Development of multi-physics numerical simulation model to investigate thermo-mechanical fatigue crack propagation in an autofrettaged gun barrel

Nveed Hussin ,Fisl Qyyum ,Rifft Asim Psh ,Msood Shh

a Department of Mechanical Engineering,University of Engineering and Technology,Taxila,Pakistan

b Institut für Metallformung,Technische Universit?t Beragakademie,Freiberg,Germany

Keywords: Steel Autofrettage Gun barrel rack propagation Thermo-mechanical fatigue Numerical simulation Residual stress dissipation

ABSTRACT In this research,a detailed multi-physics study has been carried out by numerically simulating a solid fractured gun barrel for 20 thermo-mechanical cycles.The numerical model is based on thermal effects,mechanical stress fields and fatigue crack mechanics.Elastic-plastic material data of modified AISI 4340 at temperatures ranging from 25 to 1200 °C and at strain rates of 4,16,32 and 48 s-1 was acquired from high-temperature compression tests.This was used as material property data in the simulation model.The boundary conditions applied are kept similar to the working gun barrel during continuous firing.A methodology has been provided to define thermo-mechanically active surface-to-surface type interface between the crack faces for a better approximation of stresses at the crack tip.Comparison of results from non-autofrettaged and autofrettaged simulation models provide useful information about the evolution of strains and stresses in the barrel at different points under combined thermo-mechanical loading cycles in both cases.The effect of thermal fatigue under already induced compressive yield due to autofrettage and the progressive degradation of the accumulated stresses due to thermo-mechanical cyclic loads on the internal surface of the gun barrel (mimicking the continuous firing scenario) has been analyzed.Comparison between energy release rate at tips of varying crack lengths due to cyclic thermo-mechanical loading in the non-autofrettaged and autofrettaged gun has been carried out.

1.Introduction

When a cannon is fired,the single base propellant gases reach temperatures of around 2300°C [1].However,the metal temperatures at the bore reach approximately 1100°C in a few milliseconds and decline to half this value by the time the projectile reaches the muzzle[1].Although the temperature fluctuation does not penetrate very far from the surface,it does create a hard and brittle surface according to earlier researches[2].This compressive thermal yielding of the inner gun surface in short time was observed to be the reason for cracking during operation [3].The thermomechanical model of the transient thermal damage in cannon bores was developed using the finite difference method[4].The gun barrel undergoes excessive mechanical and thermal load during use,especially in the war zone when the firing is carried out continuously [5].Analysis of in-wall temperatures by carrying out appropriate experimentation and development of simulation models was carried out for 120 mm ammunition [6].The thermomechanical erosion in the gun barrels was also analyzed [7].The influence of additives on the temperature,heat transfer,wear,fatigue life and self-ignition characteristics of a 155 mm ammunition was investigated experimentally [8].

To make the inner surface of the gun barrels resistant to applied thermal cycles and pressure,and to reduce the probability of crack initiation and propagation,autofrettage is used to introduce residual compressive stresses on the inner surface[9].Recently,some new methods of autofrettage have been proposed by researchers such as thermal autofrettage [10-12] and rotational autofrettage[13-15] for strengthening gun barrels.During autofrettage,the internal surface is pressurized mechanically or hydraulically beyond the yield point of the material,and the pressure is released.This loading beyond the yield point of the material permanently deforms the inner surface and hence due to external load compressive hoop stresses accumulate on the surface.This has been depicted as a practical solution [16,17] which helps in increasing the work-life of the barrel as the mean load in the fatigue cycle is compressive.Theoretical and experimental investigations in this area of solid mechanics attract considerable interest because of the frequent application of autofrettage.Investigations of autofrettage on the base of the perfect plasticity model [18] and isotropic hardening model[19,20]made it possible to estimate the level of residual stresses and their distribution.Recently,reseachers.

Although an extensive research in the past decade has been carried out by numerically and experimentally analyzing the crack initiation and crack propagation behavior in different materials under complex thermo-mechanical loading conditions [21,22],yet the studies are old and there has been no recent study(to the best of authors knowledge)bringing together the developed techniques for better understanding and improved life of gun barrels.The concept of residual compressive stress being more beneficial is the basis of this study.This has been shown in the author’s previous works where it was found that the damage in thermal fatigue is proportional to the magnitude of compressive yield during the heating phase [23-25].This compressive yield is directly responsible for an increased crack opening during the cooling cycle,during which the crack growth occurs.Very recently,several studies were published by Perl and Saley [26-29] analyzing the effect of internally and externally cracked gun barrels.They analyzed the effect of varying autofrettage load,crack morphology and applied pressure on the mechanics of failure occurring.The effect of thermal fatigue under already induced compressive yield needs to be established,which is the primary focus of this study.Apart from that,the progressive degradation of the accumulated stresses during the service life of autofrettaged gun barrels is still not wholly understood especially the dynamics of crack initiation and propagation behavior.It is essential to analyze and understand this behavior to operate such critical components in a safe limit.Such a study will also pave the way for enhancing the service life of gun barrels in the future.

In this research,a detailed multi-physics numerical simulation model is developed to study the solid and cracked gun barrel for 20 applied thermo-mechanical cycles.The numerical model is developed by coupling mechanical,thermal and fatigue crack mechanics all together to get a better insight into this complicated failure mechanism.Elastic-plastic material data of modified AISI 4340 at temperatures ranging from 25 to 1200°C and at strain rates of 4,16,32 and 48 s-1was acquired from high-temperature compression tests and was assigned to the simulation model.The boundary conditions in the numerical simulation model were identified from real tests of the gun barrel during continuous firing.Comparison of results from non-autofrettaged and autofrettaged simulation models provide useful information about the evolution of strains and corresponding stress evolution in the barrel at different points of combined thermo-mechanical loading cycles.The effect of thermal fatigue under already induced compressive yield is analyzed.Energy release rate at tips of varying crack lengths due to cyclic thermo-mechanical loading in non-autofrettaged and autofrettaged gun barrel is compared.The effect of thermal fatigue under already induced compressive yield due to autofrettage and the progressive degradation of the accumulated stresses due to thermo-mechanical cyclic loads on the internal surface of the gun barrel(mimicking the continuous firing scenario) is analyzed.

2.Numerical simulation modeling

The goal of this study is to analyze the interplay of mechanical and thermal properties of the material and the applied thermomechanical cyclic load combined with the fatigue crack mechanics.The numerical model can be a helpful tool to understand the physics and the evolution of stresses,strains,and crack propagation behavior with and without autofrettage.Therefore,in this study,a numerical simulation model is developed to investigate the stresses and strains produced in the gun barrel.Five different cases are selected to study the effects of thermomechanical cycling under different conditions:

1.Thermomechanical fatigue cycling in a non-autofrettaged barrel.

2.Thermomechanical fatigue cycling in a non-autofrettaged barrel at increasing crack lengths.

3.Autofrettage process.

4.Thermomechanical fatigue cycling in autofrettaged barrel.

5.Thermomechanical fatigue cycling autofrettaged barrel at increasing crack lengths.

Commercial finite element modelling (FEM) software ABAQUS 6.12 was used for modeling.The model is defined as a dynamicexplicit model and for the solution the solver provided with ABAQUS has been used.A detailed temperature-dependent elasticplastic material model has been used in the study.The material properties were identified by mechanical testing of standard material specimens at varying temperatures and strain rates.The material is defined as homogenous isotropic section with von-Mises yield criterion [30]:

Where,yield stress(σY)is calculated by solvingEq.(1),in which σr,σz,and,σθare radial,axial and hoop stresses,respectively.Yielding in the material occurs when σYreaches the defined yield strength of the material for a given temperature.

2.1.Material data

Chemical composition of the modified AISI 4340 steel was measured at 3 different points on a flat sample using spark emission spectrometer OBLF QS 750.The average of the material chemical composition measurements is shown in Table 1.The chemical composition shows slightly higher alloy concentrations for Ni,Cr,and Mo than standard AISI 4340 steel,these alloying elements are added for improving the performance of the material due to applied thermo-mechanical loads during service.

Table 1 Chemical composition of modified AISI 4340 steel.

Thermo-mechanical elastic-plastic detailed material data of AISI 4340 steel is incorporated into the model.The properties are provided in Table 2.The density of the material was measured using the Archimedes method.Young’s modulus and Poison’s ratio of the material at room temperature were measured by testing 3 samples on MTS 810 system as per ASTM E8M and were averaged.

Conductivity,coefficient of thermal expansion (CTE),and specific heat of the material was measured using B?HR DIL 805 A/D dilatometer.The values are presented in Fig.1.The presented values in Table 2 and Fig.1 were used in defining the simulation model.

To collect accurate material formability data at different temperatures,a compression test is carried out on MTS-810 servo-hydraulic machine according to the ASTM E9 standard.Circular specimens of diameter 8 mm and height of 20 mm are used for testing,which preheated to the desired temperature in a furnace.Tests are carried out at 25,100,200,300,400,500,600,700,800,900,1000,1100 and 1200°C for speeds of 4,16,32 and 48 strain/s.The maximum temperature achieved follows from Ref.[31].For each test condition,4 tests are carried out.Thus,a total of 208 compression tests are performed to identify the material model.

The temperature-dependent plastic material data is presented in Fig.2 which is assigned to the model.Average true stress/true strain compression test data at different temperatures are shown in Fig.2.For reference,average true stress/true strain data at different strain rates at 500°C is shown in the inset Fig.2.It can be seen here that the material presents substantial mechanical resistance under compression up to 500°C.However,at higher temperatures,ample material softening is observed.

Evolution of yield and ultimate tensile strength of the test material at 4 strain/second for different temperatures is presented in Table 3 for easy comparisons of the presented data.This data is extracted from the test results shown in Fig.2.

Table 2 Mechanical properties of modified AISI 4340 steel.

Table 3 Evolution of yield and ultimate tensile strength of the test material at 4 strain/second for different temperatures.

2.2.Thermomechanical model

Considering the long gun barrel,it is assumed that there would be almost no deformation in the longitudinal,i.e.z-axis.Therefore,any solution in this direction is ignored (plain strain condition).This assumption simplifies the problem to 2D,extensively reducing the computation time and cost.A 2D geometry resembling the actual gun barrel (only high erosion zone of interest) with an internal and external diameter of 130 mm and 340 mm is modeled and sectioned (for later meshing and boundary condition definition) respectively,as shown in Fig.3.The readers can refer to Ref.[24,32-34]to get further details about the development of the adopted technique of numerical simulation modeling.

Two kinds of models are developed.First being with just 1 transient thermo-mechanical step to study the barrel under thermomechanical loading without autofrettage.Second being with the initial thermo-mechanical step in which temperature is kept at 25°C and internal pressure of 1000 MPa is applied to mimic autofrettage.This produces a hoop stress of magnitude～1000 MPa on the internal surface of the barrel.At the end of this step,internal residual stresses accumulate on the surface and up to certain depth in the cylinder (discussed in detail in section 2.5) and then thetransient thermo-mechanical step is defined to study the barrel under thermomechanical loading.Appropriate field output and history outputs are recoded for all simulations at every converged time increment.

Fig.1.Temperature-dependent thermal expansion coefficient (CTE) and Specific heat of modified AISI 4340 steel.It is observed that CTE slowly increases up to 300 °C after which it decreases sharply up to 700 °C and remains constant after that to 1000 °C.Specific heat remains constant up to 400 °C,after which it linearly increases up to 1000 °C.

Fig.2.Elastic-plastic material data of modified AISI 4340 steel used for numerical simulation.It is observed that the material softens and work hardening significantly reduces with increasing temperature,whereas,there is slight softening in the material behavior with increasing strain rate from 4 to 48 strain/sec.

2.3.Boundary conditions

The temperature on the internal surface of the barrel is defined as heat flux.The magnitude of surface heat flux is tuned to match the experimental and numerical simulation thermal curves,and the value of 5400 W/m2is used for all simulations.The initial temperature of 25°C is defined for the whole model.The pinned boundary condition is applied to the few outer nodes to restrict any movement of the model during deformation.Element type CPE4T(A 4-node plane strain thermally coupled quadrilateral,bilinear displacement,and temperature)is used to generate a total of 6000 linear plane strain elements and a total of 6200 nodes in the whole assembly.The meshed assembly of the barrel is shown in Fig.3.The model is checked for the mesh dependency,and during meshing,the aspect ratio is maintained below 1.3.The applied amplitudes of temperature and pressure to obtain desired profiles are shown in Fig.4.These profiles are similar to the actual experimentally observed data,which is already published elsewhere[35].

Fig.3.Constructed meshed part showing the geometry,applied boundary conditions and points/paths where the results are plotted.

Fig.4.Pressure and temperature amplitude with the magnitudes defined to accurately match the applied boundary conditions with experimental observations of [31].

2.4.Thermomechanical cyclic loading in barrel without autofrettage

This model is developed to study the evolution of stresses and strains in the barrel,which is produced in absence of residual stresses on the internal surface.The 20-cycle thermomechanical load with temperature varying between 25 and 970°C and pressure varying between 340 and 100 MPa in each cycle is applied on the internal surface of the barrel simultaneously in transient thermomechanical step.

Considering it to be a reasonable condition,the stresses produced in this model are used as a benchmark of how much thermomechanical stresses are produced if the barrel is put in service without any pretreatment.

2.5.Thermomechanical cyclic loading in barrel with autofrettage

The idea is to compare how the residual stresses on the internal surface of the barrel behave under similar thermomechanical cyclic loading conditions.A two-step simulation model is developed to study this behavior:

-In the 1st step,the barrel is internally pressurized to 1000 MPa and then relieved to no pressure using a smooth load amplitude.This internal load is almost 3 times the yield strength of the material,which is generally advised for autofrettage [36].It is shown in Fig.5.Due to this load,the internal surface of the barrel yields and compressive residual stresses accumulate to a depth of 33 mm as shown in Fig.6.These residual stresses are expected to improve the life of components during thermomechanical loading.It is observed in Fig.6 that the hoop stress is compressive 1000 MPa on the surface and at 40 mm depth becomes zero.The value of hoop stress is around +300 MPa at 55 mm depth whereas after 60 mm it is more than 200 MPa in tension.Equivalent plastic strain after autofrettage is also plotted in Fig.6.It is observed that after the autofrettage,the barrel deforms plastically up to 50 mm depth,after which no plastic strain is present.

-In the 2nd step,10 cycles thermomechanical load is applied on the internal surface of the barrel in the transient thermomechanical step to study the behavior of barrel under such loading conditions after internal compressive stresses are accumulated [37].The cyclic pressure and temperature applied in the simulation are shown in Fig.4.

Fig.5.Evolution of hoop stresses due to applied internal pressure for autofrettage.It is observed that at the end of the autofrettage process,1000 MPa of compressive stresses and 1.1% of plastic strain are accumulated on the internal surface of the barrel.

Fig.6.Hoop stress and plastic strain distribution along the barrel radial path after autofrettage.It is observed that the compressive hoop stress is accumulated up to 33 mm and plastic strain is accumulated up to 52 mm of wall depth.

Comparing the simulation result of non-autofrettaged with autofrettaged barrel gives a better idea of how effective the autofrettage is in resisting the later deformation.It is believed that the thermal loading has a negative impact on the accumulated residual stresses as the material yield drops significantly at higher temperatures,and hence the accumulated compressive stresses on the internal surface slowly dissipate.The model clearly shows the effects of temperature on autofrettage.

2.6.Thermomechanical cyclic loading in cracked barrel without autofrettage

Now to further investigate the behavior of the same barrel under thermomechanical cycling when it is cracked,a numerical simulation model is developed with crack lengths of 3,4,5,6,7,8,10,12,14,16,18,20,25,30,35,40 and 45 mm.The simulations are based on the sequence published by Qayyum et al.[26].A total of 34 simulations are run with varying crack lengths as mentioned above.In all models,the part is especially sectioned to spider mesh the crack tip and structurally mesh the whole part.The sectioned part and the meshed part are shown in Fig.3.A contact condition is defined on the internal faces of the crack so that they do not overlap during compressive load.This is particularly important as the compression cycle creates the conditions necessary for the fracture during cooling [38,39] and hence helps in better estimation of stress evolution.This simulation model helps in better understanding the behavior of crack in the barrel at different lengths due to the applied thermomechanical load.These results are taken as a reference to compare the results of the behavior of cracked barrel with autofrettage.

2.7.Thermomechanical cyclic loading in cracked barrel with autofrettage

The overall research focuses on understanding the crack propagation behavior in autofrettaged gun barrel due to accumulative residual compressive stresses on the internal surface of the barrel.The study specifically focuses on the resistance of the barrel to crack propagation during service.For this purpose of studying this effect,the simulation models with thermomechanical loading on the autofrettaged internal surface when there is a crack present have been developed.The challenge with modeling such a complicated problem is that it is impossible to model it in a single file due to limitations of ABAQUS.Therefore,models with 3,4,5,6,7,8,10,12,14,16,18,20,25,30,35,40 and 45 mm crack lengths are developed,but the crack faces were tied during autofrettage application so that the body behaves like a solid object.

In the next simulation step,the residual stresses from previous simulations are used as predefined fields,and thermomechanical load is applied on the same models but this time a contact condition is defined between the internal faces of the crack so that they do not overlap during compressive load.However,the crack can open up during the tensile loading field.This simulation technique helped in understanding the complex behavior of growing crack due to applied thermo-mechanical fatigue load when compressive stresses due to autofrettage are present.The model geometry with spider meshed at the crack tip and opened crack face is shown in Fig.7.

3.Results

In this research,the effect of thermo-mechanical cycling on the stress/strain evolution in the gun barrel is studied.The effect of the autofrettage process in accumulating residual stresses on the inner surface is observed.The effectiveness of these residual stresses to resist the stress/strain evolution due to applied thermo-mechanical cycling is studied.Contour integral cracks of different lengths are defined and the energy absorption at the crack tip is estimated and compared.

3.1.Un-cracked barrel without autofrettage

In order to understand the deformation behavior and the evolution of stresses and strains,it is essential to observe the behavior without autofrettage and see how the cannon deforms and how stresses and strains evolve due to applied same thermo-mechanical load.

3.1.1.Hoop stress

Effect of thermo-mechanical cyclic load on the resulting hoop stress at the inner surface of the barrel when there are no autofrettage residual stresses is shown in Fig.8.The graph is for 20 cycles.In each cycle,during the heating process,compressive stresses originate and during cooling,tensile stresses originate.The profile shape is complicated due to simultaneously applied cyclic mechanical load.It is observed that the magnitude of compressive stresses increases and the tensile stresses drops with an increasing number of cycles.The evolution of hoop stresses on the inner surface of the barrel due to thermomechanical cyclic load without prior autofrettage the first,and the last cycle is separately shown inset.The complicated hoop stress profiles in the first and last cycles can be clearly seen.It is observed that during heating,there are 2 compressive stress peaks observed and only one tensile stress peak is observed in the first cycle.However,in the last cycle very high compressive stress peak is observed whereas no tensile stress is reached.

Fig.7.Sectioned crack geometry and spider mesh which represents the schematic procedure of problem definition (from top to bottom).The part is first modeled and sectioned with all the necessary details for further appropriate spider meshing.After meshing the crack faces are assigned thermo-mechanically active surface to surface interface by defining conductive and hard contact interaction.

The double compressive stress peaks are expected due to the compression of heating as well as softening during heating.After a certain point,the surface temperature increases to a greater extent while the backing material in the depth of the barrel resists the deformation.This compression is shown as a secondary peak during the heating cycle.It is expected that this secondary peak during the first cycle would cause the most damage,as it will carry a subsequent high tensile stress during the cooling cycle.

Fig.8.Effect of thermo-mechanical cyclic load on the resulting hoop stress at the inner surface of the barrel when there are no autofrettage residual stresses.Comparison of the first and last cycle in the inset image with numbering to signify the points of interest later discussed in Fig.9.

To break down the hoop stress at maxima and minima in first and last cycles for the comparison,At points 1,2,3 and 4 mentioned in Fig.8,radial distribution on the path mentioned in Fig.3 is plotted,as shown in Fig.9.The graphs are plotted on the radial path from the inner surface of the barrel to the outer surface,depicting the stress profile due to thermo-mechanical loading along the radius.After 8 mm depth,all hoop stress profiles are observed to be similar in the magnitude.A considerable variation close to the surface in the stress profiles is observed at the maxima and minima of the loading cycles.Due to heating the inner surface,compressive stresses originate,which are highest at 2-5 mm depths and increase with increasing cycles.The compressive stresses are highest at the 20th cycle.

It is important to note that this hoop stress fluctuation on the internal surface of the barrel is due to the applied temperature and pressure cycles.The radial profile at different instances is shown in Fig.9.In this figure it is observed that the applied thermal cycling induces compressive and tensile stresses up to 8 mm depth from the surface.This cyclic variation in hoop stress is responsible for micro-crack initiation in this area which has been reported by various researchers in the past[25,40].After 8 mm,the hoop stress in the barrel is nearly equal to zero,as is represented by a dotted line.

Temperature and hoop stress have been plotted together to construct a characteristic hysteresis curve in Fig.10.This representation provides information about the energy absorption by the material for a specific cycle and information about the material stress history.Although here the cyclic stress profile is plotted against temperature,its shape is influenced by applied mechanical loading as well.

Fig.9.Hoop stress on the path is plotted at different time points of the 1st cycle max and 1st cycle min.and is compared with the last cycle max and last cycle min.in nonautofrettaged barrel,the points are shown as 1,2,3 and 4 in Fig.8 inset figure.

Fig.10.Cyclic stress profile between temperature and hoop stress with only 1st and 20th cycle for comparison and complete profile is shown in inset figure.

It is observed in Fig.10 that in the non-autofrettaged barrel,the first thermo-mechanical cycle is the one in which most energy is absorbed by the material.The hoop stress is zero in the beginning and starts to accumulate as compressive hoop stress increases on the internal surface with increasing temperature and pressure.It is maximum at a temperature of 700°C as opposed to a maximum temperature of 990°C.With decreasing temperature,the stress starts relieving but does not return to zero due to plastic deformation and hence,reaches up to 800 MPa in tension at room temperature.This tensile stress accumulation on the internal surface of the barrel is responsible for the initiation of microcracks which might later grow or result in spalling and wear of the material.Due to material softening by thermo-mechanical cycling,the hysteresis cycle is observed to grow narrow,which shows that lesser energy is absorbed in each successive cycle.This behavior is typical of cyclic softening for such class of materials under low cycle fatigue regime [41,42].This plastic strain accumulation could be fatal as it causes the void formation resulting in coalescence and eventual crack ignition in such material with second phase particles[43-45].

3.1.2.Radial stress and equivalent plastic strain

The plastic straining at the inner surface of the barrel is also plotted with time in Fig.11.It is observed that with thermal cycling,the material becomes softer and at that instance due to applied mechanical load,it deforms.Tthe magnitude increases with an increase in cycles due to thermal softening at the inner surface.Radial stress actually is a combination of thermo-mechanical cyclic load and the resulting plastic deformation.It is observed that the radial stress is highly compressive at the beginning but is slowly relieved due to thermal cycling.In the last cycle,the maximum radial stress is observed to be around -100MPa.This plastic straining on the internal surface of the barrel is responsible for stress-relieving and narrow hysteresis by the end of the 20th cycle.

In reality,the material undergoes erosion and spalling after reaching a specific plastic deformation and the new surface of the material is exposed to the applied thermo-mechanical loads.The limitation of the current numerical simulation model is that it is unable to comprehend with the removal of the layer after reaching particular plastic deformation value and shows a continuous increase in the equivalent plastic strain deformation.It is due to the absence of any failure criteria definition.However,this might not happen in reality.

3.2.Crack propagation in non-autofrettaged barrel

When a crack is introduced in the barrel,it initially acts as an uncracked ligament during the first heating cycle.Following this,during the cooling cycle,the initial yielding at the highest temperature causes tensile stresses which open the crack.The second cycle causes crack closure (forward yielding) and then during subsequent cooling (second cycle) a more severe crack opening is observed.The J-integral contour plot in Fig.12 represents the development of the crack tip stress field with increasing crack length.The J-integral evolution for individual cracks is provided in supplementary data for further reference.

Fig.11.Equivalent plastic strain and radial stress evolution with cyclic load.It shows the evolution of plastic deformation and radial stress on the internal surface of the barrel due to applied thermo-mechanical cycling.

Fig.12.Individual plots of J-integral are overlaid to plot comparison at crack tips of increasing crack length.

The rise of J-integral in the second cycle as compared to the first cycle is indicative of the residual crack opening created due to the first yielding of the cracked specimen.The numerical simulation of the contour integral shows that the J-integral increases in with increasing crack length.The averaged J-integral peak for each crack length is plotted in Fig.13 and is observed that the increase in Jintegral with increasing crack length is nearly linear.This highlights the fact that the crack initiated in the barrel with continuously increasing with every applied thermo-mechanical cycle up to failure.

Fig.13.Maximum value of J-integral at the crack tip is plotted with crack length and is observed that with increasing crack length there is greater energy absorption at the crack tip due to which the crack growth rate will linearly increase in such a configuration and loading conditions when the barrel is not autofrettaged.The dotted grey line represents the JIC value of 40 kJ/m2 for this material published earlier by Amouzouvi and Bassim [46],it limits critical crack length to 28 mm for the current case.

3.3.Un-cracked barrel with autofrettage

Due to autofrettage,the internal surface of the cannon plastically deforms,and residual compressive stresses accumulate.Due to applied internal pressure which is discussed in the simulation loading definition section,the following stress evolution is observed.

3.3.1.Hoop stress

The evolution of hoop stresses after autofrettage due to applied internal thermal and pressure is interesting.Fig.14 shows the 20 temperature cycles applied and the corresponding evolution of cyclic hoop stress.It is observed that the tensile hoop stress after the first thermal cycle at the internal surface is 700 MPa and decreases in corresponding thermal cycles.In the last cycle,it is around 0 MPa.It is also interesting to note that the compressive hoop stress on the internal surface after autofrettage is 1400 MPa,which drops to 600 MPa during the first cycle but slowly increases in corresponding cycles and reaches a value of 1000 MPa in the last cycle.

To clearly present the evolution of hoop stresses on the inner surface of the barrel due to thermomechanical cyclic load with prior autofrettage,the first and last cycles are separately shown in Fig.14 inset figures.The complicated hoop stress profiles in the first and last cycles can be clearly seen.It is observed that excessive compressive stresses are accumulated on the internal surface of the barrel due to prior autofrettage which are quickly relieved due to excessive thermal load and the hoop stress profile after autofrettage due to load history follows almost the same profile as the barrel without autofrettage.This shows that the hoop stresses on the surface do not play a very significant role in life increasing.However,the residual stresses accumulated in the barrel along with depth from 8 mm to 30 mm actually restrict the crack propagation.This phenomenon is shown and discussed in section 4 of this article.

Fig.14.Comparison of hoop stress evolution during the first and last cycle on the internal surface of non-autofrettaged and autofrettaged barrel.The points of minimum and maximum hoop stress which are of interest are labelled.

The value of compression and tension of 1st,5th,10th,15th and 20th cycle are compared in Table 4 for a better understanding of the trend.It is observed that the produced tensile stress falls and compressive stresses grow with subsequent thermo-mechanical cycling in non-autofrettaged and autofrettaged barrel.However,the result is more pronounced in the case of autofrettaged barrel.

It is observed in Fig.15 that due to applied thermomechanical cycles,the autofrettaged barrel undergoes a large cyclic load up to 8 mm depth,compressive stresses accumulate in the barrel up to 30 mm depth.After that,the barrel has tensile stresses which reach the maximum value at around 50 mm depth.This accumulation of residual stresses is responsible for restricting the crack growth inside the barrel,as is discussed later.The analysis up to this point shows that autofrettage may even have a negative effect on crack initiation.However,the crack propagation would be restricted to some extent.

Hoop stress and temperature hysteresis curve in the autofrettage barrel due to applied thermomechanical cycling are shown in Fig.16.It is observed that the area under the first cycle is even bigger which means the first cycle in autofrettaged barrel is even more damaging.Towards the last cycle,the profile becomes similar to the non-autofrettaged hysteresis curve,which has already been discussed.It is important to note that this curve has only been plotted for the surface.

Fig.15.Hoop stress on the path is plotted at different time points of the 1st cycle max and 1st cycle min.and is compared with the last cycle max and last cycle min.in autofrettaged barrel.

Fig.16.Cyclic stress profile between temperature and hoop stress and provide some interesting information about the material stress history.Although here,the cyclic stress profile is plotted against temperature,it is primarily influenced by mechanical loading as well.During first-cycle,there are a lot of compressive stresses which during later cycles more or less dissipate due to excessive thermal load.The cyclic hoop stress profile follows the same history as the barrel without frottage.

Fig.17.Equivalent plastic strain and radial stress evolution with cyclic load for nonautofrettaged and autofrettaged barrel.It shows the evolution of plastic deformation and radial stress on the internal surface of the barrel due to applied thermomechanical cycling.

3.3.2.Equivalent plastic strain

Radial stress on the internal surface of the barrel during autofrettage and after that due to thermo-mechanical cycling is plotted in Fig.17.Radial stress actually is a combination of thermomechanical cyclic load and the resulting plastic deformation it is observed that the radial stress is highly compressive at the beginning but is slowly relieved due to thermal cycling and in the last cycle,maximum radial stress is observed to be around 200 MPa.It is observed that radial stresses dissipate slower in autofrettaged barrel.

Plastic strain accumulation on the internal surface of the barrel is plotted in Fig.17.High internal pressure on the internal surface of the barrel during autofrettage results in plastic deformation.Thermal cycling after that further assists the plastic deformation and accumulation of permanent deformation on the internal surface.It is observed that the PE accumulation due to thermal-cycling after autofrettage is 250% higher than the prior case without autofrettage load,which is very damaging.

Plastic strain profile in Fig.17 show that the high plastic distortion of the internal surface of the barrel after autofrettage is very prone to fast damage even in the first cycle.Therefore,in a lot of studies,the internal lining of the barrel after autofrettage is suggested [47] which has been shown to protect the internal surface from damage[30-33]potentially.Apart from the damage of a thin internal layer,we are also interested in investigating the possible crack propagation behavior in the barrel during multiple cycles.

3.4.Cracked barrel with autofrettage

The basic contour plot in Fig.18 represents the development of the crack tip stress field with increasing crack length in autofrettaged barrel.The dotted line represents zero marks and it is clear from the graph that the steady-state of crack lengths up to 20 mm are below zero i.e.the crack tip is under compressive load and even if a crack of this length is present in the material,it will not propagate due to residual compressive stresses acting on the crack tip.In ABAQUS v6.12 solver (which was used in the current analysis) J-integral is calculated by formulation proposed by Lei et al.[48].However,its correctness depends on the way stress field is applied and the choice of step used during calculations.

The negative J-integral appears commonly in numerical simulation due to the fully reversed cyclic plasticity condition created at the crack tip.This is a particular property of the cyclic J-integral.During the loading cycle the material deforms plastically in the tensile direction.However,when the material is released,even though the far field stresses remain elastic we can find a residual plastic compression at the crack tip.This shows up as the negative J-integral during simulations.It should be kept in mind that this is actual work done at the crack tip and should not be ignored.Moreover,the magnitude of the ΔJ is considered to be the driving factor behind crack growth.The readers are encouraged to consult Ph.D.Thesis of Masood Shah [49] for further details on this.

The averaged J-integral peak for each crack length is plotted in Fig.19 and is observed that the value of J-integral is zero up to 8 mm crack length for the first cycle and after that increases linearly with increasing crack length.Whereas for the steady-state condition,the value remains below zero up to 30 mm length and increases exponentially after that.

Fig.18.The graphs of J-integral evolution for different crack lengths are compared to present evolution of the values with applied cyclic loading.

Fig.19.J-integral peaks for the first and last cycles are plotted against crack lengths.It can be observed that the barrel will resist crack growth up to 8 mm depth due to residual stresses accumulated after autofrettage.The dotted grey line represents the JIC value of pre-strained AISI 4340 published earlier by Amouzouvi and Bassim [46],it limits critical crack length to 36 mm for the current case.

4.Discussion

Autofrettage is generally used to induce compressive hoop stresses inside thick cylinders for improving their fatigue life during operation.Although the science is known for quite some time and a lot of researchers have worked on improving the structural integrity of thick cylinders manufactured from different materials[1,47]and have developed models for estimating the optimal autofrettage loading and unloading pressures [4,31].The more significant challenge still remains in understanding the service life of these autofrettaged thick cylinders under complex thermo-mechanical loading cycles,i.e.in case of gun barrels which is a complicated problem to study.

In this research,a detailed multi-physics study has been carried out by numerically simulating the solid and cracked gun barrel for 20 thermo-mechanical cycles.The numerical model includes thermal,mechanical and fatigue crack mechanics.Elastic-plastic material data of modified AISI 4340 at temperatures ranging from 25 to 1200°C and at strain rates of 4,16,32 and 48 s-1was acquired from high-temperature compression tests and was assigned to the simulation model.The boundary conditions applied were kept similar to the working gun barrel during firing.A methodology is provided to define interfacial properties between the crack faces,which are assigned thermo-mechanically active surface-to-surface type interface for a better approximation of stresses at the crack tip.Four conditions of stress were explored in this study at 25-950°C and 100-350 MPa pressure during cycling.

i.Thermomechanical loading without autofrettage without crack.

ii.Thermomechanical loading with autofrettage without crack.

iii.Thermomechanical loading in cracked non-autofrettaged barrel with increasing crack length.

iv.Thermomechanical loading in cracked autofrettaged barrel with increasing crack length.

The principle idea was to determine the effect of autofrettage under cyclic thermomechanical loading.It is observed that the general tendency of loading is toward an increase in compressive stress in the uncracked barrel.This may partially be due to residual heating from the preceding thermo-mechanical cycle,but it results in the softening effect of the plastically loaded skin of the barrel.This manifests as the reduced hardness of the barrel internal skin.It is also noted that the maximum stress variation due to thermomechanical cycling is up to 8 mm depth and after that the variation is minimal.This is the active depth for crack initiation sites.In practice,it is observed by other researchers that during firing the erosion of the barrel also happens close to this 8 mm depth in barrels with similar geometries [50].

The very high tensile stress is also seen after the first cycle.It is known through different studies that the first cycle is the most detrimental to thermal fatigue loading,i.e.Oudin et al.[51] conducted a test on engine pistons to determine that the first cycle is the most damaging.Therefore,it is also a common practice to preheat a gun barrel before the first fire in a cold area of operation.

In non-autofrettaged cracked barrel due to applied thermomechanical cycling a reduction of J-integral value is observed with thermal cycling due to the softening of material ahead of the crack tip.Comparison of calculated stress intensity factor (KI) of non-autofrettaged and autofrettaged barrel with varying crack length is shown in Fig.20.However,it must be noted that theKIvalue exceedsKIC(100 MPa m1/2[52,53])at～32 mm and the barrel will probably explode if the crack length reaches this length.The Jintegral value beyond the 32 mm crack length is only academic showing these is no crack arrest condition.The crack arrest region is seen in pure thermal cycling in the works of Shah et al.[54]and Qayyum et al.[25].

The effect of autofrettage on the cracked and uncracked barrel is studied.Due to autofrettage,the internal surface of the barrel plastically deforms and residual compressive stresses accumulate to a certain depth (see Fig.6).The magnitude of accumulated residual compressive hoop stress is nearly 1000 MPa (80% of yield strength of the material is locked).As shown in Fig.14 and Table 4,on the internal surface the tensile hoop stress in autofrettaged barrel at the end of the first cycle is up to 700 MPa as compared to 800 MPa in non-autofrettaged barrel.In the 20th thermomechanical cycle the tensile stress is entirely relieved.

Table 4 Comparison of max.tensile and compressive stresses at on the internal surface of autofrettaged and non-autofrettaged barrel at 1st,5th,10th,15th and 20th cycle.

Fig.20.Comparison of calculated stress intensity factor(KI) of non-autofrettaged and autofrettaged barrel with varying crack length.Dotted lines represent the KIC(100 MPa m1/2 [52,53]) of the material at room temperature and the corresponding crack length.

Why has there been no real effect on the skin of the barrel,even autofrettage at 1000 MPa compression? The answer lies in the compression part of the first cycle.Where the hoop stress went from 800 MPa compression to 1400 MPa compression.This variation of 600 MPa cancels out the effect of autofrettage at the skin of the barrel in the first thermal cycle.The stress assumes a similar profile at the end of the 20th cycles,as in the non-autofrettaged barrel.The effect of autofrettage on the increase in life is known.So how does it affect when the skin is stress relieved as soon as the first cycles?It is observed that there is a significant difference in the values.The plotted trend lines nicely represent that in nonautofrettaged barrel,the average stress is around 150 MPa tensile,whereas in autofrettaged barrel,the hoop stress profile cycles around 600 MPa compressive residual stress.It is shown in Fig.21.Although the stresses on the surface relieve,these internal residual stresses are responsible for restricting the crack initiation and propagation in the barrel.

We know that compression stress exists right up to 40 mm depth.Thus,in this case,if we observe the J-integral up to a 7 mm crack length,we can see compression at the crack tip,measuring the crack will not grow as quickly as in non-autofrettaged barrel.This phenomenon of negative J-integral values has been discussed in detail earlier [49].However,beyond 8 mm crack length,J-integral values become positive.For a better understanding of this residual stress accumulation and its behavior,comparisons of hoop stress profile in non-autofrettaged and autofrettaged gun barrel at 7.4 mm depth are shown in Fig.21.

Fig.21.Evolution of hoop stress in non-autofrettaged and autofrettaged barrel at 7.4 mm depth due to applied thermo-mechanical cyclic load on the surface.

5.Conclusion

A detailed multi-physics numerical simulation model was developed to analyze the effect of carrying crack lengths on the fatigue service life of a non-autofrettaged and autofrettaged gun barrel in service.The high-temperature elastic-plastic material properties of the gun barrel material modified AISI 4340 were measured from testing.The numerical simulation model was run in plane strain condition by applying nearly the same thermomechanical cycling that an actual gun barrel undergoes during continuous firing.The results were analyzed in detail and were compared to get a much better insight into the factors which are responsible for gun barrel failure.The study also helped in analyzing the crack initiation and propagation behavior in the complex loading conditions during the service life of the gun barrel.From this study,it is concluded that:

-Autofrettage does not practically strength the inner surface of the barrel at the extreme of operation conditions experienced by the barrel.It is observed that the stress amplitude at the skin is increased even.However,the autofrettage causes residual compressive stress is not only the skin but the bulk of the barrel.

-This residual compression plays a vital role at the crack tip,limiting arresting the progression of the crack through the thickness.Here seen that up to 8 mm of crack length the autofrettage has a beneficial advantage at increases the life of the barrel.

-This effect is practically manifested as spalling and erosion of barrel along the length of the barrel instead of radial fracture.It increases the life of the barrel by managing the direction of the damage in a preferential direction.

-Crack initiation in autofrettaged barrel occurs just as in regular barrels,but due to a very high compressive stress regime,the propagation of cracks is restricted to a high degree.

-If due to spalling or erosion,some cracks appear on the internal surface,due to thermal softening and very high-stress concentration on the crack tip during firing,they can slowly start growing after reaching 6 mm crack length in autofrettaged barrels.

-After autofrettage,the radial stresses are not primarily affected.However,the plastic straining in the material is accelerated during service;this affects the material softening behavior which is a negative prospect of autofrettage process.

-The developed model and methodology are helpful and can be adopted to analyze complex metaphysical phenomena.

Declaration of competing interest

The authors whose names are listed immediately below certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria;educational grants;participation in speakers’ bureaus;membership,employment,consultancies,stock ownership,or other equity interest;and expert testimony or patent-licensing arrangements),or non-financial interest (such as personal or professional relationships,affiliations,knowledge or beliefs)in the subject matter or materials discussed in this manuscript.

Acknowledgment

The authors acknowledge the DAAD Faculty Development for Ph.D.Candidates (Balochistan),2016 (57245990) -HRDI-UESTP’s/UET’s funding scheme in cooperation with the Higher Education Commission of Pakistan (HEC) for sponsoring the stay at IMF TU Freiberg,Germany.The authors gratefully acknowledge the support of staff at Fracture Mechanics and Fatigue Laboratory,Mechanical Engineering Dept.UET Taxila,Pakistan,for preparing samples and in supporting the high-temperature testing.