Abstract

We propose a systematic experimental procedure and quantitative analyses to investigate the effect of cyclic loading, and time-recovery, or aging, on the mechanical properties and microstructure of particle-binder composites. Specifically, we study three compositions that differ in aluminum content from the mock sugar formulation of plastic-bonded explosive PBXN-109. Cast cylindrical specimens are subjected to high-amplitude quasi-static cyclic compressive loading, before and after a 4-week time-recovery period, and their microstructures are analyzed using micro-computed tomography (CT). For quantitative analysis, we develop a procedure for identifying the spatial distribution of primary components of the formulation, including pore space, from micro-CT images. The study shows that the stress–strain response is highly nonlinear, without a distinct yield point, and exhibits hysteresis and cyclic stress softening, or Mullins effect, with cyclic stabilization. Specimens without aluminum exhibit considerable gain in stiffness and strength after the time-recovery or aging period, owing to the development of increased sucrose particle–particle interactions during the first cyclic loading. In contrast, specimens with aluminum micro-sized powder exhibit permanent loss of stiffness and strength, owing to large ductile plastic flow and irrecoverable damage. Further insight from micro-CT analysis is gained by observing that, for all compositions, the majority of microstructural changes occur near the specimen core. Specifically, affine radial deformation of the soft and debonded binder, as it is compressed by the non-affine longitudinal motion of stiffer sucrose crystals, is observed in the formulation without aluminum, whereas non-affine rearrangement of the binder toward the specimen core, and affine radial flow of sucrose particles away from the core due to ductile macroscopic deformation of the specimen, is observed in the formulations with aluminum content.

1 Introduction

Particulate composites, or particle-binder composites, consist of hard particles, such as carbon, tungsten carbide, silica, gravel, and others, in the matrix of a soft material, such as polymers and soft metals. These materials are used across a wide range of applications in different contexts. For example, particle-reinforced composites (such as filled elastomers and concrete, i.e., gravel embedded in a cement matrix) are used to enhance the mechanical properties of the matrix. Since hard particles increase the load-bearing capacity of the material, these composites provide an improvement in strength and wear properties over single material systems. However, they also exhibit significant inelastic phenomena in their mechanical response [1], such as Mullins effect [2,3] and cyclic hysteresis [4] in filled rubbers [5,6].

Plastic-bonded explosives (PBX) are another type of particle–binder composites that are widely used in ammunition for defense applications. These materials consist of explosive crystals, such as cyclotrimethylenetrinitramine (RDX) and cyclotetraminetetranitramine (HMX), in the matrix of a binder that primarily includes (i) a soft polymer, such as hydroxyl-terminated polybutadiene (HTPB) [7] and Estane [8], (ii) a plasticizer, such as bis(2,2-dinitropropyl)acetal (BDNPA), dioctyl adipate (DOA) and bis(2,2-dinitropropyl)-formal (BDNPF), and (iii) small concentrations of antioxidants, bonding agents, wetting agents, cross-linkers, stabilizers, and catalysts [9]. The polymer acts as a safety envelope for the explosive crystals, inhibiting their movement and excitation to low amplitude loads and thus reducing their sensitivity to weak impact loads [1012]. The plasticizer improves processability, mechanical properties, and also lowers sensitivity [13]. Another major ingredient in PBX may include a metal, such as aluminum, which is often used as a fuel or booster to enhance blast effects [9,14].

Characterization and mechanistic understanding of the mechanical behavior of PBX are of particular interest to the defense community. PBXs are designed to detonate in response to a very specific external stimulus. However, these composites may be subjected to diverse loading conditions during their operational life (ranging from high to low strain-rate compression and tension, impact, mechanical vibrations and cyclic loading, among others) that in turn may alter their microstructure and their mechanical response, rendering them unpredictable and unsafe [15]. Several experimental studies have been performed to analyze the mechanical response and microstructural changes in PBX under different loading conditions. The most commonly used techniques for imparting static and dynamic loads are (i) low strain-rate uniaxial unconfined compression and tension [8,16,17], (ii) low-frequency base excitation [18,19], (iii) Brazilian test for measuring tensile strength [20,21], (iv) high strain-rate compression and impact loading using split Hopkinson pressure bars [2224], and (v) dynamic mechanical analysis [15,25]. The most common destructive and non-destructive techniques for observing microstructural changes are (i) speckle photography [10] and (ii) scanning electron microscopy (SEM) [11,26], which have been used to study fracture surfaces in PBX following a Brazilian test, (iii) high-speed synchrotron X-ray phase contrast imaging [27], which has been used to study in situ deformation and failure in PBX under dynamic compression [28], and (iv) micro-computed tomography (CT), which is an increasingly popular and sophisticated non-destructive imaging technique capable of characterizing the material in a three-dimensional (3D) space [2935]. In the context of PBX, in situ micro-CT has been used to observe microstructural changes in material specimens during large uniaxial unconfined compression [36,37], with observations of ductile plastic flow to extensive cracking and damage mechanisms like crystal-binder delamination and transgranular fracture. When coupled with optical deformation measurement techniques like digital image correlation [38] and digital volume correlation [39], micro-CT has been used effectively to obtain strain fields and quantify microstructural changes in polymer-bonded sugar during large unconfined uniaxial compression [40,41].

The stressstrain response of particle–binder composites, specifically polymer-bonded sugar under large quasi-static monotonic compression, is typically characterized by four stages (see, e.g., Ref. [37]). Initially, a nonlinear increase in the slope of the stress–strain curve is observed, which is attributed to the gradual mating of the machine platen-specimen surface. This is followed by a predominant elastic deformation of the material, where the response curve remains largely linear. The third stage commences with a nonlinear decline in the slope of the curve, attributed to predominant inelastic deformation and damage accumulation in the form of particle-binder delamination and slipping, which results in nucleation of cracks, and it continues until the material reaches its ultimate compressive strength. Subsequently, the material rapidly loses its strength, owing to an extensive transgranular crack formation during the fourth stage.

It is worth noting that, since the majority of inelastic deformation and damage accumulation without loss of structural integrity occurs within the first three stages of deformation, it is relevant to study periodic or cyclic loading within the range of moderate to high strains and to characterize its effect on microstructure and mechanical properties of particle-binder composites. Furthermore, since a material specimen may be subject to such loading conditions multiple times over its operational life, it is important to study the effect of time-recovery, degradation, or aging (see, e.g., Ref. [42]). A systematic study of the simultaneous effect of cyclic loading and time-recovery on microstructure and mechanical properties of particle-binder composites is not available in the open literature, and thus, it is the central focus of this work.

This paper specifically focuses on the qualitative and quantitative characterization of changes in mechanical behavior and microstructure of particle–binder composites due to the application of high-amplitude quasi-static cyclic compressive loading, before and after a 4-week time-recovery or aging period. Three compositions that differ in aluminum content from the mock sugar formulation of PBXN-109 are cast into cylindrical specimens and used in this study. Microstructural changes in the spatial distribution of the primary components of the formulation, including pore space, are characterized from micro-CT images.

The paper is organized as follows. Section 2 describes the experimental procedures, including specimen preparation, experimental apparatus, and the proposed cyclic testing and aging (recovery) procedure. Section 3 provides an analysis of the simple monotonic compressive response of mock specimens to identify optimum strain levels for cyclic loading. Section 4 presents a detailed analysis of cyclic loading and time-recovery effects on the mechanical response of mock specimens. Section 5 describes the postprocessing procedure for the identification of spatial distribution of different components in the formulation, including pore space, from micro-CT images. Section 6 provides a detailed quantitative analysis of changes in microstructure due to cyclic loading and time-recovery or aging. Finally, a summary and concluding remarks are presented in Sec. 7.

2 Experimental Procedures

2.1 Material Preparation and Specimen Geometry.

Cylindrical specimens of mock PBX material with dimensions of approximately 1-in. height and 1-in. diameter were prepared following the procedure by Range et al. [43].

The specimens were prepared by Ms. Allison Range2. The formulations used are variations of PBXN-109 formulation [44], with sugar used as a substitute for RDX to render the specimens inert. The base formulation is summarized in Table 1.

The main constituents of the mock specimens are an HTPB binder with equal quantities of R45-HT resin (Cray Valley, Exton, PA, USA) and DOA plasticizer (Sigma-Aldrich, St. Louis, MO, USA), sucrose particles, and spherical aluminum powder (Valimet Inc., Stockton, CA, USA). The sucrose particles are sieved to a diameter range of 106–355 µm to be comparable in size with RDX particles. The aluminum powder has an average particle size of 25 µm. To prepare the specimens, a mixing and casting procedure as described in Ref. [43] was followed. The cast specimens were then cured in an oven for 7 days at 60 °C. Experiments on the specimens were performed 1–2 months after curing.

To gain insight into the role of additive content, i.e., of aluminum powder, in the evolution of microstructure and mechanical properties during cyclic loading and aging, the base formulation was varied to produce specimens of different formulations based on the following two parameters:

  1. Solids loading, which is the weight percentage of solids (sucrose crystals and aluminum powder) in the overall mixture.

  2. Additive content, which is the weight percentage of aluminum powder in the solids.

Specimens of three different formulations were studied, all of whom have 85% solids loading but differ in additive content, namely 85-00 (0% additive in solids, 0% in total, Fig. 1(a)), 85-15 (15% additive in solids, 12.75% in total, Fig. 1(b)) and 85-30 (30% additive in solids, 25.5% in total, Fig. 1(c)). The nominal specimen heights (h), diameters (D), and test performed for 85-00, 85-15, and 85-30 are provided in Tables 2, 3, and 4, respectively.

2.2 Experimental Setup.

Compressive loading experiments were performed on a MTS Criterion C43 universal testing machine (Fig. 2). The specimens were compressed between metal platens lubricated with WD-40 dry lube. Strain-controlled tests were performed at room temperature (72 °F), and at a strain-rate of 0.001 s−1 to produce a quasi-static loading response. A load cell of 500 N capacity was used to obtain the force measurements.

2.3 Micro-Computed Tomography.

Micro-CT scans were performed on a Bruker Skyscan 1272 instrument (Fig. 3). The specimens were fixed to the mounting fixture using adhesive putty, with care being taken to mount them in a straight, upright position. Table 5 presents the optimum scan settings identified for each of the specimen formulations. The settings were optimized for scan quality (i.e., good contrast, low beam hardening, and ring artifacts), image quality (i.e., good resolution and low noise), and scan time. Due to the specimens being larger than the detector's field of view at the selected resolution, the scan was carried out at two camera positions successively to capture the entire width of the specimens. However, the length captured was still limited by the length of the field of view. Therefore, the middle portion of the specimen of length 10.92 mm was scanned to adequately capture the loading and aging effects for the entire specimen (Fig. 4).

2.4 Testing Procedure for Assessing Cyclic Loading and Time-Recovery Effects.

Figure 5 presents the standardized testing procedure employed to identify the dependence of mock energetic specimens on loading history and time-recovery or aging. A period of 24 h was selected for re-scanning the specimens after cyclic compressive testing, to allow sufficient time for scan completion and to simultaneously capture the material's dependence on cyclic loading. Before resuming the testing cycle, a resting period of 4 weeks was allowed for the specimens to capture the aging effect, during which the specimens were stored in ultraviolet (UV)-protective resealable bags in a temperature-controlled (72 °F) room.

3 Identification of Optimal Strain Levels for Cyclic Loading

To understand the basic deformation characteristics of mock specimens and to establish a standard procedure for identifying optimal strain levels for cyclic loading, monotonic compressive loading–unloading tests were performed for each specimen composition.

First, two specimens of each composition were compressed until a chosen maximum strain level of 30% (see Fig. 6), and the resulting nominal stress–strain response was studied. It was observed that the 85-00 specimen exhibits (i) an initial nonlinear behavior (machine–specimen mating), followed by (ii) an approximately linear increase in stress (predominant elasticity), succeeded by (iii) a nonlinear response with decreasing slope (predominant plasticity) until a peak stress level (ultimate compressive strength), after which the (iv) stress decreases rapidly. These observations are in agreement with the previously observed behavior of polymer-bonded sugar under monotonic compression [37]. In contrast, 85-15 and 85-30 specimens exhibit a predominantly nonlinear response followed by ductile plastic flow, reaching the peak stress at a much larger ultimate compressive strength; interestingly, the ultimate strength is not reached under 30% strain for 85-30 specimens. These observations are similar to those reported for aluminized explosives under high strain-rate compression loading [45]. The increase in compressive strength with the addition of aluminum is attributed to an increase in inter-particle interactions and, consequently, in load-bearing capacity due to a more densely packed solid phase (i.e., crystals and aluminum powder). We additionally note that the presence of sucrose–aluminum contact interactions reduces sucrose–binder interfacial area, which in turn arrests extensive interfacial debonding and larger crack formation, leading to a ductile macroscopic deformation. These observations are supported by the deformed configuration of the loaded specimens (see Fig. 7), where extensive crack formation and debonding is observed in the 85-00 specimens while lateral expansion and plastic material flow is observed in 85-15 and 85-30 specimens.

Second, optimal strain levels for cyclic loading of each specimen composition were selected on the basis of the following two principles:

  1. The maximum strain level should ensure sufficient elasto-plastic deformation while preventing the material from reaching its ultimate compressive strength. Therefore, for experimental convenience, we chose the maximum stress level to be the nearest integer function of 80% of the averaged strain at ultimate compressive strength, i.e., εmax,cyclic=0.8ε¯ultimate.

  2. The minimum strain level should allow for sufficient cyclic deformation while accounting for cyclic stress softening (Mullins effect) and ensuring that the specimen remains in compression after cyclic stabilization. Therefore, we adopted an upper bound equal to two-thirds of the maximum strain level, i.e., ɛmin,cyclic ≤ 2ɛmax,cyclic/3.

For 85-00 and 85-15 formulations, the values of strain at ultimate strength and the maximum cyclic strain level obtained from the tested specimens are reported in Table 6.

Next, two specimens each of the 85-00 and 85-15 formulations were loaded to their respective maximum strain levels and subsequently unloaded to a stress-free state (see Fig. 8). A suitable value of the minimum strain level, lying between its upper bound (depicted by dashed-dotted lines in the figure) and the residual strain (i.e., 1% for 85-00 and 2% for 85-15), was to be chosen. We adopted a minimum strain level of 5% for 85-00 and 8% for 85-15, depicted by dotted lines in the figure. Finally, we verified (and show in the later sections) that, for these chosen optimal strain levels, the specimens remained in compression during cyclic loading until the attainment of a stabilized cycle, despite the occurrence of stress softening.

For the 85-30 formulation, an ultimate compressive strength was not reached within 30% deformation, and therefore, the selection of optimal strain levels for cyclic loading was made based on the identification of the minimum strain level first. To this end, compressive loading–unloading tests were performed on three specimens for three different maximum strains, namely 30%, 15%, and 9%. The unloading stress–strain curves for each of these tests are presented in Fig. 9. It is evident from the figure that the specimens exhibited large residual strains; interestingly, the residual strain corresponding to a maximum strain of 30% is slightly larger than the two-thirds bound, i.e., 20%, while the residual strains for 15% and 9% are slightly smaller than their respective two-thirds bounds, i.e., 10% and 6%. Consequently, cyclic compression tests were performed using 10%–15% and 6%–9% minimum–maximum strain levels (see Fig. 10). The former test exhibited a stress-free state before cyclic stabilization, while the latter remained in compression until cyclic stabilization (see inserts in the Fig. 10). Therefore, we adopted optimal strain levels of 6%–9% for cyclic loading of the 85-30 formulation.

A detailed analysis of the cyclic response of each mock energetic formulation is presented in Sec. 4.

4 Cyclic Loading and Time-Recovery Effects on the Mechanical Response

Figure 11 presents the compressive cyclic nominal stress–strain response of 85-00-05, 85-15-05 and 85-30-07 specimens when loaded between the optimal strain levels selected in Sec. 3, i.e., 5%–9%, 8%–16%, and 6%–9%, respectively. Figures 11(a), 11(c), and 11(e) show the response during initial cyclic testing of virgin specimens, while Figs. 11(b), 11(d), and 11(f) show the response of the same specimens after a 4-week time-recovery or aging period. Nominal heights of the specimens measured before cyclic testing after the 4-week period were 26.05 ± 0.02 mm for 85-00-05, 26.95 ± 0.01 mm for 85-15-05, and 25.56 ± 0.01 mm for the 85-30-07 specimen. When compared with the nominal heights of the specimens in their virgin (untested) configuration (Tables 2, 3 and 4), a residual strain of ∼1% in 85-00-05, ∼1.57% in 85-15-05, and ∼2% in 85-30-07 specimen is identified after the recovery period.

Investigation of the mechanical response of the specimens suggested that all three formulations exhibited a response characterized by the following attributes: (1) highly nonlinear stressstrain response without a distinctive yield point, (2) hysteresis, and (3) cyclic stress softening with eventual stabilization, which was observed at the 20th cycle for all formulations. It is worth noting that for all compositions there are significant differences between the cyclic response before and after the time-recovery period. These differences can be quantified by the following parameters defined for a given cycle (see Fig. 12):

  1. Peak and valley stresses: the maximum σmax,i and the minimum σmin,i values of stress in the ith (i = 1, …, 20) cycle. These stresses are related to the material strength.

  2. Cumulative dissipated energy density: the difference between the energy supplied to the material (i.e., the area under the loading path) and the energy recovered after a cycle (i.e., the area under the unloading path), accumulated throughout the cyclic loading. This energy quantifies cyclic hysteresis and accumulated damage in the material.

  3. Apparent stiffness: the slope of the line connecting peak and valley stresses in each cycle, i.e., for the ith cycle, Eapparent,i = (σmax,iσmin,i)/(ɛmax,cyclicɛmin,cyclic).

In the following sections, a thorough analysis of cyclic loading and time-recovery effects on the overall stressstrain response, damage, strength, and stiffness of mock energetic specimens is presented.

4.1 Effects on the Overall StressStrain Response.

It is interesting to note that in the cyclic stressstrain response of all specimen formulations, the first loading path of the virgin material is quite different from the subsequent reloading paths, including those recorded after the time-recovery period. Excluding the initial response (until about 3% strain) which is related to gradual mating of machine platen-specimen surface, the rate of strain hardening is observed to be decreasing with compression during the first loading path of the virgin material and increasing with compression during subsequent reloading. A similar observation has been made previously for cyclic compression of aluminized RDX-based PBX in HTPB binder [46], where the authors have described the initial loading response as sharp oval shaped, and the subsequent response as crescent shaped. This type of strain hardening has not been observed in other particle–binder composites such as filled elastomers [1,5,6], where the initial loading response is similar to the response observed during subsequent reloading.

Another interesting observation is made regarding the effects of loading history and time-recovery on the first loading path of the aged material. For the 85-15 and 85-30 formulations, the first loading path of the aged specimen is similar in shape to both the loading response of the stabilized cycle before aging and the subsequent reloading paths. However, for the 85-00 formulation, the shape of the first loading path of the aged specimen remains similar to the first loading path of the virgin material. This permanent change in the mechanical behavior of aluminized specimens may be attributed to the loading history memory effect observed in plastic or ductile (polycrystalline) materials [4750].

4.2 Effects on Damage, Strength, and Stiffness.

All three formulations show a nonlinear decline in peak and valley stresses (and thus in strength), and apparent stiffness with cyclic loading, for tests performed both before and after the time-recovery period (see Figs. 13 and 14). The rate of decline, however, eventually reduces to zero, and cyclic stabilization is observed after the 20th cycle. This weakening and softening of the material, also known as the Mullins effect, is attributed to the accumulation of damage. This accumulated damage is also evident from the evolution of cumulative dissipated energy (see Fig. 14), and it is typically attributed to mesoscale physical processes such as bond rupture at polymer–filler interfaces [51], molecular slipping on the filler-particle surface [52], filler aggregate rupture [53], among others, and so on (see, e.g., Ref. [54] and references therein for a review of physical interpretations of the Mullins effect).

The effect of loading history and time-recovery on material behavior is evident for the 85-00 formulation. It shows a large increase in peak stress (∼45%), valley stress (∼66%), and apparent stiffness (∼41%) consistently for all cycles, and a ∼100% increase in both the value of energy dissipation during the first cycle and its rate of accumulation during subsequent cycles. In contrast, aluminized formulations (i.e., 85-15 and 85-30) show a small reduction in peak stress and apparent stiffness during the initial four—five cycles after the time-recovery period. The 85-15 specimen also shows a small reduction in valley stress, which is consistent with observations of other values. However, the 85-30 specimen shows a consistent increase in valley stress, although the values may be too small to be captured accurately due to limitations of the load cell. Interestingly, the rates of energy dissipation before and after aging are the same, but the values are offset by a significant drop (∼58% for 85-15 and ∼56% for 85-30) observed in the first cycle after the time-recovery period.

The observation of stiffening and strengthening of the 85-00 formulation is consistent with the proposition of an increase in sucrose particle–particle interactions during the initial loading of the virgin material. Formation of inter-particle force chains has been previously observed during high strain-rate compression of polymer-bonded sugar with high solids loading (>80%) [55] and during quasi-static compression of metal-matrix composites [56]. Specifically, it is argued that the force network within the filler particles is the main load-bearing mechanism under compression, while the binder matrix has a bulk effect on load transfer and a confining effect on the initial relative movement of filler particles [56,57]. Furthermore, if particle–binder interfacial debonding occurred at moderate to large deformations, then radial displacement of the soft polymer matrix would follow and, in turn, inter-particle interactions would be facilitated. Even if binder rearrangement was partially reversed during the time-recovery period, the increased proximity of the sucrose particles, as compared to the virgin material, would result in an apparent increase in load-bearing capacity due to a more densely packed solid phase. In sharp contrast, large ductile plastic flow and higher irrecoverable damage are observed for formulations with aluminum micro-sized powder, i.e., for the 85-15 and 85-30 formulations. As it was noted earlier, a stiffer force network is formed due to sucrose–aluminum–sucrose interactions, which results in higher material strength, but plastic irreversible deformation occurs during cyclic loading, which results in a lack of recovery during the aging period.

5 Identification of Spatial Distribution of Formulation Components at the Mesoscale

A systematic procedure for identifying the spatial distribution of the primary components of each formulation, including pore space, from micro-CT scans, was developed. To this end, the volumetric CT data were first reconstructed using nrecon (Bruker micro-CT) software to obtain multiple cross-sectional 8-bit gray-scale images along the length of the scanned portion of the specimen. A total of 1092 volumetric image slices (each image of 10-µm height, constituting the total scanned height of 10.92 mm) were obtained. To avoid low-quality images at the top and bottom edges due to cone-beam imaging geometry [58] of the Bruker micro-CT instrument, 68 image slices from the top and 70 image slices from the bottom were discarded; these values were recommended by the software. The effective analyzed height was then equal to 9.54 mm. The images were produced by attributing a gray value of 0–255 to each pixel depending upon its attenuation coefficient, which in turn depends on the material density. Figure 15 shows a schematic of the histogram of attenuation coefficient values obtained from the middle cross section of the 85-15 specimen. From the histogram, a range of attenuation values are chosen which is mapped to the gray-scale. Limits of this range, also known as contrast limits, were chosen as follows: the lower limit was selected as 0, which is the first peak in the histogram corresponding to air, while the higher limit was selected as 4%–5% more than the maximum attenuation (density) observed in the histogram to assure complete visualization of all the material within the specimen.

In addition to choosing the contrast limits, corrections were applied to the gray-scale images to reduce the effect of tomography artifacts like beam hardening and ring artifacts, while Gaussian smoothing [59] was applied to reduce image noise. Since these values affect the fundamental gray distribution of the images, they need to be the same for multiple scans of the same specimen for accurate qualitative and quantitative comparisons.

The gray-scale images were then further analyzed in the CT-analyzer (Bruker micro-CT) software, which is capable of performing a wide range of morphometric analyses on reconstructed scan data sets. A major function of the tool is thresholding, which is the binarization of a gray-scale image so that a certain range of gray values are assigned the value 0 (black), while the rest of the gray values are assigned the value 1 (white). Using thresholding, it is possible to distinguish materials of different densities (gray values) in the scanned image. However, automatic thresholding algorithms in CT-analyzer are at most capable of segmenting two-component material systems. Furthermore, due to partial volume effects in CT images [60] and extensive mixing of different materials in particle–binder composites, distinct peaks corresponding to each component are not observed in the gray-scale histogram of mock specimens, making segmentation of multiple components and porosity solely from the gray-scale histogram infeasible.

It bears emphasis that the porosity of the specimen and the volume fraction of its primary components, i.e., binder, sucrose, and aluminum can be analytically computed from the measurement of specimen’s weight and volume, since mass fractions and densities (binder: 0.907 g/cc, sucrose: 1.59 g/cc, and aluminum: 2.7 g/cc) of the components are known. While weight could be easily measured with good accuracy using a balance, the volume was best approximated using a combination of CT-analyzer tools and physical measurements. A useful feature of the software is the ability to wrap the boundary of the volume of interest (VOI) tightly around the boundary of the specimen. Figure 16 shows the region of interest (ROI) obtained using this “Shrink Wrap” feature for a cross-sectional image slice of the 85-15-05 specimen. The VOI was then constructed from ROIs created along the entire length of the scanned portion, and its volume (VVOI) was computed by CT-analyzer’s 3D morphometric analysis tool. The volume of the entire specimen (V) was then approximated as V = VVOI(h/hVOI), where h is the measured specimen height and hVOI = 9.54 mm.

Once specimen porosity and volume fractions of its components were known, gray-scale ranges corresponding to the measured values were obtained from the 3D voxel gray-scale histogram available from CT-analyzer. Figure 17 shows representative gray-scale ranges in the 3D voxel histogram of the 85-15 specimen, arranged in the order of increasing density of components (porosity < binder < sucrose < aluminum). Table 7 provides weight and volume measurements, and it compares the porosity and volume fraction of each component obtained from the true density of the specimen and the CT-analyzer voxel histogram. The table also reports the threshold gray-scale ranges identified in the analysis.

Finally, the obtained gray-scale ranges were used to binarize the gray-scale images into four sets of binary images, each containing voxels representing an individual component. These binary image sets were then color-coded in matlab® [61], assigning a specific color to each component (porosity-black, binder-red, sucrose-green, and aluminum-blue), and combined to finally obtain a single set of color-coded slices. Figure 18 shows a schematic of the described postprocessing procedure for the identification of the individual specimen components and reconstruction of their spatial distribution. Additionally, Fig. 19 provides 3D-rendered volume images of the gray-scale and the corresponding color-coded representations of the microstructure of each formulation.

6 Cyclic Loading and Time-Recovery Effects on the Microstructure

In previous sections, it has been established that cyclic loading and time-recovery introduce evident and permanent changes in the macroscopic mechanical response of mock energetic materials. This behavior is attributed to mesoscale deformation and relaxation mechanisms that emerge from the presence of a hard phase (sucrose) and a soft phase (binder) in the microstructure. The soft phase becomes progressively more compliant due to accumulated damage and particle–binder interfacial debonding. It undergoes homogeneous or affine deformation under compressive load, it recovers during unloading, and it relaxes over time. In sharp contrast, the hard phase experiences a collective rearrangement or non-affine deformation under compressive loading, and it only partially recovers during unloading and over time due to the irreversible nature of this collective rearrangement. The presence of a ductile third phase (aluminum) may significantly affect these mesoscale deformation and relaxation mechanisms by forming permanent sucrose–aluminum contacts and reducing the sucrose–binder interfacial area—i.e., by arresting interfacial debonding and thus effectively stiffening the soft phase.

In this section, the proposed mesoscale deformation and relaxation mechanisms are investigated by characterizing changes in the spatial distribution of the primary components of each formulation, including pore space, before and after each cyclic test and, thus, before and after the time-recovery or aging period.

The spatial volume distribution of components in a formulation is determined from the three-dimensional color-coded micro-CT volume images (see Fig. 19) using matlab® [61]. Distributions in the radial direction are of primary interest, and thus, the VOI is first segmented into three equal discs along its height, next into 20 angular sectors of equal volume and, lastly, into 20 radial rings (see Fig. 20). The proposed tessellation is such that the volume of the resulting elements increases linearly with radial distance, from 0.179 mm3 at r = 0.6mm to 7.013 mm3 at r = 12mm—radial distances beyond 12 mm are excluded from the analysis to eliminate any inconsistencies that may result from imperfections in the specimen geometry. The volume fraction of sucrose, binder, aluminum, and pore space within each tessellated volume, and the mean and standard deviation values along the angular direction, are then readily available and compared for scans taken before and after virgin material testing and testing after 4 weeks in Figs. 2224, 2528 and 2932 for specimens 85-00-05, 85-15-05, and 85-30-07, respectively. In the figures, plots are depicted for each segmented disc (top, middle, and bottom), with the volume fraction plotted against radial position, and standard deviation along the angular direction plotted as error bars. For additional qualitative comparison of microstructural changes, color-coded axial cross sections of the specimens are provided in Fig. 21. It is evident from the figures that the majority of the volumetric changes are observed near the core or center of the specimen. This observation is consistent with in situ micro-CT studies in polymer-bonded sugar under quasi-static uniaxial monotonic compression [40,41], which shows that accumulation of damage, in the form of debonding, starts from the core region and then propagates to the outer region. A detailed characterization of each formulation is presented next.

The 85-00 formulation shows an increase in the volume fraction of sucrose, and a reduction in that of binder and porosity, near the core upon the first cyclic loading test (see Figs. 21(a) and 21(b); circle and square plots in Figs. 2224). This behavior is consistent with an affine radial deformation of the soft and debonded binder as it is compressed by stiffer sucrose crystals and, conversely, a non-affine longitudinal motion of the sucrose crystals to form a denser contact network. Furthermore, this formulation exhibits a partial reversal of such microstructural rearrangement during the 4-week time-recovery or aging period (see Figs. 21(b) and 21(c); square and triangle plots in Figs. 2224). This reversal or recovery is facilitated by a relaxation over time of residual strains in the binder. Finally, the second cyclic test reveals trends of binder displacement similar to those observed during the first test but extended over a larger region; i.e., it indicates gradual propagation of damage away from the core [37] (see Figs. 21(c) and 21(d); triangle and diamond plots in Figs. 2224).

In the 85-15 formulation (see Figs. 21(e) and 21(f), and Figs. 2528), the spatial distribution of specimen components is highly inhomogeneous, as observed in the radial and longitudinal distributions of sucrose and binder, and the angular distribution of aluminum and porosity. However, the average microstructural rearrangement during the first cyclic test is observed to be similar to the 85-00 formulation, although, compared to 85-00, the increase in the volume fraction of sucrose and the reduction in the volume fraction of the binder and the porosity are observed much closer to the specimen core. This observation indicates an influence of the aluminum particles on the microstructure evolution. By forming additional contacts with sucrose, aluminum particles tend to reduce the sucrose–binder interfacial area and resist particle–binder debonding, thereby reducing the compliance of the binder. The effects of these contact interactions are observed prominently following partial recovery, where the deformation mechanisms of binder and sucrose are completely reversed during the 4-week cyclic test (see Figs. 21(g) and 21(h); triangle and diamond plots in Figs. 2528). It is observed that the volume fraction of sucrose decreases near the core, while the volume fraction of the binder increases. This is the indication of an affine radial motion of the sucrose crystals away from the specimen core, and a non-affine rearrangement of the binder toward the core. A similar crystal motion has been observed previously by Manner et al. during in situ micro-CT analysis of the uniaxial compression of a PBX formulation (ref. [36] for more info on the formulation and the compression study) that undergoes ductile macroscopic plastic deformation.

In the highly aluminized 85-30 formulation (see Figs. 21(i) and 21(j); circle and square plots in Figs. 2932), the prominent effects of aluminum-sucrose interactions are apparent from the first cyclic test itself, where affine motion of the sucrose crystals and aluminum particles (clearly observable in Figs. 31(a), 31(b) and 31(c) due to higher aluminum content) away from the specimen core and non-affine rearrangement of the binder towards the core are observed. The same microstructural rearrangement continues during the recovery period and during the 4-week cyclic test (see Figs. 21(k) and 21(l); triangle and diamond plots in Figs. 2932).

7 Summary and Discussion

A systematic experimental procedure and quantitative analyses have been proposed to investigate the effects of cyclic loading and time-recovery (or aging) on the mechanical properties and microstructure of particle-binder composites. Cast cylindrical specimens of mock sugar formulations of PBXN-109 differing in the amount of aluminum content (85-00 with no aluminum, 85-15 with 12.75% w/w aluminum, 85-30 with 25.5% w/w aluminum) were subjected to quasi-static cyclic compressive loading. The microstructure of each specimen was imaged before and after cyclic loading using the micro-computed tomography. This procedure was repeated after a 4-week time-recovery period. The stressstrain response of each specimen was quantified using four parameters per loading–unloading cycle, namely peak and valley stresses (related to material strength), cumulative dissipated energy density (related to cyclic hysteresis and accumulated damage), and apparent stiffness. The spatial distribution of primary components in each specimen, including pore space, was postprocessed from micro-CT images.

The cyclic compressive response of PBXN-109 mock formulations is very similar to that of filled elastomers. Namely, it exhibits a highly nonlinear elasto-plastic response without a distinct yield point, hysteresis, and progressive stress softening (or the Mullins effect) with cyclic stabilization. However, in sharp contrast to filled elastomers, these mock energetic composites exhibit an initial loading path noticeably different from subsequent loading paths. This nonlinear, path-dependent behavior observed in the macroscopic response is supported by changes in the mesoscopic spatial volume distribution of formulation components (i.e., binder, sucrose, aluminum, and pore space) observed using micro-CT. The quantification of these effects, before and after cyclic loading and time-recovering, confirmed that this behavior can be attributed to mesoscale deformation and relaxation mechanisms that emerge from the presence of a hard phase (sucrose) and a soft phase (binder) in the microstructure. The soft phase becomes progressively more compliant due to accumulated damage and particle–binder interfacial debonding. It undergoes homogeneous or affine deformation under compressive load, it recovers during unloading, and it relaxes over time. In sharp contrast, the hard phase experiences a collective rearrangement or non-affine deformation under compressive loading, and it only partially recovers during unloading and over time due to the irreversible nature of this collective rearrangement. The presence of a ductile third phase (aluminum) may significantly affect these mesoscale deformation and relaxation mechanisms by forming permanent sucrose–aluminum contacts and reducing the sucrose-binder interfacial area, i.e., by arresting interfacial debonding and thus effectively stiffening the soft phase. In all formulations, and due to the loading conditions, the majority of microstructural changes occurred near the core of the cylindrical specimens. Furthermore, in formulations with aluminum content, the non-affine nature of the deformation field appeared to be anisotropic (i.e., different longitudinal and radial behaviors were observed).

We close by pointing out that the work presented in this paper serves as the foundation of an ongoing study of the long-term effects of repetitive cyclic loading and time-recovery on the microstructure and mechanical properties of particle–binder composites. This study entails an execution of the proposed testing and recovery procedure over several months. The outcome of such a study will provide compelling affirmation of the observations and analyses presented in the paper, leading to a better understanding and characterization of the complex mechanical behavior of particle–binder composites.

Footnote

2

Allison Range, Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA. Email: range@purdue.edu

Acknowledgment

The authors wish to acknowledge Professor Patricia Davies, Professor Jeff Rhoads, Professor Steve Son, Professor Stuart Bolton, Jelena Paripovic, Allison Range, Nick Cummock, and Tim Manship for their feedback and interesting discussions. The authors would also like to acknowledge the assistance of Dr. Dhananjay Pai for his help with micro-CT experiments.

This research is supported by the Air Force Research Laboratory through Grant No. FA8651-16-0287 entitled “Near-Resonant Thermomechanics of Energetic and Mock Energetic Composite Materials,” (Grant No. FA8651-16-D-0287) entitled “Near-Resonant Thermomechanics of Energetic and Mock Energetic Composite Materials, Part II”, and (Grant No. FA8651-17-S-0003) entitled “Exploring the Thermomechanics of Energetic and Mock Energetic Composite Materials Under Quasi-Static and Near-Resonant Excitations.”

Distribution Statement A: Approved for public release; distribution unlimited. 96TW-2019-0454

Nomenclature

     
  • h=

    specimen height

  •  
  • r=

    radial distance from the specimen center

  •  
  • D=

    specimen diameter

  •  
  • V=

    specimen volume

  •  
  • hVOI=

    height of the volume of interest (VOI)

  •  
  • Eapparent,i=

    apparent stiffness for the ith compressive cycle

  •  
  • VVOI=

    volume of the volume of interest (VOI)

  •  
  • σmax,i=

    peak stress for the ith compressive cycle

  •  
  • σmin,i=

    valley stress for the ith compressive cycle

  •  
  • ɛmax,cyclic=

    maximum cyclic strain level

  •  
  • ɛmin,cyclic=

    minimum cyclic strain level

  •  
  • ɛultimate=

    strain at ultimate compressive strength

References

1.
Dargazany
,
R.
,
Khiêm
,
V. N.
, and
Itskov
,
M.
,
2014
, “
A Generalized Network Decomposition Model for the Quasi-Static Inelastic Behavior of Filled Elastomers
,”
Int. J. Plast.
,
63
, pp.
94
109
. 10.1016/j.ijplas.2013.12.004
2.
Mullins
,
L.
,
1948
, “
Effect of Stretching on the Properties of Rubber
,”
Rubber Chem. Technol.
,
21
(
2
), pp.
281
300
. 10.5254/1.3546914
3.
Mullins
,
L.
,
1969
, “
Softening of Rubber by Deformation
,”
Rubber Chem. Technol.
,
42
(
1
), pp.
339
362
. 10.5254/1.3539210
4.
Netzker
,
C.
,
Dal
,
H.
, and
Kaliske
,
M.
,
2010
, “
An Endochronic Plasticity Formulation for Filled Rubber
,”
Int. J. Solids Struct.
,
47
(
18–19
), pp.
2371
2379
. 10.1016/j.ijsolstr.2010.04.026
5.
Bergström
,
J. S.
, and
Boyce
,
M. C.
,
1998
, “
Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers
,”
J. Mech. Phys. Solids
,
46
(
5
), pp.
931
954
. 10.1016/S0022-5096(97)00075-6
6.
Miehe
,
C.
, and
Keck
,
J.
,
2000
, “
Superimposed Finite Elastic–Viscoelastic–Plastoelastic Stress Response With Damage in Filled Rubbery Polymers. Experiments, Modelling and Algorithmic Implementation
,”
J. Mech. Phys. Solids
,
48
(
2
), pp.
323
365
. 10.1016/S0022-5096(99)00017-4
7.
Akhavan
,
J.
,
2011
,
The Chemistry of Explosives
, 3rd ed.,
Royal Society of Chemistry
,
London, UK
.
8.
Idar
,
D. J.
,
Peterson
,
P. D.
,
Scott
,
P. D.
, and
Funk
,
D. J.
,
1998
, “
Low Strain Rate Compression Measurements of PBXN-9, PBX 9501, and Mock 9501
,”
AIP Conf. Proc.
,
429
(
1
), pp.
587
590
. 10.1063/1.55704
9.
Daniel
,
M. A.
,
2006
, “
Polyurethane Binder Systems for Polymer Bonded Explosives
,”
Tech. Rep. DSTO-GD-0492, Defence Science and Technology Organisation, Edinburgh (Australia) Weapons Systems DIV
.
10.
Palmer
,
S. J. P.
,
Field
,
J. E.
, and
Huntley
,
J. M.
,
1993
, “
Deformation, Strengths and Strains to Failure of Polymer Bonded Explosives
,”
Proc. R. Soc. London, A
,
440
(
1909
), pp.
399
419
. 10.1098/rspa.1993.0023
11.
Rae
,
P. J.
,
Goldrein
,
H. T.
,
Palmer
,
S. J. P.
,
Field
,
J. E.
, and
Lewis
,
A. L.
,
2002
, “
Quasi-static Studies of the Deformation and Failure of β-HMX Based Polymer Bonded Explosives
,”
Proc. R. Soc. London Ser. A: Math. Phys. Eng. Sci.
,
458
(
2019
), pp.
743
762
. 10.1098/rspa.2001.0894
12.
Ragaswamy
,
P.
,
Lewis
,
M. W.
,
Liu
,
C.
, and
Thompson
,
D. G.
,
2010
, “
Modeling the Mechanical Response of PBX 9501
,”
14th International Detonation Symposium
,
Coeur d'Alene, ID
,
Apr. 11–16
.
13.
Yılmaz
,
G. A.
,
Şen
,
D.
,
Kaya
,
Z. T.
, and
Tinçer
,
T.
,
2014
, “
Effect of Inert Plasticizers on Mechanical, Thermal, and Sensitivity Properties of Polyurethane-Based Plastic Bonded Explosives
,”
J. Appl. Polym. Sci.
,
131
(
20
), p.
40907
. 10.1002/app.40907
14.
Prakash
,
V.
,
Phadke
,
V. K.
,
Sinha
,
R. K.
, and
Singh
,
H.
,
2004
, “
Influence of Aluminium on Performance of HTPB-Based Aluminised PBXs
,”
Def. Sci. J.
,
54
(
4
), pp.
475
482
. 10.14429/dsj.54.2061
15.
Drodge
,
D. R.
, and
Williamson
,
D. M.
,
2016
, “
Understanding Damage in Polymer-Bonded Explosive Composites
,”
J. Mater. Sci.
,
51
(
2
), pp.
668
679
. 10.1007/s10853-013-7378-6
16.
Funk
,
D. J.
,
Laabs
,
G. W.
,
Peterson
,
P. D.
, and
Asay
,
B. W.
,
1996
, “
Measurement of the Stress/Strain Response of Energetic Materials as a Function of Strain Rate and Temperature: PBX 9501 and Mock 9501
,”
AIP Conf. Proc.
,
370
(
1
), pp.
145
148
. 10.1063/1.50735
17.
Thompson
,
D. G.
,
Idar
,
D. J.
,
Gray
,
G. T.
,
Blumenthal
,
W. R.
,
Cady
,
C. M.
,
Roemer
,
E. L.
,
Wright
,
W. J.
, and
Peterson
,
P. D.
,
2002
, “
Quasi-static and Dynamic Mechanical Properties of new and Virtually-Aged PBX 9501 Composites as a Function of Temperature and Strain Rate
,”
12th International Detonation Symposium
,
San Diego, CA
,
Aug. 11–16
.
18.
Paripovic
,
J.
, and
Davies
,
P.
,
2013
, “
Identification of the Dynamic Behavior of Surrogate Explosive Materials
,”
ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. Volume 8: 22nd Reliability, Stress Analysis, and Failure Prevention Conference; 25th Conference on Mechanical Vibration and Noise
,
Portland, OR
,
Aug. 4–7
.
19.
Paripovic
,
J.
, and
Davies
,
P.
,
2016
, “
A Model Identification Technique to Characterize the Low Frequency Behaviour of Surrogate Explosive Materials
,”
J. Phys.: Conf. Ser.
,
744
, p.
012124
. 10.1088/1742-6596/744/1/012124
20.
Grantham
,
S. G.
,
Siviour
,
C. R.
,
Proud
,
W. G.
, and
Field
,
J. E.
,
2004
, “
High-Strain Rate Brazilian Testing of an Explosive Simulant Using Speckle Metrology
,”
Meas. Sci. Technol.
,
15
(
9
), pp.
1867
1870
. 10.1088/0957-0233/15/9/025
21.
Williamson
,
D. M.
,
Palmer
,
S. J. P.
,
Proud
,
W. G.
, and
Govier
,
R.
,
2007
, “
Brazilian Disc Testing of a UK PBX Above and Below the Glass Transition Temperature
,”
AIP Conf. Proc.
,
955
(
1
), pp.
803
806
. 10.1063/1.2833246
22.
Idar
,
D. J.
,
Thompson
,
D. G.
,
Gray
,
G. T.
,
Blumenthal
,
W. R.
,
Cady
,
C. M.
,
Peterson
,
P. D.
,
Roemer
,
E. L.
,
Wright
,
W. J.
, and
Jacquez
,
B. J.
,
2002
, “
Influence of Polymer Molecular Weight, Temperature, and Strain Rate on the Mechanical Properties of PBX 9501
,”
AIP Conf. Proc.
,
620
(
1
), pp.
821
824
. 10.1063/1.1483663
23.
Li
,
J.
,
Lu
,
F.
,
Qin
,
J.
,
Chen
,
R.
,
Zhao
,
P.
,
Lan
,
L.
, and
Jing
,
S.
,
2012
, “
Effects of Temperature and Strain Rate on the Dynamic Responses of Three Polymer-Bonded Explosives
,”
J. Strain Anal. Eng. Des.
,
47
(
2
), pp.
104
112
. 10.1177/0309324711428836
24.
Kerschen
,
N. E.
,
Guo
,
Z.
,
Sun
,
T.
,
Claus
,
B.
,
Mares
,
J.
,
Fezzaa
,
K.
, and
Chen
,
W.
,
2017
, “
Visualization of PBX Response to Impact Loading
,”
32nd Technical Conference of the American Society for Composites
,
Purdue University, West Lafayette, IN
,
Oct. 23–25
.
25.
Thompson
,
D. G.
,
Deluca
,
R.
, and
Brown
,
G. W.
,
2012
, “
Time-Temperature Analysis, Tension and Compression in PBXs
,”
J. Energ. Mater.
,
30
(
4
), pp.
299
323
. 10.1080/07370652.2011.569831
26.
Chen
,
P.
,
Huang
,
F.
, and
Ding
,
Y.
,
2007
, “
Microstructure, Deformation and Failure of Polymer Bonded Explosives
,”
J. Mater. Sci.
,
42
(
13
), pp.
5272
5280
. 10.1007/s10853-006-0387-y
27.
Hudspeth
,
M.
,
Sun
,
T.
,
Parab
,
N.
,
Guo
,
Z.
,
Fezzaa
,
K.
,
Luo
,
S.
, and
Chen
,
W.
,
2015
, “
Simultaneous X-ray Diffraction and Phase-Contrast Imaging for Investigating Material Deformation Mechanisms During High-Rate Loading
,”
J. Synchrotron Radiat.
,
22
(
1
), pp.
49
58
. 10.1107/S1600577514022747
28.
Parab
,
N. D.
,
Roberts
,
Z. A.
,
Harr
,
M. H.
,
Mares
,
J. O.
,
Casey
,
A. D.
,
Gunduz
,
I. E.
,
Hudspeth
,
M.
,
Claus
,
B.
,
Sun
,
T.
,
Fezzaa
,
K.
,
Son
,
S. F.
, and
Chen
,
W. W.
,
2016
, “
High Speed X-Ray Phase Contrast Imaging of Energetic Composites Under Dynamic Compression
,”
Appl. Phys. Lett.
,
109
(
13
), p.
131903
. 10.1063/1.4963137
29.
Yang
,
B.-h.
,
Wu
,
A.-x.
,
Miao
,
X.-x.
, and
Liu
,
J.-z.
,
2014
, “
3D Characterization and Analysis of Pore Structure of Packed Ore Particle Beds Based on Computed Tomography Images
,”
Trans. Nonferrous Met. Soc. China
,
24
(
3
), pp.
833
838
. 10.1016/S1003-6326(14)63131-9
30.
Al-Raoush
,
R.
, and
Papadopoulos
,
A.
,
2010
, “
Representative Elementary Volume Analysis of Porous Media Using X-Ray Computed Tomography
,”
Powder Technol.
,
200
(
1–2
), pp.
69
77
. 10.1016/j.powtec.2010.02.011
31.
Koloushani
,
M.
,
Hedayati
,
R.
,
Sadighi
,
M.
, and
Mohammadi-Aghdam
,
M.
,
2018
, “
CT-based Micro-Mechanical Approach to Predict Response of Closed-Cell Porous Biomaterials to Low-Velocity Impact
,”
J. Imaging
,
4
(
3
), p.
49
. 10.3390/jimaging4030049
32.
Mukunoki
,
T.
,
Miyata
,
Y.
,
Mikami
,
K.
, and
Shiota
,
E.
,
2016
, “
X-Ray CT Analysis of Pore Structure in Sand
,”
Solid Earth
,
7
(
3
), pp.
929
942
. 10.5194/se-7-929-2016
33.
Yu
,
X.
,
Peng
,
G.
, and
Lu
,
S.
,
2018
, “
Characterizing Aggregate Pore Structure by X-Ray Micro-Computed Tomography and a Network Model
,”
Soil Sci. Soc. Am. J.
,
82
(
4
), pp.
744
756
. 10.2136/sssaj2017.11.0385
34.
He
,
Y.-j.
,
Mote
,
J.
, and
Lange
,
D. A.
,
2013
, “
Characterization of Microstructure Evolution of Cement Paste by Micro Computed Tomography
,”
J. Cent. South Univ.
,
20
(
4
), pp.
1115
1121
. 10.1007/s11771-013-1592-x
35.
Yermukhambetova
,
A.
,
Tan
,
C.
,
Daemi
,
S. R.
,
Bakenov
,
Z.
,
Darr
,
J. A.
,
Brett
,
D. J. L.
, and
Shearing
,
P. R.
,
2016
, “
Exploring 3D Microstructural Evolution in Li-Sulfur Battery Electrodes Using In-Situ X-Ray Tomography
,”
Sci. Rep.
,
6
(
1
), p.
35291
. 10.1038/srep35291
36.
Manner
,
V. W.
,
Yeager
,
J. D.
,
Patterson
,
B. M.
,
Walters
,
D. J.
,
Stull
,
J. A.
,
Cordes
,
N. L.
,
Luscher
,
D. J.
,
Henderson
,
K. C.
,
Schmalzer
,
A. M.
, and
Tappan
,
B. C.
,
2017
, “
In Situ Imaging During Compression of Plastic Bonded Explosives for Damage Modeling
,”
Materials
,
10
(
6
), p.
638
. 10.3390/ma10060638
37.
Chen
,
L.
,
Han
,
D.
,
Bai
,
S.-L.
,
Zhao
,
F.
, and
Chen
,
J.-K.
,
2017
, “
Compressive Behavior and Damage Evaluation of a PBX Substitute Material
,”
Mech. Adv. Mater. Struct.
,
24
(
9
), pp.
737
744
. 10.1080/15376494.2016.1196779
38.
Bruck
,
H. A.
,
McNeill
,
S. R.
,
Sutton
,
M. A.
, and
Peters
,
W. H.
,
1989
, “
Digital Image Correlation Using Newton-Raphson Method of Partial Differential Correction
,”
Exp. Mech.
,
29
(
3
), pp.
261
267
. 10.1007/BF02321405
39.
Bay
,
B. K.
,
Smith
,
T. S.
,
Fyhrie
,
D. P.
, and
Saad
,
M.
,
1999
, “
Digital Volume Correlation: Three-Dimensional Strain Mapping Using X-Ray Tomography
,”
Exp. Mech.
,
39
(
3
), pp.
217
226
. 10.1007/BF02323555
40.
Hu
,
Z.
,
Luo
,
H.
,
Bardenhagen
,
S. G.
,
Siviour
,
C. R.
,
Armstrong
,
R. W.
, and
Lu
,
H.
,
2015
, “
Internal Deformation Measurement of Polymer Bonded Sugar in Compression by Digital Volume Correlation of In-Situ Tomography
,”
Exp. Mech.
,
55
(
1
), pp.
289
300
. 10.1007/s11340-014-9856-4
41.
Hu
,
Z.
,
Luo
,
H.
,
Du
,
Y.
, and
Lu
,
H.
,
2016
, “Correlation of Microscale Deformations to Macroscopic Mechanical Behavior Using Incremental Digital Volume Correlation of In-Situ Tomography,”
Advancement of Optical Methods in Experimental Mechanics
, Vol.
3
,
H.
Jin
,
S.
Yoshida
,
L.
Lamberti
, and
M.-T.
Lin
, eds.,
Springer
,
Cham, Switzerland
, pp.
125
137
.
42.
Goldrein
,
H. T.
,
Rae
,
P. J.
,
Palmer
,
S. J. P.
, and
Lewis
,
A. L.
,
2001
, “Ageing Effects on the Mechanical Properties of a Polymer Bonded Explosive,”
Ageing Studies and Lifetime Extension of Materials
,
L.
Mallinson
, ed.,
Springer
,
Boston, MA
, pp.
129
136
.
43.
Range
,
A. R.
,
McMindes
,
N. R.
,
Tucker
,
J. B.
, and
Rhoads
,
J. F.
,
2018
, “The Influence of Formulation Variation and Thermal Boundary Conditions on the Near-Resonant Thermomechanics of Mock Explosives,”
Fracture, Fatigue, Failure and Damage Evolution
, Vol.
7
,
J.
Carroll
,
S.
Xia
,
A.
Beese
,
R.
Berke
, and
G.
Pataky
, eds.,
Conference Proceedings of the Society for Experimental Mechanics Series, Springer
,
Cham, Switzerland
, pp.
47
55
.
44.
Lochert
,
I. J.
,
Dexter
,
R. M.
, and
Hamshere
,
B. L.
,
2002
, “
Evaluation of Australian RDX in PBXN-109
,”
Tech. Rep. DSTO-TN-0440, Defence Science and Technology Organisation, Edinburgh (Australia) Weapons Systems DIV
.
45.
Rumchik
,
C. G.
, and
Jordan
,
J. L.
,
2007
, “
Effect of Aluminum Particle Size on the High Strain Rate Properties of Pressed Aluminized Explosives
,”
AIP Conf. Proc.
,
955
(
1
), pp.
795
798
. 10.1063/1.2833243
46.
Tang
,
M.-f.
,
Pang
,
H.-y.
,
Lan
,
L.-g.
,
Wen
,
M.-p.
, and
Ming
,
L. I.
,
2016
, “
Constitutive Behavior of RDX-Based PBX with Loading-History and Loading-Rate Effects
,”
Chin. J. Energ. Mater.
,
24
(
9
), pp.
832
837
. 10.11943/j.issn.1006-9941.2016.09.002
47.
Jiang
,
Y.
, and
Sehitoglu
,
H.
,
1994
, “
Multiaxial Cyclic Ratchetting Under Multiple Step Loading
,”
Int. J. Plast.
,
10
(
8
), pp.
849
870
. 10.1016/0749-6419(94)90017-5
48.
Zhang
,
J.
, and
Jiang
,
Y.
,
2004
, “
A Study of Inhomogeneous Plastic Deformation of 1045 Steel
,”
ASME J. Eng. Mater. Technol.
,
126
(
2
), pp.
164
171
. 10.1115/1.1647125
49.
Jiang
,
Y.
, and
Zhang
,
J.
,
2008
, “
Benchmark Experiments and Characteristic Cyclic Plasticity Deformation
,”
Int. J. Plast.
,
24
(
9
), pp.
1481
1515
. 10.1016/j.ijplas.2007.10.003
50.
Paul
,
S. K.
,
Sivaprasad
,
S.
,
Dhar
,
S.
, and
Tarafder
,
S.
,
2011
, “
Key Issues in Cyclic Plastic Deformation: Experimentation
,”
Mech. Mater.
,
43
(
11
), pp.
705
720
. 10.1016/j.mechmat.2011.07.011
51.
Blanchard
,
A. F.
, and
Parkinson
,
D.
,
1952
, “
Breakage of Carbon-Rubber Networks by Applied Stress
,”
Ind. Eng. Chem.
,
44
(
4
), pp.
799
812
. 10.1021/ie50508a034
52.
Houwink
,
R.
,
1956
, “
Slipping of Molecules During the Deformation of Reinforced Rubber
,”
Rubber Chem. Technol.
,
29
(
3
), pp.
888
893
. 10.5254/1.3542602
53.
Kraus
,
G.
,
Childers
,
C. W.
, and
Rollmann
,
K. W.
,
1966
, “
Stress Softening in Carbon Black-Reinforced Vulcanizates. Strain Rate and Temperature Effects
,”
J. Appl. Polym. Sci.
,
10
(
2
), pp.
229
244
. 10.1002/app.1966.070100205
54.
Diani
,
J.
,
Fayolle
,
B.
, and
Gilormini
,
P.
,
2009
, “
A Review on the Mullins Effect
,”
Eur. Polym. J.
,
45
(
3
), pp.
601
612
. 10.1016/j.eurpolymj.2008.11.017
55.
Ravindran
,
S.
,
Gupta
,
V.
,
Tessema
,
A.
, and
Kidane
,
A.
,
2019
, “
Effect of Particle Mass Fraction on the Multiscale Dynamic Failure Behavior of Particulate Polymer Composites
,”
Exp. Mech.
,
59
(
5
), pp.
599
609
. 10.1007/s11340-019-00493-4
56.
Liu
,
B.
,
Huang
,
W.
,
Wang
,
H.
,
Wang
,
M.
, and
Li
,
X.
,
2014
, “
Study on the Load Partition Behaviors of High Particle Content B4C/Al Composites in Compression
,”
J. Compos. Mater.
,
48
(
3
), pp.
355
364
. 10.1177/0021998312472220
57.
Topin
,
V.
,
Delenne
,
J. Y.
,
Radjaı
,
F.
,
Brendel
,
L.
, and
Mabille
,
F.
,
2007
, “
Strength and Failure of Cemented Granular Matter
,”
Eur. Phys. J. E
,
23
(
4
), pp.
413
429
. 10.1140/epje/i2007-10201-9
58.
Scarfe
,
W. C.
, and
Farman
,
A. G.
,
2008
, “
What is Cone-Beam CT and How Does it Work?
,”
Dent. Clin. North Am.
,
52
(
4
), pp.
707
730
. 10.1016/j.cden.2008.05.005
59.
Shapiro
,
L. G.
, and
Stockman
,
G. C.
,
2001
,
Computer Vision
,
Prentice Hall
,
Upper Saddle River, NJ
.
60.
Wellington
,
S. L.
, and
Vinegar
,
H. J.
,
1987
, “
X-Ray Computerized Tomography
,”
J. Pet. Technol.
,
39
(
8
), pp.
885
898
. 10.2118/16983-PA
61.
MATLAB®, Version 9.4.0, The MathWorks Inc., 1984-2018, Natick, MA
.