAJn0BKvguHDwzZbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSb qefm0B1jxALjhiov2DaebbnrfifHhDYfgasaacH8YjY=vipgYlh9vq qj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9q8 qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqa amaabaabaaGcbaqcfaieaaaaaaaaa8qacaWGsbWdamaaBaaajuaiba WdbiaadohaaKqba+aabeaapeGaeyypa0JaaGimaiaac6cacaaI1aGa aGynamaakaaapaqaa8qacaWGsbWdamaaBaaajuaibaWdbiaadogaaK qba+aabeaapeGaamOua8aadaWgaaqcfasaa8qacaWG0baajuaGpaqa baaapeqabaaaaa@5AC6@

Important aspect while numerical modeling of impact interaction is selecting the algorithm of contact boundaries calculation. Generally, the existing program software use algorithms “element-node” and “node-node” to define the possible penetration of one body into another. The given work suggests the algorithm of “element-element” type,6 proved oneself to be appropriate while solving the three dimensional objectives and enabling to use possibilities of parallel computing to a maximum extent.

Numerical results

The results of numerical simulation of ogival projectile interaction with reinforced concrete barrier are given below. The properties of the investigated materials are given in Table 1. The projective is given as a shell made from titanium and filled with explosive substance. The thickness of reinforced concrete barrier is 1 m. Layers of reinforcement are placed close to front and back surfaces of the barrier. The initial projectile velocity is 300 and 700 m/s. Two types of interaction have been considered: the standard one, and under the angle of 5 degrees to normal line.

Material

ρ, kg/ m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaCyWdiaahYcacaWHGaGaaC4AaiaahEgacaWHVaGaaCyBa8aa daahaaqabKqbGeaapeGaaC4maaaaaaa@3E5C@

Speed of
Sound
Cs, m/s

Poisson's
Ratio

Tensile
Strength, MPa

Compressive
Strength, MPa

Young's
Modulus Е,
Gpa

Concrete

2450

4500

0,2

1,75

22

26

Titanium

4500

6900

0.32

400

400

146

Steel

7850

5930

0,3

400

400

204

Table 1 Properties of materials

Figure 2 shows in section the standard penetration of projectile with initial velocity of 700 m/s. The areas of reinforcement in the barrier are marked in red color. Through penetration of barrier and concrete fracture has been observed. Concrete is destructed due to the action of tensile stresses. Because of low tensile strength of concrete it is destructed almost through the entire surface. One can also note destruction of explosive substance close to the projectile nose in the central area. Destruction of explosive substance is conditioned by the action of unloading wave distributed from the lateral surface of projectile. In this case titanium shell is deformed insignificantly almost completely saving its initial shape.

Figure 2 Configuration of projectile and barrier. υ 0 =700 m/s,α=0° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyXdu3damaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaG4naiaaicdacaaIWaGaaeiiaiaad2gacaGGVaGaam4Cai aacYcacqaHXoqycqGH9aqpcaaIWaGaeyiSaalaaa@4706@ degrees. A) t=0.2 ms, B) t=1.2 ms, C) t=3 ms.

Figure 3 shows normal penetration of projectile with the velocity of 300 m/s. In general the fracture pattern is similar to Figure 2. Larger level of destruction in the barrier and the explosive substance is conditioned by longer process of interaction thus it has lead to the moment of barrier striking (Figure 3C) and larger damage cumulation.

Figure 3 Configuration of projectile and barrier. υ 0 =300 m/s,α=0° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyXdu3damaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaG4maiaaicdacaaIWaGaaeiiaiaad2gacaGGVaGaam4Cai aacYcacqaHXoqycqGH9aqpcaaIWaGaeyiSaalaaa@4702@ .A) t=0.5 ms, B) t=3 ms, C) t=6 ms.

(Figure 4) (Figure 5C) show computation configuration of the projectile and barrier by interaction under angle of 5 degrees for the initial velocities of 300 and 700 m/s, correspondently. The same as for normal interaction (Figure 2 (Figure 3) one can observe consistent reinforced concrete barrier striking. When impacting under angle localization and geometry of destruction areas as well as destruction in explosive substance is changed. The areas of destruction in the explosive substance are formed not only in the central part of the projectile but also close to the titanium shell in the nose part on the right (Figure 4B) (Figure 5B). This can be explained by the fact that during impact under angle this part of projectile is free at the initial moment of interaction, thus the unloading wave is being formed here, further leading to tensile stresses occurrence which destruct explosive substance.

Figure 4 Configurations of projectile and barrier. υ 0 =700 m/s,α=5° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyXdu3damaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaG4naiaaicdacaaIWaGaaeiiaiaad2gacaGGVaGaam4Cai aacYcacqaHXoqycqGH9aqpcaaI1aGaeyiSaalaaa@470B@ . Time periods A) t=0.5ms, B) t=1.5ms.

Figure 5 Configurations of projectile and barrier. υ 0 =300 m/s,α=5° MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=gjVeeu0dXdPqFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyXdu3damaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaG4maiaaicdacaaIWaGaaeiiaiaad2gacaGGVaGaam4Cai aacYcacqaHXoqycqGH9aqpcaaI1aGaeyiSaalaaa@4707@ . A) t=0.7ms, B) t=4.6ms.

Conclusion

Resulting from the conducted studies the following conclusions can be made:

A. The model has been suggested for reinforced concrete barriers behavior under impact. The conditions of interaction on the projectile and reinforced concrete barrier destruction have been studied. Wide-parameter numerical studies on selection of optimal structural solutions are possible to be conducted based on the developed methodology.

Acknowledgements

The work has been conducted with the financial support of the President of the Russian Federation No. MK-413.2017.1.

Conflicts of interest

Author declares that there is no conflict of interest.

References

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