Full-field quantitative visualization of shock-driven pore collapse and failure modes in PMMA

Porous materials, such as energetic (e.g., polymer bonded explosives) and structural (e.g., foams) materials, feature in many applications. When such materials are dynamically loaded (e.g., shocked), a pore collapse phenomenon occurs, the physics of which has great implications for both these material classes. Generally, in energetic materials, the main concern is with mechanically induced hot spot generation via stress waves, which likely dictates the ignition response of the material^1–5 and may lead to detonation. This has often been linked to pore collapse.^6–8 Thus, it is desirable to understand the mechanical response of pore collapse, which may provide insights into the hot spot generation mechanisms and aid in preventing accidental detonations.

For structural materials, porosity may be either intentional, as is the case for metallic foams and architected metamaterials, or unintentional through defects in manufacturing processes such as in additive manufacturing. Additionally, in these examples, the length-scale of porosity ranges from nanometers through centimeters, making the material response of porous media a rich topic of study. The primary issue with these structures lies in the localized material and structural failure, which is often driven by stress concentrations in the neighborhood of pores. Some porous media can be treated as continuum materials through the investigation of the effective or homogenized response when the pore scale is far smaller than the application scale. However, for the most effective prediction of failure in these structures, it is critical to also characterize the failure response of the material and to understand it from a fundamental, local perspective at the individual pore scale.

The focus of this study is to quantitatively characterize the deformation and failure response surrounding a single collapsing spherical pore. This is accomplished through the generation of shock-induced pore collapse via plate impact experiments, and measurements are conducted with high speed imaging and internal digital image correlation (DIC) to capture the in situ full-field deformation response. Previous studies have extensively investigated the macroscopic dynamic response of porous materials through pointwise measurements, in addition to several works at the local scale that use qualitative imaging techniques or post-mortem analysis to understand the physics of pore collapse at the local scale. The rich literature, discussed below, leaves a substantial need for full-field quantitative analysis of the pore collapse problem, which is addressed in this study.

The macroscopic, or continuum, response of porous materials under dynamic loading has been studied at great length. In general, porous materials have been shown to possess favorable qualities such as shock disruption via micro-inertial effects^9,10 and shock attenuation due to the large plastic work required to fully close pores.¹¹ However, the spall strength has been shown to depend on the initial porosity, in some cases decreasing (compared to its fully dense counterpart),¹² and others increasing.^11,13 This dependence has been shown to depend on the bulk material properties and initial porosity. The many mechanisms involved in their shock response makes the design with porous materials a complicated task. Theoretical approaches, assuming symmetric collapse of pores, originated with the work of Herrmann, which enabled the introduction of porosity to the equation of state (EOS) for materials in what is known as the $P−α$ theory.¹⁴ Carroll and Holt subsequently developed an analytical form for the $α$ parameter¹⁵ through the analysis of a thick spherical shell. Following this work, many modifications were made to the $P−α$ theory. For example, Butcher et al. incorporated the influence of deviatoric stress, work-hardening, initial void size, and material viscosity,¹⁶ which revealed the effect of micro-inertia and viscosity on delaying the void collapse. Furthermore, extensive analytical, numerical, and experimental investigations have followed to understand the macroscopic response of porous solids under dynamic loading conditions.^9,10,17,18

While the macroscopic (continuum) response to shock loading, measured through traditional interferometric techniques, has been well investigated to characterize the equation of state and shock structure, studies on the deformation and failure at the local length scale of individual pores have been sparse. However, the advances in high speed imaging technologies in the last few decades have begun to enable full-field investigation at the local scale. Early experiments were limited to qualitative characterization of the collapse of cylindrical holes in transparent gels^19,20 through high speed imaging and later were extended to quantitative characterization via particle image velocimetry.²¹ These authors observed jetting at higher stresses and also clear evidence of shock shielding and amplification effects when various arrays of holes were loaded.

Recently, access to advanced x-ray synchrotron sources has enabled internal imaging of opaque structural materials during plate impact/shock experiments. Investigation, using x-ray phase contrast imaging (PCI), of additively manufactured lattice structures under shock revealed insights into the jet initiation (simple cubic) and mitigation (face centered tetragonal) phenomena, which were controlled by the details of the lattice structures.²² Further, the role of material parameters, lattice length scales, and impact velocity on jetting transitions was systematically investigated via experiment and simulation.²³ Similar experiments demonstrated that a pore in the shape of an elongated triangle behaved as a shock diode: promoting jetting when impacted from one direction while disrupting the shock from the other direction.²⁴ These experiments also investigated the shock shielding properties of a series of holes.

A close experimental analog to the spherical pore collapse phenomenon is the hole closure experiment, in which specimens with cylindrical holes are dynamically loaded via plate impact. Glazkov et al. used these experiments, with pre- and post-mortem hole size measurements, to infer material response at high strain rates.²⁵ Lind et al. performed similar hole closure experiments in copper,²⁶ now using PCI to capture the evolution of the area of the hole, in situ during closure. They primarily used the rich and complicated loading state in these experiments to calibrate material models at high strain rates and large strains. However, the inverse analysis also offers insights into the physics of pore collapse, as the rate of closure and the final collapsed volume were shown to be highly dependent on the strain rate hardening of the material. Further analysis of these experiments could inform parameters for analytical continuum models such as the $P−α$⁠. Follow-up experiments on tantalum²⁷ served a similar role for model parameter calibration while also observing possible shear localization (an important failure and hot-spot generation mechanism) near the closed hole through post-mortem electron backscatter diffraction (EBSD) analysis.

To investigate the physics of pore collapse, Escauriza et al. conducted plate impact experiments on PMMA with a spherical pore.²⁸ Using PCI, they captured the pore collapse evolution—observing a transition from strength dominated (low pressure) to hydrodynamic (high pressure) regimes, the development of cracks at low pressures, and jetting instabilities at high pressures. Complementary numerical simulations predicted the development of adiabatic shear bands during these collapse events.²⁹ Recently, Lovinger and Kositski carried out cylindrical pore collapse experiments in Ti–6Al–4V specimens, leveraging plate impact with a soft-catch setup, which enabled post-mortem analysis of the specimens.³⁰ Their work revealed the first definitive evidence of shear localization in pore collapse via adiabatic shear banding and shear cracking. They further explored the role of pore size and spacing on failure initiation and connectivity, respectively. While the above-mentioned works have all been limited by sample fabrication and experimental resolution to pores near the mm scale, a new technique has been recently developed, which uses PCI to capture the collapse of spherical pores of the order of several tens of micrometers.³¹ This technique promises future understanding of pore collapse at very fine spatial scales, which are relevant to many commonly occurring pores. These experimental efforts and complementary computational endeavors have enriched our understanding of the pore collapse phenomenon. However, to understand the details of shear localization and failure thresholds, a need still remains for quantitative full-field measurements during the pore collapse phenomenon under shock compression.

To quantitatively characterize the deformation field around a pore under shock compression, it is proposed to perform plate impact experiments on PMMA samples with a single, embedded, spherical pore, in conjunction with high speed digital image correlation (DIC)³²—a non-contact quantitative imaging technique. Plate impact experiments require measurements to be performed at the center of the target plate, such that the uniaxial strain condition is maintained throughout the measurement, until the time at which release waves arrive. Hence, the embedded pore must be located at the center of the sample, and DIC performed internally at the midplane of the pore. This is essential to capture the true deformation response and failure mechanisms during the pore collapse event, as opposed to alternative approaches of making surface measurements and leveraging computational tools to infer the internal response. Further, this approach enables a more physically relevant investigation of spherical pores undergoing collapse, as opposed to cylindrical holes.

In this study, the internal DIC technique³³ for shock compression experiments is implemented to study pore collapse at shock stresses ranging from 0.4 to 1.0 GPa. The internal DIC method is briefly summarized, and details regarding sample preparation, plate impact experiments, and high speed imaging are described in Sec. II. Results for the experiments and initial DIC analysis are presented in Sec. III. Simulations are then conducted, replicating the loading conditions of the experiments in Sec. IV, and the physics governing failure mechanisms are discussed and supported with theoretical models in Sec. V. Last, concluding remarks are given in Sec. VI.

Digital image correlation (DIC) is a non-contact, full-field, quantitative imaging technique, which measures the displacement field of a body undergoing deformation by correlating, or pattern matching, a unique grayscale speckle pattern, which is applied to the specimen prior to deformation.³² Images taken before and throughout the deformation then contain the necessary information for correlation to extract the displacement field with respect to the reference (undeformed) configuration, which can be converted to full-field strain measurements through discrete differentiation. The technique is traditionally restricted to surface measurements; however, in this work, an internal DIC framework is employed to capture deformation around a collapsing pore under shock compression. The concept of internal DIC has been successfully implemented previously in the quasi-static regime³⁴ and in dynamic laser-induced cavitation experiments in gels, conducted under a microscope.³⁵ Recently, this technique was extended to be used in full-scale dynamic experiments, demonstrating its feasibility and accuracy for application to both split-Hopkinson (Kolsky) pressure bar and plate impact shock experiments. Details of the development of the internal DIC technique are presented elsewhere³³ and are only briefly summarized here. The optical technique involves embedding a speckle pattern at the internal plane of the transparent sample. By visualizing through the transparent sample (shifting the camera’s focal plane to the internal plane), in-plane deformation is captured in the images (i.e., the direction of compression is in the same plane as the speckle pattern, in this case, horizontally). Here, normal plate impact experiments were conducted on PMMA samples, coupled with ultra-high speed imaging and digital image correlation (DIC)^32,36,37 to extract full-field quantitative strain measurements surrounding the pore during deformation.

Specimen preparation for plate impact experiments is a stringent process, requiring precise, detailed steps to ensure the specimens are suitable for the experiments. The target specimens for all experiments in this study, depicted in Fig. 1(a), were manufactured from PMMA stock material obtained from E&T Plastics (Long Island City, New York). To embed the internal DIC speckle pattern and spherical pore, the target is manufactured from two cuboidal half-samples with rectangular cross section of nominal dimensions, $36×18$ mm $2$ and 17 mm thick. Both half-samples are carefully lapped together in the impact and visualization directions to ensure identical dimensions and create ideal bonding surfaces. The visualization surfaces (all surfaces that are parallel to the speckle pattern plane) are then mirror polished to maximize transparency, and matching hemispheres of radius 0.4 mm are milled into each half-sample such that they precisely align to their counterpart to generate a nearly perfect spherical pore when combined. Next, the DIC speckle pattern is applied to the inner surface of one half-sample by airbrushing a thin layer of transparent paint and blowing toner powder onto the wet paint, before adding a second layer of transparent paint to seal the speckle pattern.

After the paint is allowed to dry for at least 24 h, the half-samples are glued together with a two-part epoxy, EpoxAcast 690 (SmoothOn, Macungie, Pennsylvania). Confining plates are used to precisely align the two hemispheres to generate a pristine spherical pore, and weights are applied to the sandwiched sample to squeeze out excess glue and, thus, maximize the glue bond strength. Once the sample has settled for approximately 10 min, the weights are removed, the sample is carefully disassembled, and any glue that has filled the pore is removed. The assembly is then reset with confinement and weights, without additional glue, to ensure the pore is filled with air, rather than glue. Finally, after the glue cures for 24 h, the now-intact sample is removed, re-lapped in the impact direction to meet specifications for the experiment, and the alignment of the pore is inspected under a microscope. Figure 1(b) shows an example of the pore alignment. Additionally, because of the precise restrictions on the half-sample geometry and careful alignment during gluing, residual stresses arising from the gluing process are estimated to be orders of magnitude smaller than the imposed shock stress in the experiments. Hence, these effects are considered to be negligible in the outcome of the experiments. The final restrictions for the target sample are less than 30  $μ$m variation in thickness and less than 1  $μ$m surface flatness as measured by Fizeau rings under monochromatic light with a quartz optical flat. After the target is assembled, the final specimens are nominally $36×36$ mm square plates, which are 17 mm thick.

Flyer plates made of PMMA or aluminum (⁠ $7075$ alloy) are more straightforward to manufacture, starting with machined cylindrical samples of 35 mm diameter and 13 mm thickness. They are lapped until the thickness variation is less than 10  $μ$m and the surface flatness is less than 0.5  $μ$m. These restrictions on the target and flyer plates ensure planar shock wave generation upon impact. Finally, the flyer plate is glued into the projectile and inserted in the gun barrel.

Electrical shorting pins are used in these experiments to trigger diagnostics and to measure the angle of inclination (tilt) between the flyer and target at impact. These pins are inserted into holes in the target plate, which correspond with the perimeter of the flyer, and are glued in place, as shown in Fig. 1(c). They are then sanded down, and lapped until flush with the impact surface. Finally, the target plate is glued to the target holder, the shorting pins are wired into a logic circuit box that is connected to a digital oscilloscope, and the target holder is mounted onto a six-degree-of-freedom gimbal which is moved into the vacuum chamber for alignment.

Normal plate impact experiments were performed using a powder gun facility at Caltech, which consists of a 38.7 mm diameter keyed barrel of 3 m length. Before running the experiment, the projectile is placed at the end of the gun barrel and brought close to the target, which is mounted to the gimbal in the vacuum chamber. Careful translational and rotational alignment, using an auto-collimator, is performed to ensure perfect alignment between the impact surfaces of the two plates. This is critical to minimize the impact tilt between the flyer and target plates during the shot and ensure plane wave propagation. The high speed camera is also optimally configured and aligned parallel to the target visualization surface closest to the camera, after which the field of view and focal plane are set. A schematic of the full experimental setup is provided in Fig. 2(a), and an image of the setup inside the vacuum chamber is shown in Fig. 2(b). The experiment begins when the gunpowder charge is ignited behind the projectile, propelling it down the barrel where it impacts the target plate, situated in the vacuum chamber at the end of the barrel. Upon impact, shorting pins trigger the high speed imaging diagnostics to begin recording. Simultaneously, a planar shock wave is generated in both the target and flyer, and the camera captures images of the shock propagation and the in-plane deformation of the target plate. To measure the impact velocity, a laser system composed of two precisely spaced laser gates captures the moment at which the projectile breaks each laser plane. In some experiments, which do not use a conductive flyer (Pore-0.4), the laser gate system is also used to trigger the camera to record.

The timing of several release waves must also be calculated, including those initiating from the flyer free surface, the target free surface, and the perimeter/boundary of the target plate because of traction-free conditions. The design can be visualized through a distance–time $(x−t)$ diagram which plots the propagation of all relevant stress waves in the experiment. Further discussion of wave propagation and $x−t$ diagrams can be found elsewhere.³⁸ An example $x−t$ diagram is shown in Fig. 2(c) for the 0.41 GPa shock stress pore collapse experiment. Upon impact at $time, t=0$⁠, a normal shock wave is generated in both the target (forward-propagating) and flyer (backward-propagating), which induces pore collapse after passing the pore at 2.03  $μ$s. The “measurement window” begins when the shock wave passes the camera field of view at 2.54  $μ$s and ends when the field of view is disturbed by the arrival of either a free surface release, boundary release, or “failure wave” observed previously in PMMA.³⁹ These waves alter the uniaxial strain loading conditions, occlude the field of view, or both. This measurement window can be maximized for a given shock stress by using a thick flyer and target (to delay the arrival of free surface release waves), and optimizing the distance of the pore away from the impact surface to balance competition between boundary wave and failure wave arrival. Flyer diameter and target width could also be used to delay boundary wave arrival, but are limited by the size of the powder gun barrel. In the example used in Fig. 2(c), the measurement window is closed by the arrival of boundary release waves at 5.20  $μ$s, producing a predicted window of 2.66  $μ$s for Pore-0.4. Unfortunately, the failure wave speed increases with shock stress, making experiment design difficult at higher stresses by significantly shortening the maximum measurement window.

To capture clear images of the speckle pattern during the dynamic event, precise visualization setup is necessary. The high speed imaging setup is composed of the HPV-X2 camera (Shimadzu, Kyoto, Japan) set to capture images at $10000000$ frames/s (100 ns inter-frame time, 50 ns exposure time) along with the CAVILUX Smart pulsed, incoherent laser light source (Cavitar, Tampere, Finland), which is synchronized to the camera exposures and pulsed for 50 ns durations. To perform high magnification imaging with sufficient resolution to capture fine features of deformation at the internal plane, the camera is equipped with a $0.7−4.5$x zoom lens and 2x adapter tube (Navitar, Rochester, New York), which achieved a typical field of view of approximately $2.8×1.8$ mm (⁠ $400×250$ pixels). For such a small field of view, the camera is also mounted to a five-degree-of-freedom optical stage, which allows precise alignment of the camera’s field of view. The high magnification lens also produces a highly light-starved environment, which is remedied by configuring the laser light source in a backlit setup to maximize illumination. Additionally, the large curvature of the zoom optic generates image distortions, which are addressed through standard distortion correction procedures. Images are taken of the rigid body translation of the target at several vertical and horizontal distances, and a correction function is computed, which regains uniform displacement fields. This correction function is subsequently used for the experimental images as well. Finally, as was discussed in previous work,³³ some optical distortions arise resulting from the change in refractive index across the shock front. This is mitigated by aligning the camera lens to the target visualization surface closest to the camera to maximize the parallelism between the two.

Four pore collapse experiments were conducted at shock stresses ranging from 0.4 to 1.0 GPa. The shock conditions were controlled primarily by varying the impact velocity, as well as changing the flyer plate material as-needed. For all the experiments, the pore size was kept constant at 800  $μ$m diameter while the shock stress was varied. Details of the specimen geometries and loading conditions are summarized in Table I. The naming convention adopted for numbering the experiments indicates the experiment type and nominal shock stress (e.g., Pore-0.6 refers to the pore collapse experiment under nominally 0.6 GPa shock stress).

For each experiment, the entire shock loading process is captured via high speed imaging, beginning with the ambient unshocked state and ending with the arrival of release waves, which concludes the measurement window. A representative set of images for all four experiments is shown in Fig. 3, and Fig. 3(a) illustrates the evolution of Pore-0.4. From left to right, one can see the unshocked (ambient) condition, the shock wave (dark band) passing the pore (propagating from left to right), and the shocked state. Though not shown here, it is also possible to briefly visualize the radial wave reflection off of the pore surface. Similarly, Figs. 3(b)–3(d) show one frame of the shocked state for each of the remaining experiments, as the details of evolution are generally difficult to ascertain from the deformation images alone. DIC analysis will reveal more insight to the evolution of the shocked state. In general, the deformation images reveal a clear progression from nearly indiscernible collapse in Pore-0.4 to significant collapse in Pore-1.0. They also show the development of fracture at the pore interface in Pore-0.8, which will be discussed further in Sec. III C.

After capturing deformation images with a clear speckle pattern after shock arrival, the images were processed via DIC, computing the displacement field and subsequently the Lagrangian strain measures. While blurry regions (and fracture, in the case of Pore-0.8) obscured the speckle pattern in Pore-0.8 and Pore-1.0, the image quality in Pore-0.4 and Pore-0.6 enabled excellent correlation. To analyze the strain concentrations around the collapsing pore, line slices are taken vertically and horizontally through the center of the pore. A schematic of this concept is shown in Fig. 4(a) along with an example using full-field DIC data from Pore-0.4. The results for longitudinal (⁠ $ε 11$⁠) and lateral (⁠ $ε 22$⁠) strain along both the vertical and horizontal lines are presented in Fig. 4(b). The plots summarize the shocked material response (unshocked measurements of zero strain are omitted). Data for each experiment are visualized by the strain averaged over the shocked time instances at each spatial location along the line slice (solid curve) and the shaded envelope of all measurements in time, which is bounded by the maximum and minimum (dashed curves). Each plot includes the experimental results combined from all time instances after the shocked state was achieved, along with the static, elastic solution for uniaxial strain compression of a spherical pore. This is determined using the elastic solution for static uniaxial stress loading imposed on a spherical pore, first derived by Southwell.⁴² The solution is superposed to incorporate the shock stress and lateral confining stress [ Appendix A, Eq. (A5)] associated with the plate impact experiment’s uniaxial strain loading condition. Elastic parameters for PMMA under dynamic compression with confinement are used;⁴³ further details can be found in  Appendix A.

The line slice plots reveal good overall agreement between experimental and theoretical results, demonstrating the ability to accurately capture strain concentrations as large as 16%. In particular, Pore-0.4 is expected to be nearly completely elastic because the nominal 0.41 GPa loading is well below the Hugoniot elastic limit of 0.8 GPa.⁴⁴ The resulting strain measurements reinforce this assumption, as they coincide very closely with the elastic solution. The quality of the DIC data is particularly good along the vertical line, while more noise is present along the horizontal line. This likely arises due to diffraction and the shock reflections off the pore at the front and back, leading to a complex deformation state. While agreement with the elastic solution provides confidence in the experimental result, deviations from the elastic solution are more instructive. In particular, the longitudinal strain along the horizontal line reveals a significant discrepancy from the elastic solution. The strain near the back of the pore (positive radial position) coincides with the theory, but material near the front of the pore (surface upon which the shock impinges first) undergoes much more significant compressive strain in both Pore-0.4 and Pore-0.6. This is a clear indication of the shielding effect generated by the pore when a shock wave interacts with it. That is, the front of the pore is subjected to the full extent of the shock stress, but the pore diffracts (reflects) the wave because of the traction-free boundary condition at the pore surface (owing to the impedance mismatch between PMMA and void). This leaves the region immediately behind the pore shielded from the shock, thus experiencing much less of the imposed loading. This phenomenon is consistent with the prior observations of stress wave loading in gels.²¹ 

Turning briefly to the collapsing pore itself, rather than the near-field deformation, it is worth noting that the collapsed geometry is close to an ellipsoid rather than a spheroid. This occurs, especially in this strength-dominated, low pressure regime because the shock stress is rather large compared to the lateral confining stress, leading to significantly larger deformation in the horizontal (shock) direction than the vertical (transverse) direction. An estimate for the lateral confining stress is given in Eq. (A5) using elastic theory, though this only holds near the elastic regime where minimal inelastic deformation develops. This difference becomes less severe at higher stresses (e.g., Pore-1.0) where more substantial vertical collapse begins to develop. Yet, even at very high pressures (hydrodynamic regime), as reported by Escauriza et al. for spherical pores in PMMA, the collapse in the shock direction is more drastic than the lateral directions, leading eventually to the development of jetting.²⁸ 

Additionally, it is possible to calculate the volume of the pore during its collapse. This is accomplished by thresholding the deformation images to produce a black and white image, from which the pore boundary can be identified using the MATLAB image processing toolbox.⁴⁵ The pore boundary is then fit to two partial ellipses, for the left and right sides, as depicted in Fig. 8(a). Assuming axisymmetric deformation, the volume can be computed as the sum of two ellipsoidal caps. The calculated volume evolution, normalized by the initial volume of the pore, for all four experiments is plotted in Fig. 4(c). The two lower stress (⁠ $0.41$ and 0.62 GPa) experiments appear to reach a final collapsed volume within the first 1  $μ$s after shock arrival, while the two higher stress experiments may still be evolving. The final measured normalized volume is plotted against the shock stress in Fig. 4(d). However, because of the limited time for measurements, it is difficult to establish an exact trend for the final collapsed volume. Additional data at higher stresses would be essential to establish a relationship between shock stress and collapsed volume. The work of Escauriza et al. on pores of a larger scale provides a substantial discussion of the pore volume evolution at stresses up to 17 GPa, and observes that complete collapse occurs near 1.25 GPa.²⁸ 

Because adiabatic shear bands (ASBs) are very fine structures, with a thickness in PMMA of approximately 20  $μ$m,⁴³ even the high magnification imaging used here is insufficient to capture the full details of these bands. Instead, using DIC, it is possible to capture the filtered deformation field. This is why the DIC measurements show bands of $∼$6% strain compared to expected strain of $∼$100% in ASBs. To confirm that the results shown here are consistent with the filtered deformation measurement of ASBs, a subset size analysis is performed. This analysis investigates the actual strain value by comparing DIC results with various correlation parameters. These correlation parameters (subset size, step size, and strain filter) influence the overall filtering influence of the DIC postprocessing, which is summarized by the virtual strain gage length,^41, $L VSG$ [Eq. (2)]. Results of the subset size analysis indicate that the bands are indeed physical features and suggest that the actual strain magnitude is of the same order of magnitude as is expected for shear bands. Details of the analysis can be found in  Appendix B. Recent work has also identified shear localization in hole closure experiments via post-mortem analysis for tantalum²⁷ and Ti–6Al–4V.³⁰ 

As mentioned above, Pore-0.8 reveals fracture initiation at the pore surface, beginning in the first visible image after the shock passes the pore, which can be seen in Fig. 6(b). While the emergence of a crack prevents DIC analysis for this experiment, the raw deformation images enable characterization of the dynamic crack evolution during pore collapse. Escauriza et al. have previously observed a similar fracture during pore collapse of PMMA;²⁸ however, it was unclear whether boundary release waves or wave reflections from the glue interface could have applied necessary tensile loading to initiate mode I fracture. Here, the experimental design enforces nominally uniaxial strain loading conditions during the measurement window. Though the pore introduces a non-uniform stress state, theoretical and computational analyses indicate that the development of localized tension, which could drive mode I fracture, is not possible. Hence, one can conclude the crack observed here arises from shear-driven fracture, which will be further addressed in Secs. IV and V.

The crack is identified visually by the black band, which propagates forward with increasing time increments. For each image after the shock passes, the crack path is traced and displayed in Fig. 6(b) along with a magnified view of the crack in the inset, where the crack can be seen most clearly. Additionally, the length of the crack as a function of time is shown in Fig. 6(a). The average crack tip speed, $V crack≈959$ m/s, is found to fall between the Rayleigh wave speed and shear wave speed of PMMA (⁠ $935$ and 1000 m/s, respectively⁴⁷). However, the experimental resolution for crack speed measurement is insufficient to distinguish between crack propagation at or just above the Rayleigh wave speed. Most likely, the crack speed has approached near to the Rayleigh wave speed, consistent with the understanding that crack speeds between the Rayleigh and Shear wave speeds are unstable.^48,49 Further analysis of these results is necessary to unravel the mechanisms driving the fracture initiation point, path, as well as the modality of the fracture itself. This will be discussed in Sec. V.

The initial observations from the experimental results, presented above, indicate the intriguing presence of two failure mode transitions during pore collapse of PMMA which, until now, have not been conclusively observed in situ. The physics governing these transitions will be investigated by carrying out numerical simulations in Sec. IV along with theoretical modeling in Sec. V.

Numerical simulations were performed using the finite element software, Abaqus/Explicit,⁵⁰ to augment our understanding of the physics involved in these experiments by providing access to information otherwise unavailable in the experimental data.

Simulations were set up with a 2D axisymmetric assumption, mirroring the loading conditions imposed in experiments Pore-0.4, Pore-0.6, and Pore-0.8. Simulation of the conditions in Pore-1.0 has not been conducted because the experiment did not provide information about the deformation or localization behavior with which to compare. Identical geometries were used for the pore size while shrinking the target and flyer plate lateral dimensions to improve computational times. Because of the smaller lateral dimension, zero-displacement boundary conditions were applied to the lateral boundaries to maintain uniaxial strain and prevent boundary release waves. Additionally, the flyer plate and target plate geometries were created such that they shared nodes at the impact surface, thereby ensuring perfect contact and zero tilt during the simulated impact. Finally, the model was meshed with quadrilateral elements (CAX4R) in such a way that mitigates computational cost and preserves relevant physics near the pore. This was done by refining the mesh $( 5 μ m e l e m e n t s i z e )$ near the pore interface such that the adiabatic shear bands observed in experiments could be captured while coarsening the mesh $( 20 μ m \ element \ size )$ in the remainder of the model geometry. Further details are available in  Appendix C. Because of the inherent length scale for localized shear features in finite element analysis (FEA), a mesh convergence study was performed. From this study, reasonable convergence was obtained with respect to the number, mode, and spacing of adiabatic shear bands for all simulations with element size at or below 5  $μ$m. Additionally, axisymmetric simulations were performed because full 3D simulations are prohibitively expensive to perform with the converged mesh size. To confirm that the axisymmetric assumption is reasonable, 3D simulations were performed with a coarse mesh $( 20 μ m )$ near the pore. This produced deformation and shear band patterns that were axisymmetric. Hence, 2D axisymmetric simulations are used throughout this study in place of full 3D simulations.

Results from the numerical simulations for shock stresses corresponding to the experiments are presented in Fig. 7 with the time shifted such that the shock wave arrives at the pore surface when $t=0$⁠. Equivalent plastic strain (shear strain) is plotted for several time instances to provide a comparison to the localized shear response observed in experiments. Upon initial observation, it is clear that the initiation of adiabatic shear bands (ASBs) is almost entirely absent for shock stress of 0.41 GPa [Fig. 7(a)], matching the response from experiment Pore-0.4. At 0.62 GPa shock stress [Fig. 7(b)], a substantial number of distributed ASBs develop. The right-most image in Fig. 7(b) reveals the maximum temperature in the simulated shear bands to be $∼$400 K, which corresponds with the prescribed melting temperature in the thermal softening portion of the plasticity model. These features indicate that the first failure mode transition, from diffuse strain concentration in Pore-0.4 to localized shear via adiabatic shear banding in Pore-0.6, is governed simply through thermo-viscoplastic mechanics in the form of the Johnson–Cook model, which was implemented in FEA and replicated the transition. Given that the target material, PMMA, is amorphous, one would expect this to be the case, as the development of ASBs in PMMA cannot be affected by microstructure as has been suggested for polycrystalline metals.⁵⁷ 

Finally, the model at 0.78 GPa shock stress [Fig. 7(c)] evolves toward large deformation, growth of many shear bands, and development of what appears to be a dominant shear band with a very similar initiation point and trajectory as the crack observed in Pore-0.8. This dominant shear band suggests that the subsequent fracture observed experimentally is indeed shear-driven and occurs along the ASB path—enabled by the weakened material state which was realized after undergoing significant shear deformation.⁵⁸ The FEA model also reveals a triangular feature resulting from the intersection of two ASBs in Fig. 7(c) at $t=0.7μ$s. This feature is very similar in location and geometry to that observed experimentally in Pore-0.8 where the crack initiates, which is highlighted in Fig. 7(d). Again, this reinforces the ability of the FEA to capture the dominant ASB modes, which lead to shear-driven fracture in the experiments. This fracture of a triangular chip has also been recently observed by Lovinger and Kositski³⁰ in cylindrical hole closure via post-mortem analysis.

Regarding the actual distribution and curvature of the ASBs in the 0.62 GPa simulation, it is not clear why the patterns fail to match the experiments. One possible reason is that at a given material location, two orthogonal directions of maximum shear exist, but the mesh may give slight preference toward one direction over the other, leading to the predominant occurrence of shear bands which curve toward the shock loaded direction. While the exact ASB distribution for Pore-0.6 is not captured, and the FEA implementation used here does not attempt to incorporate fracture to compare directly with Pore-0.8, the qualitative replication of failure modes provides helpful insight to the physics of the failure mode transitions. Most notably, the simulations confirm that the transition to adiabatic shear banding is governed by thermo-viscoplastic mechanics and reinforces the idea that the eventual failure via fracture is indeed shear-driven and is enabled by the weakened material along the dominant ASB. This discussion will be continued along with further investigation through theoretical modeling next, in Sec. V, with the goal of understanding the ASB trajectories and driving mechanisms for fracture.

The experiments conducted in this study, along with numerical simulations, have uncovered a distinct transition in failure modes with increasing shock stress. First, shear localization develops via many distributed adiabatic shear bands, and second, fracture occurs along a dominant adiabatic shear band which is enabled by material softening in the shear band. In these experiments, on a single, $800μ$m diameter pore in PMMA, the first transition was found to occur between $0.41$ and 0.62 GPa shock stress, while the second is between 0.62 and 0.78 GPa. However, these transitions are certain to be material dependent and may have additional dependence on (i) the pore shape and pore size and (ii) the configuration of an array of pores. With the transitions observed in experiments and supported by simulations, one can now turn to the topic of the driving mechanisms for failure mode transition, initiation locations, and propagation/arrest.

As has been discussed previously, it is clear that the transition to failure via adiabatic shear banding is governed by thermo-viscoplastic mechanics. In particular, even at relatively low shock stresses, the presence of a pore creates large stress concentrations, inducing significant plastic deformation leading to inelastic (plastic) heating and thermal softening.⁵⁸ These conditions are ideal for the development of adiabatic shear bands. However, the physics governing the transition from distributed ASBs to the dominance of a single ASB and subsequent fracture is less obvious. Toward understanding this transition, the symmetry of the pore collapse is analyzed, having visually observed a substantial asymmetry in the raw deformation images for Pore-0.8 [Fig. 3(c)]. This is accomplished by fitting a partial ellipse to each side of the pore, as was described in Sec. III A and shown in Fig. 8(a) for one image from Pore-0.8. Plotting the ratio of minor axes in Fig. 8(b), a clear difference emerges between the lower pressure (Pore-0.4 and Pore-0.6) and higher pressure (Pore-0.8 and Pore-1.0) experiments. The lower pressure experiments exhibit fairly symmetric collapse, marked by a ratio of minor axes of the two ellipses (left and right) close to 1. However, the higher pressure experiments display substantial asymmetry, as the left side (on which the shock impinges first) compresses more than the right (which is shielded), leading to a ratio well below 1. It is noteworthy that the ratio does fluctuate, first dipping significantly below 1 in the first frame after the shock has passed the pore (the first data point after losing view of the pore), then rising as the right side compresses after being shock loaded, and finally settling back down as the pore approaches its final steady state shape. This asymmetric collapse behavior is another unique feature of shock compression of pores, compared to the classical symmetric theories,^14,15 static compression, or other transient loadings.³⁵ This large asymmetry coincides directly with the initiation of fracture in Pore-0.8, which nucleates and propagates a very short distance at $t=0.5μ$s as shown in Fig. 6(a), corresponding to the time at which the axes ratio first is measured to be well below 1.

The asymmetry is driven in part by the transient nature of shock loading, in which the left side of the pore is deformed before the right side is fully loaded, leading inherently to a time shift in the amount of deformation on the right side compared to the left. These transients also would drive the failure initiation location preferentially to the left, rather than the center where the maximum shear should develop in quasistatic loading. However, an additional mechanism, and the reason for which the shocked pore remains asymmetric even after reaching a steady state, are the kinematics imposed by wave interactions with the pore surface. In particular, upon arrival at the pore surface, the shock wave propagates forward, diffracting around the pore and leading to significant longitudinal stress at the top and bottom of the pore. However, at the front of the pore (left side), the shock wave reflects off the pore owing to the free surface (traction-free boundary condition). Associated with this release, and consistent with classical shock physics,⁶⁰ the particle velocity at the front surface of the pore could be doubled compared to that of the top or bottom. This phenomenon arising from wave interactions is the same one by which jets develop at much higher shock stresses, where momentum carries the front surface forward and impinges on the back surface of the pore. Along the surface of the pore between the front and top/bottom, where the surface is neither parallel nor perpendicular to the shock loading direction, a multiaxial response is generated with a partial release. In other material systems, such as copper,²⁶ these kinematics are accommodated through large plastic deformations. Even in PMMA, in the hydrodynamic regime, it is known that spherical pores do eventually develop jets²⁸ at sufficiently high stresses, likely enabled by shock heating which melts the PMMA. In the most extreme case of the complete absence of strength in the matrix material (i.e., fluids), this phenomenon has been well studied and clearly described through the shock bubble interaction problem.^61–63 Here, in the strength-dominated regime, however, because of the more brittle nature of PMMA, the kinematic frustration generates large concentrations of shear strain. This leads to shear localization and eventual fracture, instead of developing more diffuse plastic deformation and uniform softening.

Understanding the influence of asymmetric collapse resulting from shock compression helps to explain the transition from distributed ASBs in Pore-0.6 to dominant ASB and shear fracture in Pore-0.8. Interestingly, Pore-1.0 also shows substantial asymmetry, indicating that shear fracture could be present; however, the experimental images are occluded, preventing any conclusive claims. Lovinger et al. observed post-mortem shear fracture in Ti–6Al–4V³⁰ during hole closure at various shock stresses, indicating that this phenomenon is more widespread and not limited only to PMMA or a very specific shock stress. However, in PMMA, it is also possible that the brittle-to-ductile transition, which occurs under significant confinement⁴³ may mitigate the fracture response at higher shock stresses. Coupled with material softening resulting from shock heating at higher shock stresses and the possible suppression of crack propagation via confining stress, another transition from shear fracture may be possible beyond 0.8 GPa shock stress. Finally, it is noteworthy that this fracture behavior could be highly dependent on the spatial scale. The transient loading mentioned above should become negligible for very small pores, and shear localization may become irrelevant when the pore scale approaches the inherent length-scale of ASBs in PMMA (⁠ $20μ$m⁴³). Alternatively, at large pore scales (e.g., engineered structures and metamaterials), the effect would likely become more pronounced.

The next question to address is that of shear band spacing and paths, which should provide insight to the physics behind ASB distribution and self-organization which is observed in Pore-0.6. To this end, the shear bands are tracked in the reference (Lagrangian or undeformed) configuration. This is done by manually estimating one point along each shear band from the DIC data, computing the location of the nearest local maximum shear strain, and tracking the local maxima (shear strain peaks) along the length of the shear bands. The final traced shear bands are overlaid on the Tresca shear strain [Eq. (3)] plot in Fig. 9(b). From there, one can calculate the spacing between each shear band and its neighbors as a function of radial position. These results are presented in Fig. 9(a) along with a comparison to theory.

Note that the flow stress is defined here as $τ 0= σ 0/2$⁠, where $σ 0$ is the yield stress at room temperature under uniaxial stress loading. All relevant parameters are summarized in Table IV. The parameters related to thermal softening, $τ 0$ and $a$⁠, are fit to the same dataset (yield stress as a function of temperature) used to calibrate the plasticity model for the numerical simulations.⁵² Thermal properties, $k$ and $c$⁠, are taken from the literature.^29,69 To estimate the applied strain rate, the numerical simulation framework from Sec. IV is used to simulate the pore collapse event at 0.62 GPa shock stress. The maximum strain rate (near the pore surface) in the simulation is taken to be the estimate for the applied strain rate in the experiments. Because our interest here is in the applied strain rate, neglecting the strain rates inside shear bands, this simulation is conducted without thermal softening, to prevent shear localization.

Comparing the experimental results from Pore-0.6 to the Grady–Kipp model, one finds close agreement near the pore surface. This suggests that the momentum diffusion mechanism generally captures the physics which govern the spacing of ASBs at their most densely packed location (e.g., the pore surface) during symmetric collapse. This conclusion seems reasonable, at least for the analysis of ASBs that initiate, propagate outward from the pore surface, and attain sufficient strain magnitude such that they are captured via DIC. It remains possible that other ASBs initiate near the pore surface, but the loading and unloading mechanism allows some ASBs to fully develop while others die out. Such an existence of very small ASBs that are unable to compete with the growth of neighboring ASBs is well known.⁶⁵ A numerical study⁶⁷ by Zhou et al. distinguished between initiated ASBs and developed ASBs, finding the Grady–Kipp model to predict the spacing of developed shear bands, while other models^70,71 performed better when considering the spacing between all initiated shear bands. Finally, considering the Pore-0.8 GPa experiment, one dominant ASB develops, and fracture occurs along this shear band [Figs. 6 and 7(c)]. However, this does not preclude other ASBs from developing. This cannot be captured in the experiment because the fracture prevents DIC analysis of the remaining deformation field. Additionally, one would expect that fracture would serve to unload the material even more effectively than shear bands, leading to the conclusion that other ASBs could possibly arrest in the presence of a dominant shear fracture, such as the one seen in Pore-0.8.

Having considered throughout this section the topics of ASB and crack initiation as well as the spacing between shear band initiation sites, one can now turn to the topic of shear band paths. Recalling the elastic solution introduced along with the line slices in Fig. 4, the full-field solution is considered, with a particular interest in the direction of maximum shear which would direct the shear band paths. Details of the elastic solution are provided in  Appendix A. Figures 9(b)–9(c) summarize the relevant results, first reproducing the representative experimental image of distributed shear bands in Pore-0.6 [Fig. 9(b)]. Then, the Tresca shear strain [Eq. (3)] deduced from the elastic solution is plotted in Fig. 9(c) for a far field loading, $σ 0=0.62$ GPa, along with corresponding lateral confining stress superposed. Additionally, the direction of maximum shear, the bisection of the first and third principal directions, is indicated by white arrows.

Qualitative agreement between the dynamic experimental and elastostatic theoretical Tresca strain field is clear to visualize, with major concentrations of similar magnitude on the top and bottom of the pore. The primary interest, however, is in the directions of maximum shear. To compare with experimental traces of the shear band patterns, the average angular spacing between the eight bands at the pore surface is calculated and found to be 16 $°$⁠. Thus, initiation points are estimated to be evenly spaced 16 $°$ apart, and symmetric across the $X 2$ axis. Streamlines are computed, initiating at these sites and following the path of maximum shear, and are plotted as blue overlaid curves. While not a perfect match to the experiments, it is interesting to see elastic theory capture the qualitative response of such a complex phenomenon. Considering the differences in the initial shock compressed loading state to this static elastic solution, as well as the dynamic evolution involving the development of shear bands and subsequent unloading of the neighboring regions, it is expected that the actual response will not follow the elastic solution exactly. Still, it offers a good comparison and insights for the localization behavior of pores under extreme loading.

Such a failure criterion [Eq. (8)] has been considered before in numerical models for the fracture of brittle materials⁷³ and to understand failure of brittle materials under confinement.⁷⁴ It is assumed that when the driving force goes to zero, the shear crack would stop propagating forward. This driving force is plotted as the color map in Fig. 10(b), along with a black contour curve where the driving force is zero. Hence, the shear fracture arrest can be predicted to occur near the intersection of the streamline and the zero-driving force contour. As it turns out, this location coincides very closely with the experimental result of Pore-0.8. This finding indicates that the driving force [Eq. (8)] is indeed a good estimate for shear fracture behavior in these experiments. It also clarifies the mechanism for the crack arrest observed in Pore-0.8 and possibly in the recent results on hole closure in Ti–6Al–4V.³⁰ 

In summary, plate impact pore collapse experiments were conducted at shock stresses of 0.4–1.0 GPa, using the recently developed internal DIC technique to perform quantitative measurements of deformation in shock compression experiments.³³ These experiments led to the first in situ observation of shear localization during pore collapse via adiabatic shear banding and also confirmed the previous in situ²⁸ and post-mortem³⁰ observations of crack nucleation from the pore surface. From these insights, two failure mode transitions were observed as the shock stress increased: first, from diffuse strain concentration to failure via adiabatic shear localization, and second, to dynamic fracture. Numerical simulations demonstrated that thermo-viscoplastic modeling qualitatively captures these failure mode transitions. Further, they confirmed the fracture at 0.78 GPa to be shear-driven fracture (mode-II) and indicated the development of a dominant shear band at high pressures, which enabled fracture through the weakened material inside the shear band.

Analysis of the pore asymmetry evolution during collapse demonstrated a correlation between large asymmetry in the collapsed pore shape (which arises from wave interactions with the pore) and nucleation of dynamic fracture. It is proposed that the wave interactions and subsequent asymmetric collapse create shear strain concentrations which lead to the development of a dominant ASB. The ASB effects a partial stress (energy) release and eventually gives way to shear fracture that provides an additional stress (energy) release. It also accommodates the large deformations imposed by the shock wave interactions with the pore and surrounding material. The physics governing the distribution of ASBs at 0.62 GPa are clarified through a comparison of ASB spacing with the theoretical model proposed by Grady and Kipp,⁶⁸ which fits well to the experimentally measured spacing of developed shear bands at the pore surface. Thus, the fundamental mechanism in the model: unloading of nearby material through the development of ASBs, which is captured through a momentum diffusion model,⁶⁸ should also govern the number and spacing of developed shear bands at the pore surface.

Static elastic theory helped elucidate the physics that determines the paths and arrest of ASBs and cracks. The direction of maximum shear is found to effectively replicate the paths which the ASBs and the crack follow, providing a simple and reasonable method for predicting and understanding the failure paths during pore collapse. Finally, using a simple estimate [Eq. (8)] for the driving force for shear fracture, the arrest location of the crack tip can be accurately replicated.

Future work will aspire to extend the fundamental understanding of the deformation and failure for heterogeneous materials beyond a single pore. Extending the experimental technique, and leveraging the insights gained in this work, one could investigate the interactions between multiple pores in various configurations. Additionally, the influence of pore size on the pore collapse phenomenon and its associated deformations and failure modes is likely to yield intriguing results. Synthesis of these types of experiments and implementation in multiscale modeling may also greatly enhance our understanding of the underlying mechanisms for the continuum response of porous materials. Additionally, understanding the role of hard inclusions—the fundamental building block of particulate composites—on the neighboring matrix material is of similar interest. Finally, implementing the technique presented here with phase contrast imaging at a synchrotron x-ray source could minimize the issues associated with optical distortions, enable investigation of longer loading periods and higher stresses and enlarge the list of material candidates for study.

The research reported here was supported by the DOE/NNSA (Award No. DE-NA0003957), which is gratefully acknowledged. The authors acknowledge the Army Research Laboratory (Cooperative Agreement No. W911NF-12-2-0022) for the acquisition of high speed cameras.

The authors have no conflicts to disclose.

Barry P. Lawlor: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Vatsa Gandhi: Conceptualization (supporting); Methodology (supporting); Writing – review & editing (equal). Guruswami Ravichandran: Conceptualization (supporting); Formal analysis (supporting); Funding acquisition (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Numerical simulations were conducted in Abaqus/Explicit⁵⁰ to capture the deformation and localization response at shock stresses corresponding to experiments Pore-0.4, Pore-0.6, and Pore-0.8 and to provide additional insights to the experimental results. The numerical modeling framework is described in Sec. IV. Here, additional details are provided for the mesh and equation of state implemented for the simulations.

Figure 13 depicts the mesh that was used in the simulations, where the mesh was partitioned to provide a fine mesh near the pore surface and a coarse mesh in the far field. The region within a 1.2 mm radius from the center of the pore was meshed with nominally 5  $μ$m element size, while the outer region uses a nominally 20  $μ$m element size. A view of the partitioned mesh and the refined mesh is provided in Fig. 13.

The equation of state (EOS) was defined through a $U s− u p$ relation [Eq. (5)], which, as described in Sec. IV, was segmented into two linear relations: EOS 1 and EOS 2. The EOS was segmented to capture the non-linear $U s− u p$ response, which has been well characterized in PMMA,^53,75,76 while still interfacing with the finite element solver through the traditional linear relation. Figure 14 shows the two fits along with the tabular data to which they are fit, the intersection between the fits, and the particle velocities of each experiment for reference.