A metallography and x-ray tomography study of spall damage in ultrapure

We characterize spall damage in shock-recovered ultrapure Al with metallography and x-ray tomography. The measured damage profiles in ultrapure Al induced by planar impact at different shock strengths, can be described with a Gaussian function, and showed dependence on shock strengths. Optical metallography is reasonably accurate for damage profile measurements, and agrees within 10–25% with x-ray tomography. Full tomography analysis showed that void size distributions followed a power law with an exponent of γ = 1.5 ± 2.0, which is likely due to void nucleation and growth, and the exponent is considerably smaller than the predictions from percolation models.


I. INTRODUCTION
Damage of ductile metals under high strain rate loading has long been studied with planar shock loading.2][3][4][5] However, in situ, real time measurements are largely limited to free surface or target-window interface velocities, since the technical challenges still render such measurements a formidable task on void evolution in ductile metals.][18] This non-destructive technique can resolve individual voids at micron scale and thus true damage characteristics in shock-spalled samples, but it is under exploited.Void size distributions and spall damage profiles along the shock direction are important constraints on developing damage models for high-strain rate loading.Given the simplicity and convenience of 2D metallography methods, it is also of interest to quantify the errors in damage profile measurements against true values.Here we use 2D optical metallography and 3D x-ray tomography to examine damage in shock-recovered, ultrapure Al samples, to obtain 1D damage profiles and a wenge@aps.anl.govb sluo@pims.ac.cn.3D void distributions, and gain insights into void nucleation and growth processes.The damage profiles for different shock strengths are characterized with both methods, and the 2D metallography is reasonably accurate for describing damage zone width and the maximum damage.Our knowledge on void size distribution is extremely limited, and the only experimental report is on laser-shocked Ta single crystals. 18Our measurements show power-law void size distributions with an exponent of γ = 1.5 ± 2.0.Section II addresses experiments and data analyses, followed by results and discussion in Sec.III, and summary and conclusion in Sec.IV.

A. Plate impact shock wave experiments
Plate-impact experiments were performed on a light gas gun with a 100-mm bore diameter.The flyer plate (u flyer ) velocity was measured with magnetic coils, and the target free surface velocity, with a velocity interferometer through a small hole about 5 mm in diameter at the back of the recovery chamber.The accuracy in velocity was better than 1%.Both samples and flyer plates were made from an ultrapure Al rod (99.999% purity).The thickness ratio of the flyer plates and targets can be varied to control the spall plane location (x 0 ) in the target.The diameters of the flyer plates and targets were 48 mm, except for shot Al022b (38 mm).Their thicknesses were 3 mm and 6 mm, respectively.These dimensions were chosen to minimize the edge effect resulting from radial release, which was negligible in the central region of the soft-recovered samples.Impact-induced shock waves propagated in both directions, which were then reflected as release fans from the flyer-sabot interface or target free surface.The interactions of the release fans led to tension and spallation in the targets.The targets (samples) were soft-recovered for optical metallography and x-ray tomography.The experimental parameters are listed in Table I.More details on shock loading and recovery can be found in Ref. 19.

B. Damage evaluation with 2D optical metallography
The soft-recovered samples were cut into two even halves along the shock direction with electrical-discharge machining.The cut surface of one half was ground using fine sandpapers, and then polished to mirror finish with diamond slurry (1μm) on a semi-automatic polishing machine.The polished crosssections were examined with an optical metallurgical microscope.Figs. 1 and 2 show some typical micrographs, where the voids are displayed as black dots.
Damage characteristics from 2D metallography were obtained as follows.A micrograph was divided evenly into multiple zones along sample thickness direction [Fig.3(a)].The zone width was sufficiently small for a good spatial resolution, but wide enough to contain a reasonable amount of voids for a statistical analysis of voids.The number of zones was between 20 and 30.The void size-number distribution (histogram) in each zone were obtained from image processing.For each zone, we applied the widely used Schwartz-Saltykov method of stereoscopic metallography for estimating the volume fraction of 3D voids from a measured histogram of 2D voids. 11,12,14,15 Weollow Hilliard and Lawson's nomenclature and derivation 15 to discuss briefly the Schwartz-Saltykov method, assuming spherical particles (voids here).The particle sizes and intercept sizes are divided into an equal number (k) of bins.The largest intercept size is 2Rmax; the bin size = 2R max /k, a basic size units for both particles and intercepts.We use subscripts i and j to index discrete diameter sizes for intercepts and particles, respectively.Each bin contains intercepts in the interval [(i − 1) ,i ], or particle sizes in [( j − 1) , j ].The number of intercepts per unit area (A) in the ith bin is NA(i), and the volumetric (V) counterpart is NV (j).
The number of intercepts with size i per unit area due to size j particles is Here P(i, j) is the probability that an intercept of classification i results, given that the cutting plane intercepts a particle of size j; NA(j) refers to the intercepts with particles of size j; 2Rj is the mean caliper diameter of bodies of size j, and F i is the integrated density function for intercepts.The above equation is further reduced to with which can be rewritten in matrix and vector forms as where [α i j ] is a lower triangular matrix.This equation states that 3D size distributions of particles can be obtained from 2D size distributions of intercepts on a cutting plane.For shock recovered samples, the collective damage within a zone is defined as the volume fraction of particles or voids The 1D damage or porosity profiles, D(x), along the sample thickness or shock direction were then obtained.Here x is the distance from sample free surface.

C. 3D x-ray tomography
High resolution x-ray tomography was conducted on the "soft-recovered" samples using the tomography system at the beamline 2BM of the Advanced Photon Source (APS), Argonne National Laboratory [Fig.1(b)].The beam spot size at 2BM upstream was about 3.5 mm × 15 mm.The energy range of the x-ray beam at 2BM was selectable between 5 and 20 keV by the use of a Kohzu double-crystal monochromator.To characterize the spatial distribution of damage along the thickness direction, a rectangular cuboid (6 mm × 2 mm × 2 mm) was cut from the middle section of one half of the shocked sample.Because the cuboid was rotated during tomography, the thickness traversed by x-rays was between 2 mm to 2.828 mm, corresponding to a transmission of 18% and 10% at 20keV, respectively; the thickness along the x-ray direction should be below 3 mm for a reasonable signal-to-noise ratio.
The sample was mounted on a standard kinematic mount and rotation was achieved with a high-precision rotation stage.The x-rays were transmitted through the sample and detected with a CdWO4 scintillator, which converted the x-rays to visible light.The light was collected by a 2048 × 2048 pixel CoolSnap K4 CCD camera through a microscope objective lens.The system was capable of capturing high resolution images (spatial resolution ∼1μm) while concurrently obtaining 180 of projection data.The exposure time was between 80 and 200 ms per projection.A total of 1504 projections with an angular step size of 1 8 • for each sample were collected.Once the tomographic data was collected, a group of slices were created using inverse Fourier transformation, and imported to ImageJ 20 to adjust image contrast.These images were composed and rendered into a 3D representation of the sample by the software Amira TM (Indeen-Visual Concepts GmbH).Some examples of reconstructed and rendered images are shown in Figs.Each sample with a thickness of 6 mm included 4,000 slices in the thickness direction.All slices were divided into 200 groups with an equal distance (30μm) in the thickness direction (x).Therefore, each group included 20 slices.The slices in each group were used to calculate porosity and 200 damage values (D) were calculated for each sample to obtain porosity, i.e., the relative void volume, or damage D within a 30μm thickness along the sample thickness direction, and thus the damage profiles, D(x).

III. RESULTS AND DISCUSSION
Six plate impact shock-recovery experiments were conducted at different flyer velocities (u flyer = 149.6m/s-236.3m/s),intended to cover incipient to intermediate spall damage.The soft-recovered samples were examined with 2D metallography and 3D x-ray tomography, and results are shown in Figs.2-6 and Table I.
Continuum-level simulations are useful for developing damage models and predicting shock responses.An important constraint from experiments is the damage profiles along the sample thickness direction, i.e., D(x) at different shock strengths.Figure 4 shows such profiles obtained both from 2D optical metallography and 3D x-ray tomography (referred to as 2D and 3D, respectively, for simplicity).At the lowest u flyer (149.6 m/s, Al010b), the 2D profile is asymmetric while the 3D profile is highly symmetric [Figs.4(a) and 4(b)].Both profiles display multiple small peaks, which could be due to isolated voids or multiple spallations.At higher shock strengths, the general features for both 2D and 3D profiles are all similar: the profiles are smooth with a Gaussian shape.For example, the voids in Fig. 2 (149.6 m/s) are smaller and more isolated compared to those in Fig. 3 (196.9m/s), and thus D(x) is less smooth in the former case.With increasing shock strength, the void size increases from μm level to 100's μm, so do void quantity and D, for instance, at the spall plane.For a given shot, both damage and void size decrease as the location under consideration moves away from the spall plane (nominally at x = 3 mm) as seen, e.g., from Fig. 3(a) for u flyer = 196.9m/s.
To quantify the damage profiles D(x), we fit the measurements with a Gaussian function, where D max denotes the maximum damage, and x 0 is the mean and σ is the variance of the damage distribution.Physically, x 0 and σ indicate spall plane location and characteristic damage zone width, respectively.We also calculate the full-width at half maximum of the damage zone as FWHM = 2σ √ 2 ln 2. The fitting results are shown in Fig. 4, and the parameters listed in Table I.It seems that spall damage profiles can be well described with a Gaussian.Such profiles were also observed in molecular dynamics simulations of single crystal Cu. 21igure 5 shows the damage profile parameters for different shock strengths or peak stresses.The 3D results [Figs.5(b), 5(d), and 5(f)] suggest that to the first order approximation, the maximum damage and damage zone width increase, while the spall plane location decreases, with increasing strength.The broad damage width at the incipient spall may be due to the tensile stress profile (u flyer = 149.6 m/s), and the decrease in x 0 can be caused by compression.The 2D results are consistent, except that the damage zone width does not show a clear trend, and one reason could be the large measurement scatter inherent in this 2D method.
An important advantage of 2D metallography lies in its simplicity in measurements and data processing.This 2D optical method deduces 3D size distributions from surface measurements via, for example, the Schwartz-Saltykov method, does not require such large x-ray facilities as synchrotrons, and is widely used in materials characterization including spall damage. 1 The 3D x-ray tomography measurements, on the other hand, represents the "true" damage measurements.We thus compare these two methods and quantify the relative errors in D max , σ , and x 0 measurements with 2D metallography, with respect to the true values obtained with 3D tomography (Fig. 6).The relative error of a quantity Q is defined as|Q 2D − Q 3D | /Q 3D .For D max , the 2D values appear to lie close to or below the 3D values, and the average relative error is 23%, excluding the outlier at the lowest impact velocity (149.6 m/s).There are four 2D data points are smaller than the 3D results.Part of the reason could be the inherent under-estimation by the Shwartz-Saltykov method. 14For σ , the 2D values are equal to or slightly larger than their 3D counterparts except an outlier at 170.0 m/s; the average relative error is 14% without the outlier.For x 0 , the 2D values are in excellent agreement with the 3D values within 3%.Despite its simplicity, the 2D metallography method is a reasonable first-order characterization technique for ductile damage, and can predict the 1D damage profiles within an uncertainty of 10-25%.However, caution should exercised given the possibility of outliers in 2D measurements, and sufficient statistics are desirable for reducing the scatter.In addition, sample preparation for metallography is also important since optical measurements are more susceptible to surface morphology than x-ray tomography with penetration capabilities.
The 3D tomography measurements also allow us to characterize void size distributions.We developed a code for such image analysis, and the code was validated against known microstructures and a commercial software Avizo TM .Figure 7 shows such distributions within a 0.3 mm slab centered on the spall plane, for different shock strengths.The voxel size is 1.5 × 1.5 × 1.5 μm 3 .The number FIG. 6.Comparison of the values of D max (a), σ (b) and x 0 (c) measured with 2D metallography and 3D x-ray tomography, and corresponding relative errors (e-f).Also refer to Eq. ( 7).The lines in (a-c) denote y = x, and those in (e-f), the mean relative errors calculated without the outliers.of voids (ns) of a given size s, decreases with increasing size, and then ns(s) flattens, although the scatter in measurements does exist.The ns-decreasing segment can be described with a power law The fitting parameters n 0 and γ are plotted in Fig. 8 for different shock strengths (the exponents are also listed in Table I).The flat regime represents high connectivity of voids, i.e., void coalescence.3][24] However, γ ∼1.4 was reported for laser-spalled single-crystal Ta, 18 in accord with our result.Zheng et al.I.
suggested that the critical-sized nuclei form via percolation of subcritical nuclei during homogeneous melting, which could be analogous to formation of critical voids.However, γ is much smaller in our experiments, indicating a strong deviation from 3D percolation.Soulard et al. 18 proposed three power laws for void nucleation, nucleation-growth, and nucleation growth-coalescence from their MD simulations, with decreasing exponents in that order.Other MD simulations observed decreasing exponent (γ ∼1−3.5) at different times of spallation. 25Damage in our experiments ranges from incipient to full spallation, with slightly increasing exponents (except for an "outlier" at u flyer = 215.4m/s).The cause for the differences between simulations and experiments is unclear.The full spectrum of phenomenology and physical meaning behind the "power laws" remain to be explored.Spall damage depends both on loading (duration and shape) and microstructure, 5,7,8,26 and thus systematic investigations of the dependence of damage on these factors are also desirable.Hopefully, a "scaling" law and scale invariance in spall damage can be obtained considering different materials, microstructures and loadings.

IV. SUMMARY
We have characterized the spall damage in ultrapure Al induced by shock loading at different shock strengths with 2D optical metallography via the Schwartz-Saltykov method and 3D synchrotron x-ray tomography.The profiles can be well described with a Gaussian function and show appreciable dependence on shock strengths.The results from the 2D method are in accord with those from 3D tomography within 10-25%.Full tomography analysis shows that void size distributions follow a power law with an exponent of γ = 1.5±0.2,considerably smaller than the predictions from percolation models.

FIG. 1 .
FIG. 1. 2D and 3D void distributions in shock-recovered pure Al sample Al010b (u flyer = 149.6 m/s): (a) An optical micrograph of a cross-section.(b) A 0.6 mm thick slice from x-ray tomography, cut perpendicular to the shock direction.The arrows indicate shock direction.The box dimensions in (b) are 0.6 mm ×2 mm × 2 mm.

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FIG. 2. 2D and 3D void distributions in shock-recovered pure Al sample Al006b (u flyer = 196.9m/s): (a) An optical micrograph of a cross-section.(b) A 0.6 mm thick slice near the spall plane and (c) a 0.45 mm thick slice far from the spall plane, cut perpendicular to the shock direction.The box dimensions in (b) and (c) are 0.6 mm × 2 mm × 2 mm and 0.45 mm × 2 mm × 2 mm, respectively.The arrows indicate shock direction.

FIG. 3 .
FIG. 3. (a) Illustration of 2D metallography for measuring damage profiles (the micrograph is for Al022b).(b) Schematic for 3D x-ray tomography at the APS (Advanced Photon Source) beamline 2BM.CCD: charge coupled device.The arrow in (a) indicates shock direction.