Influence of Stress Wave on Dynamics Damage Character of Ship-build Low-carbon Steel Based on Low-velocity Taylor Impact Bar

LI Xiao-bin,LI Ying,ZHENG Yuan-zhou

(School of Transportation,Wuhan University of Technology,Wuhan 430063,China)

1 Introduction

Warships in the war are susceptible to mines,torpedoes and anti-ship missile attacks.When the explosion occurred in the near-field under water or in the cabin,the damage characteristics of the warships have become hot topics in research.Especially,the characteristics of the warship materials are the key points of the research.Review the predecessors’references,the damage model of general metal is related with the stress status:when stretch the metal,some holes appear,propagate and penetrate in the metal inside,and the metal presents tenacity ability in macroscopic;when shear the metal material,the material mainly presents shear failure,even adiabatic shear.

The Taylor test can get high strain rate(104～106)per second,in which a deformable flatnosed cylinder is fired against a fixed,rigid wall and was originally proposed to determine dynamic yield stresses of materials[3].Taylor test results are used to validate or to estimate coefficients for phenomenological strength models needed for simulating dynamic loading processes using inverse identification procedures[4].

Fig.1 Typical penetration and crack under air blast[1,2]

The work of Wilkins and Guinan[5]is probably the first one to numerically simulate and study the Taylor rod impact test.After this,a number of authors have studied the problem of Taylor rod impact test numerically.Here,only some latest references which have studied the fracture in Taylor rod impact test are discussed.When Taylor bar runs fast,cracks and fragmentation phenomenon are very common.Couque(1998)[6]observed several spiral cracks which formed on the lateral surface of the cylinder in symmetric Taylor tests on swaged tungsten alloys.

In the present paper,the fundamental theory of simulating taylor bar impact test is investigated.SHPB,notched tension tests were taken to investigate the constitutive relation and fracture characteristics.Three distinct failure macro-modes are identified:large plastic deformation,confined fracture inside the specimen and shear cracking.Stress triaxiality effect of failure mechanisms is discussed.

A developed ductile fracture criterion by Bao and Wierzbicki[7]is used to predict crack formation and growth in the Taylor test.The initial impact velocity of the cylinder ranges from 150 m/s to 250 m/s.Stages and failure modes are investigated with the stress wave effect.

2 Fundamental theory

A thermo-elasto-plastic constitutive relationship between the incremental stress and the strain tensors is developed based on the von Mises yield function and the isotropic hardening due to strain and strain rate and the softening due to temperature.

During the plastic deformation,the plastic work is transformed into heat.Further,the work spent in overcoming the surface friction is also transformed into heat.Both these phenomena induce a thermal strain inside the body.Therefore,the incremental logarithmic strainis decomposed as the sum of three parts:the incremental elastic strain,the incremental plastic strainand the incremental thermal strain

The incremental thermal strain depends on the incremental temperaturetDT through the following relation:where α is the coefficient of thermal expansion of the material.

The effective incremental Cauchy stressis related to the incremental Cauchy stress tensortdσijas

Then,using the principal of strain equivalence the incremental elastic stress-strain relationship can be expressed as follows:

Again using the principal of strain equivalence,the von Mises yield function of damaged solid subjected to thermo-elastic-plastic deformation is described by:

Stress triaxiality is defined as:

Fig.2 Stress triaxiality effect on fracture model

3 Experiment of constitutive relation and damage characteristic

3.1 SHPB(Split Hopkinson Pressure Bar)test

To get the stress-strain relationship of slow-carbon steel,a series of SHPB tests were taken.As shown in Fig.3,yield stress increases with the strain rate:yield stress is 420 MPa as strain rate is 900 s-1,yield stress is 497 MPa as strain rate is 3 500 s-1,yield stress is 612 MPa as strain rate is 5 500 s-1.As shown in Fig.4,yield stress reduces with the strain rate:yield stress is 436 MPa at temperature 200 °C,yield is 314 MPa at temperature 400 °C,yield stress is 242 MPa at temperature 600°C.

Fig.3 Results of effect of strain rate of steel Q235

Fig.4 Results of effect of tempreture of steel Q235

3.2 Notched tension test

Material failure at complex stress state is complicated physical and mechanical phenomenon.According to the research,stress triaxiality is an effective way to evaluate the influence of the stress state.Fracture strain is defined as:

where,d0is unfractured tension specimen,and dfis unfractured tension specimen.

Radius of notched tension specimen is 3 mm,while the radius of notch are 1 mm,2 mm,4 mm and 6 mm.According to Bridgman’s[8]analysis,relevant stress triaxiality is 1.25,0.89,0.65 and 0.55.

Fig.5 Fracture apperance of tensile specimen of different stress triaxiality

Fig.6 Stress triaxiality effect on fracture strain of ship-build low-carbon steel

4 The numerical simulation model

Generally,in the Taylor test a deformable flat-nosed cylinder is fired against a fixed rigid wall(see Fig.7).The cylindrical projectile is of the diameter d=6 mm and the length l=30 mm.The friction coefficient between the front surface of the projectile and the rigid wall is assumed to be μ=0.1.Smoothed Particle Hydrodynamics method(SPH)was used to simulate the failure phenomena of cylinder.A 3-D solid finite element model was built rather than an axisymmetric model(see Fig.8).

Fig.7 Schematic of a cylindrical projectile impacting a rigid wall

Fig.8 Finite element model of the projectiletarget system

The Johnson-Cook hardening relation was selected to model the taylor bar and the target.There are more sophisticated hardening relations,but Johnson-Cook’s is probably the most widely used among those accounting for equivalent plastic strainequivalent plastic strain rateand temperature effect.Since numerous efforts have been made in the past to determine their parameters for a large number of metallic materials,it has been implemented in many FE explicit codes.The relation is stated through the following multiplicative equation[3]:

where the reference plastic strain rateTrand Tmare the room temperature and the material melting temperature respectively,and A,B,n,C,and m are five material constants.The JC model accounts for isotropic strain hardening,strain rate sensitivity,and thermal softening in the uncoupled form.The first term of the right hand side of Eq.(1)represents the quasi-static stress-strain relation at room temperature;the second term signifies the strain-rate hardening;the third term means the temperature dependence of the stress-strain relation.

The material parameters in the JC model for ship-build low-carbon steel were listed in Tab.1.

Tab.1 Material parameters for ship-build low-carbon steel

5 Result and analysis

5.1 Stress wave in the progress of Taylor bar test

The process of Taylor bar impacting the target and damaging can be divided into four stages as Fig.9:

(a)Initial impact stage,Taylor bar impact to the rigid target and produce compression wave,the pressure of the compression wave can be calculated by equation:

where v0is impact velocity,ρ0and ρ1are density of the bar and target,C1and C2are sound spread in the the bar and target.

(b)Convert stage,the compression wave spreads to the free surface,and then converts to extension wave,the extension wave gathers to the axis of the Taylor bar,since it is non-vertical part of the powerful extension wave will change into shear wave.

(c)Rebound stage,the bottom of Taylor bar rebounds the target,and a new free surface formation,the compression wave spreads into this surface,and then negative pressure zone will appear,this stage will last a long time.

(d)Damping stage,the stress wave spreads back and forth in the Taylor bar and damps gradually.

Fig.9 Stress wave effect of the Taylor bar front end

Transmission and reflection will occur when the compression wave impact to the free surface.When the compression wave incidence from A medium to B medium,the angle of the incidence is 90°,the value of the reflect wave can be calculated by the Eq.(12):

where,σRis the reflect wave,σIis the incident wave,CAand CBare sound spread in the medium,ρAand ρBare density of the medium.

At incidence angle 90°,σR/σI=-1,that is to say,the reflect stress is equal to the incidence stress;when the angle of incidence is not equal to 90°,compression wave will reflect to be extension wave and shear wave,which will cause shear damage of the boundary[9].

5.2 Large plastic deformation

As shown in Figs.10-12,rare deformation can be seen in the tail part of Taylor bar,while obvious plastic deformation can be seen in the front part.The front part changes to a‘mushroom’when the bar crashes a rigid wall at high speed.The maximums of vertical deformation are 2.8 mm,4.6 mm and 7.0 mm at the speed 150 m/s,200 m/s and 250 m/s,respectively while the maximums of horizontal deformation are 0.94 mm,2.4 mm and 4.7 mm.However,the relative deformation of vertical deformation is smaller than that of horizontal deformation.

Fig.10 Vertical deformation

Fig.11 Horizontal deformation

5.3 Confined fracture(ductile damage)

In the whole impact progress,central point in the impact end is under tension and shear momentlythen enters a extended tension period),see B section in Fig.2.At the same time,ductile damage forms rapidly,while microvoids form,gather and become macroscopic,as shown Fig.14.

Fig.12 Bar deformation for different initial velocities

This failure mode is caused by void nucleation,coalescence,and growth,which were demonstrated experimentally by e.g.Worswick and Pick[5].Since the failed elements are embedded by the unfractured material,this type of failure mode is termed ‘confined fracture’in the present paper.The mechanism of the void growth(tensile failure)is caused by tensile wave reflected from the transient gap formed in the interface soon after the impact.This kind of damage is similar to the spall mechanism.The formation of the cavity is mainly because of the propagation of the mini-cavity in the negative area,these cavities combine into a whole and a larger scale of damage is occurred.

Fig.13 Variation of stress triaxiality and damage of central point in the impact end

Fig.14 Ductile damage of central point in the impact end

5.4 Shear crack(shear damage)

As shown in Figs.15-16,since compression translates rarefaction wave at two free sides,central point in the impact end is under tension momentlyUnder combined obliqueincidence transmission compression wave and rarefaction wave,the central point bears tension combined shear,and then shear damage forms,see A section in Fig.2.

Shear damge,forming narrow shear zones,is different from ductile damage,microvoids gathering,in microcosmic mechanism.There may be dynamic recrystallization and metal phase transitio,and even adiabatic shear band.

Fig.15 Variation of stress triaxiality and damage of central point in the impact end

Fig.16 Ductile damage of central point in the impact end

6 Conclusions

Through numerical simulation of the Taylor test,deformations of different impact velocity;impact damages of the axis centre;the reasons of impact damage are analyzed.Typical conclusions are as follows:

(1)According to the stress wave,there are four stages in the process of Taylor bar impacting the target:initial impact stage,convert stage,rebound stage and damping stage.

(2)Three failure modes generate in low speed Taylor impact bar:large plastic deformation,ductile damage and shear damage,which stress triaxiality have an obvious effect on.Obvious plastic deformation is found in the front of the bar with a shape of‘mushroom’;and the relative deformation of vertical deformation is smaller than that of horizontal deformation.

(3)Stress wave affects failure damage models obviously.Rarefaction wave from two free sides converges central point in the impact end,stress triaxiality is greater than 1/3,then ductile damage formed.Oblique-incidence transmission to free side of compression wave produces shear wave and rarefaction wave,central point in the impact end formed shear damage even shear crack.

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