Effect of interface behaviour on damage and instability of PBX under combined tension-shear loading

Quan-zhi Xia,Yan-qing Wu,Feng-lei Huang

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

Keywords: PBX explosives Interface behaviour Damage Instability Crack propagation

ABSTRACT To study the effect of interface behaviour on the mechanical properties and damage evolution of PBX under combined tension-shear loading,the present work establishes the numerical model of a PBX three-phase hybrid system,which introduces a nonlinear plastic damage cohesion model to study the mechanical response and damage process.The parameters in the model were fitted and calibrated.Taking the crack growth rate as the feature,the damage state in each stage was determined,and the damage instability criterion was given.The effects of interfacial tensile strength and shear strength on the damage process of PBX were studied.On this basis,serrated and hemispherical structures interface of PBX has been developed,which affects the damage process and instability during the loading process.The results indicate that damage state response of PBX experiences the process of stable load bearing,unstable propagation,and complete failure.At the critical moment of instability,the overall equivalent effective strain of material reaches 3024 με and instability loading displacement reaches 0.39 mm.The increase of interfacial tensile strength and shear strength significantly inhibits the damage of PBX.The effect of interfacial shear strength on critical instability of PBX is approximately 1.7 times that of the interfacial tensile strength.Further,interface opening along the normal direction is the main damage form at the interface.Serrated and hemispherical rough interfaces can significantly inhibit propagation of cracks,and the load bearing capacity is improved by 22% and 9.7%,respectively.Appropriate improvement of the roughness of the interface structure can effectively improve the mechanical properties.It is significantly important to have a better understanding of deformation,damage and failure mechanisms of PBX and to improve our predictive ability.

1.Introduction

Polymer-bonded explosives (PBXs) are widely used in strategic and advanced conventional weapons because of their good mechanical strength,high-energy and low-sensitivity performance[1,2].PBX is a highly filled multiphase composite.It has many interfaces and extremely complex structure due to its specific components and the complexity of its moulding process.The transition region where the physical and chemical properties of materials abruptly change is an important part of composites.Studies have shown that interface debonding is the main reason for the cracking of PBX parts during transportation and storage [3,4].When the explosive is under external complex loads,weak interface action causes the interface to be easily subjected to tensile and shear loads,resulting in interface debonding damage,which degrades the mechanical properties of the explosive and reduce the bearing capacity of the material.Therefore,the study of the interfacial properties between explosive particles and the binder under combined tension-shear loading is important for the evaluation and improvement of the mechanical properties and reliability of PBX.

The mechanical properties of PBX cannot be separated from the interface behaviour,such as force transmission during loading and the evolution of the fine structure occur through the interface.Furthermore,the structure,content,and strength of the interface considerably affect the fine structure and macroscopic mechanical properties of PBX materials[5].On the premise of determining each component of a PBX system,the interfacial bonding between explosive particles and the binder has become the key factor affecting the mechanical properties of PBX[6,7].Liu Yong et al.[8]measured the contact angles of TATB and different binders;they calculated the interface property parameters and analysed the interface properties using X-ray photoelectron spectroscopy,confirming that the interface effect on PBX mechanical properties is not negligible.Xiao et al.[9,10]studied the interaction between explosive crystal and binder by the molecular dynamics method.The interface structure and strength were found to be the key factors affecting the elastic modulus and ductility of materials.Duan et al.[11]conducted tensile strength tests on two explosives and found that the tensile strength at the interface was between the intrinsic tensile strength of both explosives;this confirmed the effect of the interface on the properties of PBX from another perspective.When PBX is subjected to an external load,the strength of the interface between explosive crystal and binder largely determines the properties of the PBX material.Therefore,in the production and design of explosives,the interface effect is often enhanced by adding a coupling agent.Li et al.[12]found that the interfacial tension between TATB modified by a coupling agent and fluoropolymer was considerably reduced,interface bonding work increased,and the coupling agent could form a strong interaction with TATB.Wu et al.[13]used the insulation ratio test and in situ observation of tensile scanning electron microscopy(SEM)to study the effect of the bonding agent on the propellant interface,and they found that the bonding agent formed a high-modulus intermediate phase around the ammonium nitrate particles,improving the tensile properties of the propellant and effectively solving the interface dewetting problem.Therefore,the key to enhance the mechanical properties of PBX is to improve the interfacial strength[14,15].

Moreover,the interface is closely associated with PBX damage and crack propagation as well as determines the failure mechanisms of PBX.The formation and propagation of cracks often degrade the mechanical properties of PBX,or even lead to its failure,thereby affecting the reliability and stability of the whole weapon system.Chen [16]showed that the main damage form of PBX is interface debonding.Interface damage leads to the formation of holes and cracks.New holes may also become a new hot spot source of PBX under abnormal conditions and reduce its safety properties.Chen et al.[17]studied the interface action between explosive crystals and different binders and proved that irreversible damage is more likely to occur in the material under weak interface action.Li et al.[18]studied the fracture damage characteristics of PBX under tension by using the Brazilian test,an indirect tensile loading method,and obtained the evolution process of fracture damage morphology.It was found that the interface is the weakest part of PBX explosive,debonding is the primary damage mode,and the crack tends to expand along the direct interface of large particle crystals.Therefore,the interface effect cannot be ignored in the damage process of PBX and is difficult to study at this stage.At present,scholars worldwide have extensively studied the mechanical properties of explosive crystals and binders;However,the effect of interface on the mechanical properties and damage evolution of PBX has not been sufficiently explored.The study of the effect of interface is challenging owing to the high energy sensitivity of the explosives crystals to micro-mechanical tests associated with the PBX interface.The PBX interface structure is complex and affected by many internal and external factors.The changes in the interface structure and properties can cause different degrees of damage,which in turn affect the overall mechanical properties of PBX.Currently,few relevant studies focus on this topic and there is lack of statistical law characterization of PBX mechanical properties and damage features.The relevant research of PBX damage at home and abroad mainly describes the phenomenon in the process of damage,but the accumulation of PBX damage cannot be quantitatively described,and there is no accurate definition of all stages in the process of PBX damage.In this work,the crack expansion rate is extracted as the feature,which determines the damage state of various stages,and gives the PBX damage instability criterion under combined tension-shear loading,which can effectively determine the damage of different degrees.

Based on the above discussion,this study mainly investigates the effect of interface action on the mechanical properties and damage evolution of PBX.The present work establishes the numerical model of a PBX three-phase hybrid system,which introduces a nonlinear plastic damage cohesion model to study the mechanical response and damage process.The parameters in the model were fitted and calibrated.Taking the crack growth rate as the feature,the damage state in each stage was determined,and the damage instability criterion was given.The effects of interfacial tensile strength and shear strength on the damage process of PBX were studied.On this basis,serrated and hemispherical structures interface of PBX has been developed,which affects the damage process and instability during the loading process.

2.Simulation model

2.1. Model description

In the process of loading,storage and transportation,weapons will be subject to various loads from the outside environment,and the internal PBX will be subjected to complex loads.In this case,owing to the weak interface effect,the interface is easily subjected to tensile and shear loads,resulting in interface debonding damage,which degrades the mechanical properties of the explosive and reduces the bearing capacity of the material.In this study,based on the tensile and shear test designed by China Academy of Engineering Physics,the tensile and shear loading test model of PBX is established.Fig.1 shows the tension-shear experiment device and various component.

Fig.1.Fixture device of tension-shear loading experiment;1-Arc connecting piece;2-Specimen bonded joint;3-Sample mounting fixture seat;4-Sample [19].

The sample size is Φ 20 mm×40 mm,the internal particles are TATB crystals,and the matrix is a fluoropolymer binder.The combined tension-shear loading test device can meet the test requirements of pure tension,pure shear,and tension-shear on PBX grains.When the beam of the material testing machine applies load to the fixture,the tensile load and shear load on the sample are,respectively,

whereFis the loading load of the beam,Fnis the axial tensile force of the sample,Fτ is the radial shear force of the sample,andais the angle between the axial direction of the sample and the load direction.Using the device,mechanical loading tests at six different angles,namely 0°，30°，45°，60°，75°，and 90°,are performed;here 0°is defined as pure tension and 90°as pure shear.In this study,45°is selected for tension-shear loading simulation.The test temperature is 25 ℃ and the loading rate is 0.5 mm/min.The PBX adopts a fine view structure model,in which the content of TATB accounts for 90% and that of the binder accounts for 10%.TATB particle sizes range between 0.3 and 0.6 mm and randomly distributed.All particle shapes are random polygons.The 0.1 mm triangular mesh is used for finite element meshing.The mesh element attribute is CPS3 three node plane stress triangular element.After meshing,the cohesive element is embedded in the crystal,binder,and interface.To effectively correlate material fine structure with macroscopic mechanical properties,representative volume unit (RVE) is a commonly used method,specifically in energy-containing materials [20-25],fibre-filling materials [26],and other particle-filling materials[27].To simplify the model,the finite element model of the PBX mesoscopic structure shown in Fig.1 is established,and the PBX mechanical response and damage evolution under tension-shear loading is analysed.Further,the boundary conditions are illustrated in Fig.2.According to the Tension-shear experiment,symmetric periodic boundary conditions are added to the left and right boundaries of the model,and the displacement load is added to the upper and lower boundaries.The angle of the loading load from the axial direction is 45°.

Fig.2.Microstructure model of PBXs under combined tension-shear loading.

2.2. Cohesive force model of nonlinear plastic damage

The cohesive force model,which was developed by Dugdale and Barenblatt [28],is widely used for the description of material damage and fracture damage characteristics.It is based on the principle that the cohesion unit shows an elastic-softening-failure process [29].Based on this,through expansion,the binder crack model is directly applied to the material calculation grid and embedded in the boundaries of each entity unit to realize the expansion of any crack.Because the required material parameters are simple and have clear physical significance,it can describe the cracking or expansion of the tensile crack at any location as well as the material softening behaviour caused by crack expansion.

The analysis of quasi-brittle materials such as PBX explosive shows that its failure process involves gradual transition of microcracks to macroscopic cracks and fracture.Note that the force and displacement of the cohesion unit under stretching and complex loading are not completely linear.By contrast,by the initial linear elastic deformation,through the nonlinear plastic reinforcement phase and strain softening phase,until the complex process of macroscopic crack formation,the bilinear cohesion model and elastic damage cannot accurately describe the mechanical damage evolution behaviour of the three-phase materials of PBX.

Plastic mechanics and damage mechanics are commonly used theories to describe the nonlinear damage of materials.The former can simulate material plastic deformation,while the latter can simulate the decline of material stiffness,and combine the two into plastic-damage mechanics,which can more fully describe the nonlinear damage process of materials.The cohesion unit combined with the physical unit can simulate the cracking process of various materials.Although,currently,there are plastic damage mechanics of the physical unit,the plastic damage model of the cohesion unit does not exist.The cohesion unit usually uses a bilinear constitutive model to characterize the linear elastic damage of the internal cohesion unit,which cannot accurately describe the plastic damage process of the PBX material.This study improves the constitutive DEM model established by Nguyen [30]to express the cohesion unit plastic damage.The relative displacement u (un,us) is first decomposed into elastic and plastic components to represent reversible and irreversible displacements in a bonding contact.

The stress state in this contact includes normal and shear contributions,which are linked to relative displacements of two particles in contact by the following equations,respectively.

where σnand σsare the normal and shear stresses in the bonding contact;unandare the total and plastic displacements in the normal direction;usandare the total and plastic displacements in the shear direction;andandare the initial normal and shear stiffness values,respectively.

Damage is controlled by plastic deformation

upis the plastic displacement anducis the softening parameter indicating the degree of corrective damage.The complete coefficient of the corresponding damage coefficientDis

To satisfy the above requirements,a hyperbolic criterion is proposed based on the yield function originally developed for single crack development in quasi-brittle materials [31,32].The proposed yielding function takes the following form:

The yield equation

According to the non-associate flow rule,the potential function is then

The plastic strain increment can be expressed as

The initial test stress is substituted into the yield equation.To calculate the plastic damage forF＞0,the following plastic damage multiplier is counted:

The plastic displacement is calculated,and the stress regression value is obtained after updating the damage coefficient.

2.3. Determination of the model parameters

The elastic-plastic model is selected for the explosive crystal and the viscoelastic-plastic model is selected for the binder.A,N,m,andfare the viscoelastic-plastic material parameters,which are obtained by fitting the test results of PBX hollow hemispherical vertex deformation [33].Other parameters are obtained from literature [34-36].The specific parameters are shown in Table 1.

Table 1 Material parameters.

Combined tension-shear loading can be decomposed into the effect of joint action of tension and shear loading.At loading angle 0°,it is pure tension,whereas at 90°,it is pure shear load.The load displacement curve of PBX under pure tension and pure shear measured by the simulation and test is shown in Fig.3,and the calibrated PBX material parameters are shown in Table 2.

Table 2 Cohesion model parameters.

Fig.3.Comparison of force and displacement curves under test and simulation conditions.

Fig.4 shows the comparison of the simulation and test results for a loading angle of 45°;the trend of the two curves is similar and the results are consistent.Therefore,the model can accurately characterize the mechanical response and damage process of PBX under combined tension-shear loading.

Fig.4.Curves of force and displacement under tension-shear loading.

2.4. Analysis of results

Fig.5 is the stress state of PBX at different times for a loading angle 45°.During the loading process,the internal stress of PBX gradually increases,and multiple stress concentration zones are formed in the binder.The stress concentration is obvious at the dense distribution position of particles.The damage and debonding phenomenon first appear at the interface.Due to the weak bonding strength,small microcracks are formed along the boundary of explosive particles,which gradually accumulate and expand.With increasing stress,the interfacial bonding crack continues to expand and spread into the binder.In the process of crack propagation,the stress concentration in the adjacent area is released,making other microcracks to close or no longer expand.Multiple cracks in the binder extend and merge with each other,and finally two obvious main cracks are formed.At this time,the material has completely lost its bearing capacity and reached complete failure.

Fig.5.Equivalent stress contours of PBX sample under tension-shear loading (MPa): (a) 5 s;(b) 15 s;(c) 30 s;(d) 45 s.

Fig.6 shows the damage distribution of PBX at different positions under tension-shear loading.The crack propagation starts along the interface and continues to expand in the binder.Fig.7 shows the internal stress distribution of the explosive crystal and binder at different times.It can be seen that in the same area,the maximum stress in the binder is greater than that in the explosive crystal,and the strain in the binder is generally greater than that in the crystal.Therefore,the binder is not only a high stress area but also a high strain area.Under the tensile-shear combined loading,stress concentration is more likely to occur in the binder.

Fig.6.Damage distribution at different locations of the PBX sample (t=28 s).

Fig.7.Strain and stress distribution in the crystal and binder (t=28 s).

3.Judgment of the damage interval

Under tensile-shear combined loading,many cracks appear in PBX,and there is a large range of damage zone formed by many microcracks at the end of the main crack.As the load increases,the scope of damage zone expands and the number of microcracks increases.When the load reaches a certain degree,microcracks are fully developed,and the shape and size of the damage zone reach a stable state.That is,when the microcrack in the damage area develops to a certain extent,the main crack reaches the critical state of damage.If the maximum length of the damage zone exceeds a certain critical value,the main crack expands unsteadily.At this time,the material is in an unstable state and no longer has the bearing capacity,which eventually leads to the failure.It is significantly important to have a better understanding of deformation,damage and failure mechanisms of PBX and to improve our predictive ability.

Fig.8 shows the variation of maximum principal stress with crack propagation under tension-shear loading;Fig.9 shows the change of crack propagation during loading,and the effective judgment of each damage interval.According to the crack propagation law and the change of the maximum principal stress during loading,the mechanical response and damage process of PBX can be divided into the following areas:

Fig.8.Curve of maximum principal stress during crack expansion.

Fig.9.Determination of damage state interval under tension-shear combined loading.

(1)In section OA,with the tensile-shear load loading,the stress in the material increases continuously;However,there is no crack in the model because the damage condition has not been reached.(2) In section AB,when the loading time is 11 s,the loading displacement reaches 0.09 mm,and microcracks begin to appear in the material.At this time,the maximum principal stress of the model reaches 5.9 MPa,and the crack begins to grow at a uniform speed and stably.The material is in the linear damage stage;the damage increases steadily,and A is the crack initiation point.(3)In section BC,when the loading time is 24 s,the loading displacement reaches 0.2 mm,the crack length is 11.2 mm,the damage begins to increase unsteadily,and the crack shows a step-by-step increasing trend.Its manifestation is that after the crack suddenly expands,there is a short slow increase state,and then suddenly increase,which is due to the energy accumulation and release at the crack tip.When the crack propagation condition is not reached,the energy at the crack tip continues to accumulate.Once the energy accumulation reaches the fracture condition,it is suddenly released and then begins to accumulate again.This law is consistent with the crack energy release criterion.At this time,although the model still has a certain bearing capacity,it is unstable.(4)In section CD,when the loading time is 42s,the loading displacement is 0.35 mm and the crack length is 24.7 mm.At this time,the crack has a short and slow propagation state.The reason is that with the crack propagation,some stress is released in local areas,eliminating a certain degree of stress concentration and reducing the crack propagation rate.This stage is short and in the critical area before critical instability.(5) In section DE,when the loading time is 46 s,the loading displacement is 0.39 mm and the crack length is 25.3 mm.At this time,the crack growth rate suddenly increases,and the material is in a complete instability state.The bearing capacity of material decreases sharply,and point D is the critical instability point.(6) In section EF,when the loading time is 52 s,the loading displacement reaches 0.44 mm and the length of propagating crack is 40.3 mm.The material has reached failure and no longer has bearing capacity.It can be seen that under the tension-shear loading,the critical damage initial displacement of PBX is 0.09 mm.The critical instability displacement is 0.39 mm,and the critical instability crack length is 25.3 mm.At the critical moment of instability,the overall equivalent effective strain of material reaches 3024 με,while the strain at the crack tip is much larger than this value.

To describe the mechanical deformation of PBX,Fig.10 and Fig.11 show the effective strain ∈11 and ∈22 of PBX samples during tension-shear loading.These results indicate that during loading,the strain at the binder is much greater than that inside the crystal,and the binder absorbs the main deformation in PBX.During the loading process,the softer binder coated the crystal and absorbed its deformation,weakened the interaction between grains,and then reduced the stress concentration of PBX.Comparing the radial strain and axial strain of PBX,it can be seen that the axial strain of the sample is much greater than the radial strain.The crystals in PBX are randomly and evenly distributed and are more prone to large deformation in the axial direction.When the large axial deformation is transmitted to the crystal interface,the opening separation damage and tangential sliding dislocation damage occur at the interface.The two kinds of damage work together to cause damage failure at the interface.

Fig.10.The effective strain ∈11 of PBX sample.

Fig.11.The effective strain ∈22 of PBX sample.

To further study the damage behaviour of the model under tension-shear loading,the normal plastic displacement and tangential plastic displacement of the cohesive element in the model at the same time are extracted.As shown in Fig.12 and Fig.13,the cohesive element is not damaged in the elastic stage,and the damage continues to accumulate in the plastic stage.Both the normal plastic displacement and tangential plastic displacement reach the maximum at the initial end of the crack.The normal plastic displacement is higher than the tangential plastic displacement,and the plastic work in the normal direction is greater.Therefore,interface opening displacement is the main damage form at the interface.

Fig.12.PBX normal direction plastic displacement distribution.

Fig.13.PBX tangential direction plastic displacement distribution.

To deeply study the damage distribution at the crystal interface,as shown in Fig.14,we select two paths,respectively,along the interface and interface normal directions.At the same time,some nodes were selected separately along these two paths to analyze the damage situation.The results are shown in Fig.15 and Fig.16.At different loading times,the damage range along the interface direction gradually increases outward,and the range of the interface debonding constantly expands.In the interface normal direction,due to the selected node 1 and node 2 are inside the crystal,no damage in the loading process.When the loading time is 30s,the damage accumulation gradually occurs at this position,but still in the initial stage of damage.This also demonstrates from another perspective that no cracks occur inside the crystal under combined tension-shear loading.The weakest part is in the middle of the interface.

Fig.14.Distribution of nodes at the crystal interface.

Fig.15.Distribution of damage along the interface direction.

Fig.16.Distribution of damage along the interface normal direction.

4.Effect of interface properties on PBX damage

The effect of interfacial mechanical behaviour on the damage properties of PBX is considerable.To study in depth the effect of interfacial strength on the mechanical properties and damage state of PBX,when the included angle between load and the sample axis is 45°,the interfacial tensile strength is 0.5 MPa,0.75 MPa,1.0 MPa,1.25 MPa,and 1.5 MPa,and the corresponding interfacial shear strength is 0.8 MPa,1.0 MPa,1.2 MPa,1.4 MPa,and 1.6 MPa.The effects of interfacial tensile strength and shear strength on PBX damage are evaluated.The calculation results are as follows.

Fig.17 and Fig.18 indicate that different interfacial tensile strengths affect the distribution of damage and instability regions of PBX materials.When the interfacial tensile strength increases from 0.5 to 1.5 MPa,the PBX crack damage initiation displacement gradually increases from 0.078 to 0.15 mm,the critical instability displacement increases from 0.29 to 0.57 mm,and the critical instability crack length increases from 22.3 to 32.7 mm.With increasing interfacial tensile strength,the crack propagation length fluctuates more obviously,and the law of energy release at the crack tip is more significant.At this time,when the PBX structure reaches failure,the full length of the crack is smaller.

Fig.17.Variation curve of maximum principal stress with crack propagation under different tensile strength.

Fig.18.Variation curve of loading displacement with crack propagation under different tensile strength.

Fig.19 and Fig.20 indicate that with the increase of interfacial shear strength,the critical instability length of crack also gradually increases.With the increase of interfacial shear strength from 0.8 to 1.6 Mpa,the initial displacement of PBX crack damage gradually increases from 0.05 to 0.18 mm.The critical instability loading displacement increases from 0.3 to 0.69 mm,and the critical instability crack length increases from 23 to 33 mm.

Fig.19.Variation curve of maximum principal stress with crack propagation under different shear strength.

Fig.20.Variation curve of loading displacement with crack propagation under different shear strength.

Fig.21 shows the relationship between different interfacial strength and critical instability loading displacement.With the increase of interfacial tensile and shear strength,the PBX damage initiation displacement and critical instability displacement also increase.The effect of interfacial shear strength on the critical instability loading displacement of PBX material is greater than that of interfacial tensile strength.The slope of curve K1 is 0.29 and that of curve K2 is 0.49.Therefore,the effect of interfacial shear strength on the critical instability of PBX is approximately 1.7 times that of interfacial tensile strength.The enhancement of both can significantly inhibit the damage propagation of PBX.

Fig.21.Variation curve of the critical unstable loading displacement with interface strength of the PBX sample.

According to the above,the PBX damage process can be divided into stable damage,critical instability state,and instability damage.The interfacial tensile strength is lower than shear strength,and the effect of interfacial shear strength on critical instability of PBX is greater.The material is more prone to damage in the axial direction.It is assumed that the included angle between the tensile and shear load and the axial direction of the material is a and the initial strain of axial damage is ε0,the critical strain of instability is ε1.If the loaded displacement isS,the axial displacement isS·cosaand the positive strain isS·cosa/L.the following damage instability criteria can be derived:

(1) WhenS·cosa/L＜ε0,the material has no damage,and it possess a stable load-bearing capacity.

(2) When ε0≤S·cosa/L＜ε1,the material begins to damage.The crack expands stably,and the bearing capacity of the material decreases gradually.

(3) WhenS·cosa/L＝ε1,the material is in the critical state of crack instability propagation.

(4) WhenS·cosa/L＞ε1,the crack is in an unstable propagation state and the material loses its bearing capacity.

5.Effect of interface structure on PBX damage

Due to there are many interfaces in PBX and the interaction between interfaces is weak,interface debonding is the most common failure form.To further enhance the interface interaction and increase the interfacial tensile strength and shear strength,this section is designed to process the crystal surface into a rough interface.To study the effect of the rough interface on the PBX mechanical properties and damage response,the rough interface structure is set to (1) a serrated interface structure and (2) a hemispherical interface structure;the PBX models shown in Fig.22 are established.

Fig.22.Serrated and hemispherical interface enhancement structures of PBX.

The calculation results are shown in Fig.23.Comparison of the load and displacement curves of PBX with the two interface structures.The hemispherical and serrated interface structures can significantly improve the critical failure load and critical failure displacement of PBX,inhibiting PBX damage extension.

Fig.23.Variation curve of displacement with load under different structures.

Fig.24 and Fig.25 show the results of the PBX numerical calculations for the different interface structures.Compared with that of the smooth interface structure,the crack critical instability length of hemispherical and serrated rough interfaces increases obviously under tension-shear loading.When the interface is smooth,the initial loading displacement of PBX crack damage is 0.09 mm,while the hemispherical structure and the serrated structure increases to 0.15 mm and 0.19 mm,respectively.Comparing the load-bearing ranges of PBX with different interface structures,when the internal interface of PBX is smooth,the critical instability loading displacement is 0.38 mm,while those for the hemispherical and serrated interface structures increase to 0.69 mm and 0.82 mm,and the critical instability crack length increases from 25 to 30.4 mm and 33 mm,respectively.Thus,the hemispherical and serrated structures are confirmed to effectively improve the bonding strength of the PBX interface and have a longer and stable load-bearing range.

Fig.24.Variation curve of maximum principal stress with crack propagation under different interface structures.

Fig.25.Variation curve of loading displacement with crack propagation with different interface structures.

In addition,the maximum load that the smooth interface structure PBX can bear is 341.3 N,whereas that of the hemispherical structure interface is 374.5 N,which is a 9.7%improvement.The maximum load that PBX with the serrated structure interface can bear is 416.3 N,which is a 22% improvement.Comparison of the properties of PBX with the two interface structures indicates that the interface interaction of the serrated structure is the strongest.The reason for this phenomenon is that the rough interface structure increases the bonding area between the crystal and the binder,it produces the self-locking effect on the interface under tension-shear loading,which significantly increases the interface strength[37-39].

For the models of different interface structures,we selected the same interface position to obtain the damage distribution in the interface direction and normal direction.The results are shown in Fig.26 and Fig.27.The serrated and hemispheric structural interface can effectively shorten the interface damage area and slow down the damage accumulation rate of the unit.The effect of inhibiting the interface damage is obvious.Compared with the hemispherical interface,the PBX with a serrated structural interface has a larger carrying range and a smaller damage interval.

Fig.26.Distribution of damage along the interface direction.

Fig.27.Distribution of damage along the interface normal direction.

To quantify the internal damage of PBX,Fig.28 shows the statistical distribution of damage units at different loading times.The smooth interface structures increase significantly,while the hemispherical and serrated interface structures increase more gently.Damage unit content also increasing over time.When the loading time is 40 s,the proportion of damaged elements in PBX of three interface structures exceeds 1.5%.

Fig.28.Damage statistics of PBX with different interface structures.

6.Effect of interface roughness on PBX damage

To further study the effect of interface roughness on the properties of PBX,the serrated structure interface PBX is tested,and the distance h between the highest point and the lowest point of the interface is varied (Fig.29).For largerh,the roughness at the interface is higher.The calculation results for h values of 0.10 mm,0.14 mm,and 0.18 mm are shown in Fig.30 and Fig.31.

Fig.29.Roughness of serrated and hemispherical interface structures of PBX.

Fig.30.Variation curve of maximum principal stress with crack propagation under different interface roughness.

Fig.31.Variation curve of loading displacement with crack propagation under different interface roughness.

From the calculation results we can conclude that the interface roughness varies will affect the PBX crack damage and instability propagation.When the interface fluctuation height differencehincreases from 0.1 to 0.18,the interface roughness increases gradually.Under tension-shear loading,the crack critical instability length increases significantly.The PBX crack damage initiation displacement increases from 0.13 to 0.25 mm,the critical instability loading displacement increases from 0.69 to 1.02 mm,and the critical instability crack length increases from 29 to 37.5 mm.It can be seen that the effect of the interface structure on the mechanical properties and damage process of PBX cannot be ignored.Appropriate improvement of the roughness of the interface structure in engineering can enhance the interaction between interfaces,thereby effectively improving the tensile and shear strength of PBX.

Fig.32 shows the statistics of PBX damage content with different interface roughness.With the increase of roughness at the interface,the damage content in PBX will decrease.Moreover,when the roughness becomes larger,the growth rate of damage element content will become slow during loading.It is also proved that properly increasing the interface roughness can effectively inhibit the damage accumulation of PBX.

Fig.32.Damage statistics of PBX with different interface roughness.

7.Conclusions

To study the effect of interface behaviour on the mechanical properties and damage evolution of PBX under combined tension-shear loading,the present work establishes the numerical model of a PBX three-phase hybrid system,which introduces a nonlinear plastic damage cohesion model to study the mechanical response and damage process.Taking the crack growth rate as the feature,the damage state in each stage was determined,and the damage instability criterion was given.The effects of interfacial tensile strength and shear strength on the damage process of PBX were studied.On this basis,serrated and hemispherical structures interface of PBX has been developed,which affects the damage process and instability during the loading process.It is significantly important to have a better understanding of deformation,damage and failure mechanisms of PBX and to improve our predictive ability.

The main conclusions are as follows:

(1) Under the tension-shear loading,the state response of PBX experiences the process from stable bearing,damage extension and failure.When the microcrack develops to a certain extent,it will reach an unstable critical state.If the damage crack length extends beyond the critical value,the crack expands unsteadily.At this time,the material is in an unstable state and no longer has stable bearing capacity,resulting in the complete failure of the structure.The initial loading displacement of PBX damage is 0.09 mm.At the critical moment of instability,the overall equivalent effective strain of the material reached 3024 με at the critical instability.The strain at the crack tip is much larger than this strain.Further,the critical instability displacement is 0.39 mm,and the critical instability crack length is 25.3 mm.

(2) The interfacial tensile strength and shear strength affect the distribution of damage and instability ranges.With the increase of interfacial tensile and shear strength,the PBX damage initiation displacement and critical instability displacement also increase.The variation curve of crack growth rate fluctuates more obviously,and the law of energy release at the crack tip is more significant.The effect of interfacial shear strength on the critical instability of PBX material is approximately 1.7 times that of interfacial tensile strength.The enhancement of both can significantly inhibit the damage propagation of PBX.Interface opening along the normal direction is the main damage form at the interface.

(3) According to the crack growth rate,the damage instability criterion under tension-shear combined loading is proposed.WhenS·cosa/L＝ε1,the material is in the critical state of crack instability propagation;whenS·cosa/L＞ε1,the crack is in an unstable propagation state and the material loses its bearing capacity.Rough structure interface can significantly improve the bonding strength of the interface,and inhibit the damage.Due to the contact interface increases and the self-locking phenomenon at the interface，it has a longer and more stable bearing range.Compared with the smooth interface structure,the bearing capacity of the hemispherical interface structure was improved by 9.7% and that of the serrated interface structure was improved by 22%.Properly improving the interface roughness can effectively inhibit the internal damage propagation of PBX.

In the future,we expect to introduce fracture into the calculations as an implicit damage.The revised model should provide better predictions.The proposed models have still to be verified and validated under a complex loading configuration based on more mesoscale measurements on crystals and the polymer binder.Crystal anisotropy,microstructural heterogeneity and viscosity of the polymer binder have a great influence on the mechanical response of PBX.Moreover,statistical works need to be done to have a better understanding the mechanical response of PBX samples at both scales.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to thank the China National Nature Science Foundation (Grant No.11872119),China Postdoctoral Science Foundation (Grant Nos.BX20200046,2020M680394),and Pre-research Project of Armament(Grant No.6142A03202002) for supporting this project.