Cheng-zong Wng ,Ai-jun Chen ,,,Zi-qing Li ,Cho-n Gong ,Shu Wng ,Wen-min Yn

a School of Science,Nanjing University of Science and Technology,Nanjing,210094,China

b Science and Technology on Transient Impact Laboratory,Beijing,102202,China

Keywords: Clay-fired brick Penetration of masonry RHT model Impact High strain rate

ABSTRACT This study investigates a kind of masonry consisting of clay-fired brick(fc=10 MPa;ρ=1.38 g/cm3)and mortar(fc=10 MPa;ρ=1.8 g/cm3).Clay-fired brick masonry connotes a traditional construction material of old architecture and public buildings.We carried out penetration experiments in which four clay-fired brick walls employing two different patterns were subjected to impact from small high-speed projectile,i.e.12.7 mm armor-piercing explosive incendiary projectile and material tests in which the static and dynamic compressive strengths of clay-fired brick and mortar were determined by quasi-static and SHPB(Split Hopkinson Pressure Bar)tests.The experimental data include hit and exit velocities,damage con figuration of clay brick masonry and mechanical properties of material at low and high strain rates,though which in fluence of thickness and bonding patterns of wall on kinetic loss of bullet,the damage patterns of masonry observed experimentally and dynamic increase of material strengths are analyzed.To keep minimum boundary inconsistency with reality,full 3D detailed finite element model consisting of two different material is established.Sharing common nodes and employing automatic tiebreak contact are combined to reduce computational time usage of large-scale model.For description of clay-fired brick and mortar Riedel-Hiermaier-Thoma(RHT)material model is employed.Material parameter set is derived based on experimental data,available literature and engineering assumptions.The numerical simulations study the mesh resolution dependency of material model,reproduce the crucial phenomena of masonry in experiment acceptably and offer more time-resolved insight into motion of bullet in the process of penetration.The feasibility of means of constructing finite element model and applying RHT model to the masonry herein and analogous constructions is explored through numerical investigation.

1.Introduction

Masonry is one of the most common building materials,which is used on account of low cost,good sound and heat insulation properties,locally available material[1].It has a long history.The first bricks were based on dried mud and were used for the first time in 8,000 BC in Mesopotamia[2].Ever since its invention,people around the world have relied on the structure in addition to its characteristics mentioned for it is robust enough for human beings to obtain a safe and tidy environment.Masonry walls consist of bricks and adhesive joints by which the bricks are connected.The bricks and joints are of varying types.Fired-clay bricks(fc=10 MPa;ρ=1.38 g/cm3)and mortar(fc=10 MPa;ρ=1.8 g/cm3)are of relevance in this paper.Even though the use of clay bricks has been shrank because of environmental impacts brought by fired-clay brick industry,there are still considerable amount of clay brick masonry buildings existing such as churches,historical constructions and civilian infrastructure in the world.The predictive response of clay brick masonry to seismic load and dynamic impacts is instructive for design con figuration[3].The ballistic response of masonry is of interest because of some military value[4].In the past decades the characteristic of concrete under dynamic loading conditions such as penetration or explosive impact has been studied thoroughly but the similar research with respect to brick,mortar and masonry was relatively much less.

What has been historically employed to investigate the dynamic properties of concrete include theoretical study,experimental investigation and numerical simulation.It has been shown that numerical simulation can provide more information of process history for sophisticated analysis than experimental investigation.Nevertheless,the data from experiments are crucial because on the one hand any modern analysis and numerical computation should necessarily be validated by those from real shooting range tests,on the other hand the material parameters which are of vital importance to obtain a reasonable and compelling result must be furnished by or refer to experimental data[5].The experimental exploration indicates that the masonry constituents,namely clay brick and mortar joint,act similarly to concrete which is brittle material as well when encountering static loadings and dynamic impacts[6-8].Many material models in simulation are proposed to describe the behavior of concrete,which are able to be applied to solution of simulating behavior of masonry reasonably.Li et al.[9]investigated the response mode of unreinforced clay brick masonry subjected to vented gas explosion through experiment and simulation,in which the brittle damage material model developed in Ref.[10]for concrete has been adopted.The potential of Holmquist-Johnson-Cook(HJC)material model was explored profoundly in Ref.[11]to apply in simulation of penetration of masonry,which was initialled by Holmquist and Johnson in 1993[12]for concrete.Based on phenomenology of concrete Riedel-Hiermaier-Thoma(RHT)[13-15]material model describing macroscopic properties of concrete was used even in simulation of adobe[16-19].

To better represent the performance of material under dynamic loadings the effect of strain rate must be taken into consideration.Experimental investigation is normally a way of studying clay brick and mortar at different strain rates and DIF(dynamic increase factor)is defined to describe the increment to properties of material[3,6,7].Numerical simulation was adopted to explore the effect of strain rate in Ref.[20].Most of literature focused on the dynamic characteristic of brick or mortar under compressive impact.The tensile DIF of masonry joint was investigated in Ref.[21]and the DIF 3.1 relative to reference strain rate of 1-1was concluded.It was found that the increment to material properties results not only from intrinsic properties of material but also structural effect associated with dynamic experiments,such as inertial effect[22]and end friction con finement[23].The contribution of fibres and water content in adobe to DIF has been illustrated in Ref.[24],which provides a reasonable explanation for dynamic strengthening phenomenon for adobe but also an appropriate interpretation for other analogous material.

The strategies to numerically model masonry wall fall into three categories in scale[8].First method is macro-model in which the inhomogeneous composite material masonry,consisting of bricks and mortar,is homogenized as one single material,which is used to capture the behavior of the whole wall and circumvent computational expense[25].The second approach is simpli fied micromodel which simpli fies the mortar joint as a surface of no thickness and the dimensions of the masonry units should be adjusted properly,as in Ref.[20,26,27].The last one is micro-model which constructs masonry with an assemblage of bricks connected by mortar joints and retains the most detail.The model can examine the failure of brick,mortar joint and the bond between the brick and mortar,as in Ref.[9].

Analytical modeling of masonry demands the properties and interrelationships of brick and mortar;it is not easily available because of limitation of experimental tests and variation in properties and proportion of material[1].The research mentioned previously has investigated quasi-static and dynamic behaviors of brick,mortar and masonry in different perspectives.Some focused on the constitutive formula of brick,mortar and bond between them,some detailed ways of constructing model of simulation,some investigated the mode of failure of masonry subjected to quasi-static pressure,explosive blast impacts or seismic loads,which all take us further on the road of understanding masonry as a composite building structure,properties of its anisotropic constituents,namely brick and mortar,and interaction between them.Still,the knowledge and code with regard to masonry and relevant material is far less consensual than concrete or steel[25].

In this paper experimental and numerical investigation on response of clay masonry subjected to bullet impact is presented.It addresses the damage model of this class of construction consisting of clay-fired brick and mortar in experiment.Through simulation the reliable finite element model and material model are developed and explored.

In the following,the experimental campaigns including penetration and material tests are elaborated and the results and discussion are reported in Section 2.The material model and finite element model are described in Section 3.Numerical results and assessment are presented in Section 4.

2.Experimental investigation

To study the weapon resistance of masonry and the capability of the bullet to penetrate clay masonry,the shooting range tests were conducted with 12.7 mm armor-piercing explosive incendiary bullet as penetrator and clay masonry employing two different bonding patterns as target.After range tests,the parametric tests of materials were carried out to measure mechanical properties of brick and mortar individually.

2.1.Test con figuration

The Type MU15 bricks were made according to GB5101-2017[28] possessing nominal dimensions of 240mm ×115mm×53mm.Four walls with two kinds of bonding patterns were built.The clay-fired bricks were connected by Type M5 mortar with thickness of 1~2 cm.With employing different bonding pattern,the masonry is called “24 wall” or “37 wall” ,which is named after their thickness of masonry respectively though the actual values vary slightly.Fig.1 shows the maximal units to construct a wall by connecting the top and bottom of same units,the dark thick lines representing the mortar.The unit of 37 wall,showed in Fig.1(b),consists of two layers of bricks,one of which can exchange the front row of bricks with the rear one to obtain the same pattern as the other.Each layer has one or two partial bricks to avoid the continuous mortar across the whole wall without any corners,which decreases the clay brick masonry’s capacity to bear loading.It should be stated that realistic constructions are not completely same as examples in Fig.1 but follow similar methodology.The experimental specimens are shown in Fig.2.The charge format of the bullet is 16 g.Different initial velocities of the bullets were achieved by altering the charge to 14 g and 12 g.

To ensure safety of shooter and collaborators,a rope was connected to trigger of machine gun.Fig.3 shows the setup by which the shooter hiding behind the shelter can press the trigger through the rope.After the bullets were fired with zero attack angle,the velocities of which were first measured by laser velocimeter which is 8 m from muzzle.Then the bullets arrived at the masonry and high-speed camera(resolution:1024×1008;frames rate:13,500 frames/s)recorded the process of the bullets penetrating the masonry.

Fig.1.Two different bonding patterns:(a)24 wall;(b)37wall.

Fig.2.24 wall and 37 wall(left:24 wall;right:37 wall).

Fig.3.Experimental delineation.

2.2.Test results and discussion

The hit and exit velocities were estimated from high-speed videos by reading key frames and considering the distance between bullets and high-speed camera,which were calibrated by the measured velocities from laser velocimeter.The laser velocimeter is not used to measure the velocities directly because the block of the diffusing smoke and flying debris will interference with the results considerably,as Fig.4(a)showed.Fig.5(a)presents the cartridges.The projectile consists of four components:steel core,lead sheath,incendiary agent and jacket.The materials of steel core,lead sheath and jacket are T12A stool steel,Pb and F11 copper clad steel respectively.As presented in Fig.10 right,the incendiary agent is located in front of steel core,however which is omitted for simpli fication.The steel core shown in Fig.5(b)is retrieved from the masonry specimen,geometric properties of which are given in Table 1.The experimental results are shown in Table 2.The masonry in test 2 and 3 were not perforated.The depth of penetration was not measured due to dif ficulty to break clay brick wall.

Table 1 Dimensions of steel core illustrated in Fig.5(b).

Table 2 Test results.

Table 3 Compressive strengths of brick and mortar at low and high strain rates.(Q represents quasi-static or H is split Hopkinson bar;B for brick or M for mortar).

Supposing that the mass of the bullet is a constant over the process,i.e.m=48 g,de fineEby Eq.(1)as energy loss characterizing the capacity of bullet to penetrate clay masonry(or the resistance of the masonry against impact of bullet).

whereEIdenotes the initial energy of the bullet,ERrepresents the residual energy after penetration.The energy herein is considered as energy of motion of bullet only.Using hit and exit velocities derived from tests yieldsEIandERrespectively.According to test 1 and 4,the energy loss of bullets in 37 wall and 24 wall areE37≈1.689×104J andE24≈1.077×104J respectively,which demonstrates 37 wall consumed more energy than 24 wall.The ratio of energy loss in 37 wall to that in 24 wall is 1.568 that was little greater to ratio of 37 wall’s thickness to 24 wall’s that is 37.5/24≈1.563,which reveals that the bonding patterns to some extent improve resistance of brick wall against impact.However,the improvement is insigni ficant.Therefore thickness of wall is a more relevant factor predicting the resistance of clay brick masonry than bonding patterns.Moreover,it is clear the energy loss over thickness is approximately constant and with assumption of constant mass of projectile,the energy loss is proportional to difference of square of two velocities,which leads to constant average deceleration with any given thickness of wall then constant average drag force in the process of penetration,which is consistent with the inference from qualitative analysis in Refs.[29].

Fig.4.(a)The diffusing smoke and flying debris;(b)tilt of masonry wall.

Fig.5.(a)The cartridges of projectiles;(b)the steel core retrieved after experiment.

2.3.Material test

After the range tests,the densities of the materials were measured first.It has reached a consensus that the building materials under dynamic loadings usually behave differently from quasi-static loadings.Most building materials show an increase of strength and stiffness with an increase of the strain rate.To determine dynamic increase of material properties the quasi-static and Split-Hopkinson-Pressure-Bar(SHPB)tests were employed to investigate the properties of materials under static and dynamic conditions.

A number of fragments of brick and mortar were collected from masonry by breaking the walls after the range tests.The properties of fragments of brick or mortar were assumed to be the same,because they should have similar properties though realistically,the properties of brick and mortar vary slightly from fragment to another fragment.Some of them were used to measure densities,were weighed and measured volume through Archimedes method.The densities of materials were calculated according to Eq.(2)[30].

wherewdis dry weight,wsrepresents the saturated weight,wAis the weight of the saturated specimen suspended in a container of the saturating liquid(water),ρwdenotes the density of water andρbis the bulk density of the material.The average bulk densities of brick and mortar are 1.38g/cm3and 1.8g/cm3,respectively.The experimental investigation shows that geometry of the specimen has in fluence on the compressive and the tensile strength[8].To circumvent resonances,inertial and end friction effects in the types of system at high rates,the dimensions of the specimens must be a compromise between(1)maximizing the size of the specimen to have a comprehensive characterization of the materials,(2)appropriateL/D(length versus diameter)ratio to reduce the friction effects at two ends(the ratio is recommended to be 1 or more than 1[23]),(3)minimizing the size of the specimen to reduce inertial effect and non-uniform stress and strain distribution.Some of fragments of brick and mortar were used to prepare the specimens for quasi-static and SHPB tests,which were split from the fragments of brick and mortar by electrodrill then sanded to reduce friction.Moreover,the Vaseline was smeared on end of specimens to reduce end friction(Fig.6(b)).The specimens for quasi-static tests were made into cubes with size of 10×10×10 mm and cylinders with 10 mm diameter,5 mm high for SHPB tests.Because of dif ficulty to split specimens from the fragments of brick and mortar the numbers of specimens were limited.The surfaces of specimens are not perfectly smooth due to lack of appropriate apparatuses,which introduces error to some extent.The results are indicative and as reference when determining the material parameters of simulation though.

Fig.6.The specimens of(a)quasi-static test;(b)SHPB test.

Quasi-static tests were conducted on three samples of brick and mortar using MTS machine,as shown in Fig.6(a).The displacement controlled upper plate was moving with a constant velocity of 0.8~0.9 mm/s.Displacement and load values were captured by digital controller unit.Displacement was measured by crosshead movement,which may introduce error because deformation of loading unit itself and engagement between platen and sample also contribute to platen displacement.SHPB tests were performed on three samples of brick and mortar as well.Strain values are in a state of changing rapidly in dynamic tests,which can be captured by the semiconductor strain gauges.The high dynamic strain indicator recording the data through transferring to computer was connected to the wheatstone bridges consisting of semiconductor strain gauges and the basic circuit con figuration.Utilizing Eq.(3)yields the transient strain.

whereεirepresents strain att=iΔt,Uis the voltage value measured from the wheatstone bridge,Edenotes voltage of bridge box,kis sensitivity coef ficient of semiconductor strain gauges,nis magni fication of high dynamic strain indicator andsis a value depends on bridge method,s=1/2 if the half-bridge has been adopted.According to one-dimensional wave propagation theory and two-wave method,integrating the re flected strain component εrover time yields the average strainεin a specimen.

The strain rate and stress are obtained by

whereεr,εtrepresent reflection and transmission strain respectively.C,E,Adenotes the elastic wave velocity,elastic modulus and cross-sectional area of compression bar respectively.A0,l0are cross-sectional area and length of specimen.Table 3 summarizes the static and dynamic test results.What should be stated is that the strain rate is not averaged but selected when the maximum strength is reached,which is reasonable because the strain rate varied instantly in experiment and the material properties are reflected by corresponding strain rate at the point of maximum strength.All strain and stress values were transferred to true ones by considering the cross-sectional changes,which is normally negligible.Nevertheless this transverse change presenting in dynamic tests leads to inertial force and is responsible for partial increment of DIF,which is characterized by Poisson’s ratio and like an effect of refusing to be flattened,is structural and inevitable and known as inertial effect.Therefore the ratio of specimen length to diameter should be close to 1 because inertial effect is size dependent[31].The enhancement of compressive strength is obvious for both brick and mortar in Table 3.

Bene fited from research on concrete,the dynamic increase of concrete-like material such as clay-fired brick,adobe and mortar has been accepted resulting from viscoelastic effect normally related to the hardened material by water and rate dependent crack evolution[24].Normal force will appear when two plates between which is a thin viscous layer are separated or approaching,known as Stefan effect.The force is proportional to the velocity of moving plates.In dynamic tests the specimens are loaded rapidly,which results in greater force between soil particles and re flects higher strength in macro scale.Researchers have found heated or frozen(attempt to exclude the effect of water)concrete is less rate sensitive and saturated one is double in tensile strength,which supports the theory[32].Another factor is evolution of crack.When subjected to slow loading the energy accumulated in specimens has enough time to release in a path defined by minimum energy,avoiding stiffer areas and connecting cracks that are pre-existent and weaker areas.Instead in dynamic scenario the velocities of particles are high and they will encounter coarse aggregates(stiffer areas)but they are too fast and it is crucial that they have the energy to break them,which is illustrated from the fact from concrete experiments that there are increasing amount of broken coarse aggregates along the facture surface with increasing tension load rate[32].The similar phenomena were reproduced numerically in Ref.[23,31].The high confining stress solicited by impact condition contributes to strain rate strengthening effect as well,which keeps the interfaces under overall loading so as to retard emerging of cracks and causes deformation to occur both in weaker areas and stronger aggregates.The conclusion is that the aggregates are broken in dynamic situation to withstand much more stress and dissipate the energy,which is not the case in static condition[33].

3.Material model

The simulations were conducted with LS-DYNA hydrocode.The class of hydrocode decomposes the dynamic problem into partial differential equations which are solved by time integration based on conservation of mass,momentum,energy coupled with state of equation and constitutive model[34].The state of equation determining the relation between density,pressure and energy while the constitutive model catering for behavior resulting from deviatoric mechanical loading,are speci fic for certain material.The crucial mechanical phenomena such as transition from elastic to plastic regime and the onset of failure for material are included in constitutive model.RHT material model is applied in this paper[13-16].

3.1.RHT model

RHT material model consists of equation of state and strength characterization.The validation of RHT model for adobe,clay bricks and lightweight adobe masonry was explored in Refs.[16-19,26].In the following the RHT material model is reviewed.

3.1.1.Strength characterization

The material model treats hydrostatic pressurepand deviatoric stress portions,which are both split from stress tensorσ,separately[35,36].Thus two groups of formula are developed:an EOS(see Section 3.1.2)to associate pressure with thermodynamical state variables density and internal energy;and a strength model to deal with the deviatoric stress tensor

Iis the second-order identity tensor.The first term in Eq.(7)is illustrated in this section and description of EOS will be in Section 3.1.2.

The strength model de fines three limit surfaces in principle stress space(i.e.the failure surface,elastic limit surface,and residual friction surface,shown in Fig.7)to describe the behavior of material,which are defined using pressurep,effective stressσeffand lode angleθ(stress values are always positive).

Fig.7.Three limit surfaces of RHT model[37].

For a given stress state and strain rate,the failure surface is given as

which consists offcand two functions.The first demonstrates the pressure dependence for principle stress conditionsσ1=σ2>σ3and is expressed in terms of the failure stress

the superscript asterisk means the values are normalized with the compressive strength,in whichFris dynamic increment factor and

Fig.8.Typical deviatoric plane of strength surfaces for low pressures[37].

The experimental observation demonstrated that the tensile meridian gets close to the compressive meridian when pressure are increasing,which is characterized by parameterQ

Qis approaching 1 with increasing pressure,which leads to a shape change from triangle in Fig.8 to circle and re flects the brittleto-ductile transition of material.TheQ1andQ2are given as

Frcharacterizes the dependence of strain rate.

with

Subscript or superscript c/t means compression or tension,hence RHT model respects same formula for strain rate dependence in compression and expansion but theβcandβtare derived slightly differently from

The formula allows steeper increment when strain rate exceeds.With=30s-1recommended by the CEB-FIP Model Code[38],γc/tis obtained in combination of continuity requirements

The initial elastic limit surface is expressed with failure surface,scaling functionFeand cap functionFc

Feis calculated interpolating between yield surface parametersg*candg*tdetermined by uniaxial material experiments.

The cap function is used to set the elastic limit to 0 when pressure goes beyond the pore crush pressure,through which deviatoric stress is consistent with inelastic volumetric stress built into EOS,as shown in Fig.7 and Eq.(25).

G*is plastic shear modulus and calculated with original shear modulus of material multiplied by reduction factorξcharacterizing the hardening behavior.

The residual surface is defined as

Once the failure surface is reached the damage initiates.With further inelastic loading damage is accumulated and re flected by plastic strain.The damage parameter is defined as

The resulting damage surface is interpolated between failure surface and residual surface fromD=0 toD=1(asP2toP3in Fig.7).

3.1.2.Equation of state

In the RHT model,the pressure is described by Mie-Gruneisen form through a polynomial Hugoniot curve andp-αcompaction relation.For compression(η>0)or expansion(η<0),it is given as

together with

in whichAi,Bi,Tiare material constants.α0denotes the initial porosity and equals toρmatrix/ρ0.To re flect a drop inus-up(shock over particle velocity)curve distinctive for porous material,compaction path is achieved by using variableαwhich is constructed as

in whichpelis the current pore crush pressure andpcompdenotes solid compaction pressure.Nis compaction exponent.These parameters are determined by dynamic inverse planar-plate impact tests recommended in Ref.[35].

3.2.Determination of material parameters for mortar and clay brick

In this section the material parameters determination of brick and mortar is illustrated respectively.The compressive strength is average of experimental data(the value that is too small has been excluded),which for brick and mortar are very close therefore they are both set as 10 MPa in parameter set for simulation.The parameters of material are shown in Table 4.The source of data is illustrated in Fig.9.

Brick.The density of brick is 1.8 g/cm3.The shear modulus is determined through Young’s modulus from Ref.[6]and Poisson’s ratioν=0.15.The compressive strain rate exponent is derived from ratio of dynamic compressive strength to static one.Only one relation is activated by remaining default break compressive strain rate as 3×1022s-1.Utilizing Eq.(19)yields the compressive strain rate exponent.

Experimentally determinedfcdandfcare 25.6 MPa and 10 MPa respectively.andare 991.07s-1and 0.009s-1respectively.Thus compressive strain rate exponentβc=0.081.The tensile strain rate exponent andare from Ref.[26].The yield surface parameterfailure surface parameters and residual surface parameters are adopted from Refs.[16]but with minor adjustment.Residual surface parameternfis smaller than that in Ref.[16],so as failure surface parametern,which are based on the parameter set for standard concrete(C30/37).The normalized shear,tensile strength and lode angle dependence factor are derived from Refs.[16]as well.Moreover,default value ofD1which is equivalent to 0.04,is not used and the value 0.015 in Ref.[39]is chosen.The in fluence of this parameter and also residual surface parameters is illustrated in Ref.[39,40].The parameters of equation of state are obtained from default material parameters for the standard concrete(C30/37)by scaling those figures with the ratio of respective porous densities(The standard concrete possesses the density of 2.31 g/cm3).The procedure of determining parameters of EOS is adopted from Ref.[19].In addition,initial compaction pressurepelslightly below uniaxial compressive strength is selected,as is for standard concrete.The solid compaction pressurepcompis calculated by retaining the ratio of solid compaction pressure and initial compaction pressure for standard concrete.The volume faction porosity of brick 24%is chosen according to Ref.[30],hence initial porosity of brick is 1.32.The compaction exponent remains unchanged.

Mortar.Some parameters of brick are benchmarks of those of mortar.The shear modulus of mortar is set approximately half of that of brick based on engineering assumption.For mortar,fcdandfcare 27.0 MPa and 10 MPa respectively.pandc0are 692.53 s-1and 0.005 s-1respectively.Hence,the compressive strain rate exponent βc=0.084 is determined.The initial porosity of mortar is retained from standard concrete material parameters because they are similar material.Moreover,other parameters of mortar are obtained by using similar methods as brick and based on the standard concrete parameters.Some minor adjustments are included considering the experiment data or different characterization between them though overall similarity is assumed of their properties.

Fig.9.Acquisition of material parameters.

3.3.Material model and parameters for projectile

12.7 mm armor-piercing explosive incendiary projectile consists of steel core,lead sheath,incendiary agent and jacket.The in fluence of incendiary agent on penetration is very small,which is validated through experiments[41].Thus it is omitted in the simulation.The numerical model is shown in Fig.10.The material of steel core,lead sheath and jacket are T12A stool steel,Pb and F11 copper clad steel,all of which are simulated using simpli fied Johnson-Cook material model.Johnson-Cook material model is capable of capturing behavior of metal subjected to large deformation,high strain rate and high temperature.The thermal effects and damage are ignored in the simpli fied Johnson-Cook,hence the maximum stress is limited to compensate thermal softening.The damage here is not of paramount importance on account of completeness of bullets after penetration(Fig.5(b)).The model is suitable for situation where the strain rates change over a large range.The constitutive formula is given as

Fig.10.The geometric model and mesh for 24 wall and projectile.(a):The material location is illustrated by respective color.Area I of 24 wall has finest mesh,area II takes the second place and area III has the coarsest;(b):from outside to inside are F11,Pb and T12A.

whereσyis the flow stress;Ais the yield stress under reference strain rate;Bandnare parameters characterizing strain hardening of material;cis the material constant re flecting the strain rate dependence of the material;denotes the effective plastic strain and is obtained by removing the elastic strain from the total strain;is the strain rate;andis the normalized effective strain rate calculated from,whereε0=1/s.The material parameters of projectile are adopted from Refs.[41],shown in Table 5.

Table 5 Simpli fied Johnson-Cook constitutive material parameters used in simulation for bullet components([41];http://www.varmintal.com/aengr.htm).The parameters are arranged in LS-DYNA format.

Table 6 The exit velocities and damage using three different mesh resolutions.

4.Simulation

The simulations are performed using hydrocode LS-DYNA.To keep minimum boundary inconsistency with reality,full 3D detailed finite element model consisting of two different material is established.The base of masonry wall is a block of wood and also included in the finite element model.As shown in Fig.10(a),the masonry wall is divided into three areas I,II and III.Area I and II share common nodes but area II is discretized with coarser mesh,which is as a transition from Area I to Area III.The interaction between Area II and Area III is re flected by automatic tiebreak contact algorithm,which is chosen because of high compatibility of different mesh resolution on interface being contacted.The existence of Area II contributes to avoiding much too sharp mesh inconsistency between Area I and Area III and itself is helpful to reduce computing time with gradually larger element size.Within three Areas sharing common nodes is employed to represent interaction of brick and mortar.The three components of 12.7 mm armor-piercing explosive incendiary projectile are constructed individually sharing common nodes.The gravity is loaded in global model.

4.1.Contact algorithm applied in simulation

Choosing appropriate contact algorithm is crucial to obtain more accurate and realistic results for simulation.The masonry consists of brick and mortar,interaction between which is very complicated.Sharing common nodes and utilizing contact algorithm are combined to solve the problem.The contact between Area II and Area III are simulated with automatic TIEBREAK surfaceto-surface contact algorithm,which allows the transmission of both tensile and compressive forces resulting in a TIE.The separation of the slave node from the master is resisted by a linear contact spring for both tensile and compressive forces until failure after which the tensile coupling is removed.Post failure in all TIEBREAK contacts allows the node to interact with the segment as in traditional compression only contacts.The debonding of contact is assumed to be governed by failure criterion[36].

whereσnandσsare normal and shear stress on the interface,respectively;NFLSandSFLSare normal and shear strength of the interface,which are calculated from minimum of compressive and shear strength of brick and mortar due to weaker strength of interface(see Section 4.3.2).The eroding surface-to-surface contact algorithm is selected to simulate the interaction between penetrator and target,namely three components of bullet and masonry wall consisting of brick and mortar,which is crucial to de fine erosion of elements reaching material failure and allows the remaining interior elements to continue to contact after the failure of outer elements during impact process.The friction coef ficient of components of bullets with brick and mortar is considered as 0.15[16],and value 0.3 is used between Area II and Area III.

Fig.11.Damage con figuration of 24wall(test 4):(a)front view;(b)rear view.

4.2.Investigation on mesh resolution with 24 wall(test 4)

The hydrocode simulation in this section is performed with parameter set,finite element model and contact algorithm presented above.The RHT model shows mesh resolution dependency,which is illustrated in Ref.[13,16].Considering relative simpler structure and lower demand for computer power,three different mesh resolutions are investigated using 24 wall compared to test 4 first.The steel core of bullet and Area I which are directly subjected to impact are discretized with the finest mesh varying with different mesh resolutions.The Area III remains the element edge size same as thickness of mortar with the lead sheath and jacket keeping element edge size as 0.1 cm on account of their small thickness.It should be stated that the bullet in simulation impacts the target with estimated pitch 2.0375°though the bullets were fired with zero attack angle,which is based on observation from high-speed camera that the masonry wall in test 4 is placed aslant(Fig.4(b)).Fig.11 shows the damage picture of 24 wall in front view and rear view.The results of simulation are listed in Table 6 and Fig.12.For the sake of convenience in writing,simulations using minimum element edge size 2,1.8 and 1.5 mm are denoted by S2,S1.8 and S1.5.

Table 7 Results of simulation of 37 wall(test 1 and 2).

Fig.12.The upward deviation of trajectory.

Fig.13.Damage con figuration of 37wall(test 1):(a)front view;(b)rear view.

Fig.14.Damage con figuration of 37wall(test 2):(a)front view;(b)rear view.

The exit velocities of S2,S1.8 and S1.5 are 146,161and 158 m/s respectively compared to 139.1 m/s in experiment.The RHT material model displays softer characteristic when the mesh is finer[13].Therefore greater exit velocity and less energy loss of bullet are predictable if higher mesh resolution is adopted.Nonetheless,it is not the case because the exit velocity experiences a drop from S1.8 to S1.5.This phenomenon will be explained later.The damage con figuration is analyzed first.The front and rear views with damage scalar are shown in Table 6.Comparing simulation results to Fig.11 shows a similar failure mode to experiment in case of front view of masonry wall.The scope of crater is comparable.In addition to the overall similarity in simulation using different mesh resolutions,with increase of mesh resolution the delicate cracks are appearing.The depth of crater is not measured in experiment thus it is not examined.As for rear view,an obvious deviation is observed in Fig.11.The hit position at Fig.11(a)is at mortar but the exit position of bullet occurs at middle of the upper brick,which means that the bullet experienced an upward displacement.The front and rear views are only a reflection of initial and final states of bullet,however,the detailed movement during the process is unknown in experiment.To investigate the trajectory,side view of masonry wall is obtained by removing corresponding part.Fig.12 presents the initial and final states of bullet in simulations,these snapshots are not in an identical plane due to lateral de flection of bullet.The trajectories show a small increase of slope when mesh resolutions becomes higher,as S2 to S1.8 then S1.8 to S1.5.In RHT material model,tensile failure will occur when the tensile stress falls under a speci fic value,meanwhile minimum strain to failure is predefined in tension,which demonstrates failure stress and strain are independent from element size therefore the fracture energy is mesh dependent[37].The phenomenon can be illustrated by

whereet,fminandσtis tensile failure strain and stress respectively.Leqdenotes characteristic length of element andGFis fracture energy.For given failure stress and strain the fracture energy of bigger element will be higher than that of smaller one because characteristic length is highly dependent on size of element,which can partially explain why RHT model shows mesh dependency.As shown in Fig.12,with pitch of 2.0375°and finer mesh resolution the exit position is getting closer to the middle of the upper brick,which means more de flection of the bullet.More de flection of bullet make it lose more energy when travelling through the masonry wall because of more contact area with target.This effect coupled with less energy loss of bullet introduced by mesh resolution results in a drop in S1.5.The dominant factor is less energy loss of bullet introduced by mesh resolution from S2 to S1.8,hence the exit velocity increases a bit.However from S1.8 to S1.5,more de flection introduced by mesh resolution takes the power,hence the velocity drops a little.There are more tiny visible debris behind the masonry wall when mesh resolution is higher.Though it is far behind the phenomenon shown in Fig.4(a)it makes simulations more realistic.The higher mesh resolution is deemed necessary.Nonetheless,only three cases of mesh resolution are included because full 3D finite element model is very expensive and on account of computer power limitation.The finest mesh resolution herein is kept in following simulations.This phenomenon that the numerical results(deviation of bullet)are more similar to what was observed experimentally seem to be obtained by applying higher resolution should be taken care of,what is demonstrated from which is the in fluence of mesh resolution on RHT model.The phenomenon does not necessarily mean that the numerical simulation is more accurate because there are many other factors,two of which are small yaw angles of projectiles and slight tilt of brick wall undetected in experiment thus not being included in simulation but having quite an in fluence on trajectory of bullet.Besides,the deviation of bullet is clear to be affected by some random factors,such as irregular voids in masonry walls,variation in material properties and uneven size distribution of mortar because brick and mortar are arranged by hand.The size of brick and mortar varies in reality,which is hard to synchronize with in simulation.The precise hit position is not easy to locate due to acute damage of masonry wall.Therefore the deviation of bullet in reality is elusive to some extent and of many possibility especially for clay brick masonry wall that is not well standardized and consists of site dependent material[24].Considering these facts,the agreement of simulation to experiment is remarkable.

Fig.15.The trajectory of bullet.

4.3.Simulation of 37 wall

The finite element model with minimum element edge size 1.5 mm presented above and parameter set in Table 4 are used to simulate experiments of 37 wall.The charge of test 1 and test 2 are 16 g and their initial velocities are close(Table 2).Their hit positions are at mortar layer located at center of the front of wall(Fig.13(a)and Fig.14(a)).Nonetheless,their results are different.In contrast with perforation of test 1,the masonry wall in test 2 is not perforated.The comparison of Figs.13(a)and Fig.14(a)reveals that the damage con figuration of front of masonry walls is distinctive,which can partially attribute to variation of material properties in different masonry walls.It appears that the masonry wall in test 1 is more brittle and of weaker strength thus the impact energy can concentrate on penetration ef ficiently by pulverizing the brick and mortar in front of bullet and cohesion of the area being impacted and its periphery can be broken rapidly,which result in smaller crater diameter and perforation.Conversely,the material properties in test 2 seem stronger though they are from same lot,however,small quantitative can cause a qualitative change.The comparison of simulation with experiment includes damage con figuration and trajectory of bullet.

4.3.1.Test 1 and 2

Only one simulation is carried out because of the high similarity of test 1 and test 2.The mesh of wall is graded similarly as shown in Fig.10.The initial velocity in simulation is 842.4 m/s.As shown in Table 7,the size of crater of front view of masonry wall is comparable to that in Fig.14(a)and a hairline crack is observed in the top right-hand corner,which is observed in Fig.14(a)as well.

Fig.16.Damage con figuration of 37wall(test 3):(a)front view;(b)rear view.

But the tiny cracks in bottom right-hand corner in simulation are not observed in experiment.Concerning trajectory plot in Table 7 bullet is not shown because of lateral de flection of bullet,which makes it dif ficult to show more complete trajectory and meanwhile bullet.The rear view of experiments in Figs.13(b)and Fig.14(b)both demonstrate pulverization of big block of brick though the position is not exactly same.The hit positions of test 1 and test 2 are very similar at mortar but the exit position in test 1 is upper brick and that of test 2 is nether brick,which seems to be symmetrical.The hairline cracks in Figs.13(b)and Fig.14(b)are more or less captured by simulation.The pulverization of brick is re flected by outline of damage in rear view in Table 7 but underestimates the experimental results.The shape of trajectory is like S,which is close to that in Fig.15,along which the energy of bullet will be consumed more quickly than a straight one.

Table 8 Results of simulation of 37 wall(test 3).

4.3.2.Test 3

In test 3,the hit velocity in experiment is 768.5 m/s.The damage con figuration in experiment is presented in Fig.16.The hit position is interface of brick and mortar,which can be observed from Fig.16(a).The crater diameter becomes relatively smaller and the depth of crater is deeper compared to test 2 and 4.The damage model of rear view,i.e.pulverization of big block of brick,is validated once more,which attributes to the stress state of brick wall during impact.The rear of masonry wall is subjected to tensile stress before bullet arrives at the rear of masonry wall because of spreading of shock wave.The failure emerges on the interface of brick and mortar because the spreading velocities of stress wave in two material are not same,which causes complex stress state on interface and the strength of interface is much weaker(we can know the fact from Figs.11,Fig.13,Figs.14 and 16 that all the damage con figurations are with hairline cracks on the interface of brick and mortar).Then the brick is impacted by concentrated power from relatively small bullet.Combined with brittle property of brick the phenomenon occurs.The results of simulation are shown in Table 8.The simulation predicts approximate damage con figuration of front view to experiment.Regarding rear view,the pulverization of brick is not captured exactly.It seems that the simulation overestimates the strength of the material.However both in front and rear views,the hairline cracks in simulation,which are similar to experiment,are outlined along the edge of brick or mortar similar in experiment.Fig.16 demonstrates that bullet has an upward de flection,which is not captured by simulation.The trajectory of simulation shows downward de flection in Table 8.Still considering the reason stating in end of Section 4.2,the numerical results are acceptable.

5.Conclusions

The response of construction consisting of clay-fired brick and mortar subjected to quasi-static or dynamic loading is important to ensure safety in some historic sites,public buildings and so on.To this end,the penetration experiments were carried out using four clay brick masonry walls with two different patterns as target and 12.7 mm armor-piercing explosive incendiary bullet as penetrator.The experimental investigation shows that the average drag force in the process of penetration is constant,which is basic assumption of some theoretical ballistic formula.Then static and dynamic material tests were conducted on specimens of brick and mortar,which demonstrates the clay-fired brick and mortar show an increase of compressive strength at high strain rate.

The simulations are performed using LS-DYNA hydrocode.The RHT model incorporating strength model in which pressure,lode angle dependency of material and damage are described in combination with porous equation of state is adopted.With assistance of some references and experimental results the parameter set is obtained.To keep minimum boundary inconsistency with reality,full 3D detailed finite element model consisting of two different material is established.To deal with the problem of computational time usage of large-scale model,sharing common nodes and employing automatic tiebreak contact are combined.Using developed parameter set and finite element model,the in fluence of mesh resolution on RHT model is investigated with test 4,which reveals that deviation of bullet in simulation is affected by less energy loss and more reflection of bullet introduced by mesh resolution.The experimental observation demonstrates:(1)much weaker strength of the interface of brick and mortar;(2)damage model of crater approximating to circle in front of clay brick masonry;(3)damage model of pulverization of big block of brick in rear.The numerical results reproduce the experimental phenomena with respect to exit velocity,damage con figuration and trajectory acceptably.The way of constructing the finite element model and the material model are validated suitable for this class of construction.We perceive our work contributes to the understanding of clay-fired brick wall that is a complicated structure and relevant exploration in experiment and simulation is of value for future investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgement

The work presented in this paper is funded by Opening Project of Science and Technology on Transient Impact Laboratory(Grant No.614260601010517).This support and the good cooperation are gratefully acknowledged.