Effect of a viscoelastic target on the impact response of a flat-nosed projectile

Hu Liu·Jialing Yang·Hua Liu

1 Introduction

With the increasing requirement of high speed vehicles in the areas of aerospace and transportation,dynamic material characterization has attracted more attention because these structures are exposed to severe loading conditions such as explosion and impact.In 1948,a simple one-dimensional analysis model was proposed by Taylor[1]and Whiffin[2]to estimate the dynamic behavior of steel specimens at high strain rate,which is referred to as Taylor cylinder impact.In this test,a cylindrical flat-nosed projectile strikes a rigid anvil target normally and the material strength can be predicted by measuring the final deformed shape of the projectile.

In order to predict the dynamic behavior of materials more precisely,much effort has been made to improve the classic Taylor model.The study of Hawkyard[3]proposed an energy equilibrium equation to replace the momentum equilibrium across the plastic wavefront,and the mushrooming deformation of the projectile predicted by the energy-based approach was in good agreement with experiments[2].Also,Lee and Tupper[4]modified the theoretical model taking into account the strain hardening effect and elastic wave propagation in the projectile.In their work,Hashmi and Thompson[5]used the Taylor cylinder impact to determine the dynamic behavior of materials which considered both the strain-hardening and strain-rate effects.Then Hutchings[6]extended the classic Taylor theory to analyze the dynamic response of polymer materials at high strain rate.The study of Wang et al.[7]carried out the Taylor experiment to predict the dynamic behavior of porous materials.Work by Wang et al.[8]adopted the Taylor impact method to investigate the dynamic constitutive relation of cellular materials.It should be noted that most investigations were proposed by firing a cylindrical flat-nosed projectile onto a rigid wall,while the elastic or viscoelastic effect of the target bar was seldom considered.In fact,no target bar is perfectly rigid,and the energy dissipation of the target affects the deformation behavior of the projectile.

Many researchers have recognized this problem and some improvements were made.For example,Critescu and Bell[9]introduced a symmetric Taylor test to eliminate the assumption of regarding the target bar as an ideally rigid material in the classic Taylor model.Experimentally,the symmetric Taylor impact configuration was performed by striking a stationary rod target with an identical rod projectile of the same material and diameter.This method was also employed by Willmott and Radford[10]to observe the deformation and fracture behavior of glass materials,and the strain signals in the target bar were recorded by using gauges.Undoubtedly,because the projectile and target bar were made from the same material,the target bar also underwent plastic deformation,which made it difficult to obtain accurate strain signals.

In their study,Yang et al.[11]proposed an improved model by using a flat-nosed projectile striking against a stationary semi-infinite output bar,and the stiffness of the target bar was much higher than that of the flat-nosed projectile to ensure that no plastic deformation occurred during the impact process.A theoretical model combining plastic shock wave propagating in the flat-nosed projectile and elastic wave traveling in the output bar was derived in their study.The effect of the elastic target bar on the plastic behavior of metallic materials could be accurately predicted by Yang’s model.Actually,this strategy is widely adopted and called the Taylor-cylinder Hopkinson impact test[12,13].Compared with the traditional Spite Hopkinson Pressure Bar(SHPB)test,this technique achieves much higher strain rates(in the range of 103–107s?1),and the strain signals recorded in the target bar can provide a validation of the theoretical prediction.

Recently,the dynamic behavior of soft materials such as rubber and polymer has aroused wide concern.Taylor and SHPB tests are carried out to determine their dynamic behavior.For instance,Turgutlu et al.[14]studied the mushrooming deformation of low density polyethylene samples by using a flat-ended polymeric cylindrical projectile striking against a rigid target bar normally.Also,Sarva et al.[15]conducted the direct Taylor impact to examine the dynamic mechanical behavior of polycarbonates under high-rate loading conditions experimentally and numerically.Employing SHPB tests,Wang et al.[16]analyzed the high strain-rate mechanical behavior of low impedance materials.In their experiment,the polymeric input and output bars were used to provide accurate strain signals,and they also discussed the influence caused by the viscoelastic Hopkinson bars.Work by Zhao et al.[17,18]studied the dynamic behavior of low impedance materials at high strain rate by using the PMMA viscoelastic Hopkinson bar.The wave propagation attenuation and dispersion characteristics in viscoelastic Hopkinson bars were analyzed by Bacon[19].Subsequently,the viscoelastic Hopkinson bars are used to test low stiffness materials at high strain rate,and the viscoelastic behavior of Hopkinson bars has also aroused worldwide concern[20–22].Most recently,Othman[23]set up a polymeric SHPB experiment to test soft materials,and a complex Young’s modulus model was proposed to analyze the viscoelastic behavior of polymeric Hopkinson bars.Work of Butt et al.[24–26]proposed a parametric identification method to determine the viscoelastic material parameters of PMMA Hopkinson bar.Furthermore,Harrigan et al.[27–29]replaced the traditional metal bars with much weaker polymer bars to test the low stiffness materials in SHPB experiments.As mentioned,the viscoelastic effect of the device should be taken into consideration when using the SHPB technique to obtain the constitutive relation of soft materials.Similarly,the Taylor-cylinder Hopkinson impact test can also be used to evaluate the stress–strain relation of soft materials at high stain rate.The only difference between the classic Taylortest and the Taylor-cylinder Hopkinson impact is that the rigid anvil in the Taylor test is replaced by an output target bar in the Taylor-cylinder Hopkinson impact[12,13].Meanwhile,in order to avoid the wave impedance mismatch between the flat-nosed projectile and the target bar,much weaker materials of the target bar are used to ensure the accuracy of the recorded strain signals.Unfortunately,many materials used in the target bar may exhibit viscosity properties,but most studies ignored the potential influence caused by the viscoelastic target bar.

The aim of this paper is to investigate the effect of viscoelastic target on the impact response of the flat-nosed projectile at high rate.An extended theoretical model is proposed by using a flat-nosed projectile striking against a semi-infinite viscoelastic target bar normally.The present model offers guidelines on how to make use of Taylor cylinder Hopkinson impact to identify the dynamic material parameters more accurately.

2 Problem formulation and basic equations

A cylindrical flat-nosed projectile striking axially against a semi-infinite stationary target bar is considered,as shown in Fig.1a.The cylindrical target bar is made of viscoelastic material instead of rigid material in the classic Taylor model.Similar to the classic Taylor impact,the flat-nosed projectile is regarded as a homogeneous continuum,which obeys the rigid,perfectly plastic constitutive law,and the yield strength of the flat-nosed projectile isσY.The length and initial velocity of the flat-nosed projectile isL0andU0,respectively.Letρ0andρ1denote the densities of the flat-nosed projectile and the target bar,respectively.A0andA1are the initial cross-sectional areas of the flat-nosed projectile and target bar,respectively.As shown in Fig.1b,when the flat-nosed projectile strikes against the target bar,the impinging layer of the projectile yields instantly,and a plastic shock wave will travel leftwards with a speedV.The remaining region of the flat-nosed projectile stays un-deformed,and the velocity of this un-deformed region is denoted asU1.The deformed region of the flat-nosed projectile and the impinging layer of the target bar will move with a common velocityU2.A viscoelastic wave will propagate rightwards in the semiin finite target bar as long as the target bar is struck by the flat-nosed projectile.LetY,σ,andεdenote the traveling distance of viscoelastic wave,the stress and strain in the viscoelastic target bar,respectively.During the impact process,the cross-sectional area across the shock front of the flatnosed projectile increases fromA0toA,and denoteXandHthe lengths of the un-deformed and deformed regions of the flat-nosed projectile,respectively.The impact process is terminated at instanttfas the plastic wave front in the flatnosed projectile stops moving.DenoteH f,X f,andZ fas the final deformed length,the un-deformed length and total length of the flat-nosed projectile,respectively,as illustrated in Fig.1c.

From kinematic consideration,the governing equations can be written as

Fig.1 Impact process of the flat-nosed projectile striking a viscoelastic bar.a Before impact;b intermediate phase;c final deformation

The conservation of mass for the flat-nosed projectile across the plastic wave front is

In the classic Taylor model,it is assumed that the crosssectional areaA0of the flat-nosed projectile increases toAimmediately when passing through the plastic wave front.In his work,Qian[30]noticed that a very short period was actually needed to complete the area expansion process and the momentum conservation equation across the plastic wave front modified by Qian is adopted here,which reads

One can see that the only difference is that the coefficient 1 in the classical Taylor model is replaced by 2/3 in Qian’s model[11,30].

Assume that the initial length dl0of an element along the striking direction changes to dlafter compression.The volume of the element is assumed to be unchanged during the compression process,therefore,the longitudinal compressive plastic strain at any point in the deformed section of the projectile can be determined by

Substituting the above equation into Eq.(4),the following relationship can be obtained

Substituting Eq.(7)into Eqs.(1)and(2),respectively,one has

Substituting Eqs.(6)and(7)into Eq.(5),leads to

As depicted in Fig.1b,a viscoelastic wave is initiated and travels towards the rear end of the viscoelastic bar as the target bar is struck by the flat-nosed projectile.The equation of motion at the any cross sectionYalong the axis of the target bar is

The continuity equation can be expressed as

whereσ(Y,t),Ub(Y,t),andε(Y,t)represent the stress,particle velocity and strain at any cross sectionYin the time domain.The viscoelastic behavior of the target bar can be determined by

whereE(t)is the relaxation function,which can be characterized by using a simple three-element linear viscoelastic solid,as sketched in Fig.2.In this model,EeandE vdenote the stiffness of the springs andηis the viscosity of the dashpot.

The Laplace transformation gives

wheresis the conversion form in the Laplace domain with respect to timet.ˉσ(Y,s)andˉε(Y,s)are the axial stress and strain in Laplace domain,respectively.ˉE(s)is the complex Young’s modulus which consists of two lineal springs and a dashpot,as shown in Fig.2,and it can be given as

Fig.2 A simple three-element model to represent the viscoelasticity of the target bar

whereE e,E v,andηare the characteristic parameters of the viscoelastic material.

Combining Eqs.(11)and(12)to eliminateUb(Y,t),one can get the equation of longitudinal motion in the time domain,and employing the Laplace transform to this equation gives

The following equation can be derived from Eqs.(14)and(16)by eliminatingˉε(Y,s)

The solution to Eq.(17)can be expressed as

whereA(s)andB(s)represent the Laplace transforms of the stresses at the cross sectionY=0 due to the waves traveling along the positive and negative directions,respectively.λ(s)represents the wave propagation coefficient,and can be defined by

Both force and velocity are continuous at the interface between the flat-nosed projectile and the target bar,hence,one has

The stress in the rear end of the target bar should vanish,i.e.,

Substituting Eqs.(21)and(22)into Eq.(18),we can obtain the stress at any cross sectionY

Together with Eq.(20),one has

Combining with Eq.(15),this equation can be rewritten as

whereαgives the stiffness ratio between the two linear springs,βdenotes the relaxation time.

Employing the inverse-Laplace transformation toˉG b(s),one obtains

The total length of the flat-nosed projectile during the impact process can be calculated as

The following non-dimensional variables are introduced

wherexandhare the non-dimensional lengths of the undeformed and deformed regions of the flat-nosed projectile,respectively;u1andu2are the non-dimensional velocities corresponding to the un-deformed and deformed regions of the flat-nosed projectile;τdenotes then on-dimensional time;nais the cross-sectional area ratio of the target bar to the flat nosed projectile;nmis the material parameter ratio of the flat-nosed projectile to the target bar;ξis the ratio ofnatonm;γgives the non-dimensional relaxation time.It is found that the impact response of the flat-nosed projectile will be influenced by the material parametersnm,α,andγ.When the non-dimensional relaxation timeγ→ ∞,the current model will reduce to the flat-nosed projectile striking against an elastic bar model,while the non-dimensional relaxation timeγ→∞and the material parameter rationm→0,the current model can be degenerated into the classic Taylor model.

The non-dimensional governing equations(3),(8)–(10),and(29)can be recast to

These equations can be numerically solved by using the finite difference method.

3 Validation of the theoretical model

3.1 Verification with finite element(FE)simulation

In this section,an FE model is established to simulate the flat-nosed projectile striking onto the viscoelastic target bar.The flat-nosed projectile and the target bar are made from low-density polyethylene and epoxy,respectively.The detail material parameters are shown in Table 1.The flat-nosed projectile is modeled as a cylindrical flat-nosed projectile with an initial velocityU0=50 m/s.The diameter and length of the projectile are set asd0=28 mm andL0=0.2 m,respectively.A long stationary output bar with lengthL=2 m and diameterd=40 mm is employed to make sure that no wave reflects from the un-impacted end of the target bar during the impact process.The explicit nonlinear program ABAQUS/Explicit is used.Both the projectile and the target bar are modeled by an 8-node solid element with reduced integration(C3D8R).An element size of 0.5 mm for the flat-nosed projectile and 2 mm for the target bar are used.The present element size has been verified to be effective to produce reliable results via a mesh sensitivity analysis.The surface-to-surface contact with out friction is defined between the projectile and the target bar.The detailed FE model is shown in Fig.3.

Table 1 Material parameters for the low-density polyethylene projectile and the epoxy target bar

Fig.3 FE model for the flat-nosed projectile striking the viscoelastic target bar.a Whole model;b an enlarged view

Fig.4 Stress and deformation distribution of the projectile at the instant as the impact terminates

Table 2 Comparisons between the theoretical and FE predictions

3.2 Verification with previous literature

In this section,two degenerated models are given to validate the present model.In the first example,the non-dimensional relaxation timeγis set as∞in the present model.To compare with Yang’s elastic target bar model[11],the parametersnm=0.0324,na=2,andu0=0.5 are used.As shown in Fig.5,it is clear that the changing trends for the plastic straine,the deformed and un-deformed lengths of the flat-nosed projectile(handx)predicted by the two models are identical,indicating the present theoretical model can be reduced to Yang’s elastic target bar model when the viscosity of the target bar is neglected.

The second example is the case that the parametersγ→∞andnm→0 in the present theoretical model.The dynamic responses of this degenerated model and the classic Taylor model(rigid target bar model)are compared in Fig.6.The modified momentum conservation equation is adopted in the classic Taylor model,i.e.,the coefficient 1 in the classic Taylor model is replaced by 2/3.The initial velocity of the flat-nosed projectileu0=0.5 is adopted in the comparison.One can find that the plastic straine,the lengths of the deformed and un-deformed regions of the flat-nosed projectile(handx)predicted by the two models have the same trend,which indicates that the present model can be degenerated into the rigid target bar model as the non-dimensional relaxation timeγ→ ∞and material rationm→0.

Fig.5 Dynamic response for the present model with γ → ∞ and elastic target bar model in Ref.[11].a e versus τ;b x and h versus e

Fig.6 Dynamic response for the present model with γ → ∞ and nm → 0 and the rigid target bar model in Ref.[1].a e versus τ;b x and h versus e

In order to illustrate further the differences among the viscoelastic,elastic,and rigid target bar models,plastic strains,deformations and velocity histories are compared.A low-density polycarbonate cylindrical flat-nosed projectile striking on the epoxy target bar is considered here,and the material parameters are the same as those in Table 1.The non-dimensional initial velocityu0and the cross-sectional area rationaare set as 0.5 and 2,respectively.As depicted in Fig.7,the plastic straine,the length of the un-deformed region of the flat-nosed projectilex,and the velocities of the un-deformed and deformed region of the flat-nosed projectileu1andu2decrease versus time,while the length of the deformed regionhincreases with time during the impact process.These characteristic variables for the rigid,elastic,and viscoelastic bar models at two selected instants are also compared in Table 3.It can be found that the viscoelastic bar will affect the dynamic response of the flat-nosed projectile significantly.From Fig.7 and Table 3,we can find that compared with the rigid target model,the plastic strainebecomes smaller,the length of the un-deformed region of the flat-nosed projectilexbecome longer,and the velocities of the un-deformed and deformed region of the flat-nosed projectileu1andu2become larger for the elastic and viscoelastic target bar models.Meanwhile,the viscosity of the target bar will enlarge the residual velocity and reduce the plastic deformation of the flat-nosed projectile.

Fig.7 Dynamic responses for the rigid,elastic,and viscoelastic target bar models.a e versus τ;b x versus e;c h versus e;d u1 versus e;e u2 versus e

4 Numerical results and discussions

The coupled effect of the viscoelastic target bar is investigated in this section.The non-dimensional initial velocityu0=0.5 and the cross-sectional area ratio of the flat-nosed projectile to the target barna=2 are used in the following numerical examples.

Figure 8 depicts the relationship between the final impact responses of the flat-nosed projectile and the material rationm.The stiffness ratio of the two elastic springsαand the non-dimensional relaxation timeγare set as 1 and 0.01,respectively.As shown in the figure,the initial maximum straine0,the final length of the deformed region of the flat-nosed projectileh f,and the duration of impactτfdecrease by increasing the value ofnm,whereas the final length of the un-deformed region of the flat-nosed projectilex f,the final total length of the flat-nosed projectilez f,and the residual velocity of the flat-nosed projectileu fshow an opposite trend.This is because the impact process terminates at the instant when the projectile and the target bar reach a common velocity,i.e.,the residual velocity.Asn mincreases,the elastic effect of the target bar becomes more evident,which leads to a larger residual velocity of the projectile and the target bar.Accordingly,the plastic energy dissipation of the projectile decreases as the input energy of the projectile is specified,which results in a smaller deformed length of the flat-nosed projectile.In addition,the duration of impact also decreases with increasingn mbe cause the impact process will terminate at a larger value of residual velocity.The changing trend exhibited in Fig.8 is consistent with Yang’s analyses[11].

Table 3 Comparisons of rigid,elastic and viscoelastic bar models at two selected instants

Fig.8 Final dynamic responses of the flat-nosed projectile as a function of the material ratio nm.a e0;b x f;c h f;d z f;e u f;f τf

The final deformation,the initial maximum plastic strain,the residual velocity,and the impact duration versus the parameterαare sketched in Fig.9.The material rationm=0.0324 and the non-dimensional relaxation timeγ=0.01 are adopted.It is obvious that the final deformation(includingh f,x f,andz f),the duration of impactτf,and the residual velocity of the flat-nosed projectileu fare sensitive toαwhenαis small,whilst they tend to be constant whenαis larger than 2.Initially,an increase can be observed for the length of the deformed region of the flat-nosed projectileh f,and the duration of the impactτf,while the final undeformed and total length of the flat-nosed projectilex fandz f,the residual velocity of the flat-nosed projectileu fshow an opposite trend.Form Fig.9a,the stiffness ratioαshows an insignificant effect on the initial maximum plastic straine0.

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Fig.9 Final dynamic responses of the flat-nosed projectile as a function of the stiffness ratio α.a e0;b x f;c h f;d z f;e u f;f τf

Figure 10 plots the final responses of the flat-nosed projectile versus the parameterγ,which can reflect the viscous effect of the target bar directly.Asγincreases,the viscosity of the target bar decreases.The other two parameters are defined asα=1 andnm=0.0324,respectively.As can be observed,the influence of the non-dimensional relaxation timeγon the initial maximum plastic straine0is insignificant.Whereas the final deformation(includingh f,x f,andz f)of the flat-nosed projectile,the residual velocityu fand the duration of impactτfare sensitive to the non-dimensional relaxation time parameterγ.Asγincreases,there is an increase in length of the deformed region of the flat-nosed projectileh f,and duration of impactτf,while a decrease in residual velocityu f,final un-deformed lengthx fand final total lengthz fof the flat-nosed projectile can be found.This is because if a viscoelastic target barisused,the material viscosity will cause viscous friction among molecules at a micro level and energy dissipation at a macro level,and leads to wave attenuation and dispersion during propagation.As the in put energy of the projectile is fixed,the residual velocity of the projectile and target bar increases as the viscosity of the target bar increases.Similar to the analysis in Fig.8,both the plastic energy dissipated by the projectile and the duration of impact will decrease when the residual velocity becomes larger.In this case,we can find the deformed length of the projectile and the duration of impact increase with increasingγin Fig.10.The result approaches an elastic case asγ→∞.

Fig.10 Final dynamic responses of the flat-nosed projectile as a function of the non-dimensional relaxation time γ.a e0;b x f;c h f;d z f;e u f;f τf

5 Conclusions

An extended Taylor impact model based on the classic Taylor impact theory in conjunction with the wave propagation analysis in the viscoelastic target bar is proposed.The influence of the viscoelastic target bar on the Taylor impact of the flat-nosed projectile is investigated in detail.The FE simulation is also carried out to verify the present theoretical model and an excellent agreement is achieved.The time histories of the flat-nosed projectile based on the rigid,elastic,and viscoelastic target bar models are compared to illustrate the influence of the viscoelastic target bar.Several special cases are displayed,which demonstrate that the current theoretical model can be degenerated into the elastic and rigid target bar models.The final impact responses of the flat-nosed projectile are also analyzed in detail.Some concluding remarks can be drawn:

(1)The present model can be deduced to a flat-nosed projectile striking onto an elastic bar model as the viscosity of the output target bar is neglected.Meanwhile,the classic Taylor model can also be treated as a special case of the present model.

(2)The material ratio of the target bar to the flat-nosed projectile,the stiffness ratio of the two springs,and the relaxation time of the target bar show important influence on the final dynamic responses of the flat-nosed projectile.

(3)The viscoelastic effect of the target bar can enlarge the residual velocity and reduce the impact duration and the plastic deformation of the flat-nosed projectile significantly.

From the present study,it is suggested that the viscoelastic effect of the output bar should not be neglected and a careful estimation is needed in the Taylor-cylinder Hopkinson impact test of low-density materials.

AcknowledgementsThe work was supported by the National Natural Science Foundation of China(Grant 11472034).

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