Wei Li ,Peng Wng ,Go-peng Feng ,Yong-gng Lu ,Jun-zheng Yue ,Hui-min Li ,

a Institute of System Engineering,CAEP,Mianyang 621900,Sichuan,China

b Institute of Mechanics,Chinese Academy of Science,Beijing 100190,China

Keywords: Pre-formed holes Cylindrical shell and square plate Deformation mechanism Failure mechanism

ABSTRACT The deformation and failure mechanism of cylindrical shells and square plate with pre-formed holes under blast loading were investigated numerically by employing the Ansys 17.0 and Ls-Dyna 971.To calibrate the numerical model,the experiments of square plates with pre-formed circle holes were modeled and the numerical results have a good agreement with the experiment data.The calibrated numerical model was used to study the deformation and failure mechanism of cylindrical shells with pre-formed circle holes subjected to blast loading.The structure response and stress field changing process has been divided into four speci fic stages and the deformation mechanism has been discussed systematically.The local and global deformation curves,degree of damage,change of stress status and failure modes of cylindrical shell and square plate with pre-formed circular holes are obtained,compared and analyzed,it can be concluded as:(1)The transition of tensile stress fields is due to the geometrical characteristic of pre-formed holes and cylindrical shell with arch con figuration;(2)The existence of preformed holes not only lead to the increasing of stress concentration around the holes,but also release the stress concentration during whole response process;(3)There are three and two kinds of failure modes for square plate and cylindrical shell with pre-formed holes,respectively.and the standoff distance has a key in fluence on the forming location of the crack initiating point and the locus of crack propagation;(4)The square plate with pre-formed holes has a better performance than cylindrical shell on blast-resistant capability at a smaller standoff distance,while the in fluence of pre-formed holes on the reduction of blast-resistant capability of square plate is bigger than that of cylindrical shell.

1.Introduction

For the conventional warhead or the metal container storing flammable explosive gas,fragments and blast loading will be incurred under initiation or accident stimuli.Under the combined action of these two kinds of loading,what the shell structures such as vehicles,airplanes,buildings,cabins of ship and a large number of potentially containers subjected to is not the accumulation of effects of fragments and blast loading separately,but the damage effect of blast loading applied on the perforated structures or on the contrary.According to Forsn et al.[1],the mechanism and the degree of damage of the coupled effect will be more complicated and serious,respectively.

The coupled effect of some structures under the combined action of fragments and blast loading has been studied by experiments and numerical simulations.He et al.[2]carried out a blast experiment to study the coupled effect of protective door under the combined loading of fragments and blast wave.Nystrom et al.[3]investigated the combined action of blast and fragment loading on the reinforced concrete wall through numerical simulation method.The damage effect of the wall subjected to blast loading,fragment loading,and combined blast and fragment loading were compared,which indicates that the coupled effect has a larger damage than others.Zhang et al.[4]investigated the deformation failure modes and damage mechanisms of steel-GFRP-steel sandwich plates under blast waves with high-velocity fragments and found that the damage degree under the combined action of these two loadings is much higher than that under blast waves only.Liu et al.[5]compared the damage effect and obtained the damage modes of helicopter rotor under the combined action of fragments and blast loading with that under the action alone of them by numerical simulation.Hou et al.[6]conducted experiments adopting cast TNT and prefabricated fragments to study the damage characteristics,failure modes and the protective mechanism of sandwich bulkhead under the impact of shock and high-velocity fragments and pointed out that the main failure mode of the front plate is the combination of large deformation and large amounts of perforation holes while that of the back plate is only large deformation.

Because of the complexity on the response process and mechanism of the combined action,it is dif ficult to control the in fluent factors in experiment.Considering the fragments arriving the structures earlier than blast loading at relatively farther distance[7,8]in according to the attenuation rules of blast wave and fragments velocity in air,the approach of structures with pre-formed holes subjected to blast loading is widely adopted to decouple the complex coupled effect.Considering plastic hinge formation and extensional effects,Schleyer et al.[9]applied an energy approach to solve the large deformation problem of square plates with either square or circular holes under transverse pulse pressure loading and conducted experiments to validate this approach.Assuming that the fragments perforate the square plates before the pressure load arrives,Rakv?g et al.[10]carried out experiments that plates with pre-formed holes are subjected to controlled pressure pulses and analyzed deformation modes.In addition,the difference of deformation around the pre-formed holes between standard Lagrangian and Fully coupled FSI simulations was discussed.Hou et al.[11,12]studies the deformation rules with change of the number and diameter of holes and proposed a formula to calculate the midpoint deformation of plate with holes.Carrying out experiments and adopting numerical simulation for the response of square plates with pre-formed holes under blast loading,Li et al.[13]investigated the difference of deformation,failure and damage in three kinds of hole shapes,quantitatively.Wu et al.[14]conducted experiments and numerical simulation to investigate the stress concentration,deformation,energy changes and failure of a metal cylindrical pipeline with pre-formed holes under blast loading,in addition,the capability to resist blast loading with change of hole-spacing and stand-off distance were analyzed.Aune et al.[15]et al.studies the dynamic response of the steel plates with and without pre-formed holes under the blast loading generated by a shock tube facility,the deformation is quanti fied and measured by 3D-DIC,which is used to compare the failure with numerical simulation results,in addition,the crack propagation was observed by using adaptive mesh re finement and element erosion.

In order to obtain the deformation and failure mechanism of cylindrical shells and square plates with pre-formed holes,the present research is concerned with the distribution and quantitation of stress concentration,local and global deformation,degree of damage,change of stress status and failure modes generated on cylindrical shells and square plates with pre-formed holes under blast loading,which not only represents an extension of the previous works on the perforated structures but also describes the deformation and failure mechanism from multi-viewpoints.The commercial software Ls-Dyna 971 was applied to investigate the dynamic responses and deformation of cylindrical shells and square plates with pre-formed holes under blast loading.To calibrate the accuracy of the numerical model,the numerical results of a series of square plates with pre-formed circular holes were compared with the experimental results recorded by Li et al.[13].The calibrated models were then employed to perform the simulations of the cylindrical shells with pre-formed circular holes.The comparison of structure response process,stress nephogram,local and global deformation,degree of damage,change of stress status and failure modes between cylindrical shell and square plate with pre-formed circular holes under blast loading have been discussed in this study.In addition,the stages of stress concentration are classi fied and the deformation and failure mechanism are analyzed.Details are presented in the following sections.

2.Numerical model calibrations

The commercial software Ansys 17.0 and Ls-Dyna 971 was used to establish the numerical model and perform numerical analysis[16].A non-linear,explicit finite element numerical method of Ls-Dyna 971 was adopted to model the large inelastic deformation and failure.Reliable and accurate numerical predictions of structural response to blast load have been approved[17-20].So the numerical simulation of the structural responses of the cylindrical shells with pre-formed holes under blast loading was carried out by this software.

To validate the accuracy and reliability of the numerical model in this paper,a square plate with pre-formed circular holes tested by Li et al.[13]was used to calibrate the model.A total of nine explosion tests were conducted in the experiments,three of them were related to circular holes under different charges of 550 g,650 g and 750 g,respectively.The same structural parameters were adopted in these three tests including a square plate with the length of 1000 mm and the thicknesses of 4 mm,the core loading area of 600 mm×600 mm,nine holes with the diameter of 80 mm,the distance of 150 mm between centers of two adjacent holes and the distance between charge and the square plate is 250 mm(defined as standoff distance in the following),as shown in Fig.1.There are 76 bolts with diameter of 20 mm used to fix the square plates with pre-formed circular holes between the supporting frame and the cover plate.The supporting frame and the cover plate have the same breadth and thickness of 100 mm and 30 mm,respectively.The maximum midpoint displacement,the middle and 1/4 locus deformation curves of square plates were measured by laser scanning reverse forming technique and the reverse postengineering software Geomagic Qualify.Selected test data were used to calibrate the numerical model established in this study.

Fig.1.The schematic diagram of the experiment settings.

2.1.Element,mesh,boundary conditions and contact modeling

Since the square plate with circular holes,the supporting frame and the cover plate are all symmetric about two middle planes which are perpendicular to each other,a quarter of the plates and frames were modeled,as shown in Fig.2.The numerical models have been established by the commercial software Ansys 17.0.After carrying out a mesh convergence analysis,a feature element size of 5 mm was selected to be optimal for the shell and solid elements,which balanced the numerical element sensitivity requirement,the accuracy of the results and the computational ef ficiency.The square plate with pre-formed circular holes was modeled by Belyschko-Tasy four-node shell elements[21],the supporting frame and the cover plate were modeled by eight-node solid elements.

Fig.2.Finite element model of square plate with pre-formed holes.

To simulating the action of bolts used in the experiments to fix the square plate,two approaches of boundary conditions were considered in this numerical simulation.The first approach(BC.1)is more close to represent the experimental conditions,so the nodal constraints were used at positions of bolts in the square plate,the supporting frame and the cover plate,as showed in Fig.3.The second approach(BC.2)is a simpli fication of the first approach.The nodal constraints were only used at positions of bolts in the square plate,as showed in Fig.4.

Fig.3.Boundary condition 1(B.C.1).

Fig.4.Boundary condition 2(B.C.2).

The*CONTACT_AUTOMATIC_SINGLE_SURFACE model embed in Ls-Dyna 971 was adopted to prevent the penetration between the square plate,the supporting frame and the cover plate,which is a single surface contact defined wholly by the slave side with a contact static and a contact dynamic friction value of 0.28 and 0.20,respectively.In addition,shell thickness is also considered in this contact model,which is suitable to simulate the structures comprised by shells.

2.2.Material model

According to the experiments of Li et al.[13],Q345 steel was adopted as material of square plates.With quite well describing of the hardening effect,strain rate effect and temperature softening effect of metallic materials[22],the Johnson-Cook strength model of Ls-Dyna 971 named*MAT_JOHNSON_COOK was used to represent the mechanical behavior of Q345 and to model the dynamic response of square plate with pre-formed circular holes under blast loading.The model is expressed as Eq.(1):

whereA,B,C,nandmare the constants of the material and given in Table 1,σEis the equivalent stress,εEis the effective plastic strain,is the strain rate,0is the reference strain rate in quasi-static experiment,Tis the current temperament,Tris the room temperature and theTmis the melting temperature of the material.

The damage model embed in Johnson-Cook model is based on the concept of cumulative damage generated by plastic strain.Damage factorDis defined as Eq.(2):

whereΔεis the increment of effective plastic strain for every integration cycle.εfis the effective fracture strain when the fracture occurs,at the moment,the damage factorDis 1.The effective fracture strainεfis expressed as Eq.(3):

whereD1,D2,D3,D4andD5are the damage constants of the material and are given in Table 1,ηis the stress triaxiality factor.

The keyword*EOS_GRUNEISEN was adopted as the equation of state of Q345 steel and the pressure for compressed materials as Eq.(4)[16],:

whereCis the intercept of thevs-vpcurve;S1,S2andS3are the coef ficients of the slope of thevs-vpcurve;γ0is the Gruneisen gamma;ais the first order volume correction toγ0;andμ=ρ/ρ0-1.

The Johnson-cook parameters of Q345 steel listed in Table 1 were obtained by Li[13]who adopt MTS universal material testing machine and Split Hopkinson Tension Bar(SHPB)to carrying out quasi-static tension test and dynamic tensile test for measuring the static and dynamic tensile mechanical properties of Q345,respectively.These parameters have been used to simulate the numerical models established by Li[13],and the maximum midpoint displacement of numerical simulation has a good agreement with the experimental results.In addition,the maximum midpoint displacement and the local deformation around the holes are obtained by numerical simulation carried out by this paper and agree well with the experimental results reported by Li[13],which has described in section 2.4.As a result,the Johnson-cook parameters of Q345 steel listed in Table 1 has been demonstrated that the material parameters used in the simulations truly represent the actual physical properties of the material investigated.

Table 1 Johnson-cook parameters of Q345 steel[13].

2.3.Blast load modeling

In this study,blast loading was generated by the CONWEP(Conventional Weapons Effects Program)empirical model implemented as the*LOAD_BLAST[16]in Ls-Dyna 971.The CONWEP algorithms developed by Kingery and Bulmash[23]accounts for angle of incidence by combining the re flected pressure(normalincidence)value and the incident pressure(side-on-incidence)value,which is a quite accurate method to load air-blast free-field blast wave.The pressure can be calculated by Eq.(5):

whereθis the angle of incidence;Pincidentis the incident pressure;Preflectedis the re flected pressure.

The keyword*LOAD_BLASTcan provide the blast pressure on the certain surfaces predefined by the keyword*LOAD_SEGMENT_SET.The parameters required for*LOAD_BLAST include equivalent mass of TNTexplosive,the spatial coordinates of the detonation point,the initiative time and the type of blast.According to the experiments of Li et al.[13],the mass of charge are 550 g,650 g and 750 g and are all with the standoff distance of 250 mm.

3.Result and discussions

Fig.5 compares the displacement time history of center point of square plate with pre-formed holes between two kinds of boundary conditions for 550g charge with the standoff distance 250 mm.The results indicate that there is almost no difference of displacement time history curves between B.C.1 and B.C.2,while the computational ef ficiency of B.C.2 is higher than that of B.C.1.Therefore,the B.C.2 was employed in the subsequent models.In addition,due to the clamped frame boundary,the weakening of shock wave and the increasing deformation energy of square plate,the displacement of center point begins to rebound and a de finitely decreasing occurs at 2 ms.

Fig.5.Comparison of displacement time history of center point between two kinds of boundary conditions.

In order to validate the agreement between the numerical results and experimental data reported[13]of the square plate with pre-formed circular holes under blast loading,the permanent displacements of max-deformation point(center point)are compared for three cases with different weight charge(550 g,650 g and 750 g)with the standoff distance 250 mm.As shown in Fig.6,the numerical results of permanent displacement of max-deformation point agree well with the experimental data of that and the maximum errors were all less than 5%.

Fig.6.Comparison of permanent displacement of max-deformation point between the experimental data and the numerical results.

Because of the local deformation occurring around the circular holes,it is not enough to validate the numerical models by comparing the permanent displacement of max-deformation point between the numerical results and experimental data.In this study,the comparison is further performed by considering the deformation curves of the plate center locus and 1/4 locus which is shown as red solid line in Fig.7.The deformation curves of the center locus and 1/4 locus of the square plate with pre-formed holes under blast loading of 750 g TNT charge with the standoff distance 250 mm are illustrated in Fig.8.

Fig.7.The schematic diagram of center locus and 1/4 locus.

The numerical results and experimental data of deformation curves of the center locus are compared as shown in Fig.8(a).There are three gaps on the deformation curves,which represent the projections of the pre-formed circular holes on the center locus.The local deformation occurring around the circular holes and on the rest of square plate both have a good agreement between the numerical results and experimental data.The numerical results of deformation curves of the 1/4 locus also agree well with the experimental data,as shown in Fig.8(b).

The comparison above indicates that the numerical model established in this study gives reliable predictions of square plate with pre-formed circular holes under blast loads and can be applied in subsequent calculation and analysis.

4.Numerical simulation

The calibrated numerical model above was adopted to carry out the simulation of dynamic response,deformation and damage of cylindrical shell with pre-formed circular holes under external blast loading.The response and damage quantities such as the permanent displacement and the displacement of the center locus and 1/4 locus,the degree of damage,the dynamic stress concentration factor and the change of stress status around the pre-formed circular holes and failure modes were analyzed and compared with that of the square plate with pre-formed circular holes to examine the capacities of blast-resistance.

4.1.The con figurations of cylindrical shell with pre-formed circular holes

In order to compare the capacities of blast-resistance between cylindrical shell and square plate with pre-formed circular holes,keeping the same size of holes,the number of holes,the holespacing,the loading area and the thickness of shell/plate,the equivalent cylindrical shell with pre-formed circular holes were considered in this study.The schematic diagram of equivalent structure were shown in Fig.9.

Fig.8.Comparison of permanent de flection at the center locus and 1/4 locus between the experimental data and the numerical results.

Fig.9.The schematic diagram of cylindrical shell with pre-formed holes.

4.2.Modeling

The 750 g TNT charge was ignited with the standoff distance 250 mm above the cylindrical shell as shown in Fig.9.Blast load was modeled by adopting the CONWEP empirical model implemented in Ls-Dyna.The cylindrical shell with pre-formed circular holes was modeled with Belyschko-Tasy four-node shell elements of 5 mm size,the supporting frame and the cover plate were modeled by eight-node solid elements of 5 mm size.The material properties,type of elements,mesh method,blast load modeling,boundary conditions and contact modeling are same as the calibration simulation,as shown in Fig.10.

5.Simulation results and discussion

The simulation results presented and discussed about the comparison between cylindrical shell and square plate with preformed circular holes in six sections include:the structure response process and the deformation mechanism; the deformation curves,degree of damage,dynamic stress concentration factor,change of stress status and failure modes.These were individually described in Section 4.1-4.6.

Fig.10.Finite element model of cylindrical shell with pre-formed holes.

5.1.The structure response process and the deformation mechanism

Fig.11 show the structure response process and stress field of square plate and cylindrical shell with pre-formed circular holes under blast loading.The operating condition of them in the simulation were 750 g TNT charge and 250 mm standoff distance.According to the characteristic of the structural response of cylindrical shell with pre-formed circular holes,the process is divided into four speci fic stages,that is,First stage-In control of axial tensile stress field;Second stage-The transition of tensile stress field;Third stage-In control of circumferential tensile stress field;Fourth stage-Becoming uniform stress field.

5.1.1.First stage(60μs-150μs)

Fig.11.The structure response process and stress field of square plate and cylindrical shell with pre-formed circular holes under blast loading.

750 g TNT charge is ignited att=0μs.Because of the same standoff distance,the blast loading begins to interact with the square plate and cylindrical shell on material around the center hole at the same timet=60μs.As shown at t=110μs,before the blast loading arriving at the cover plate,the stress-zone generated by the blast loading occurs on the square plate with a circular shape while on the cylindrical shell with a shape similar to rectangle,which is due to the geometrical difference of cylindrical shell between axial and circumferential direction.In addition,the stress concentration zone of square plate is uniformly distributed around the center hole,however,because of blast loading firstly arriving at middle line of cylindrical shell in axial direction,the stress concentration zone of cylindrical shell is only shown along axial direction around the center hole.As shown att=150μs,after arriving at the cover plate,with the blast loading acting on these two structures progressively,the stress-zone on the square plate is represented as a shape similar to rectangle covering up all holes exactly,at the same time,the zones of re flected stress appear near the cover plate.There is a signi ficant stress concentration zone around the center hole and there are eight relatively week stress concentration zones between other adjacent holes.For cylindrical shell,the loading area is covered with stress-zone completely.The stress concentration zones appear on the regions of non-holes along circumferential direction and the regions between holes along axial direction,as a result,there are two stress concentration bands along circumferential direction between three rows of holes.This phenomenon can be attributed to the generating of plastic hinge along circumferential direction more easy than that along axial direction because of the arc con figuration and the existence of pre-formed holes,therefore,the axial tensile stress field is generated and broken off by holes.

5.1.2.Second stage(150μs-290μs)

Fromt=150μs tot=210μs,the stress concentration zone of square plate is changing to the region between the center hole and the outer holes.However,with the deforming downwards,the intensity of the circumferential tensile stress field increases quickly and the global tensile stress filed comes into the stage of transition.Accordingly,the stress concentration zones of cylindrical shell are distributed between the adjacent holes and between holes and the cover plate along axial direction,at the same time,stress on the region of non-holes along axial direction begins to dissipate,as shown att=210μs.As a result,there are three stress concentration bands along axial direction and broken by holes.As demonstrated att=290μs,the stress concentration zones of square plate change to the nine regions around nine hole,respectively.However,with cylindrical shell deforming downwards progressively,the axial tensile stress field is being translating to the circumferential tensile stress field,consequently,the middle stress concentration band along axial direction is becoming to extent along circumferential direction until intersecting with the other two stress concentration bands along axial direction.

5.1.3.Third stage(290μs-840μs)

Fromt=290μs tot=840μs,a global sunken deformation with a circular shape tending to square shape is generated for the square plate,while that with an elliptical shape tending to square shape is generated for the cylindrical shell.This is an interesting trend because of the existence of pre-formed holes.As shown att=840 μs,the stress concentration zone of square plate is located at the region between the adjacent holes and the intensity of the stress concentration zone between outer holes is bigger relatively.The stress concentration zone of cylindrical shell is only located between holes along circumferential direction,which indicates that this stage has been in control of circumferential tensile stress field.

5.1.4.Forth stage(840μs-4000μs)

Fromt=840μs tot=1600μs,with the global sunken deformation increasing rapidly,the intensity of stress concentration is dissipating due to the energy of blast loading translating to the energy of plastic deformation.The stress concentration zone is only around the holes in a small range att=1600μs and disappears completely att=4000μs,which indicates that this stage has been in control of uniform stress field.

5.2.The local and global deformation curves

The same con figuration,parameters,blast loading,boundary and mesh method as Figs.1 and 9 but without the pre-formed holes are adopted as the control experiment performed by the method of numerical simulation.A typical displacement time histories at the max-deformation point of the square plate and cylindrical shell with and without pre-formed holes are given in Fig.12 the peak deformation of cylindrical shell and square plate with pre-formed holes are 73.9 mm and 61.5 mm,in addition,the permanent deformation of cylindrical shell and square plate with pre-formed holes are 70.5 mm and 58.7 mm,while the peak deformation of cylindrical shell and square plate without pre-formed holes are 65.3 mm and 52.7 mm,besides,the permanent deformation of cylindrical shell and square plate with pre-formed holes are 61.9 mm and 47.8 mm,that is,the peak and permanent displacement of the cylindrical shell with pre-formed holes are 13.2%and 13.9%more than that without pre-formed holes,likewise,the peak and permanent displacement of the square plate with pre-formed holes are 16.7%and 22.8%more than that without pre-formed holes,which indicates that the square plates have a better performance than that of cylindrical shell with or without pre-formed holes,whereas the in fluence of pre-formed holes on the reduction of blast-resistant capability of square plate is bigger than that of cylindrical shell.This may be caused by being in control of single direction tensile stress field for the cylindrical shell,while all direction tensile stress field for the square plate,as described at section 4.1.Because of the local deformation occurring around the circular holes,it is not enough to comprehensively describe the deformation by comparing the displacement time histories at the max-deformation point.For demonstrating the difference of the local deformation between these two structures with pre-formed holes,the permanent deformation curves at the center locus and 1/4 locus of them as shown in Fig.13.In addition,the center and 1/4 locus are considered in axial and circumferential direction of cylindrical shell,respectively,on account of the difference of curves between these two directions.

Fig.12.Comparison of displacement time histories at the max-deformation point between the square plate and cylindrical shell.

Fig.13 shows the permanent local deformation at the center locus and 1/4 locus of the square plate and cylindrical shell.As shown in Fig.13 for the center zone of these two structures,the maximum and minimum local deformation at the center locus are belong to cylindrical shell along axial direction and square plate,respectively,however,that at the 1/4 locus are belong to cylindrical shell along circumferential direction and square plate,respectively.For the zone near the cover plate,the maximum and minimum local deformation at the center locus are belong to cylindrical shell along axial direction and cylindrical shell along circumferential direction,respectively,which is the same as that at the 1/4 locus.This phenomenon of local deformation indicates that the local deformation of cylindrical shell is bigger than that of square plate for a majority of zones besides the zone along the circumferential direction near the cover plate.Comparing Fig.13(a)and(b),the deformation of square plate at the center locus is bigger than that at the 1/4 locus,which indicates that a local sunken with an approximate spherical surface has been generated and the center hole keeps a circular shape.In addition,the deformation of cylindrical shell along axial direction at the center locus is bigger than that at the 1/4 locus,however,there is almost no difference between the deformation of cylindrical shell along circumferential direction at the center locus and the 1/4 locus,which indicates that the shape of the center hole is similar to a space shape of saddle.

In order to explore the reason leading to the phenomena above,the existence of pre-formed holes and the different con figuration of cylindrical shell between the axial direction and the circumferential direction are presented in this study as two viewpoints.The local deformation between the square plate and cylindrical shell with pre-formed holes is shown in Fig.14,which has the same phenomena as the cylindrical shell and square plate with preformed holes and indicates that the different con figuration of cylindrical shell between the axial direction and the circumferential direction is key leading to the phenomena mentioned above and the existence of pre-formed holes just has an effect to increase the global and local deformation.Besides,the comparison of the phenomena between Figs.13 and 14 also indicates the smoothness of space surface of cylindrical shell and square plate after permanent deformation,otherwise the shape of wave would appear on them.Comparing Figs.13(a)and Fig.14(a),there is an interesting phenomenon that the shape of the curve between the center hole and the adjacent holes at the center locus along axial or circumferential direction is raised towards the source of TNT charge for cylindrical shell with pre-formed holes,however this phenomenon is not existent for cylindrical shell without preformed holes and for square with or without preformed holes,which may result fromthe combination of the local effect of pre-formed holes and the different con figuration of cylindrical shell between the axial direction and the circumferential direction.

Fig.13.The local deformation between the square plate and cylindrical shell with pre-formed holes.

Fig.14.The local deformation between the square plate and cylindrical shell without pre-formed holes.

5.3.Degree of damage at typical points

According to the structure response process,the distributing of stress concentration and failure mode studied in section 4.1 and 4.6,the typical points is chosen at the edge of pre-formed holes in order to analyze the degree of damage and marked from “A” to “E” as shown in Fig.15.Because of the different con figuration along axial and circumferential direction of cylindrical shell,the subscript “0” , “1” and “2” are defined to represent the typical points on the square plate,along axial and circumferential direction of cylindrical shell,respectively.The rule of the subscript will be used in section 4.3-4.5.

Fig.15.The typical points for analyzing the degree of damage.

As shown in Fig.16,the degree of damage at five typical points is defined as the damage factor D in Eqs.(2)and(3).When the damage factor D reaches 1,the material fails and the crack will be generated and propagating.The operating condition in the simulation were that 750 g TNT charge and 250 mm standoff distance.Fig.16(a)-(e)shows the damage factor history at pointA-Eseries on square plate and cylindrical shell.The damage factor history curve can be divided into three stage:elastic deformation,plastic accumulation,keeping constant.The earliest point comes into the stage of plastic accumulation is pointA1,B1,C1,D2andE2,which indicates that the con figure difference of cylindrical shell between axial and circumferential direction easily leads to unequally distributed deformation relative to square plate at the forepart of deformation.In addition,the line between pointA1,B1,andC1is along axial direction and the line betweenD2andE2is also along axial direction,which indicates that the plastic deformation begins along axial direction of cylindrical shell and corresponds to the phenomenon mentioned in section 4.1.1.The damage factor history curve of these five point series interlace with each other in the stage of plastic accumulation,which exhibits the transition of tensile stress field in manner of damage changing and corresponds with the description in second stage of structure response process.The damage factor history curves at pointA,DandEseries reach the stage of keeping constant at about 1 ms earlier than that at pointBandCseries at about 2.5 ms-4.5 ms.The final damage factor at pointA-Eseries are listed in Table 2 and shown as Fig.16(f),it can be seen that the degree of damage at pointAandBon square plate is similar to that on cylindrical shell along circumferential direction and larger than that on cylindrical shell along axial direction,and the degree of damage at pointCon square plate is similar to that on cylindrical shell along axial direction and smaller than that on cylindrical shell along circumferential direction,while the largest and smallest degree of damage at pointDandEare both on square plate and on cylindrical shell along circumferential direction,respectively,which indicates that when the standoff distance of 750 g TNT charge is 250 mm,the failure of square plate and cylindrical shell with pre-formed holes easily appears at pointDandEseries,and this suppose has been validated in section 4.6.

Fig.16.The comparison of damage factor at five typical points between square plate and cylindrical shell with pre-formed holes.

Table 2 The final damage factor of typical points.

5.4.Dynamic stress concentration at typical points

Although the phenomenon and transformation of stress concentration has been observed in Fig.11 and discussed in section 4.1,it is still necessary to obtain the quantitative analysis of stress concentration in order to investigate and compare the in fluence of pre-formed holes on the blast-resistance between square plate and cylindrical shell.The dynamic stress concentration factorKd(t)is adopted to represent the in fluence of pre-formed holes at any momenttand defined as Eq.(6),[13].

whereσdw(t)is the dynamic stress of structures with pre-formed holes at any momentt,σdn(t)is the dynamic stress of structures without pre-formed holes at any momentt.

Fig.17.The comparison of dynamic stress concentration factor at five typical points between square plate and cylindrical shell with pre-formed holes.

The operating condition in the simulation were that 750 g TNT charge and 250 mm standoff distance.Fig.17(a)-(e)shows the dynamic stress concentration factor history(the solid line)and the integral curve(the dash line)at five typical pointA-Eseries on square plate and cylindrical shell which have been mentioned in section 4.3.At the initial moment,the max dynamic stress concentration factor appears at the pointA1-D1andE2and reaches 1.9,3.5,3.1,2.2 and 3.6,respectively.Afterwards,the peak value of dynamic stress concentration factor appears obviously after 1.6 ms at all point series and reaches 5.1,5.9,4.4,5.5 and 3.9,respectively.In addition,from the initial moment to 1.6 ms,the dynamic stress concentration factor is below 2 with slight fluctuation.However,the stress concentration zone begins to disappear after 1.6 ms and the stress field becomes into the control of uniform stress field with large deformation,which has been observed and mentioned in section 4.1.Accordingly,it can be concluded that there is a small in fluence of pre-formed holes on dynamic stress concentration at the stages with sharp change of stress fields(from First stage to Third stage mentioned in section 4.1),while a great in fluence of pre-formed holes on dynamic stress concentration at the stage with uniform stress field(Forth stage mentioned in section 4.1),which indicates that the difference in large deformation between with and without the pre-formed holes is the key to the in fluence on degree of stress concentration.

As shown in Fig.17,the dynamic stress concentration factor history curves all fluctuate nearby the value of 1(the green solid line),which indicates that the existence of pre-formed holes not only lead to the increasing of stress concentration around the holes,but also release the stress concentration.In order to express the total contribution of stress concentration and stress dissipation due to pre-formed holes,The Integrated dynamic stress concentration factorIkd(t)at any momenttwas defined as Eq.(7).

whereKd(t)is the dynamic stress concentration factor at any momenttand defined as Eq.(6).

As shown in Fig.17(a)-(e),the green dash line is the integration of the green solid line and represents the total contribution that stress concentration and stress dissipation balance out.For pointBandDseries,the integrated dynamic stress concentration factor curves of all points are above the green dash line.For pointAandCseries,the integrated dynamic stress concentration factor curves of points on square plate are below the green dash line and that of points on cylindrical shell are above.For pointEseries,the integrated dynamic stress concentration factor curve of point along axial direction of cylindrical shell is below the green dash line.It can be concluded that the total contribution of stress concentration and stress dissipation due to pre-formed holes at the pointsA0、C0andE1are tenting to status of stress dissipation and others are tenting to status of stress concentration,which indicates that the status of stress concentration and stress dissipation occur around the pre-formed holes of the square plate and cylindrical shell,simultaneity.

5.5.Change of stress status at typical points

In order to study and compare the in fluence of pre-formed holes on the ductile fracture between square plate and cylindrical shell,the stress status change is analyzed and discussed.The value of stress triaxialityηand the deviatoric state parameterξcan be used to describe the material stress status and can correspond to the stress status one by one as listed in Table 3[24-27],which are defined as Eqs.(8)and(9).And the fracture strain decreases with increasing of stress triaxiality[28].

Table 3 The relationship between stress triaxiality and the stress status.

Whereσ1,σ2andσ3are the principle stresses.Because of the relationship of Eq.(9)between the stress triaxialityηand the deviatoric state parameterξ,as a matter of convenience,the stress triaxialityηis adopted as one and only parameter to analyze and discuss the stress status change.

The operating condition in the simulation were that 750 g TNT charge and 250 mm standoff distance.Fig.18(a)-(e)shows the stress triaxialality history at five typical pointA-Eseries on square plate and cylindrical shell which have been mentioned in section 4.3.At the initial stage(0 ms-0.5 ms),namely,before Third stage mentioned in section 4.1,the stress state of typical points on cylindrical shell reciprocate sharply between the pure pressure and pure tentile,while that on square plate keep the pure tentile or reciprocate between pure tentile and pure shear,which indicates that the square plate fractures more easier than cylindrical shell at the typical point series around the pre-formed holes at the initial moment.At the stage from 0.5 ms to 1.2 ms,namely between Third stage and Fourth stage mentioned in section 4.1,the stress triaxiality curves of a majority of typical pointA-Eseries fluctuate nearby the value of 0.33,except that of the pointC1reciprocate between-0.33 and 0 and that of the pointA1andB1reciprocate between 0 and 0.33 from 0.8 ms to 1.2 ms,which exhibits the transition of tensile stress field in manner of stress status changing and also indicates that the fracture happens mainly at this stage.At the stage from 1.2 ms to 2.5 ms,namely Fourth stage in section 4.1,the stress state of typical points A-E series fluctuate between pure tentile and pure pressure with one or two wavelength,which is due to the free oscillation of these two kinds of structures with almost permanent deformation.

5.6.Failure modes and crack propagation process

In order to obtain the failure modes,crack propagation process and the critical equivalent TNT value(the minimal equivalent TNT value when failure occurs),the standoff distance of 100 mm,150 mm,250 mm,300 mm,400 mm and 500 mm are selected to detonate the charge,in addition,the adaptive mesh re finement and element erosion are adopted to present the crack propagation on the plate and shell[15].

In order to obtain the critical equivalent TNT value with a certain standoff distance,the mass of TNT charge is gradually increased 100 g by 100 g until failure occurred on the square plate and the cylindrical shell.As listed in Table 4,the critical equivalent TNT value have been obtained with standoff distance from 100 mm to 500 mm.At the standoff distance of 100 mm,the critical equivalent TNT value applied on square plate is equal to that on cylindrical shell.At the standoff distance from 150 mm to 400 mm,the critical equivalent TNT value applied on cylindrical shell is 10%,6.7%,5.6%and 8%smaller than that on square plate.While,at the standoff distance of 500 mm,the critical equivalent TNT value applied on cylindrical shell is 10%bigger than that on square plate.The tendency above indicates that cylindrical shell provides a better performance of blast-resistance than square plate at a bigger standoff distance,comparing that at a smaller standoff distance,which may be attributed to that the cylindrical shell is in control of single direction tensile stress field at a smaller standoff distance due to the geometrical characteristic of arch con figuration while in control of relative uniform stress field at a bigger standoff distance.

Table 4 The critical equivalent TNT value.

As shown in Fig.19,the shapes of fracture under the critical equivalent TNT value corresponding to the different standoff distance are obtained and are classi fied as three(S1,S2 and S3)and two(C1 and C2)kinds of failure modes for square plate and cylindrical shell with pre-formed holes,respectively.That is,the failure mode S1(standoff distance from 100 mm to 150 mm)-fracture of the center hole;the failure mode S2(standoff distance from 250 mm to 300 mm)-fracture of the outer holes;the failure mode S3(standoff distance from 400 mm to 500 mm)-fracture of the boundary;the failure mode C1(standoff distance from 100 mm to 150 mm)-fracture of the midline holes along circumferential direction and the failure mode C2(standoff distance from 250 mm to 500 mm)-fracture of the side holes along circumferential direction.Although the fracture shapes of the cylindrical shell at the standoff distance from 250 mm to 500 mm all belong to the same failure mode C2,the degree of fracture decreases with the increasing of standoff distance.Considering the in fluence of blast intensity on the failure modes,some events with gradual value of TNT charge above the critical equivalent TNT value for every standoff distance have been performed and compared,it can be concluded that the blast intensity has an in fluence on the degree of fracture and failure but not on failure modes.

Fig.18.The comparison of stress status change at five typical points between square plate and cylindrical shell with pre-formed holes.

Fig.19.The comparison of failure mode between cylindrical shell and square plate with pre-formed circular holes.

Fig.20.The crack propagation of different failure modes for square plate with circular holes.

Fig.21.The crack propagation of different failure modes for cylindrical shell with circular holes.

In order to analyze the crack propagation of different failure modes,an event has been chosen from every failure mode and the process of the crack propagation has been shown in Fig.20 and Fig.21.For failure mode S1(standoff distance of 100 mm and under 500 g TNT),cracks initiate at the pointA0mentioned in section 4.3 and propagate along the line(A0B0)between the adjacent holes until run-through,subsequently,the petal-like broken hole is formed and expanded outwards.For failure mode S2(standoff distance of 250 mm and under 1500 g TNT),cracks initiate at the pointD0and propagate along the line(D0E0)between the adjacent holes until run-through,subsequently,the square-like plate with the center hole is perfectly separated from the original structure and flying towards the direction of blast loading.For failure mode S3(standoff distance of 500 m m and under 3000 g TNT),cracks initiate at the midpoint of the inner boundary of the cover plate and propagate along this boundary until intersecting with each other,subsequently,the entire loading area is separated from the cover plate and flying towards the direction of blast loading.For failure mode C1(standoff distance of 100 mm and under 500 g TNT),cracks initiate at the pointA2mentioned in section 4.3 and propagate along the line(A2B2)between the adjacent holes until runthrough,subsequently,the elliptic broken hole is formed by linking the three holes along circumferential direction and expanded outwards.For failure mode C2(standoff distance of 250 mm and under 1400 g TNT),cracks initiate at the pointD1mentioned in section 4.3 and propagate along the line(D1E1)between the adjacent holes until run-through,subsequently,two elliptic broken holes are formed by linking the three holes along circumferential direction and expanded outwards.According to the stress nephogram shown in Figs.20 and 21 and the discussion in section 4.4,the crack initiation points and the edges of the crack in the propagation path mentioned above are both the locations of stress concentration,which contributes to the failure.

According to the failure modes and the process of crack propagation described above,there are two conclusions obtained as follows:On one hand,the point at which the crack initiates is further from the center of structure with the standoff distance increasing,which is because the time sequence and the space distributing of the shock wave applied on each parts of structure are different and decided by the standoff distance and have a key in fluence on the forming location of the crack initiating point and the locus of crack propagation;On the other hand,comparing the failure modes under different standoff distance between square plate and cylindrical shell,the local fracture/failure mode appears on both square plate and cylindrical shell at a smaller standoff distance from 100 mm to 150 mm.With increasing of standoff distance,there are two kinds of failure modes appearing on the square plate at the standoff distance from 250 mm to 300 mm and from 400 mm to 500 mm,respectively,however,there is only one kind of failure modes appearing on the cylindrical shell at the standoff distance from 250 mm to 500 mm,which may be attributed to that the generating of plastic hinge along circumferential direction of cylindrical shell is more easy than that along axial direction because of the arc con figuration of cylindrical shell and the existence of pre-formed holes.Accordingly,the occurring of crack initiating point at pointD1and the crack propagation along the locus ofD1E1(axial direction)are more easy than other points and locus,after crack run through the adjacent holes,the stress concentration zone along circumferential direction and near the cover plate will dissipate and there will be no more cracks occurring on the cylindrical shell.

6.Conclusion

This study presents the deformation and failure mechanism of cylindrical shells and square plate considering the in fluence of preformed holes under blast loading by employing a numerical model validated.The structure response and stress field changing process has been divided into four speci fic stages from the viewpoint of stress concentration and the deformation mechanism has been discussed systematically.The local and global deformation curves,degree of damage,change of stress status and failure modes of cylindrical shell and square plate with pre-formed circular holes under blast loading are obtained,compared and analyzed,it can be concluded as:(1)The transition of tensile stress fields is due to the geometrical characteristic of pre-formed holes and cylindrical shell with arch con figuration and is demonstrated by damage factor and stress triaxialality;(2)Comparing with structures without preformed holes,the existence of pre-formed holes not only lead to the increasing of stress concentration around the holes,but also release the stress concentration during whole response process;(3)There are three and two kinds of failure modes for square plate and cylindrical shell with pre-formed holes,respectively.and the standoff distance has a key in fluence on the forming location of the crack initiating point and the locus of crack propagation;(4)The square plates with pre-formed holes has a better performance than cylindrical shell on blast-resistant capability at a smaller standoff distance,while the in fluence of pre-formed holes on the reduction of blast-resistant capability of square plate is bigger than that of cylindrical shell,which may be mainly due to the geometrical characteristic of cylindrical shell with arch con figuration.Because of the multiformity of fragment perforation distributing and number,to obtain the deformation and failure mechanism at a much wider range,we need to investigate further on the in fluence of the number and distribution of pre-formed holes in our subsequent research.

Declaration of competing interest

We declare there is no competing interest.

Acknowledgments

The reported research is financially supported by The National Natural Science Foundation of China under Grant No.11902310 and No.11802292. The financial contributions are gratefully acknowledged.