Le Tang.Chao Ge.Huan-guo Guo.Qing-bo Yu.Hai-fu Wang

Beijing Institute of Technology.China

Keywords: Al-PTFE Granular composites Mesoscale simulation Dynamic response Force chains

ABSTRACT Force chains based mesoscale simulation is conducted to investigate the response behavior of aluminumpolytetrafluoroethylene (Al-PTFE) granular composites under a low-velocity impact.A two-dimensional model followed the randomly normal distribution of real Al particles size is developed.The dynamic compressive process of Al-PTFE composites with varied Al mass fraction is simulated and validated against the experiments.The results indicate that,force chains behavior governed by the number and the size of agglomerated Al particles.significantly affects the impact response of the material.The failure mode of the material evolves from shear failure of matrix to debonding failure of particles with increasing density.A high crack area of the material is critical mechanism to arouse the initiation reaction.The damage maintained by force chains during large plastic strain builds up more local stresses concentration to enhance a possible reaction performance.In addition.simulation is performed with identical mass fraction but various Al size distribution to explore the effects of size centralization and dispersion on the mechanical properties of materials.It is found that smaller sized Al particle of composites are more preferred than its bulky material in ultimate strength.Increasing dispersed degree is facilitated to create stable force chains in samples with comparable particle number.The simulation studies provide further insights into the plastic deformation.failure mechanism.and possible energy release capacity for Al-PTFE composites.which is helpful for further design and application of reactive materials.

1.Introduction

Reactive materials (RMs) have potential applications in both military and civilian fields owing to their dual function of mechanical properties and energetic performances [1,2].Al-PTFE granular composites is a special class of reactive material which is prepared by mixing.molding and sintering of metal powders into polymer binder [3].Unlike conventional energetic materials.such as explosives or propellants,Al-PTFE composites have high energyreleasing capability under dynamic loading while remain inert in ambient conditions.It can be fabricated into reactive elements or reactive fragments which would react upon high rate intensive loading to undergo bulk disintegration [4].Thus.the potential chemical energy is usually not self-sustaining,and released mainly depending on the mechanical behaviors of materials.

The Al-PTFE reaction has been described as initiation at lowvelocity loads [5,6].and opening cracks driven by quasi-static compression [7,8]or drop weight tests [9,10]are deemed as prerequisite for Al-PTFE material to go through reaction.However,the mechanical responses of granular materials are governed by the grain properties and microstructure[11].In this case,initiation sites within the materials are ascribed to the stress distribution or force chains formation.Force chains served as a link across scales and a pivotal mechanism.solving the mechanics of granular materials,have been studied by several methods.However.most of the proposed methods are more applicable to materials of discrete solid particles.which refers to rock or soil particulate matter.whereas little research is made to consider the effects of force chains on the mechanical and chemical properties of RMs.

The mechanical performance of Al-PTFE composites attracts significant attention due to its relationship with the performance of reactivity.Attempts are made to tailor and optimize structures ofAl-PTFE composites to meet the requirements of desirable mechanical properties while facilitating the chemical energy release.The strongest effect is obtained by altering the material compositions such as mass fraction [12]or particle size [13].It should be emphasized that the plastic deformation.failure behaviors.and even chemical reaction of these materials subjected to dynamic loading are extremely complex[14],and whichever picture is most associated with the microscale characteristics of the granular composites.Direct observations with current research methods,such as theoretical analysis or experiments,are limited in the timeresolved problem and contribute little to explore the processes leading to initiation reaction in real time.As such.simulation methodology is needed and used as an effective technique to reveal the detailed deformation mechanism at particle level.which is account for initiation phenomena observed in experiments.

Some simulation work has studied the mechanical response of granular composites focusing on the effects of volume fraction and matrix damage parameters[15],the metallic particle size[16],and shear locations [17].However.particular sizing techniques used in these simulations implicitly assumed that particles are desired equal geometry.Particle shape.as one of the most important microscopic characteristics.significantly affects the material strength and reaction sensitivity[18].Further improved simulation methods followed a real morphology distribution.such as those based on microstructure images[19],statistical sense[20],electron backscatter diffraction [21].and representative volume elements[22].were conducted to investigate the compressive behavior of granular composites.Though some advancements have been achieved on modelling methods.such as extended and scaled boundary finite element method [23,24].these researches mainly focus on numerical analysis and do not have definitive descriptions on how the particles behavior affect the mechanical properties,and it also not explains the fundamental reaction mechanism associated with the compression process.

In order to reveal the reaction mechanism in the microscopic level.detailed analysis of agglomerated particle behaviors of composites is needed to understand the complex compressive process.In this paper.a mesoscale simulation of Al-PTFE granular composites based on real particle size distribution under a lowvelocity impact is presented and validated by the experimental results.The simulation is developed to explore how the dynamic response is affected by varying the mass fraction and Al size distribution,which also determines the energy-releasing capacity.The overall strength.failure mode.and fracture behavior of Al-PTFE composites are the primary point of investigation.The stressstrain curves and Von Mises stress of Al-PTFE are calculated to describe the mechanical performance.in which force chains are introduced to explain the unusual phenomena as observed in the simulation.The crack-induced initiation mechanism of the material is demonstrated through the combination of simulation and theoretical analysis.

2.Material and methods

2.1.Real particle distribution model

The simulation of how Al particles influence the shock compression response of Al-PTFE granular composites (AP) is conducted by using the mesoscale distribution model.The 2D model covering an area of 3000 μm × 2000 μm is developed by a Lagrange method in ANSYS finite element software.and an adaptive mesh with a maximum refinement of 30 μm/cell is utilized.In the model,PTFE particles are considered as a matrix and Al particles are randomly filled.There are two aspects considered to generate the model:the determination of particle size and the arrangement of particle location.In order to derive the distribution of Al particles size possibly upon raw material (air atomized Al powder.type FLPA250) rather than representative elements of an equal size.a normal distribution curve is well fitted based on the real variation of Al particles diameter [25,26].Fig.1 shows both the real distribution and fitted distribution of Al particles diameter.A normal distribution curve of Al size with specific mean μd= 119 μm and standard deviation σd=36 μm is obtained from the red fitted curve.Additionally.considering that sintered PTFE material undergoes a melting and then recrystallizing process.the particular characteristics of that are ignored and this material is regarded as a compacted/homogeneous matrix.Thus,the Al-PTFE model is created by randomly placing Al particles into a PTFE matrix.The automatic single surface contact is applied to model the interaction between the two components.This automatic contact could search all outer surfaces and react when penetration is generated.Also the benefit lays on the exemption of the defining of the contact surfaces during impact loading.And thus the cost of calculation is greatly reduced.When the penetration between two surfaces exceeds 40% of the thickness of the contact element.debonding between Al particles and the PTFE matrix occurs.

Simplified measures are adopted in the model.The real Al particle microstructure is defined as an ideal sphere in order to have a greater control over the mesh resolution.and the porosity of the composites is ignored.which provide great convenience to obtain the certain number and locations of particles for samples.When placed particles volume V closely reach the given Al content or area fractions V0of the sample constituents.and no overlap is allowed with the particles already placed.the mesoscale model is generated.Two steel plate in the area of 3750 μm × 500 μm are situated on the upper and lower side of Al-PTFE models forming a sandwich-like structure to measure average stress in real time during compressive process.In addition,the process is idealized as the upper plate moving down at a velocity of 5 m/s towards Al-PTFE model,which is determined by the appropriate constant velocity in drop-weight test where drop height is 1.3 m,and a fixed boundary condition is assigned to the lower plate on the bottom.A typical model is shown in Fig.2.In addition.the consistency of the simulation is proved by checking the coefficient of variation from a sample with different size distribution.

2.2.Material model

The material model is composed of two parts: the strength model and the equation of the state(EOS).In this study,due to thelarge deformation for the material subjected to large strains,standard Johnson-Cook strength model and Gruneisen EOS are adopted to describe the physical and mechanical properties for Al-PTFE materials.Additionally.the linear elastic EOS is used for the steel plate and rigid EOS is used for the rigid plate.

Fig.1.Distribution of Al particles diameter.

Fig.2.Al-PTFE composites mesoscale simulation model.

In the strength model.the equivalent plastic stress σ is defined as

Due to the relatively weaker PTFE matrix.most of the plastic strain is accommodated by the soft matrix with practically undeformed metal particles.When the matrix generates severe plastic deformation such as macrocracks,which is below the overall failure behavior in samples,deformation or fracture of Al particles bonding between PTFE under dynamic loading is minimal.Therefore.the failure criterion is used only for sintered PTFE matrix rather than Al particles.which is based on the equation

where σ*is the ratio of pressure divided by the effective stress,and the damage parameter is defined as

Table 1 Parameters of strength models for Al and PTFE [14].

Table 2 Parameters of EOS for Al and PTFE [17,20].

3.Results and discussion

3.1.Impact-induced mechanical response

In order to investigate the influence of the Al mass ratios on the impact response of AP material.four models with Al contents of 20%,30%,40%and 50%,respectively,were developed.The mass and area fraction of each type of models were incorporated with the calculations and experiments.Descriptions of the simulation scheme and number of Al particles used in the models are listed in Table 3.It is interesting that although the mass fraction of samples is displaced by the area fraction V0of models,and the number of Al particles in the experimental samples are quantitatively more than that in simulation models.the considered mass ratio of models reflect obviously different behaviors as observed in experiments.

The curves of average engineering stress σ versus global strain ε for four models with different Al contents under the same loading are shown in Fig.3 (the initial parameters of models are shown in Table 3).The relationship between mechanical properties including yield strength σsand ultimate compressive strength σb.and Al content for Al-PTFE samples are shown in Fig.4.The results show that the difference in mass ratio of Al has a clear influence on the shock compression behavior of granular composites.When the mass fraction of Al increases from 20%to 50%,the σsvalue increases gradually while the σbincreased first and then decreased,which is in good agreement with the experimental results[27].Specially,the maximum σbof 64.2 MPa is obtained in the sample AP30at the strain of 0.51.Additionally.according to the fitted curve of relationship between σband the material with different mass fractions,the maximum value is most likely achieved at the content of 35 wt%Al under dynamic loading.which is also validated against the experiments [27].However.due to fewer packing particles in the simulation and slightly different loading conditions in experiments,the global stress of samples is sacrificed compared with the experimental samples.which lead to a reduced compressive strength shown in the simulation.

The simulation for shock process and Von Mises stress distributions of four models are presented within state 1-3 in Table 4.There are three compressive stages distinctly presented as mechanical response of the samples: nonlinear elastic deformation,severe plastic deformation.and macrocracks propagation.Due to the soft PTFE matrix.mechanical properties of the material are governed by the collection of Al particles.However,state 1 makes it clear that mainly the matrix resists the early loading,and the stress increases promptly upon fast impact.The material has a temporally linear elastic deformation lasting for a few microseconds.and followed by a phased transition process since the interacted squeezeexists between the metal particles in matrix.

Table 3 Details of Al-PTFE simulation schemes.

Fig.3.Stress-strain curves of different Al content.

Fig.4.Relationships between strength and Al content.

Upon the continuous load,severe plastic deformation appears in the material with no increase in stress.At this stage.the material shows softening state due to the high-pressure and hightemperature.As the deformation continues to develop.the overall stress increases drastically.The rising stress should be attributed to the adaptive capacity resisting further deformation after yielding.which is created by the behavior of agglomerated Al particles.As shown in state 2.several mesoscale force chains consisting of granular system are formed and linked throughout the top to the bottom of the samples.Then the different increases in the stress-strain curves could be compared to the corresponding specific force chain configurations.as shown in Fig.3.These specific configurations appear quasilinear.which bears and transmits the main compressive load.instead of the matrix.There is stress concentration distinctly shown in the interface of particles which forms the force chains.especially in particles with large size difference.When the stress is greater than the debonding strength between particles and the matrix.local microcracks occurs.It is fairly obvious in the sample AP50because more particles can create more surface crack in matrix.Here the stress concentration does not affect the orientation of force chains.but prevents the further propagation of the cracks and enhances the material toughness in stress direction.As a result.compressive strength does not immediately decrease with the increasing of samples strain.

The evolution of force chains.as governed by the number and the modes of contacted particles and their spatial distribution,plays a primary role in the mechanical properties of the material.When the lateral support from the matrix and the contact moment between particles reach plastic threshold.particles rearrangement is observed to lead to two possible scenarios.The first scenario is that the rearranged particles activate new strong force chains upon further deformation.which is suggestive of a higher ultimate strength in the material.In contrast,the second scenario is that the unstable force chains disintegrate and macrocracks propagate in matrix,resulting in a dramatic decrease in strength(e.g.,AP50)[28].Consequently,a lower ultimate strength is observed in the sample AP20since less Al particles to maintain force chains compared to other samples.However.excessive Al particles would damage the stability of the force chains and accelerate the formation of cracks,though the yield strength of material increases.These considerations suggest that the metal particles number governed by the mass ratio determines the quantity and stability of force chains,affecting the mechanical properties of composites.

The mechanical properties of the interphase zone between metal particles and matrix are essentially different from the matrix.The interfacial thickness and strength varied considerably with different mass ratios.which plays an important role in the failure performance.Particular attention is paid to the effect of the interphase zone on the failure behaviors of the Al-PTFE composites.There are two typical failure modes shown in samples under impact.One failure mode is shear failure of matrix when the Al mass ratio is relatively small.where a reinforced interphase zone existing.When the Al mass ratio ranges from 20%to 30%,the matrix supports force chains rather than metal particles.A stronger interphase zone which largely depends on the interfacial tensile strength exists,and the interfacial slip is prohibited,enhancing the strength of the material.Thus,the shear failure of matrix is caused by tensile stress.The other failure mode is debonding failure of particles since excessive Al particles damaged the continuity of the matrix [29].which significantly reduced the interphase adhesion.As shown in state 3.the jagged grids on the interface of particlematrix were slipping and debonding under loading.where the locations are sites of severe plastic deformation and cracks development.Therefore.in addition to shear failure.debonding failure between metal particles and the matrix is the main failure mechanism for the samples with increased densities.

3.2.Crack-induced initiation mechanism

Initiation of RMs is always associated with the formation of cracks.The cracks running speed is underestimated in the simulation more than 2 km/s.According to Wang’s numerical model[30],the temperature rise is beyond the reacting value of 3300 K for Al-PTFE composites [31].Alternatively.the maximum contact pressure between force chain particles during fast cracks propagation is more than 400 MPa.which reaches a typical pressure of real contact area under a friction condition at molecular level[32].In fact,the pressure of real contact interface is probably higher than the mesoscopic pressure due to point contact.Sufficient exothermic could be achieved at the cracks tips which accounts for the disintegration of PTFE.and the decomposed products could diffuse through the oxide passivation shell to react with the Al particles[33].and subsequently initiate reaction.Furthermore.the propagation of cracks could tear the inertia alumina shell to expose the Al[34].which also promotes the initiation reaction.

As shown in state 3 of Table 4,typical shear cracks are developed in samples at approximate 45°from the direction of compression.The sample AP20shows great ductility and cracks only shown at the edge.Besides edge cracks.several shear cracks are crisscrossed inthe middle of the sample AP30.For the sample AP40and AP50.the material shows apparent brittleness where a heavily crack through the bulk of the sample.The simulations provide the necessary idea to explain crack-induced initiation reaction of Al-PTFE composites as observed in impact experiments(Table 5)[27].The remnants of the specimen increase with increasing density.suggesting the reaction is more difficult to be triggered due to less open cracks generated.Referring to Table 5,when the mass ratio arrived at the equilibrium ratio (Al/PTFE.26.5%/73.5%).full reaction occurred frequently.which is supported by the cracks behavior of sample AP30,shown in Table 4.Several force chains maintained the cracks damage during large strain buildup more local stresses and enhance a possible chemical reaction.Partly reaction mainly occurred in the specimen contained 40% of Al.When the mass fraction raised to 50%.nearly 90% mass fraction of the specimen remained intact.The main possible reason for this phenomenon can be concluded form the sample AP50.Although the excessive Al particles can significantly increase the contact area.an inert Al2O3shell with a thickness of 50 nm inevitably forms on the Al particles[33].This means a higher temperature needed to hot Al expand through the oxide shells.but it is limited in AP50due to less local stress formed and more oxidizer existed.In contrast.few force chains disintegrated and no cracks reached the bulk of the sample AP20.As such.it seems reasonable to predict that partly reaction mainly occur in specimen with less mass ratio of Al.since the reaction will be confined at the middles and subjected to quench at the cracks surface.It also demonstrates that the mass ratio affects initiation as well as the propagation of reaction.Therefore.a high crack area of the material is critical for its chemical reaction performances.With the increase of the Al content,the sample is more difficult to form open cracks and the reaction is more difficult to betriggered.

Table 4 Compressive response of samples with different Al mass ratios.

Table 5 Remnants of Al-PTFE in impact experiments [23].

3.3.The effect of central tendency of the particle size distribution

Varied distribution of particle size directly determines the number and the spatial dispersion of particles for composites with the identical mass ratio.which further affects the force chains skeletons and even the mechanical behaviors of the materials.Thus.two sets of distribution of Al particle size were used in simulation with the same mass fractions(Al/PTFE,26.5%/73.5%),as an attempt to explore the effect of Al particles size on Al-PTFE composites.Particle size following the mean distribution and standard deviation distribution pattern.which could be well characterized by the central tendency,μ,and the dispersed degree,σ.was selected.The distribution curves of Al particle size of four different central tendencies (μ = 79 μm,119 μm,159 μm,199 μm),while the distribution curves of Al size of four different dispersed degrees (σ = 0 μm.24 μm.48 μm.72 μm).Simulated microstructures were then set up based on the corresponding distribution of Al particle size and random particle arrangements.under the loading condition of impact velocity of 5 m/s.

The initial particle arrangements and failure states of samples used in the simulation.exploring how differences in the relative central tendency of Al size affect the mechanical behavior of material,are shown in Table 6.It is obvious that the particle number of samples clearly decreases as the central tendency of Al size increases.The relationship of average engineering stress σ at the certain boundary condition versus the global stress ε are calculated and described in Fig.5.The results show that the given samples have similar yield behaviors.but different failure responses.The ultimate strength decreases firstly and subsequently increases with the rising of the central tendency.Particularly.the maximum ultimate strength corresponds to the sample with the minimum central tendency of Al size.

Some insights into the varied strength behavior observed here can be obtained from the Von Mises stress of the samples in Table 6,which illustrates the failure state before macrocracks produced.Table 6e shows several strong force chains propagated throughoutthe matrix.and the force chains are reactivated upon further deformation.As depicted in Fig.5.the sample 1 has two stress spikes in stress-strain curves.which provides evidence for the sample with an excellent mechanical performance.In addition,interfacial debonding failure is dominated due to its more particles existing.A comparison of stress distribution where force chains skeleton of Table 6e and that of Table 6f indicates that force chains are closely related to the number of metal particles.A lower ultimate strength shows in sample 2 with a decreased number of Al particles and force chains.Table 6g shows that a weaker force chains developed accounting for the lower stress spikes for sample 3.The same mass fraction of samples.when a fraction consists of fewer number of particles,is less conductive to activating the force chains skeleton that is able to affect the global strength of the material,but one particular case stood out.A close look at the rising ultimate strength of sample 4 suggests that strength is also related to the expected better interfacial bonding between particles and the matrix.when rather few metal particles existing in material.Table 6f shows that mainly the PTFE matrix resisted the impact load rather than those groups of agglomerated particles.The dominant failure mode of the sample evolves to matrix shear failure.where the matrix disperses metal particles and prevents them from forming force chains to maintain damage throughout the bulk of the sample.Though high strength is achieved.it does not provide local stress area as strong as in the first sample,and has unfavorable effects on the formation of initiation reaction.Thus.smaller sized Al-PTFE composites are more preferred than its bulky material.

Table 6 Initial arrangements and failure states of samples with different central tendencies of Al size.

Fig.5.Stress-strain curves of different central tendencies.

3.4.The effect of dispersed degree of the particle size distribution

The initial mesostructures and failure states for samples with four different dispersed degrees of Al size to investigate the effects of size dispersion.is shown in Table 7.It should be noted that the particle number of Al are comparable,but the dispersion of particle size is clearly increased.Fig.6 shows the stress-strain curves for the samples under the given impact consition.The results indicate that the variation of particle size contributes little to the yield behavior of the samples with the same mass fraction,which coincides with that behavior observed in the samples with varied central tendencies of Al particle.However.the dispersion of Al size can significantly affect the ultimate strength of this granular composites that the strength decreases firstly and subsequently increases as the dispersed degree rises.

Table 7 shows the corresponding Von Mises stress for the samples before macrocracks formed.It is difficult to qualitatively interpret the different strength from these quite different samples since each has a rather different particle size distribution but comparable particle number.especially by experimental methods.Nevertheless.simulations performed here makes it clear that whether strong force chains can be formed is an effective way to determine material strength.Although a higher ultimate strength is shown in both sample 1 and 4.there are two possible mechanical mechanisms observed from the results to explain the enhanced compressive behaviors.According to Table 7e.strong force chains are generated in sample 1 due to the identical Al size.which is conductive to maintain the stabilization of this mesostructure feature.In contrast to sample 2 and sample 3.larger particles are prior to result in stress concentration than smaller particles under the same loading,so larger particles are more subjected to produce fracture in matrix [35].In this case.the material with wide dispersion of Al size has a lower ultimate strength.which is consistent with the weaker force chains formed in Tables 7f and 7g.Nevertheless.much smaller particles are beneficial to restructure stable force chains to some extent as shown in Table 7h.and the force chains propagates through the cracks (depicted in Fig.6).With smaller Al size.smaller repulsion stress will be felt by a particle dislocation and a higher applied stress will be needed to propagate dislocations through in matrix.Again.the enhanced strength may be accompanied with a possible improved reactivity.It should be noted that while the simulations are obtained from four samples of given dispersed degrees.the effect of size dispersion on Al-PTFE composites under impact can be explained.Besides the ideally identical size grading.increasing dispersed degree contributes to develop stable force chains with comparable particle number.

Table 7 Initial arrangements and failure states of samples with different dispersed degrees of Al size.

Fig.6.Stress-strain curves of different standard deviation.

4.Conclusions

This work investigates the influence of the mass fraction and Al size distribution on the mechanical response of Al-PTFE granular composites under low-velocity impact.The major conclusions can be summarized as follows:

(1) The model followed the real Al particles size distribution is developed to analyze the influence of aggregated particles behavior of the material.

(2) The mass fraction has a significant influence on the impact response.With increasing density,the ultimate compressive strength of the material increases first and then decreases,and failure mode evolves from shear failure of matrix to debonding failure of particles.

(3) The mesoscale force chains behavior associated with local concentrated stresses is responsible for the crack-induced initiation reaction as well as the propagation of reaction.The sample with a high crack area has more potential to the energy release capacity.

(4) The particle size distribution plays a key role in the mechanical performance of the composites.The lower central tendency of Al size certainly improves the ultimate compressive strength of the material.A higher strength is observed in samples with similar size grading or great size dispersion under the identical mass fraction.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No.U1730112).