Blast response of continuous-density graded cellular material based on the 3D Voronoi model

Xu-ke Lan,Shun-shan Feng,Qi Huang,Tong Zhou

State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology,Beijing,100081,China

Keywords:Gradient Blast response Cellular material 3D Voronoi model

ABSTRACT One-dimensional blast response of continuous-density graded cellular rods was investigated theoretically and numerically.Analytical model based on the rigid-plastic hardening(R-PH)model was used to predict the blast response of density-graded cellular rods.Finite element(FE)analysis was performed using a new model based on the 3D Voronoi technique.The FE results have a good agreement with the analytical predictions.The blast response and energy absorption of cellular rods with the same mass but different density distributions were examined under different blast loading.As a blast resistance structure,cellular materials with high energy absorption and low impulse transmit is attractive.However,high energy absorption and low impulse transmit cannot be achieved at the same time by changing the density distribution.The energy absorption capacity increases with the initial blast pressure and characteristic time of the exponentially decaying blast loading.By contract,when the blast loading exceeds the resistance capacity of cellular material,the transmitted stress will be enhanced which is detrimental to the structure being protected.

1.Introduction

Cellular materials,such as honeycomb,foam,corrugated plate and metal hollow sphere,have been widely used in aerospace and defense industries as an energy absorption device to mitigate shock and impact by progressive local crushing of its micro-structure under dynamic loading[1].Cellular materials are attached to the protect structure as sacrificial layers for blast and impact,the cladding is expected to attenuate the load on the structure behind it[2,3].

Over the latest decade,the dynamic responses of cellular materials with uniform density have been well studied experimentally[4-12].Based on these tests,two main phenomena were observed as follows:(a)when the impact velocity exceeds a critical value,the cellular material will be separated into crushed and uncrushed regions obviously by a propagating discontinuity;(b)the stress of proximal end is higher than the quasi-static plateau stress due to the dynamic enhancement.

To analysis these phenomena,a one-dimensional shock-wave model with rigid-perfectly-plastic-locking(R-PP-L)constitutive relation was firstly developed by Reid and Peng based on the dynamic response of wood[4].Tan et al.extend this shock-wave model to metal foam and honeycomb[5].To depict the deformation behind the shock front at different velocities more accurately,various constitutive relations were developed considering the plastic hardening for different cellular materials.An elastic perfectly plastic-rigid(E-PP-R)idealization and an elastic-plastic rigid(E-P-R) idealization were employed by Lopatinikov et al.to consider the effect of elastic[13-15].Meanwhile Zheng et al.proposed a linearly hardening plastic-locking model(R-LHP-L)as a supplement for the R-PP-L model to capture the transition mode model[16].Moreover,a dynamic rigid-plastic hardening(R-PH)idealization was proposed by Zheng et al.[17].This stress-strain idealization can well characterize the dynamic compression behavior of cellular materials under high velocity impact[18].

Blast responses of cellular materials with uniform density were well investigated,and provide a useful guide for designing protective structures against blast loading[3,19].According to previous study,layered sacrificial cladding were observed as highly effective for energy absorption,with predictable behavior under blast loading.However experimental evidence has highlighted the fact that the presence of this “protective”layer can result in an enhanced loading of the structure,the influence of using cellular material as a protective layer remains debated[20-22].To explain the increase in energy and impulse transferred to pendulum when foam panel exist,Hansen proposed an analytical solution based on shock-wave theory[21].A systematic study was carried out,but the propagation mechanism of shock wave was beyond the scope of their study.Ma and Ye investigated the deformation of foam subjected to blast loading by proposing an analytical Load-Cladding-Structure(LCS)model[23,24].This study focused on the deformation behavior of the protected structure only,the crushing process of the foam was not considered.Aleyassin et al.focused on the attenuation/enhancement boundary based on the R-P-P-L model,a new method of accounting for fluid-structure interaction is derived[25].However,the using of R-P-P-L model may overestimate the energy absorption ability of cellular material.Karagiozova et al.developed an analytical model to reveal the characteristic features of foam compaction[26].The model is accurate in predicting the front of the propagation shock but failed to capture the impulse transmitted to the protected structure.Although the blast response of cellular materials has been studied experimentally,annalistically and numerically,the mechanism of cellular materials subject to blast loading needs further investigation.

In attempt to find an optimal design,investigations on the blast response of graded cellular materials have been done in recent years.Liu studied the blast resistance of sandwich-walled hollow cylinder with graded aluminum foam core[27].It was found that the introduce of graded foam core can increase the energy absorption efficiency.MA and Ye derived the energy absorption capacity of double-layer foam cladding under blast loading based the R-P-P-L model[28].However,the process of compaction wave propagation was lake of investigation due to the complex behavior of shock-wave propagation caused by different gradient distribution.With the development of manufacture,cellular material with continuous-density variation can be made[29].Although some analytical and numerical studies have been carried about the impact response of cellular material with continuous-density variation[30-34],seldom results about blast response can be found.And the numerical models used in these studies were limited to 2D Voronoi structures,they were not accurate enough compared with the 3D Voronoi structure.

The present study focused on the blast response and energy absorption of cellular materials with continuous-density gradient.To clarify the effect of density distribution an analytical model was proposed.Meanwhile,a numerical model with continuous-density gradient is constructed based on the 3D Voronoi technology.Blast responses of cellular rods with continuous-density gradient were investigated.And the energy absorption capacity was examined.

2.Analytical analysis

2.1.Problem formulation

Consider a density-graded cellular rod sandwiched by two rigid plates,as shown in Fig.1 Modeling of the cellular rod under blast loading.The front plate at the proximal end is regarded as a rigid mass,and the plate at the distal end is regarded as a rigid wall.The compressive blast loading p(t)acts directly on the front plate.According to Fleck et al.[35].when the distance between the explosive source and the sacrificial layer is much larger than the size of the protected structure,the spherical wave generated by the blast loading can be approximated as a plane wave which decays exponentially as follow:

Where p0is the initial peak pressure of blast loading,andτis the characteristic time.The impulse of the blast loading is p0τ.

The liner density gradient of cellular rod is defined as g= ρdistal- ρproximal,where ρdistal and ρproximal are the relative densities of distal end and proximal end.

2.2.Cellular material model

The rate-independent R-PP-L model has been widely used to characterize the nominal stress-strain relation of the cellular material,as shown in Fig.2.This model is simplyenough with only two parameters,and it can characterize the shock wave in cellular material well.However,the R-PP-L model failed to consider the plastic hardening,which exists in the real stress-strain curves of cellular materials.Cai predicted the blast response of 2D graded Voronoi structure by using a rigid-plastic hardening(R-PH)model,which can be extended to 3D Voronoi structure[36].In our study,this R-PH model is adopted to model the cellular material,written as:

whereσ0is the initial crushing stress and C is an empirical fitting parameter,which characterizes the strain hardening behavior.The two material parameters are related to the relative densityρof cellular material.Quasi-static compression tests were performed in section 3 to obtain the two parameters.

2.3.Analytical model for blast response of graded cellular rods

The analytical model is mainly used to calibrate the simulation model.Although there may occur two shock waves,for the sake of simplicity,only the case of a single shock wave is considered.

As shown in Fig.3,the compressive shock front travels along the cellular rod with a Lagrange coordinate: φ（t）.According to R-PH model,the speed of elastic wave is in finite while the shock front travels at a finite speed given by˙φ（t）.It means that the shock front never catches the elastic wave.When the shock front travels across one position in the cellular rod,the particle speed,strain and stress of this position will jump from ｛0，0，σ0（ρ（φ））｝ to ｛v（t），εb（t），σb（t）｝ ,where σ0（ρ（φ））is the initial crushing stress.According to the continuum-based stress wave theory,the conservation relations of mass and momentum across the shock front are given by

Whereρsis the density of base material.Combining Eq.(3)and Eq.(4)gives the stress behind the shock front:

Combining Eq.(2)and Eq.(5)gives the strain behind the shock front:

Mass conservation law gives the mass of crushed part as:

Assumption that the crushed part moves with the front-plate at the same speed.The momentum conservation law for the front plate and compacted part gives that:

Combining Eqs.(3),(5),(6)and(8)gives the governing equations:

With the initial conditions,φ（0）=0 and v（0）=0,the governing equations can be solved numerically by using the Runge-Kutta method.

3.Cell-based finite element modeling

3.1.3D graded Voronoi structure

The 2D random Voronoi structure has been widely used to simulate cellular materials with uniform and graded density.The 3D random Voronoi structure can better simulate the microstructure of cellular materials[18,37].However,only few studies using cellular structures with continues-density gradient,generated by using 3D random Voronoi technology can be found in the literature[38].ar.In order to generate 3D Voronoi structure with continues density gradient,the cellular rod(20×20×80 mm3)was cut into 8 sections along the long axis,and each section has a volume of 20×20×10 mm3.Referring to Zheng et al.[18],the density of one section can be changed by verifing the cell-wall thickness or the cell size,in this study the later method is adopted which can better simulate the microstructure of cellular materials.

Cellular model with N nuclei and uniform cell-wall thickness is constructed in the volume of Vfoam(20×20×10mm3).The relative densityis related to the cell-wall thickness h by

Where ρ0is the density of the foam,ρsis the density of the cell-wall material,Ajis the area of j-th cell-wall surface.In geometry,when the cell-wall thickness h is given(0.05mm in this study),ΣAjcan be controlled by N.Therefore,the relative densityis a function of nuclei number N.Date fitting from N=50 to N=600 gives the relation ofto N,as shown in Fig.4.Based on the liner density distribution,the nuclei of each section can be obtained.Combing all the nuclei of these sections can give a distribution of nuclei in the cellular rod.Then cellular rod with liner density gradient can be constructed,as shown in Fig.5.

3.2.Numerical model

Numerical simulations are performed by using ABAQUS/Explicit software.The cellular rod is sandwiched by two rigid plates.The pressure of blast is load on the rigid plate at the proximal end,while the rigid plate at the distal end is fixed.The cell-wall material is assumed to be elastic-perfectly plastic with density ρs=2700kg/m3,Young's modulus Es=69GPa,Poisson ratio νs=0.3 and yield stress. σys=170MPa.The relative density of cellular material is set as 0.04.The cell-wall is meshed by using ABAQUS shell elements S3R and S4R,as shown in Fig.6,Through a mesh sensitivity study[18],the characteristic size of shell elements is set to be about 0.3mm.To save computational time,shell elements of type S3R with sharp angle are eliminated.General contact is applied to all possible contacts during crushing with a friction coefficient of 0.02.

A numerical compression test at low constant velocity is also performed to obtain a quasi-static nominal stress-strain relation.The quasi-static cellular specimens with different densities(by varying the cellsize) are constructed in a volume of 20×20×30 mm3.The specimens are sandwiched between two rigid walls.One wall is fixed and the other wall travels at a constant velocity of 1 m/s(the corresponding nominal strain rate is 33.33/s)compressing the cellular specimen.The quasi-static nominal stress strain relations of cellular material models is fitted by using the RHP idealization,referring to Zheng[18]for detail.Repeating the compression tests on samples with different density gives the initial crushing stress and the hardening parameter in terms of relative density as follows:

4.Comparison between theoretical and numerical results

FE simulations for cellular materials subject to blast loading with an initial peak pressure 16.9MPa and characteristic time τ=0.05ms are carried out and the results are presented in comparison with the results from the theoretical models.

Fig.7 shows the velocity history of the front-plate under blast loading.The velocity history can be obviously divided into three main stages.In stageⅠ,the high blast pressure acts on the front plate and drives the front-plate to the maximum velocity in a very short time.In stageⅡ,the high velocity front-plate compacts the cellular rod and the velocity decreases gradually till to the crush end.In stageⅢ,the front-plate and the fully crushed cellular rod impact on the protected structure which leads to a sharp velocity decrease.The inertia stress of cellular material rebounds the frontplate to achieve a negative velocity.The velocity predicted by theoretical analysis is compared with the FE results,and good agreement is achieved.

The reaction stresses on the blast end and fixed end are compared as depicted in Fig.8.There exists a perturbation in FE results,because the finite element model reflects the mesostructure of cellular materials which is different from the theoretical hypotheses.According to the Assumption of the R-HP model,the speed of elastic wave is in finite,so the stress of fixed end does not start from zero but starts from initial crushing stress σ0（ρ（φ））,as mentioned in section 2.3.However,the FE results agree well with the theoretical analysis in the mean value.

5.Results and discussion

5.1.Deformation mode

The macroscopic deformation of cellular material is complicated under blast loading,andthe related studies are limited,but it can be classified into three modes.In modeⅠ,the deformation initiates at the weakest cells and localized in crushing bands,which are randomly distributed at a low impact velocity.In modeⅡ,the crushing bands concentrate near the impact end because the inertia effect becomes crucial at a moderate impact velocity.In modeⅢ,the crushing bands highly localized at the impact end and progressive cell crushing is observed to propagate like a shock wave at a high impact velocity.Zheng cataloged the deformation into the Quasi-static mode,Transitional mode and Dynamic model.Fig.9 demonstrates the deformation process of cellular bar at an initial blast peak pressure of 16.9MPa.As discussed in section 4,the motion of front-plate can be divided into three stages.In stageⅠ,the blast load drives the front-plate to the maximum speed in a very short time.According to Zhenget al.[16],the critical velocity for the occurrence of the impact-induced shock wave in cellular can be given as

ModelⅢoccurs when the impact exceeds the critical velocity.Therefore,the cellular material always appears in modelⅢin stageⅡ.As the impact velocity decreases,the deformation model changes to modelⅡ and finally changes to modelⅠ.

As shown in Fig.10,difference deformation patterns are illustrated with initial blast peak pressures of 8.45 and 16.9 MPa.When the initial last peak is 8.45MPa the cellular bar undergoes three deformation models and the deformation distributes randomly in the cellular bar.When the initial blast peak is 16.9 MPa,the crushing highly localized at the blast end and propagates to the fixed end like a shock wave.Only ModelⅢis observed,because the velocity of the front-plate exceeds the critical velocity though the deformation process.

5.2.Gradient effects

The deformation patterns of cellular bars with different density gradients are presented here(shown in Fig.11)to give an insight into the effects of gradient distribution on deformation mechanisms.Because the velocity of front plate increases rapidly in a very short period,the deformation process is similar to the phenomena in which cellular material is impacted by a rigid mass as presented in Ref.[39].Cellular rods with negative and the uniform density distribution have similar deformation mechanisms,the blast end crushes first,and the fixed end compacts slightly later.When the gradient is positive,the compaction initiates from the blast end and propagates to the fixed end.

When evaluating the protective capability,the energy absorption capability and the impulse transferred to the fixed end are two important parameters.Cellular structure with high energy absorption ability and low impulse transmission is a good choice to mitigate shock and impact.In comparison with the plastic deformation energy of cellular material,the elastic deformation energy is negligible.Therefore,the energy absorbed by the cellular rod can be obtained through the plastic deformation energy in the finite element model.Cellular bars with different density gradient are fully compacted under the blast loading with initial blast peak of 16.9 MPa and the characteristic timeτ=0.1ms,Fig.12 depicts the energy absorption capacity.It should be noted that cellular bar is fully crushed when the shock front reaches the interface of cellular bar and the fixed end,and the fully crushed cellar bar along with the front plate will impact the fixed end as a rigid body.The present study focuses on the crushing process before the fully crush is reached.

The impulse I transmitted to the fixed end can be obtained by

where ffixedis the force transmitted,which is the reaction force at the fixed end in the numerical results.Fig.13 shows the effects of gradient distribution on the transmitted impulse.

The following conclusions can be drawn in combination with Figs.12 and 13:the cellular rod with positive density gradient shows the highest energy absorption and impulse transferred,whereas the cellular rod with negative density gradient display the lowest energy absorption and impulse transferred.When the cellular materials are designed to mitigate shock and blast,the one with high energy absorption and low transferred impulse is the best choice.However,by improving density gradient variation,high energy absorption and low transferred impulse cannot be achieved at the same time,which are two conflicting objectives for blast resistance capacity of cellular materials.

5.3.Blast loading

The effects of blast loading on cellular materials are investigated.Two groups of blast loadings with different initial peak pressure p0and characteristic timeτare designed here,each group has the same impulse (group1: p0τ=0.845 MPa?ms , group2:p0τ =1.69 MPa?ms).Fig.14 demonstrates the energy absorption history of cellular rod under different blast loading.The figure illustrates that,in each group higher initial peak pressure results in higher energy absorption rate and energy absorption capacity.According to R-PH model the energy absorbed per unit volume at the shock front can be obtained by

The difference in velocity between the crushed region and uncrushed region plays a dominant role in energy absorption.

Fig.15 shows the relationship between displacement and velocity of front-plate under different blast loadings.In each group,the maximum velocity of front-plate increases with increasing blast loading while the crushed displacements share the same value.The total energy absorption increases with the crushed distance and the dynamic enhancement which is velocity dependent.Therefore,the increase of initial peak pressure and characteristic time leads to an increase in energy absorption capacity.It should be note that,when the blast energy exceeds the energy absorption capacity of cellular material,full crush will be reached.The front-plate and the crushed parts will impact the fixed end directly,which leads to a rapidly velocity decease of front-plate(as shown in Fig.15 group2)and the transmitted pressure becomes much larger than the plateau stress of the cellular material as shown in Fig.16.Li and Meng also introduced this phenomenon and indicated that intensive loading may leads to the stress enhancement in cellular material[22].Therefore,the energy absorption capacity increases with the blast loading,but when the full crush is reached the transmitted pressure will be enhanced.

6.Conclusions

The blast responses of density-graded cellular materials are investigated theoretically and numerically.The theoretical model is developed based on the R-PH model.FE models with continues density gradient are constructed through a novel method based on the 3DVoronoi technique.Numerical simulations are carried out by using the ABAQUS/Explicit software and the numerical results are verified by the theoretical predictions.The cellular rod appears different deformation models as a result of different blast loading.When the initial blast peak pressure is high enough,the shock model propagates throughout the compression process and the cellular material is fully crushed.A partial crush will happen when the blast peak is not sufficient high.The blast response and energy absorption capacity of cellular rods with density gradient are investigated to clarify the effects of the density gradient distribution.The deformation first begins at the blastend and propagates to the fixed end,then the weakest part crushes subsequently.The deformation propagates through the weakest part and finally reaches the fixed end.When cellular material is used as a blast protective device,the ability to absorb energy while controlling the loading transmitted to the protected structure makes it attractive.However,a positive density gradient achieves the highest energy absorption and transmits a relatively high impulse to the protected structure,while a negative one shows a relatively low impulse transmission but attains the lowest energy absorption.Therefore,the introducing of density distribution cannot solve the contradiction between energy absorption and impulse transmitted to the protected structure.The effect of blast loadings with different initial peak pressure and characteristic time is also examined.The energy absorption capacity increase with the initial peak pressure and the characteristic time because of the dynamic enhancement and the increasing of crushed distance.When the blast loading exceeds the resistance capacity of cellular material,a fully crush will happen,and the transmitted stress will be enhanced which is a negative factor in engineering applications.