# Rapid quantification of dynamic and spall strength of metals across strain
rates

Suhas Eswarappa Prameela<EMAIL_ADDRESS>Christopher C. Walker  Christopher
S. DiMarco  Debjoy D. Mallick  Xingsheng Sun  Stephanie Hernandez  Taisuke
Sasaki  Justin W. Wilkerson  K.T. Ramesh  George M. Pharr  Timothy P. Weihs
<EMAIL_ADDRESS>Department of Materials Science and Engineering, MIT, Cambridge,
MA, USA  Department of Aeronautics and Astronautics, MIT, Cambridge, MA, USA
Department of Materials Science and Engineering, Johns Hopkins University,
Baltimore, MD 21218, USA  Hopkins Extreme Materials Institute, Johns Hopkins
University, Baltimore, MD, 21218, USA  Department of Materials Science and
Engineering, Texas A&M University, College Station, TX, 77843, USA  Sindri
Materials Corp., West Chester, PA, 19382, USA  DEVCOM Army Research
Laboratory, 321 Colleran Road, Aberdeen Proving Ground, MD, 21005-5066, USA
Department of Mechanical and Aerospace Engineering, University of Kentucky,
Lexington, KY 40506, USA  Department of Mechanical Engineering, Johns Hopkins
University, Baltimore, MD 21218, USA  National Institute for Materials
Science, Tsukuba 305-0047, Japan  Center for Elements Strategy Initiative for
Structural Materials (ESISM), Kyoto University, Yoshida Honmachi, Sakyo, Kyoto
606-8501, Japan  Department of Mechanical Engineering, Texas A&M University,
College Station, TX, 77843, USA  These authors contributed equally to this
manuscript

###### Abstract

Over the last few decades, numerous small-scale mechanical testing techniques
have been developed to evaluate the fundamental deformation mechanisms and
failure behavior of metals. However, many of these techniques are insufficient
to accurately predict material behavior at extreme strain rates, critical for
applications such as car crashes, meteoroid impact on satellites, launch and
re-entry of rockets, and protection materials to stop bullets. The response of
metals and their microstructures under extreme dynamic conditions can be
markedly different from that under quasistatic conditions. Traditionally, high
strain rates and shock stresses are measured using cumbersome and expensive
methods such as the Kolsky bar or large spall experiments. These methods are
low throughput and do not facilitate high-fidelity microstructure-property
linkages. In this work, we combine two powerful small-scale testing methods,
custom nanoindentation, and laser-driven micro-flyer shock, to measure the
dynamic and spall strength of metals. The nanoindentation system is configured
to test samples from quasistatic to dynamic strain rate regimes (10-3 s-1 to
10+4 s-1). The laser-driven micro-flyer shock system can test samples through
impact loading between 10+5 s-1 to 10+7 s-1 strain rates, triggering spall
failure. The model material used for testing is Magnesium alloys, which are
lightweight, possess high-specific strengths and have historically been
challenging to design and strengthen due to their mechanical anisotropy. Here,
we modulate their microstructure by adding or removing precipitates to
demonstrate interesting upticks in strain rate sensitivity and evolution of
dynamic strength. At high shock loading rates, we unravel an interesting
paradigm where the spall strength of these materials converges, but the
failure mechanisms are markedly different. Peak aging, considered to be a
standard method to strengthen metallic alloys, causes catastrophic failure,
faring much worse than solutionized alloys. Our high throughput testing
framework not only quantifies strength but also teases out unexplored failure
mechanisms at extreme strain rates, providing valuable insights for the rapid
design and improvement of metals for extreme environments.

###### keywords:

Metallurgy $|$ Structure - Processing - Property Relationships $|$ Strain Rate
$|$ Dynamic Behaviour $|$ Spall $|$ Microstructure Design

††journal: Journal

###### Contents

  1. 1 Introduction
  2. 2 Materials and Methods
    1. 2.1 Thermomechanical Processing and Sample Preparation
    2. 2.2 Low to High Strain Rate Nanoindentation Experiments
    3. 2.3 Laser Driven Micro-Flyer Shock Experiments
  3. 3 Results and Discussion
    1. 3.1 Initial Microstructure Characterization Results
    2. 3.2 Nanoindentation Results - Mechanical Property and Microscopy
    3. 3.3 Mechanisms Related to Quasistatic Nanoindentation
    4. 3.4 Mechanisms Related to High Strain Rate Nanoindentation
    5. 3.5 Spall Results - Mechanical Property and Microscopy
    6. 3.6 Mechanisms Related to Spall
  4. 4 Conclusions
  5. 5 References
  6. 6 Supplementary Information
    1. 6.1 Details regarding the data from custom Nanoindentation set-up
    2. 6.2 Details regarding the strength modeling on the Nanoindentation data
    3. 6.3 Details regarding the data from custom laser spall set-up
    4. 6.4 Additional Experimental Details
  7. 7 Terminology Table
  8. 8 Supplementary References

##  1 Introduction

Traditionally, the mechanical properties of bulk structural materials across
strain rate regimes have been probed using bulk techniques. For the
quasistatic strain rate regime, this evaluation has been done through tension,
compression, bending, and torsion experiments. For the dynamic strain rate
regime, researchers have employed plate impact, shock, isentropic compression,
and Kolsky bar experiments [1]. These test protocols are useful for bulk
samples but pose challenges when rapid screening of properties is needed for
materials design and discovery for extreme environments [2]. Furthermore, some
large-scale shock/high strain rate experiments can be highly destructive,
expensive, and logistically burdensome to implement. There have been several
efforts to implement small-scale mechanical testing methods, both at the
microscale and nanoscale. Micro-tensile, micro-cantilever, and micro-pillar
tests and their counterparts at the nanoscale have allowed researchers to
carry out site-specific or volume-specific experiments and obtain the
attendant mechanical response [3, 4, 5]. These experiments have also been
coupled with various diagnostic tools such as scanning electron microscopy
(SEM), transmission electron microscopy (TEM), and other X-ray or beamline
instruments that offer insights into deformation mechanisms and in-situ
tracking of key parameters such as texture/precipitate evolution, extent of
slip behavior, and twin volume fractions [6, 7] in metals.

Over the last two decades, several efforts have pushed small-scale mechanical
testing protocols to adopt testing conditions to recapitulate those
encountered in extreme environments. For example, there are several active
efforts to push the popular nanoindentation technique to high strain rate
regimes often encountered during car crashes and ballistic impact. These
efforts have focused on improving the calibration methods, measurement
strategies, noise reduction techniques, and expanding the range of material
systems that can be tested [8, 9, 10, 11, 12]. Some studies have looked at the
impact of continuous stiffness measurement methods (CSM) on hardness
overestimation and subsequent challenges in measuring strain rate sensitivity
[13]. Managing noise levels and data analysis become especially challenging at
high strain rates. Advanced sensors and high-frequency modulation techniques
are being developed to circumvent these challenges.

There have also been several efforts to mimic shock-loading conditions through
launched particles and plates. These impact experiments cause materials to
experience deformation at high strain rates. For example, Laser-Induced
Particle Impact Test (LIPIT) experiments have been successfully implemented to
test various metallic alloys, ceramics, and other structural materials [14,
15]. In these experiments, a laser is used to accelerate micro-particles
($\sim$10-50 $\mu$m) at varying speeds ($\sim$100 ms-1 to 900 ms-1) towards a
target material. These experiments have helped to elucidate adhesion
mechanisms, cold spray mechanisms, recrystallization, plasticity, and damage
at extreme strain rates. Similarly, laser energy can be used to accelerate
thin circular metal disks to mimic plate impact and interrogate spall
behavior. Several recent studies have looked at laser-driven micro-flyer shock
experiments for testing various metallic alloys, single crystals, ceramic
carbides, and other structural materials [16, 17, 18].

In this study, we chose a Magnesium (Mg) alloy as the model material to
demonstrate rapid quantification of strength across various strain rates and
link them to attendant microstructures and plasticity mechanisms. Our testing
framework (Fig. 1) employs custom nanoindentation and laser-driven micro-flyer
shock experiments to help quantify the quasistatic, dynamic, and spall
strength of these metallic alloys. To explore the effect of heterogeneous
inclusions, such as precipitates, we apply similar testing protocols on two
different variants of the same metallic alloy, peak-aged and solutionized
samples with and without precipitates, respectively. High-throughput
experiments can accelerate the testing of these variants while shedding
insights into how microstructural features such as precipitates that are
conventionally found to be favorable for metal strengthening at quasistatic
strain rates behave in dynamic and spall regimes.

When Mg alloys are deformed at high strain rates ($\sim 10^{4-8}$ s-1) through
impact experiments, shockwaves are generated that first load the material in
uniaxial compression. When these shock waves meet free surfaces, they reflect
as rarefaction fans that can intersect within the material, resulting in
dynamic tensile loading that is nearly hydrostatic. This dynamic hydrostatic
tension causes the material to fail through a process called spallation, with
such spall failures typically driven in metals by void nucleation, growth, and
coalescence. The high specific strength of Mg alloys offers a compelling
reason to pursue directions related to enhancing spall strength [19]. The
spall strength, defined as resistance to spallation, strongly depends on
microstructural features such as grain size, texture, precipitate type, and
volume fraction. The spall strength is also a strong function of the tensile
strain rate and potentially of the degree of shock compression before the
tensile loading occurs. While there have been studies on the dynamic and spall
failure of Mg single crystals and alloys [18, 20, 21, 22, 23, 24] there is
still a lack of clarity on the microstructure property linkages. Recent spall
studies in Mg alloys have shown the importance of precipitate size and
distribution. In the case of AZ31 alloy, some large-size precipitates result
in catalytic void nucleation and accelerated spall failure in the materials
[25, 23]. Taken together, these prior studies make a strong case to employ
high-throughput techniques to quickly quantify spall strength and to use that
data along with the quasistatic and dynamic strengths of Mg alloys to
understand the microstructure-property linkages in these materials.

##  2 Materials and Methods

###  2.1 Thermomechanical Processing and Sample Preparation

The Mg-5Zn (at%) alloy, referred to as Z5, was prepared by melting and mixing
high purity Mg (99.97%, Grade II, U.S. Magnesium) and Zn (99.999%, Alfa Aesar)
in an argon atmosphere and casting into bars. The bars were solutionized at
$500^{\circ}$C for 25 hours to remove any preexisting precipitates from the
casting process. The bars were then cut into rectangular pieces of 20 mm by 10
mm by 5 mm. Some of these pieces were then peak-aged at $150^{\circ}$C for 99
hours to disperse precipitates throughout the bulk.

The solutionized and peak-aged Z5 pieces were then mechanically polished for
further characterization and testing. A field emission scanning electron
microscope (SEM, Carl Zeiss Crossbeam 1540 EsB FIB/SEM) was used for electron
microscopy studies and electron backscattered diffraction(EBSD) scans. The SEM
has an HKL EBSD system with the Channel 5 software for EBSD analysis. For
scanning transmission electron microscope(STEM) studies, thin foil specimens,
including grain boundaries, were placed by a standard lift-out technique using
a dual beam FIB/SEM, FEI Helios G4. An FEI Titan G2 80–200 STEM was used for
annular dark–field (ADF) imaging and diffraction studies.

###  2.2 Low to High Strain Rate Nanoindentation Experiments

ISE

    indentation size effect
CSM

    continuous stiffness measurements
RMS

    root means square
$h$

    indentation depth
$h_{c}$

    contact depth
$h_{c}/h$

    ratio of contact depth to depth
$\dot{h}$

    velocity
$\dot{\varepsilon}_{i}$

    indentation strain rate
$P$

    load
$\dot{P}$

    loading rate
$\dot{P}/P$

    ratio of loading rate to load
$A_{c}$

    contact area
$H$

    hardness
$\dot{H}$

    change in hardness over time
$E$

    young’s modulus

Nanoindentation is a well-established method for measuring the basic
mechanical properties of materials and provides many advantages to traditional
hardness testing [26, 27]. In this work, nanoindentation was performed using a
custom nanoindentation system made by KLA (USA) that is capable of testing in
both the quasistatic and dynamic strain rate regimes (details of the custom
instrumentation are outlined by Hackett et al. [28]). A diamond indenter tip
with the three-sided Berkovich geometry was used for all indents. The
schematic of the nanoindentation system equipped with both the quasistatic
straining protocol and dynamic straining protocol is shown in (Fig. 1 A and
B). For the quasistatic indents, a constant strain rate was achieved by
loading such that the ratio of loading rate to load ($\dot{P}/P$) was
constant. With a constant $\dot{P}/P$, the indentation strain rate
($\dot{\varepsilon}_{i}$) can be calculated as

$\epsilon_{i}\equiv\frac{\dot{h}}{h}=\frac{1}{2}\left(\frac{\dot{P}}{P}-\frac{\dot{H}}{H}\right)$
(1)

with velocity ($\dot{h}$), indentation depth ($h$), loading rate ($\dot{P}$),
load ($P$), change in hardness over time ($\dot{H}$), and hardness ($H$) [29].
When $H$ is constant, $\dot{H}=0$ and equation (1) simplifies to

$\epsilon_{i}\equiv\dot{h}/h=\frac{\dot{P}}{2P}.$ (2)

H is constant as a function of depth for the tested Mg alloys, so equation (2)
can be used. Quasi-static indents were performed to a maximum load of
$200\text{\,}\mathrm{mN}$ and at $\dot{P}/P$s of
$0.02\text{\,}{\mathrm{s}}^{-1}0.2\text{\,}{\mathrm{s}}^{-1}2.0\text{\,}{\mathrm{s}}^{-1}$.
During loading, continuous stiffness measurements (CSM) were taken by applying
a $110\text{\,}\mathrm{Hz}$ oscillation to the load such that the root means
square (RMS) amplitude of the oscillation was 10% of the load as prescribed by
Phani et al. [30]. Results for $H$ and $\dot{\varepsilon}_{i}$ were averaged
over an indentation depth of
$500\text{\,}\mathrm{nm}2500\text{\,}\mathrm{nm}$. A linear least squares fit
of contact depth vs. indentation depth was used to determine a value for the
ratio of contact depth to depth ($h_{c}/h$) in each material. This fit results
in an $h_{c}/h$ of 0.94 and 0.98 for the solutionized and peak-aged material,
respectively. Optical images of each indent were taken to confirm the residual
contact impression matched the results of the CSM measurements.

An impact testing approach was used for the high strain rate indentation, as
described by Phani et al. [31] and Hackett et al. [28]. To account for the
dynamic overload during impact, loads of
$30\text{\,}\mathrm{mN}50\text{\,}\mathrm{mN}$ were used so the final load on
the sample would be between $200\text{\,}\mathrm{mN}300\text{\,}\mathrm{mN}$,
reaching comparable indent sizes with the quasistatic testing. Due to the
nature of the impact test, a constant $\dot{P}/P$ can not be maintained. Thus
$\dot{\varepsilon}_{i}$ during these impact tests is not constant, allowing a
range of strain rates to be reported from each test. Additionally, CSM cannot
be used to measure $H$ as CSM has been shown to have problems at high strain
rates [32], and the instrument cannot apply the oscillation required for CSM
at a frequency suitable for the $<$ $1\text{\,}\mathrm{ms}$ impact test.
Without CSM, hardness is calculated with

$\displaystyle h_{c}$ $\displaystyle=\frac{h_{c}}{h}\times h$ (3)
$\displaystyle A_{c}$ $\displaystyle=\sum_{n=0}^{5}C_{n}h_{c}^{2^{(1-n)}}$ (4)
$\displaystyle H$ $\displaystyle=P/A_{c}$ (5)

where $h_{c}/h$ was first determined during the quasistatic testing, and
$C_{n}$ are a series of constants that describe the tip shape [26, 28].
However, optical imaging of the residual contact impressions showed that the
calculated contact area ($A_{c}$) when using $h_{c}/h$ from quasistatic
testing did not align with the physical size of the indent. A new $h_{c}/h$
was calculated, $0.901.0$, for solutionized and peak-aged, respectively, so
that $A_{c}$ aligned with the measured contact area for the dynamic indents. A
miniature ICP force sensor from PCB Piezoelectronics (USA) was added to the
system to measure the applied load during dynamic testing. The addition of the
piezoelectric force sensor lowers the instrument frame stiffness from
$25\text{\,}\mathrm{M}\mathrm{N}\mathrm{/}\mathrm{m}$ to
$8\text{\,}\mathrm{M}\mathrm{N}\mathrm{/}\mathrm{m}$, but a standard frame
stiffness correction can account for this correctly[26]. Hardness and
indentation strain rate were binned into 120 half-open bins between strain
rates of ${10}^{1}\text{\,}{\mathrm{s}}^{-1}$ and
${10}^{4}\text{\,}{\mathrm{s}}^{-1}$.

###  2.3 Laser Driven Micro-Flyer Shock Experiments

Laser shock methods are well-poised for high-throughput dynamic experiments;
in contrast to conventional methods, these methods attain similar energy
densities while operating more safely, at smaller scales requiring
significantly less material, and at much lower overall expense [33, 34, 35,
16, 36]. The Laser-Driven Micro-Flyer (LDMF) Shock experimental set-up, as
shown in (Fig. 1) is a subset of these laser shock methods that utilizes the
energy from a laser pulse to accelerate a micro-scale flyer plate to achieve a
high-velocity impact with a target, achieving tensile strain rates of
$\geq\mathcal{O}$(106) s-1 during spall. We determine the dynamic response of
the material through Photonic Doppler Velocimetry (PDV) measurements on the
target’s free surface during the impact [16, 37]. In contrast to traditional
plate impact experiments, which average $\sim$ 1-3 experiments per day, LDMFs
can obtain similar impact energy densities and can easily exceed 100
experiments per day. While the small scale of the LDMF experiment
significantly reduces material waste, it requires careful consideration of the
microstructure and deformation length scales at play. Even with recent
attention to several shock compression applications [36], very few
investigations optimized to study spall failure. Recently, we have developed
an LDMF system and methodology for investigating spall failure with high-
throughput while maintaining reasonably high fidelity. In this effort, there
are two critical challenges for establishing confidence and reproducibility:
(1) maintaining a high degree of flyer planarity from launch to impact, and
(2) establishing conditions such that a consistently high-quality PDV signal
is obtained. This work is among the first experimental demonstrations [38] of
our new methods to address these challenges.

The LDMF experiment consists of three sections: (1) the pulse laser and free-
space optics; (2) the launch package; and (3) the PDV diagnostics (Fig. 1 E).
The goal of the first section is to emit and manipulate the energetic,
spatial, and temporal characteristics of the driving laser pulse to achieve
optimal launch conditions (i.e., planar flyer at a desired velocity). Our
system uses a 1064 nm Nd: YAG 2.5 J 10 Hz 10 ns Spectra-Physics Quanta-Ray 350
with a beam quality (M2 value) of 15. The pulse energy drives the flyer
velocity through the laser fluence. The pulse duration must be sufficiently
long relative to the round-trip shock wave time inside the flyer to prevent
reverberations during the launch that can break up the flyer [35]. We employ
an optical cavity to lengthen the pulse duration from $\sim$10 ns to $\sim$21
ns, which is sufficient for the flyer material and thickness used in these
studies. Lastly, and perhaps most critical, the acceleration of a planar and
intact flyer requires a homogenized beam profile, and so the beam is shaped
just before the flyer launch. We utilize a custom-built Diffractive Optical
Element (DOE) by Silios Technologies to shape the beam profile to a low
variation top-hat profile. This DOE is used in series with a focusing lens to
shape the beam to a desired diameter at the effective focal length (EFL). A
beam profiler is used to measure and monitor the beam shape during each
experiment.

The second section of the experiment is the launch package, which refers to
the arrangement of the flyers and targets (Fig. 1 F). Each launch package has
lateral dimensions of 50 mm by 50 mm and contains an array of strategically
spaced flyers: here, a 7 by 7 square array yielding 49 experiments per
package. Multiple launch packages are fabricated in advance, and the spall
experiments are performed systematically through the pre-fabricated launch
packages to improve experimental throughput. The design of the launch package
is critical for consistently and reliably achieving planar impact and strong
PDV return signals at high throughput rates. It is a layered structure foil as
shown in Fig. 1 F and consists of a substrate, a flyer, a spacer, and a
target. The glass substrates are 50 mm by 50 mm by 0.625 mm borosilicate glass
from McMaster Carr, and the epoxy used for bonding is Henkel Loctite Ablestik
24. The flyers are 100 µm thick, 1.5mm diameter Aluminum from Alufoil, and the
spacer is a 240 µm thick Kapton sheet with built-in double-sided silicone-
based adhesive. The targets (samples) are prepared through double-sided
polishing of larger area foils to a thickness of 200 µm +/- 10 µm, and then 3
mm disks are created using a TEM-punch [18]. The flyer and target thickness
are determined based on wave propagation analysis so that the spall plane
occurs within the target. Their diameters are sufficiently large to avoid any
unloading wave effects on the central area of interest during the time of
interest. The flyers are pre-cut using a femtosecond laser to obtain a flyer
with a sufficiently large diameter in order to prevent edge unloading from
affecting the PDV signal and to guarantee impact planarity. The pulse laser is
operated at $\sim$ 800 mJ and is focused to a spot size of 1.85 mm with a 250
mm EFL. This yields fluences of $\sim$ 30-32 J cm-2 that drive impact
velocities $\sim$ 550-600 ms-1.

The diagnostics consist of a high-speed camera and a PDV system. The high-
speed camera (Shimadzu HPV-X) operates at 10 million frames per second. It
provides a side profile view of the launch package for qualitative information
regarding the flyer and impact planarity and a macroscopic assessment of the
developed spall damage. The PDV system measures the normal particle velocity
of the rear free surface of the target [16, 37]. It is a heterodyne system
that consists of two 1550 nm centered fiber lasers, a seed signal laser
directed towards the target, and a 2.3 GHz upshifted reference signal for
mixing. The seed laser is focused on the backside of the sample with a spot
size of 80 $\mu m$; the material response is averaged over this area, so this
length scale must be sufficiently large relative to the relevant material
length scales. During the experiment, the reflected light (return signal) is
imparted with a frequency shift based on the particle velocity. The return
signal is mixed with the reference signal to obtain the beat frequency, which
is measured and recorded using a 16 GHz LeCroy Oscilloscope. The strength of
the return signal, and therefore the quality of the spall signal, is a strong
function of the target’s surface roughness and orientation. We use a custom-
built alignment apparatus that allows fine orientation control of the launch
package for optimizing the return signal. Samples are double-sided polished to
high reflectivity with diamond lapping paper to a 1-micron mirror finish to
further maximize the return signal.

##  3 Results and Discussion

###  3.1 Initial Microstructure Characterization Results

Figure 1: Schematic of nanoindentation setup (A and C) with typical depth vs.
time plots for B) quasistatic and D) dynamic loading protocols. (E) Schematic
of laser-driven micro-flyer shock setup with expected results collected to
calculate spall strength. The overall layout of the testing system is broken
into three major sections: the pulse laser, the launch package, and the
photonic Doppler velocimetry system. (F) A side view schematic of the layered
launch package depicting a typical single impactor-target configuration. This
shows the three stages of the launch process: ablation of the epoxy,
acceleration of the flyer, and finally, impact with the target. (G) A rotated
view of a position vs. time Lagrangian diagram depicting the propagation of
waves throughout the impactor and target that leads to the spall event, as
well as the critical points measured via PDV. (H) An idealized velocity vs.
time plot calculated from the PDV measurements on the target’s free surface.
The critical points correspond to the ones shown in (G).

A Mg-5Zn (at%) alloy, referred to as Z5 hereafter, was processed in two
conditions: solutionized and peak-aged. EBSD of these two samples (Fig. 2 A
and C) showed the average grain size was around $\sim$ 205 $\mu$m for the Z5
solutionized (without precipitates) and $\sim$ 227 $\mu$m for the Z5 peak-aged
sample (with precipitates). Furthermore, STEM micrographs (Fig. 2 B and D)
showed that Z5 solutionized was devoid of any precipitates within and along
grain boundaries while the Z5 peak-aged sample had uniformly distributed
precipitates from the aging treatment. The precipitate length is around $\sim$
52 nm, and the areal density is $\sim$ 390 precipitates per $\mu m^{2}$. Data
from additional STEM TEM studies were used to verify the initial precipitate
microstructures is shown in SI Appendix, FS.7 (a) to (b).

###  3.2 Nanoindentation Results - Mechanical Property and Microscopy

A custom nanoindentation protocol (Fig. 1 A to D) was used to probe the
mechanical properties at quasistatic and dynamic strain rate regimes. The
indents across the strain rate regimes varied in size and are shown in Fig. 3
A as a function of discrete strain rates. The nanoindentation hardness was
measured between strain rates of ($10^{-2}$ to $10^{+4}$ s-1) as shown in SI
Appendix, FS.3 (e) and (f). The hardness values were then converted to
strength by a conversion factor of 1/3 [39, 40], and a graph of
nanoindentation strength as a function of strain rate was plotted in (Fig. 2 E
and G). The depth vs. time and velocity vs. time profiles of nanoindentation
experiments at both quasistatic and dynamic strain rate regimes are depicted
in SI Appendix, FS.1 (a) to (f) and SI Appendix, FS.2 (a) to (f),
respectively. The corresponding strain rate and load profiles have also been
indicated in these figure panels. Furthermore, we show the hardness vs. depth
profiles for both Z5 solutionized and peak-aged at different strain rates in
SI Appendix, FS.3 (a) to (d). The average and standard deviation for hardness
values were calculated between a depth of 2200 nm and 3000 nm. All data from
these experiments are listed in SI Appendix, Table S4-S6.

The main takeaway from these results is that the Z5 peak-aged sample is
stronger at the lower end of the strain rate regime (quasistatic) and the
strength difference persists at the dynamic strain rate regime until
$10^{+4}$/s. The Z5 solutionized sample’s average quasistatic strength is
$\sim$ 254 MPa for strain rates ranging from $10^{-2}$ s-1 to $10^{0}$ s-1.
The Z5 peak-aged sample’s average quasistatic strength is $\sim$ 301 MPa for
the same range of strain rates. The dynamic strengths, though, increase with
strain rate. In the case of the Z5 solutionized sample, the average dynamic
strength rises from $\sim$ 294 MPa at $10^{+1}$ s-1 to $\sim$ 355 MPa at
$10^{+4}$ s-1. For the Z5 peak-aged sample, the average dynamic strength rises
from $\sim$ 312 MPa at $10^{+1}$ s-1 to $\sim$ 363 MPa at $10^{+4}$ s-1.

###  3.3 Mechanisms Related to Quasistatic Nanoindentation

At quasistatic and low values of strain rates ($10^{-2}$ s-1 to $10^{0}$ s-1),
peak-aged Z5 is stronger than solutionized Z5. In the case of the peak-aged
sample, finely distributed precipitates produced during the heat treatment
process cover all the grains. When the indenter plunges into the peak-aged
samples, the array of dislocations produced during deformation has to overcome
these spatially distributed obstacles.

The critical resolved shear stress (CRSS) for doing so can be predicted as
$\Delta\tau_{Orowan}=G_{m}b/L$, where $G_{m}$ is the shear modulus of the
matrix, $b$ is the Burgers vector, and $L$ is the spacing between precipitates
[41]. A further modification [42] to capture the dependence of the geometric
configuration of precipitates can be written as

$\Delta\tau_{Orowan}=\frac{G_{m}b}{2\pi(d_{s}-2r_{p})\sqrt{1-\nu}}\text{ln}\frac{2r_{p}}{r_{0}},$
(6)

Where $r_{0}$ is the core radius of the dislocation [43], $r_{p}$ is the
average radius of the precipitates on the slip plane, $\nu$ is Poisson’s
ratio, $d_{s}$ is the spacing between the precipitates on the glide plane, and
$d_{s}=n_{s}^{-1/2}$ where $n_{s}$ denotes the number of precipitates per unit
area. For the Mg system, one can calculate the effective planar, inter-
precipitate spacing $\lambda_{e}=d_{s}-2r_{p}$ of different precipitate
morphologies in the HCP system and estimate the change in $\tau_{Orowan}$
using Eq. 6. Let $f$ denote the volume fraction of the precipitates, and
assume that $r_{0}=b$. The Orowan CRSS from the $c$-axis precipitate rods
present in the Mg-Zn alloy system is given by

$\Delta\tau_{Orowan,c}=\frac{G_{m}b}{2\pi\left[\frac{0.953}{\sqrt{f}}-1\right]d_{t}\sqrt{1-\nu}}\text{ln}\frac{d_{t}}{b},$
(7)

where $d_{t}$ is the precipitate rod diameter [44]. It follows from above that
$\Delta\tau_{Orowan,c}$ depends strongly on $f$ and precipitate diameter
($d_{t}$), which in turn depend on the effective inter-particle spacing
$\lambda_{e}$. From this equation, one can estimate the dependence of Orowon
CRSS for any precipitate size and a given volume fraction [44]. The Orowon
increment in CRSS from Eq. (7) for the peak aged Z5 alloy gives a value of
$\Delta\tau_{Orowan,c}\sim 43$ MPa.

In the solutionized Z5, the alloying elements provide strengthening
($\Delta\tau_{ss}$) via distortion of the lattice, which may be estimated as

$\Delta\tau_{ss}={B_{r}}c^{2/3}+{B_{s}}c^{2}\left(1-c\right)^{2},$ (8)

where $c=5\%$ denotes the nominal concentration of Zn, $B_{r}=43.2$ MPa is the
coefficient of random solid solution strengthening, and $B_{s}=6$ GPa is the
coefficient of strengthening associated with short-range order [45]. According
to Eq. (8), $\Delta\tau_{ss}|_{c=5\%}\sim 19$ MPa, implying that the Zn
alloying content is roughly twice as effective in precipitate form as compared
to solution form. It is worth noting that peak-aged Z5 has $c\sim 1\%$
concentration of Zn that remains in solution, which results in a small solid
solution strengthening of $\Delta\tau_{ss}|_{c=1\%}\sim 2$ MPa, according to
Eq. (8). The difference in the the quasi-static yield strength
$\Delta\sigma_{Y}$ can be estimated via the Taylor factor, i.e.
$\Delta\sigma_{Y}=M_{P-A}\left(\Delta\tau_{Orowan,c}+\Delta\tau_{ss}|_{c=1\%}\right)-M_{S}\Delta\tau_{ss}|_{c=5\%}$,
where $M_{S}\sim 4.5$ and $M_{P-A}\sim 3$ denote the Taylor factors for random
solutionized Z5 and weakly textured peak-aged Z5, respectively. Substituting
all the values, $\Delta\sigma_{Y}\sim 50$ MPa. This value is in good agreement
with the experimentally measured difference in the average yield strength
between the solutionized and peak-aged samples from nanoindentation
experiments of $\sim$ 47 MPa and as shown in Fig. 2 F.

###  3.4 Mechanisms Related to High Strain Rate Nanoindentation

Zerilli and Armstrong [46, 47] developed a constitutive model to characterize
the strain-, strain rate- and temperature-dependent response of HCP metals, by
combining the terms from their earlier BCC and FCC constitutive models [48].
Specifically, Zerilli and Armstrong concluded that overcoming Peierls–Nabarro
barriers, associated with dislocation motion, was the principal thermal
activation mechanism for BCC metals, whereas dislocation interactions, and
thus density, were the governing mechanism for FCC metals. Further, they
considered HCP constitutive behavior to combine mechanisms of both BCC and FCC
strain rate sensitivity. The current solutionized and peak-aged Z5 samples do,
in fact, exhibit such a “BCC response”, that is, a rate-dependent initial
yield strength commonly observed in BCC alloys. To this end, we employ the BCC
term of the Zerilli-Armstrong [46] constitutive model for the yield strength,
which is given by

$\sigma_{Y}=\sigma_{G}+\frac{k}{\sqrt{l}}+B\exp\left[-\beta_{0}T+\beta_{1}T\ln\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\right],$
(9)

where $\sigma_{Y}$ is the yield strength, $l$ is average grain diameter, $T$
is the absolute temperature, $\dot{\epsilon}$ is the strain rate, and
$\dot{\epsilon}_{0}$ is the reference strain rate. The model parameters are
$\sigma_{G}$, the quasi-static strength due to alloying content and pre-
existing dislocations; $k$, the Hall–Petch slope; $B$, the strain-rate
hardening modulus; $\beta_{0}$, the thermal-softening coefficient; and
$\beta_{1}$, the rate-sensitivity parameter.

The constitutive parameters required by the Zerilli–Armstrong model (Eq. 9)
were fit to the experimental results of dynamic strength using a nonlinear
regression algorithm, and are reported in SI Appendix, Table S1. The excellent
agreement between the Zerilli-Armstrong model and the experimental data is
shown in Fig. 2 H, in support of the “BCC response” assumption made in the
constitutive analysis.

As expected, $\sigma_{G}$ of peak-aged Z5 is greater than solutionized Z5,
because the Zn alloying content is more effective at strengthening in
precipitate form as compared to solute form. Both peak-aged and solutionized
Z5 are found to have the same $\beta_{0}$ and $\beta_{1}$. That said, the
strain rate hardening modulus $B$ is found to be larger in solutionized Z5 as
compared to peak-aged Z5. Given that $B$ is correlated with viscosity, this
finding would seem to indicate that the viscosity of peak-aged Z5 is lower
than the solutionized Z5. One possible explanation for this is that
significant dislocation bowing around precipitates in peak-aged Z5 results in
rapid dislocation multiplication at early deformation. The higher mobile
dislocation density then results in a lower viscosity, [49]. Another
contributing factor is that solute atoms often decrease the mobility of
dislocations (and hence increase the overall viscosity), [50]. This argument
would also support our finding that solutionized Z5 seems to be more rate-
sensitive than peak-aged Z5. As a result of this higher rate sensitivity, the
dynamic strength for these two alloys diminishes with increasing strain rate.
Extrapolating our calibrated Zerilli-Armstrong model, we expect the
solutionized Z5 to exhibit higher strength than peak-aged Z5 at sufficiently
high strain rates, e.g. $\gtrsim 10^{5}$ s-1.

Figure 2: A) EBSD map of Z5 solutionized alloy at $150^{\circ}$C. B) STEM
images of Z5 solutionized alloy at $150^{\circ}$C. C) EBSD map of Z5 peak-aged
at $150^{\circ}$C. D) STEM images of Z5 peak-aged at $150^{\circ}$C. E)
Quasistatic strength measured via nanoindentation with error bars. F)
Theoretical and experimental values of strength at quasistatic strain rates.
G) Dynamic strength measured via nanoindentation. H) Theoretical dynamic
strength predictions using the Zerilli-Armstrong model I) Spall strength
measured via laser spall with uncertainty of spall strength parallel to the
y-axis and uncertainty of strain rate parallel to the x-axis. J) Spall
strength predictions using the Wilkerson-Ramesh model

In addition to the Zerilli-Armstrong model, we also employed three other
commonly used constitutive models to describe the dynamic strengths of the
solutionized and peak-aged Z5, i.e., the standard Johnson-Cook model [51], the
modified Johnson-Cook model with quadratic form proposed by Huh and Kang [52],
and the Cowper-Symonds model [53] (see SI Appendix, FS.4 (a) to (d)).
Comparing the coefficients of determination $R^{2}$ of these fitting results,
we find that both the Zerilli–Armstrong and the Cowper-Symonds models provide
the best prediction for the dynamic strengths depending upon a broad range of
strain rates. A further check on these two models shows that they essentially
have the same mathematical formulation, both expressing the yield strength as
a power function of the strain rate.

Very few studies have reported high strain rate nanoindentation of Mg alloys,
especially in the $10^{+3}$ s-1 to $10^{+7}$ s-1 regime. Researchers have
tested pure Mg and dilute Mg alloys with nearly $\sim$ 2-3 $\mu m$ grain sizes
up to a $10^{+2}$ s-1 strain rate [10]. The small grain sizes ensured that the
primary deformation mode was dislocation glide rather than twinning [10]. The
strength at high strain rates was influenced by the solute type, consistent
with the theories of solid solution strengthening. Another study tested much
coarser grained Mg alloys with $\sim$ 100 $\mu m$ grain sizes at strain rates
up to $10^{+2}$ s-1 [54]. In this case, twins were found to form during the
early stages of nanoindentation, resulting in strain compatibility constraints
leading to cross-slip promotion within the characteristic activation volume.
Another study found that the yield point in nanoscale indents was often
identified by pop-in events, which had a strong rate dependence and much lower
activation volume [55]. In our study, given the large grain sizes, it is
reasonable to assume that twinning is activated during the early stages of
deformation indentation. Since the kinetics of twins and dislocations are
closely related, using a pseudo-slip approach to model twins is reasonable.
Therefore, the Zerilli-Armstrong model is appropriate for solutionized and
peak-aged Z5 whose plasticity is mediated by either dislocations, twins, or a
combination of both, although it was developed initially based on dislocation
motion. Furthermore, the strain rate has a strong dependence on dislocation
and twin mobility [56]. Finely spaced precipitates are found to be effective
obstacles to the motion of dislocations (affected by precipitate inter-
spacing) and the motion of twins (affected by precipitate size) [44]. Thus,
these competing mechanisms play out in the case of the peak-aged Z5 sample,
which has precipitates, as opposed to the solutionized Z5 sample with no
precipitates. This further affirms the observation of the consistently higher
dynamic strength of peak-aged Z5 over solutionized Z5 in the $10^{+1}$ s-1 to
$10^{+4}$ s-1 strain rate regime.

Figure 3: A) Nanoindentations of peak-aged and solutionized Z5 samples at
quasistatic and dynamic strain rates. $h=21\;\mu m$, $21\;\mu m,21\;\mu m$,
$59\;\mu m$, $59\;\mu m$, $29\;\mu m$, $39\;\mu m$ for each indentation at
each strain rate, from low to high strain rates respectively. High-speed video
shots and post-mortem SEM samples of B) Z5 solutionized, C) Z5 peak-aged after
undergoing spall. The scale bar of 1 mm applies to all images shown in B and
C.

###  3.5 Spall Results - Mechanical Property and Microscopy

The fully spalled samples experience approximately planar separation within
the material, parallel to the wavefront imposed by the flyer plate loading as
a result of the dynamic tensile stresses in this plane. The initially
compressive shock stress ($\Sigma_{S}$) generated in the target material from
the loading is calculated from

$\Sigma_{S}=\dfrac{1}{2}\rho_{0}U_{S}U_{B},$ (10)

where $\rho_{0}$ is the reference density, $U_{S}$, is the shock speed
estimated by assuming the linear equation of state with a parameter, $S_{1}$,
of 1.21 from Marsh et al. [57] and $U_{B}$ is the maximum compression shock
stress. The spall strengths of Z5 solutionized and Z5 peak-aged materials are
obtained from the measured rear surface particle velocities using the
following relationship:

$\Sigma^{*}=\frac{1}{2}\rho_{0}C_{0}(\Delta U_{fs}+\delta),$ (11)

where $\Sigma^{*}$ represents the spall strength, $\Delta U_{fs}$ is the
velocity drop seen in (Fig. 1 H), $C_{0}$ is the bulk wave-speed and $\delta$
elastic-plastic correction factor. The bulk wave speed was assumed to be 4540
m/s, the reference density as 1780 kg/m3, and the elastic-plastic correction
factor is zero [16]. The tensile strain rate, $\dot{\epsilon}$, is estimated
via the velocity gradient and is given by

$\dot{\epsilon}=\dfrac{1}{2C_{0}}\dfrac{\Delta U_{fs}}{|t_{c}-t_{d}|},$ (12)

where $t_{c}$ and $t_{d}$ are the times at points c and d in (Fig. 1 H).
Representative photon Doppler velocimetry spectrograms describing the
time–frequency response of the spall signal are shown in Fig. 1 G. More
detailed spectrograms can be seen in SI Appendix, FS.6 (a) and (b) along with
a compilation of the free-surface velocity traces for all spall experiments.
The calculated spall strength and strain rate values are plotted in Fig. 2 I.
The tensile strain rates for both sample sets range from 10+5 s-1 to 10+7 s-1.
The Z5 solutionized samples have an average spall strength of 1.44 $\pm$ 0.14
GPa, while the Z5 peak-aged samples have an average spall strength of 1.41
$\pm$ 0.35 GPa. Supplementary tables (SI Appendix, TS.2 and TS.3) provide a
summary of sample thickness and spall results for the Z5 solutionized and Z5
peak-aged data sets, respectively. A t-test analysis suggests that there is no
statistically significant difference in the spall strength between the two
datasets, yielding a p-value of 0.77 (SI Appendix, FS.5 (d)). To limit the
effect of strain rate dependency, which is amplified by the mismatched strain
rate range between the data sets, we also perform the t-test on a subset of
data, from 9.42x10+5 s-1 to 1.44x10+6 s-1, where both material preparations
have a significant overlap in the volume of experiments. The same analyses
show little change in mean and median in the subset of the experimental data.
Representative images from high-speed photography for the spall experiments
are shown in the left columns of Fig. 3 B and C. The post-mortem SEM images of
the 3 mm disks of both Z5 solutionized and Z5 peak-aged are shown in the right
columns of Fig. 3 B and C. For these spall experiments, the variation in
$\Delta U_{fs}$ as a function of peak shock stress is shown in SI Appendix,
FS.5 (a).

To see the trends in strength clearly, we also plot normalized effective
stress (the ratio of shock stress to quasistatic yield strength) as a function
of strain rate as shown in SI Appendix, FS.5 (b) [58]. The quasistatic yield
strength used for this ratio was taken from the nanoindentation data at
$10^{-2}$ s-1. From analyzing the spall results, we see that the spall
strengths are nominally the same between the solutionized and peak-aged
samples, with the solutionized microstructure perhaps exhibiting a slightly
higher spall strength. However, the fracture and subsequent failure of the
samples from the spall are dramatically different. We see that the damage is
much more significant in the case of the Z5 peak-aged samples, per SEM
micrographs shown in Fig. 3 B and C, likely from precipitate-mediated void
nucleation and coalescence.

The sharp contrast between the measured spall strength and the fracture
morphology highlights the importance of not relying on the spall strength
value alone to gain a complete understanding of the failure process of a
material undergoing spall. The large number of experiments in this work is
necessary to confidently identify these trends that might otherwise be lost
through traditional low-throughput methods. While the measured spall strengths
are similar, the fracture surfaces are dramatically different. In high-strain
rate applications, such as for protection materials, knowledge of both the
developed stress state and the postmortem failure morphology, are crucial to
understanding material failure.

###  3.6 Mechanisms Related to Spall

Our diagnostics provide both qualitative and quantitative understanding of the
spall failure phenomena through imaging and in-situ velocimetry, respectively,
yet the imaging techniques show the greatest difference in the failure process
in our experiments. The high-speed video and postmortem microscopy, shown in
Fig. 3 B and C, indicate that the spalled layer has completely fragmented away
from the sample in the peak-aged Z5 case. In contrast, solutionized Z5 shows
incomplete fragmentation in the spalled layer, more akin to a bulging
separation.

The observed dichotomy in the failure mechanism (Fig. 3) is likely driven by
defects or heterogeneities in the microstructure of the metallic alloy.
Assuming that the damage during spall primarily nucleates (i) along grain
boundaries or (ii) at precipitate-matrix interfaces, we can estimate
differences in the density of sites for damage via void nucleation. We assume
that the critical pressure for void nucleation sites $N$ follows a bounded
probability distribution function with a power-law exponent of $\beta$=3.
Following the Wilkerson-Ramesh spall model [59], we assume that the density of
potential nucleation sites along grain boundaries scales inversely with grain
size $l$, and (following similar scaling arguments) that nucleation sites at
second phase particles scale with the inverse cube of their mean spacing
$d_{s}$, i.e.

$\displaystyle
N(l,d_{s})=N_{1}\left(\frac{l_{0}}{l}\right)+N_{2}\left(\frac{d^{0}_{s}}{d_{s}}\right)^{3},$
(13)

where $N_{1}=1\>\mu\text{m}^{-3}$ and $N_{2}=10\>\mu\text{m}^{-3}$ are the
densities of grain boundary and particle nucleation sites for a reference
grain size of $l_{0}=1\>\mu\text{m}$, and reference mean particle spacing of
$d^{0}_{s}=10$ nm, respectively. For peak-aged Z5, $l=227\>\mu\text{m}$ and
the mean particle spacing is equal to the precipitate spacing, nominally
$d_{s}=50$ nm. For solutionied Z5, $l=205\>\mu\text{m}$ and the mean particle
spacing is taken to be a relatively large value governed by impurity content,
$d_{s}=1\>\mu\text{m}$. The lower bound of the probability distribution
function for the critical nucleation pressure is assumed to be a third of the
limit critical tensile pressure of an idealized elastic-perfectly plastic
material containing an infinitely small pre-existing void, i.e.,

$\displaystyle\mathcal{R}_{y}\equiv\frac{2}{3}\left(\sigma_{G}+\frac{k}{\sqrt{l}}\right)\left[1-\ln\frac{3}{2}\left(\frac{\sigma_{G}}{E}+\frac{k}{E\sqrt{l}}\right)\right],$
(14)

with $E$= 47.4 GPa. The upper bound of the probability distribution function
is taken as $\mathcal{R}_{eos}=$ 7 GPa, corresponding to the ideal spall
strength of a perfect Mg crystal. The solid lines in Fig. 2 J are model
predictions of spall strength according to the Wilkerson-Ramesh spall model
[59] invoking the aforementioned model parameters for solutionized and peak-
aged Z5. Considering the experimental variability, the agreement between the
model and experiments is remarkable.

Supplementary figure (SI Appendix, FS.5 (c)) shows the theoretical predictions
of the mean spacing between nucleated voids (dimples observed postmortem) on
the spall surface of solutionized and peak-aged Z5 as a function of the
experimentally measured spall strength. The dimples on the fracture surface of
peak-aged Z5 are expected to be smaller by approximately a factor of 3 than
for solutionized Z5. As such, the areal density of dimples on fracture
surfaces is anticipated to be roughly a factor of 10 higher on peak-aged Z5
than solutionized Z5, so fractures linking the nucleated voids are expected to
be significantly more prevalent in the peak-aged Z5.

While this model slightly over-predicts the difference in spall strength
between the solutionized and peak-aged alloys, when compared to the averages
from our laser-shock experiments, the model does offer a compelling
explanation for the more brittle-like failure and complete separation of the
spalled region, as observed in peak-aged Z5 (Fig. 3). The model curves shown
in (Fig. 2 J) employ a single set of nominal (average) values for precipitate
spacing and grain boundary densities. As such, the model predictions are
deterministic and make no attempt to capture the stochasticity observed in our
dynamic strength or spall strength measurements (Fig. 2 H to J). The
stochasticity is expected due to the fact that both nanoindentation and laser-
driven micro-flyer plate experiments probe relatively local, relatively small
volumes of material, which may not contain a statistically representative
volume of microstructure. As such, the locally measured properties are
themselves spatially dependent for any type of experiment that does not probe
a statistically representative volume of material. Interestingly, peak-aged Z5
microstructure (Fig. 2 C and D) appears to exhibit significantly more spatial
variability than the solutionized Z5 microstructure (Fig. 2 C and D), which
would seem to suggest that it would exhibit greater spatial variability in
properties. Indeed, our spall strength measurements (conducted at various
locations in the target plate) show significantly higher (spatial) variability
in the peak-aged Z5 as compared to the solutionized Z5 (SI Appendix, FS.5
(d)).

##  4 Conclusions

In this work, we combined two powerful small-scale mechanical testing
techniques, namely custom nanoindentation and laser-driven micro-flyer shock,
to probe the mechanical properties of Mg alloys across a large strain rate
regime ($10^{-2}$ s-1 to $10^{+7}$ s-1). We tested Z5 solutionized (no
precipitates) and Z5 peak-aged (with precipitates) across quasistatic,
dynamic, and spall regimes. We found that at low to medium strain rates (i.e.,
quasistatic and dynamic), the Z5 peak-aged sample had higher strength when
compared to the Z5 solutionized sample. We measured Orowon yield stress
increments at quasistatic strain rates and applied the Zerilli and Armstrong
constitutive model to predict the dynamic strength values. At higher strain
rates approaching $10^{+4}$ s-1, the nano-indentation experiments and
constitutive modeling showed converging strength for both peak aged and
solutionized Z5.

We observed that spall strength, a value measured at ultra-high strain rates
and captured in the initial moments of shock loading, was not influenced by
the differences in void nucleation density introduced by precipitates in the
peak-aged alloy. In contrast, our post-mortem observations of the spalled
samples showed very different failure behavior between the solutionized and
peak-aged samples. We adopted a void nucleation model to show the differences
between solutionized and peak-aged Z5 samples. The Z5 peak-aged samples failed
more significantly due to precipitate-mediated void nucleation and accelerated
spall fracture.

This demonstrated the importance of not only relying on the measured physical
quantity of spall strength, which is often the default in engineering design
work at ultra-high strain rates, but also examining the damage morphology for
a more complete understanding of material failure in extreme environments.
Experiments on the same material systems through traditional low-throughput
methods would not have readily shown these interesting trends. This study
demonstrates the potential for using high-throughput techniques to quickly map
the mechanical properties of various metallic alloys and aid in the Materials
by Design paradigm. Finally, our study also highlights an important lesson in
paying attention to how microstructures can fail differently despite having
similar strength at very high strain rates.

Data and code availability The datasets generated and codes used in the
current study are publicly available at
https://craedl.org/pubs?p=6352&t=3&c=187&s=hemi&d=https:%2F%2Ffs.craedl.org#publications.

## Acknowldegements

The authors would like to gratefully acknowledge the financial and technical
support from the Center for Materials under Extreme Dynamic Environment
(CMEDE). The research was sponsored by the Army Research Laboratory and was
accomplished under cooperative agreement numbers W911NF-12-2-0022 and
W911NF-22-2-0014. The views and conclusions contained in this document are
those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Army Research Office or
the U.S. Government. The U.S. Government is authorized to reproduce and
distribute reprints for Government purposes, notwithstanding any copyright
notation herein. The nanoindentation measurements were supported by the
Department of Energy, National Nuclear Security Administration (NNSA), under
award number DE-NA0003857.

##  5 References

## References

  * [1] Meyers, M. A.  _Dynamic behavior of materials_ (John wiley & sons, 1994).
  * [2] Eswarappa Prameela, S. _et al._ Materials for extreme environments.  _Nature Reviews Materials_ 8, 81–88 (2023).
  * [3] Weihs, T., Hong, S., Bravman, J. & Nix, W.  Mechanical deflection of cantilever microbeams: A new technique for testing the mechanical properties of thin films.  _Journal of Materials Research_ 3, 931–942 (1988).
  * [4] Rajagopalan, J.  Microelectromechanical systems (mems)-based testing of materials.  In _Handbook of Mechanics of Materials_ , 1955–1979 (Springer Singapore, 2019).
  * [5] Nix, W. D. & Lee, S.-W.  Micro-pillar plasticity controlled by dislocation nucleation at surfaces.  _Philosophical Magazine_ 91, 1084–1096 (2011).
  * [6] Barnard, H. S. _et al._ Synchrotron x-ray micro-tomography at the advanced light source: Developments in high-temperature in-situ mechanical testing.  In _Journal of Physics: Conference Series_ , vol. 849, 012043 (IOP Publishing, 2017).
  * [7] Van Petegem, S. _et al._ A miniaturized biaxial deformation rig for in situ mechanical testing.  _Experimental Mechanics_ 57, 569–580 (2017).
  * [8] Guillonneau, G. _et al._ Nanomechanical testing at high strain rates: New instrumentation for nanoindentation and microcompression.  _Materials & Design_ 148, 39–48 (2018).
  * [9] Zehnder, C., Peltzer, J.-N., Gibson, J. S.-L. & Korte-Kerzel, S.  High strain rate testing at the nano-scale: A proposed methodology for impact nanoindentation.  _Materials & Design_ 151, 17–28 (2018).
  * [10] Somekawa, H. & Schuh, C. A.  High-strain-rate nanoindentation behavior of fine-grained magnesium alloys.  _Journal of Materials Research_ 27, 1295–1302 (2012).
  * [11] Merle, B., Higgins, W. H. & Pharr, G. M.  Critical issues in conducting constant strain rate nanoindentation tests at higher strain rates.  _Journal of Materials Research_ 34, 3495–3503 (2019).
  * [12] Merle, B., Higgins, W. H. & Pharr, G. M.  Extending the range of constant strain rate nanoindentation testing.  _Journal of Materials Research_ 35, 343–352 (2020).
  * [13] Sudharshan Phani, P. & Oliver, W. C.  Ultra high strain rate nanoindentation testing.  _Materials_ 10, 663 (2017).
  * [14] Imbriglio, S. _et al._ Adhesion strength of titanium particles to alumina substrates: A combined cold spray and lipit study.  _Surface and Coatings Technology_ 361, 403–412 (2019).
  * [15] Tiamiyu, A. A. _et al._ Nanotwinning-assisted dynamic recrystallization at high strains and strain rates.  _Nature Materials_ 1–9 (2022).
  * [16] Mallick, D. _et al._ Laser-driven flyers and nanosecond-resolved velocimetry for spall studies in thin metal foils.  _Experimental Mechanics_ 59, 611–628 (2019).
  * [17] Mallick, D.  _Shock-induced failure of protection materials using laser-driven micro-flyers_.  Ph.D. thesis, The Johns Hopkins University (2019).
  * [18] Mallick, D. D. _et al._ Spall strength in alloyed magnesium: A compendium of research efforts from the cmede 10-year effort.  _Mechanics of Materials_ 162, 104065 (2021).
  * [19] Mallick, D., Williams, C. & Wilkerson, J.  A brief review of spall failure in pure and alloyed magnesium.  _Journal of Dynamic Behavior of Materials_ 6, 423–431 (2020).
  * [20] Yu, X., Li, T., Li, L., Liu, S. & Li, Y.  Influence of initial texture on the shock property and spall behavior of magnesium alloy az31b.  _Materials Science and Engineering: A_ 700, 259–268 (2017).
  * [21] De Rességuier, T. _et al._ Spall fracture and twinning in laser shock-loaded single-crystal magnesium.  _Journal of Applied Physics_ 121, 165104 (2017).
  * [22] Hazell, P., Appleby-Thomas, G., Wielewski, E., Stennett, C. & Siviour, C.  The influence of microstructure on the shock and spall behaviour of the magnesium alloy, elektron 675.  _Acta Materialia_ 60, 6042–6050 (2012).
  * [23] Farbaniec, L., Williams, C., Kecskes, L., Becker, R. & Ramesh, K.  Spall response and failure mechanisms associated with a hot-extruded amx602 mg alloy.  _Materials Science and Engineering: A_ 707, 725–731 (2017).
  * [24] Sun, X.  Uncertainty quantification of material properties in ballistic impact of magnesium alloys.  _Materials_ 15, 6961 (2022).
  * [25] Williams, C., Mallick, D. & Wilkerson, J.  A concise note on deformation twinning and spall failure in magnesium at the extremes.  _Journal of Dynamic Behavior of Materials_ 6, 432–444 (2020).
  * [26] Oliver, W. C. & Pharr, G. M.  An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments.  _Journal of Materials Research_ 7, 1564–1583 (1992).
  * [27] Oliver, W. C. & Pharr, G. M.  Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology.  _Journal of Materials Research_ 19, 3–20 (2004).
  * [28] Hackett, B., Sudharshan Phani, P., Walker, C., Oliver, W. & Pharr, G.  Advances in the measurement of hardness at high strain rates by nanoindentation.  _Journal of Materials Research_ 38, 1163–1177 (2023).
  * [29] Lucas, B. N., Oliver, W. C., Pharr, G. M. & Loubet, J. L.  Time dependent deformation during indentation testing.  _MRS Proceedings_ 436, 207 (1997).
  * [30] Phani, P. S., Oliver, W. C. & Pharr, G. M.  An experimental assessment of methods for mitigating plasticity error during nanoindentation with continuous stiffness measurement.  _Materials and Design_ 194, 108924 (2020).  URL https://doi.org/10.1016/j.matdes.2020.108924.
  * [31] Phani, P. S., Hackett, B., Walker, C., Oliver, W. & Pharr, G.  On the measurement of hardness at high strain rates by nanoindentation impact testing.  _Journal of the Mechanics and Physics of Solids_ 170, 105105 (2023).
  * [32] Merle, B., Higgins, W. H. & Pharr, G. M.  Critical issues in conducting constant strain rate nanoindentation tests at higher strain rates.  _Journal of Materials Research_ 34, 3495–3503 (2019).
  * [33] Paisley, D., Warnes, R. & Kopp, R.  Laser-driven flat plate impacts to 100 gpa with sub-nanosecond pulse duration and resolution for material property studies.  Tech. Rep., Los Alamos National Lab., N.M.(United States) (1991).
  * [34] Paisley, D. _et al._ Experimental method for laser-driven flyer plates for 1-d shocks.  In _AIP Conference Proceedings_ , vol. 955, 1337–1340 (American Institute of Physics, 2007).
  * [35] Brown, K. E., Shaw, W. L., Zheng, X. & Dlott, D. D.  Simplified laser-driven flyer plates for shock compression science.  _Review of Scientific Instruments_ 83, 103901 (2012).
  * [36] Li, F. & Dlott, D. D.  High throughput tabletop shock techniques and measurements.  _Journal of Applied Physics_ 131, 075901 (2022).
  * [37] Dolan, D.  Extreme measurements with photonic doppler velocimetry (pdv).  _Review of Scientific Instruments_ 91, 051501 (2020).
  * [38] DiMarco, C. S. _et al._ Spall failure of ecae mg-al alloys at extreme strain rates: Influence of a refined precipitate and grain microstructure.  _Metals_ 13, 454 (2023).
  * [39] Bhattacharya, A. & Nix, W.  Finite element simulation of indentation experiments.  _International Journal of Solids and Structures_ 24, 881–891 (1988).
  * [40] Nix, W. D. & Gao, H.  Indentation size effects in crystalline materials: a law for strain gradient plasticity.  _Journal of the Mechanics and Physics of Solids_ 46, 411–425 (1998).
  * [41] Orowan, E.  Symposium on internal stresses in metals and alloys.  _Institute of Metals, London_ 451 (1948).
  * [42] Hirsch, P. & Humphreys, F.  Physics of strength and plasticity.  _MIT Press, Cambridge_ (1969).
  * [43] Kocks, U. F., AS, A. & M.F., A.  Thermodynamics and kinetics of slip (1975).
  * [44] Prameela, S. E. _et al._ Strengthening magnesium by design: Integrating alloying and dynamic processing.  _Mechanics of Materials_ 167, 104203 (2022).
  * [45] Blake, A. & Cáceres, C.  Solid-solution hardening and softening in mg–zn alloys.  _Materials Science and Engineering: A_ 483, 161–163 (2008).
  * [46] Zerilli, F. J. & Armstrong, R. W.  Constitutive relations for titanium and ti-6al-4v.  In _AIP conference proceedings_ , vol. 370, 315–318 (American Institute of Physics, 1996).
  * [47] Zerilli, F. J. & Armstrong, R. W.  Dislocation mechanics based constitutive equation incorporating dynamic recovery and applied to thermomechanical shear instability.  In _AIP Conference proceedings_ , vol. 429, 215–218 (American Institute of Physics, 1998).
  * [48] Zerilli, F. J. & Armstrong, R. W.  Dislocation-mechanics-based constitutive relations for material dynamics calculations.  _Journal of applied physics_ 61, 1816–1825 (1987).
  * [49] Nguyen, T., Luscher, D. J. & Wilkerson, J.  A physics-based model and simple scaling law to predict the pressure dependence of single crystal spall strength.  _Journal of the Mechanics and Physics of Solids_ 137, 103875 (2020).
  * [50] Yi, P., Cammarata, R. C. & Falk, M. L.  Atomistic simulation of solid solution hardening in mg/al alloys: Examination of composition scaling and thermo-mechanical relationships.  _Acta Materialia_ 105, 378–389 (2016).  URL https://doi.org/10.1016/j.actamat.2015.12.038.
  * [51] Johnson, G. R.  A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures.  _Proc. 7th Inf. Sympo. Ballistics_ 541–547 (1983).
  * [52] Huh, H. & Kang, W.  Crash-worthiness assessment of thin-walled structures with the high-strength steel sheet.  _International Journal of Vehicle Design_ 30, 1–21 (2002).
  * [53] Cowper, G. R. & Symonds, P. S.  Strain-hardening and strain-rate effects in the impact loading of cantilever beams.  Tech. Rep., Brown Univ Providence Ri (1957).
  * [54] Somekawa, H. & Schuh, C. A.  Nanoindentation behavior and deformed microstructures in coarse-grained magnesium alloys.  _Scripta Materialia_ 68, 416–419 (2013).
  * [55] Somekawa, H. & Schuh, C. A.  Effect of solid solution elements on nanoindentation hardness, rate dependence, and incipient plasticity in fine grained magnesium alloys.  _Acta materialia_ 59, 7554–7563 (2011).
  * [56] Kannan, V., Hazeli, K. & Ramesh, K.  The mechanics of dynamic twinning in single crystal magnesium.  _Journal of the Mechanics and Physics of Solids_ 120, 154–178 (2018).
  * [57] Marsh, S. P.  _LASL shock Hugoniot data_ (University of California Press, 1980).
  * [58] Wu, X., Ramesh, K. & Wright, T.  The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids.  _International journal of solids and structures_ 40, 6633–6651 (2003).
  * [59] Wilkerson, J. & Ramesh, K.  Unraveling the anomalous grain size dependence of cavitation.  _Physical review letters_ 117, 215503 (2016).

##  6 Supplementary Information

###  6.1 Details regarding the data from custom Nanoindentation set-up

FS 1: Depth vs. time for nanoindentation tests at different strain rates. For
quasistatic tests (a-d), the load vs. time profiles are given to show the
exponential loading used. For impact indentation tests (e-f), strain rate vs.
time profiles are shown to indicate the non-constant strain rate during
loading.

FS 1 shows the exemplary depth-time and load-time data for the quasistatic
nanoindentation testing. A $1\text{\,}\mathrm{kHz}$ feedback control loop was
used to maintain a constant strain rate while loading to
$200\text{\,}\mathrm{mN}$. The difference between the final depths of the
Solutionized Z5 and Peak-Aged Z5 is a result of the small difference in
hardness at each strain rate. The final load at $\dot{\epsilon}=$
${10}^{1}\text{\,}{\mathrm{s}}^{-1}$ is much larger than the target
$200\text{\,}\mathrm{mN}$ due to dynamic effects brought on by the order of
magnitude increase in velocity required to maintain the constant strain rate
(seen in FS 2). Additionally, as the load is only updated in discrete steps
every $1\text{\,}\mathrm{ms}$ by the control loop, the final load step is
often larger than necessary to reach the target load in very fast tests. FS 1
(e-f) shows depth-time and strain rate-time data for high strain rate impact
nanoindentation tests. The strain rate is shown instead of load due to the
strain rate not being constant throughout the experiment. Load is also not
constant, but due to the nature of the impact test, a fixed impact force is
given, and then most of the experiment is driven by dynamic forces from the
rapid deceleration at the moment of impact.

FS 2: Velocity vs. time for nanoindentation tests at different strain rates.
For quasistatic tests (a-d), the load vs. time profiles are given to show the
exponential loading used. For impact indentation tests (e-f), strain rate vs.
time profiles are shown to indicate the non-constant strain rate during
loading.

FS 2 highlights velocity vs. time for each of the corresponding tests in FS 1.
The exponentially increasing velocity is necessary during quasistatic
nanoindentation to maintain a constant strain rate as depth increases and,
therefore contact area between the diamond tip and sample increases. The
velocity at low strain rates
(${10}^{-2}\text{\,}{\mathrm{s}}^{-1}{10}^{-1}\text{\,}{\mathrm{s}}^{-1}$) is
particularly noisy due to the relatively small velocity of the test compared
to the large $\Delta$velocity from the constant stiffness measurements (CSM)
oscillation. At $\dot{\epsilon}=$ $undef\text{\,}{\mathrm{s}}^{-1}$ (FS 2 (c))
the velocity of the test is much larger than the $\Delta$velocity created by
CSM, making the noise negligible. At a strain rate of
${10}^{1}\text{\,}{\mathrm{s}}^{-1}$ (FS 2 (d)), CSM cannot be used since the
high velocities result in a very short test and the $1\text{\,}\mathrm{kHz}$
control loop cannot produce the oscillatory frequency high enough to capture
quality stiffness measurements while maintaining the increasing velocity. For
$undef\text{\,}{\mathrm{s}}^{-1}{10}^{1}\text{\,}{\mathrm{s}}^{-1}$, the
decreasing and flat velocities (respectively) after reaching peak velocity are
a result of the dynamic forces generated by the high velocities and
accelerations and vary depending on the timing of the last force step and the
amount of time it takes the feedback control loop to determine it is at or
past the target load. For the impact tests in FS 2 (e-f), the test is designed
so that contact with the sample surface is made at the instance of max
velocity. No additional force is added to the system after this point to allow
dynamic effects to drive the loading process. This results in a decreasing
velocity throughout the entire experiment.

The measured hardness from both quasistatic and impact nanoindentation
experiments are shown in FS 3. FS 3 (a-c) shows that hardness vs. depth is
fairly constant for both the solutionized Z5 and peak-aged Z5 samples after
some initial indentation size effect (ISE). To mitigate the effects of ISE on
the reported hardness, an average hardness between
$2200\text{\,}\mathrm{nm}3000\text{\,}\mathrm{nm}$ is reported in FS 3 (e).
Hardness vs. depth is also reported for the impact nanoindentation experiments
in FS 3 (d), which shows a decreasing hardness as a function of depth even
past a depth where hardness is flat in the quasistatic results. This decrease
in hardness is explained by FS 3 (f), which shows hardness decreases as the
strain rate decreases. During a nanoindentation impact experiment, the strain
rate is not constant and decreases until the very end of the test where it
falls to effectively $0\text{\,}{\mathrm{s}}^{-1}$ at the peak load. Though
strain rates as high as $5\text{\times}{10}^{5}\text{\,}{\mathrm{s}}^{-1}$ can
be seen in FS 1 (e-f) and FS 2 (e-f), hardness is only reported for
$\dot{\epsilon}\leq$ ${10}^{4}\text{\,}{\mathrm{s}}^{-1}$ to avoid edge
effects created the strain rate being infinite at the moment of impact.

FS 3: The average and standard deviation for hardness values were calculated
between a depth of $2200\text{\,}\mathrm{nm}3000\text{\,}\mathrm{nm}$. A
representative hardness vs. depth for nanoindentation tests for Z5 (wt%)
solutionized and peak-aged at strain rate a) $10^{-2}s^{-1}$, b)
$10^{-1}s^{-1}$, c) $10^{0}s^{-1}$and d) $10^{1}$ to $10^{4}s^{-1}$, e)
Quasistatic hardness measured via nanoindentation, f) Dynamic hardness
measured via nanoindentation.

###  6.2 Details regarding the strength modeling on the Nanoindentation data

FS 4: Comparisons of dynamic strengths between nanoindentation measurements
and constitutive model fittings for solutionized and peak-aged Z5 samples: a)
JC model with $R^{2}$ of $0.65$ for solutionized and $0.64$ for peak-aged, b)
JC-quad model with $R^{2}$ of $0.92$ for solutionized and $0.88$ for peak-
aged, c) CS model with $R^{2}$ of $0.94$ for solutionized and $0.91$ for peak-
aged, and d) ZA model with $R^{2}$ of $0.94$ for solutionized and $0.91$ for
peak-aged.

In addition to the Zerilli-Armstrong (ZA) model, we also employ three other
constitutive models to describe the dynamic strength of the solutionized and
peak-aged Z5 depending upon strain rate, i.e., the standard Johnson-Cook (JC)
model [51], the modified JC model with quadratic form (JC-quad) proposed by
Huh and Kang [52], and the Cowper-Symonds (CS) model [53]. Specifically, the
JC form of the yield strength at room temperature is written as Eq [51]. In Eq
15, $\epsilon_{0}$ is the reference strain rate. The fitting parameters are
the quasi-static yield strength $\sigma_{Y0}$ and the strengthening
coefficient of strain rate $C$. The JC-quad form of the yield strength is
given by Eq 16. In Eq 16, $\sigma_{Y0}$, $C_{1}$ and $C_{2}$ are fitting
parameters. Moreover, the yield strength with the CS form can be expressed by
Eq 17. In Eq 17 $\sigma_{Y0}$, $D$ and $P$ are fitting material constants. A
comparison of the fitting results with experimental data is shown in FS. 4.
The fitting parameters are tabulated in Table 1.

$\sigma_{Y}=\sigma_{Y0}\bigg{(}1+C\ln\Big{(}\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\Big{)}\bigg{)},$
(15)

$\sigma_{Y}=\sigma_{Y0}\bigg{(}1+C_{1}\ln\Big{(}\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\Big{)}+C_{2}\ln\Big{(}\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\Big{)}^{2}\bigg{)},$
(16)
$\sigma_{Y}=\sigma_{Y0}\bigg{(}1+\Big{(}\frac{\dot{\epsilon}}{D}\Big{)}^{1/P}\bigg{)},$
(17)

Table 1: Constitutive model parameters for dynamic strengths of solutionized and peak-aged Z5. Parameter | Unit | Solutionized | Peak-aged
---|---|---|---
JC model | $\sigma_{Y0}$ | MPa | $234.5$ | $280.4$
$C$ | - | $0.03606$ | $0.02084$
JC-quad model | $\sigma_{Y0}$ | MPa | $335$ | $348.5$
$C_{1}$ | - | $-0.08496$ | $-0.05405$
$C_{2}$ | - | $0.009067$ | $0.005781$
CS model | $\sigma_{Y0}$ | MPa | $271.5$ | $306.6$
$D$ | - | $5.172\times 10^{4}$ | $8.924\times 10^{4}$
$P$ | - | $1.211$ | $1.172$
ZA model | $\sigma_{G}$ | MPa | $234.74$ | $271.71$
$k$ | $\text{MPa}\cdot\mu\text{m}^{1/2}$ | $526$ | $526$
$B$ | MPa | $0.0347$ | $0.0182$
| $\beta_{0}$ | $\text{K}^{-1}$ | $0$ | $0$
| $\beta_{1}$ | $\text{K}^{-1}$ | $0.0028$ | $0.0028$

###  6.3 Details regarding the data from custom laser spall set-up

All spall data results, including raw and analyzed PDV results, high-speed
camera footage, experimental parameter data sheets, and Python-based PDV code,
are available via the link in the Data and Code Availability section of the
main text. A reference file in the main directory summarizes the results and
maps them with their relevant PDV and high-speed camera data files. A
FileMaker relational database was created to track metadata associated with
the experimental methods and techniques, and this data was exported into Excel
sheets for reference.

Selected high-speed camera videos were chosen to clearly demonstrate the
custom laser-driven micro-flyer experiment and the spall failure results of
the two data sets. The videos were recorded from a side angle, showing the
flyer accelerating from top to bottom in the frame. During impact experiments,
the flyer is not visible to the PDV or camera. Therefore, the flyer’s velocity
and planarity are determined independently before impacting a sample. The
first video demonstrates the launch of a single flyer in the absence of a
sample to showcase the high degree of planarity achieved. The second and third
videos show characteristic impact experiments of solutionized Z5 samples,
while the fourth and fifth videos show characteristic impacts for peak-aged Z5
samples. The high planarity maintained during the impact event indicates the
planarity of the event itself. Solutionized samples demonstrate higher damage
resistance than peak-aged samples when impacted under the same conditions.

Video 1 | Flyer Launch | Link 1
---|---|---
Video 2 | Solutionized Z5 Impact 1 | Link 2
Video 3 | Solutionized Z5 Impact 2 | Link 3
Video 4 | Peak-Aged Z5 Impact 1 | Link 4
Video 5 | Peak-Aged Z5 Impact 2 | Link 5

The spall results are presented in two plots in FS 5. FS 5a displays the
pullback velocity against the peak shock stress for both sample sets. While
spall results are typically plotted against strain rate, it is not directly
controllable in laser-driven micro-flyer experiments. Instead, the impact
velocity is controlled, which determines the peak shock stress in the
material. As peak shock stress increases, metals experience strain-hardening
from the initial shock compression wave before spall failure. The correlation
is clearly observable with the peak-aged samples, while the solutionized
sample set is too narrow to draw definitive conclusions. The box plot in FS 5c
provides a more rigorous statistical analysis and a t-test. An ANOVA test
indicates that there are no statistically significant differences in the spall
strengths between the two datasets within a 95% confidence interval, which is
also evident from the comparison of the box-plot distributions.

FS 5: a)Delta U vs. Shock Stress, b)Normalized Effective Stress vs. Strain
rate, c) Mean spacing of voids vs. experimentally measured spall strength, d)
Spall strength of solutionized Z5 and peak-aged Z5 samples.

FS 5b plots the normalized effective stress against the strain rate, where the
former is calculated as the peak shock stress divided by the yield strength.
The yield strength of a material is a measure of its resistance to spall
failure, with higher yield strength indicating higher resistance to void
growth. In FS 5b, lower normalized effective stress corresponds to higher
spall resistance. These results highlight the differences between the two
datasets and are consistent with the observed variations in damage morphology.
Specifically, the peak-aged samples exhibit a higher normalized effective
stress and sustain more significant damage under the same impact conditions.
FS 6 shows a summary of the PDV results for both datasets, with essentially
the same information but for each dataset displayed in the top and bottom
frames. Within each frame, the top four figures represent how our PDV code
extracts the PDV trace and relevant data points. While a short-time Fourier
transform is used for viewing purposes, the PDV code employs direct phase
differentiation for more accurate analysis of high-frequency waves. First, the
spectrum is imported and coarsely pre-filtered to include only the expected
data range. Second, the starting point of the signal is identified, and a
finer time-based filter is applied based on the expected duration of the
signal. Third, the spall signal is isolated by filtering out the frequency
upshifted signal. Fourth, the signal is differentiated, the velocity trace is
calculated, and the critical data points are automatically determined. The
fourth frame shows the identification of the velocities at maximum compression
and tension. The PDV code, processing input parameters, and a graphical
summary of results are included in the shared data directory. Lastly, at the
bottom of each frame, a compilation of all PDV traces is presented side-by-
side for an easily viewed summary and direct comparison.

FS 6: Representative photon Doppler velocimetry spectrograms describing the
time-frequency response and free surface velocity of the spall signal from a)
solutionized Z5 and b) peak-aged Z5

Tables LABEL:fig:tb5 and 3 provide a concise summary of the spall failure
results for the solutionized and peak-aged samples, respectively. These tables
include information such as the thickness of each individual sample, the peak
shock stress, spall strength, and strain rate. The velocities at maximum
compression and tension are also included in the shared data link. The
thickness of each sample was measured using a micrometer prior to testing. The
remaining quantities were determined based on the material density, the bulk
wave speed, the equation of state, and the velocities at maximum compression
and tension. These relationships are provided in the main text and were used
to calculate the spall failure results for each sample.

FS 5 shows the theoretical predictions of the relationship between mean
spacing of nucleated voids (dimples that might be observed postmortem) on the
spall surface of the solutionized and peak-aged Z5 specimens as a function of
the experimentally measured spall strength, per the analytic model by
Wilkerson and Ramesh, 2016. In both the solutionized and peak-aged cases, the
void spacing will decrease as the spall strength increases, but the increased
density of failure nucleation sites in the peak-aged case will lead to smaller
void spacing when compared to the solutionied case for a given spall strength,
with the difference in spacing increasing as the spall strength increases.

Table 2: The results of spall experiments on solutionized Z5 alloy including thickness ($\mu$m), strain rate ($s^{-1}$), shock stress (GPa), spall strength (GPa) and pullback velocity (m/s). Shot No. | Thickness ($\mu$m) | Strain rate ($s^{-1}$) | Shock stress (GPa) | Spall strength (GPa) | Pullback (m/s)
---|---|---|---|---|---
1 | 214 | 1880410 | 3.58 | 1.61 | 398.47
2 | 211 | 1658727 | 3.27 | 1.89 | 466.90
3 | 207 | 2222355 | 3.20 | 1.66 | 411.15
4 | 215 | 2093194 | 2.68 | 1.37 | 338.79
5 | 213 | 1428659 | 3.13 | 1.78 | 440.73
6 | 201 | 1367488 | 3.06 | 1.39 | 345.03
7 | 269 | 1543896 | 3.66 | 1.48 | 366.94
8 | 242 | 706241.6 | 2.44 | 1.47 | 362.80
9 | 272 | 664606 | 2.51 | 1.56 | 386.22
10 | 300 | 922959.1 | 2.81 | 1.35 | 333.23
11 | 243 | 752198.5 | 3.21 | 1.43 | 352.94
12 | 281 | 727316.8 | 2.99 | 1.34 | 330.86
13 | 248 | 942287.7 | 3.11 | 1.46 | 360.31
14 | 285 | 1163233 | 3.10 | 1.36 | 336.67
15 | 244 | 1254023 | 3.28 | 1.39 | 344.02
16 | 281 | 966358.6 | 3.09 | 1.52 | 375.55
17 | 277 | 1180744 | 3.19 | 1.37 | 338.92
18 | 259 | 1798959 | 3.40 | 1.52 | 375.08
19 | 302 | 1506518 | 3.13 | 1.29 | 319.41
20 | 248 | 1478650 | 3.41 | 1.43 | 352.94
21 | 266 | 1332230 | 3.09 | 1.23 | 304.53
22 | 283 | 985658.9 | 3.06 | 1.33 | 329.58
23 | 270 | 1669877 | 3.33 | 1.47 | 364.66
24 | 289 | 1132115 | 2.82 | 1.50 | 372.38
25 | 232 | 1099777 | 3.32 | 1.34 | 332.03
26 | 302 | 1250070 | 3.17 | 1.41 | 348.46
27 | 278 | 1391273 | 3.19 | 1.44 | 357.35
28 | 271 | 1476892 | 3.00 | 1.29 | 320.00
29 | 242 | 1386178 | 3.33 | 1.27 | 314.82
30 | 293 | 1328424 | 3.30 | 1.40 | 346.48
31 | 242 | 1131126 | 3.28 | 1.35 | 333.15
Table 3: The results of spall experiments on peak-aged Z5 alloy including thickness ($\mu$m), strain rate ($s^{-1}$), shock stress (GPa), spall strength (GPa) and pullback velocity (m/s). Shot No. | Thickness ($\mu$m) | Strain rate ($s^{-1}$) | Shock stress (GPa) | Spall strength (Gpa) | Pullback (m/s)
---|---|---|---|---|---
1 | 203 | 2217417 | 3.67 | 1.58 | 389.85
2 | 203 | 1690616 | 3.27 | 1.45 | 359.02
3 | 198 | 2798721 | 3.35 | 1.81 | 447.58
4 | 207 | 2683489 | 3.42 | 2.21 | 546.11
5 | 206 | 1890421 | 3.19 | 1.37 | 338.15
6 | 206 | 3056042 | 3.88 | 1.99 | 493.58
7 | 193 | 2476961 | 3.42 | 1.29 | 319.65
8 | 213 | 830342.5 | 3.51 | 1.51 | 374.15
9 | 185 | 1034565 | 3.12 | 1.44 | 356.61
10 | 253 | 1182466 | 3.15 | 1.42 | 352.03
11 | 249 | 1313375 | 3.27 | 1.57 | 388.32
12 | 236 | 1028951 | 3.48 | 1.54 | 382.36
13 | 214 | 1440590 | 3.47 | 1.41 | 348.92
14 | 212 | 1133202 | 3.12 | 1.21 | 299.42
15 | 200 | 986129.5 | 2.56 | 1.36 | 335.89
16 | 203 | 979103.3 | 2.61 | 1.09 | 269.49
17 | 236 | 407457.7 | 1.73 | 0.83 | 204.18
18 | 197 | 618610.5 | 2.64 | 0.97 | 240.97
19 | 203 | 363273.1 | 1.39 | 0.48 | 117.63
20 | 199 | 1001282 | 2.53 | 1.22 | 301.16
21 | 258 | 1038576 | 3.29 | 1.49 | 368.96
22 | 207 | 862159.5 | 3.06 | 1.65 | 408.55
23 | 216 | 1411300 | 3.26 | 1.50 | 370.82
24 | 218 | 1347976 | 3.42 | 1.46 | 360.61
25 | 236 | 1387231 | 3.55 | 1.21 | 298.53
26 | 208 | 1200006 | 3.80 | 1.61 | 397.71

###  6.4 Additional Experimental Details

FS 7: STEM images of a) Z5 peak aged observed under bright field microscope
and b) Z5 peak aged observed under dark field microscope.

Standard deviation from the Nanoindentation experiments

Table 4: Standard deviation of Quasistatic hardness for solutionized Z5 and peak-aged Z5 samples. Solutionized Z5 | Peak-Aged Z5
---|---
S.No | Strain Rate (1/s) | Quasistatic Hardness_std | S.No | Strain Rate (1/s) | Quasistatic Hardness_std
1 | 0.01 | 0.044385 | 1 | 0.01 | 0.097712
2 | 0.1 | 0.063733 | 2 | 0.1 | 0.091532
3 | 1 | 0.065156 | 3 | 1 | 0.057564
Table 5: Standard deviation of Dynamic hardness for Solutionized Z5 sample. S.No | Strain Rate (1/s) | Dynamic Hardness_STD | S.No | Strain Rate (1/s) | Dynamic Hardness_STD
---|---|---|---|---|---
1 | 10.5368 | 0.01615 | 61 | 1006.83789 | 0.09357
2 | 11.87996 | 0.08946 | 62 | 1058.67786 | 0.10125
3 | 13.71746 | 0.06307 | 63 | 1112.73409 | 0.10189
4 | 15.63303 | 0.0629 | 64 | 1168.54589 | 0.10428
5 | 16.9499 | 0.10939 | 65 | 1224.95179 | 0.10851
6 | 19.38332 | 0.09094 | 66 | 1285.555 | 0.11345
7 | 21.26067 | 0.06215 | 67 | 1345.54054 | 0.11029
8 | 23.5918 | 0.06152 | 68 | 1409.77242 | 0.11027
9 | 26.25462 | 0.06137 | 69 | 1479.21582 | 0.11207
10 | 29.31402 | 0.15121 | 70 | 1550.52019 | 0.11519
11 | 31.36326 | 0.07808 | 71 | 1621.00242 | 0.12046
12 | 35.12035 | 0.07173 | 72 | 1692.96066 | 0.11825
13 | 38.91717 | 0.09452 | 73 | 1769.79048 | 0.11893
14 | 42.93214 | 0.10059 | 74 | 1850.61845 | 0.11277
15 | 47.28611 | 0.11147 | 75 | 1935.71226 | 0.12544
16 | 51.71428 | 0.12663 | 76 | 2020.78274 | 0.12857
17 | 57.09975 | 0.09618 | 77 | 2109.09653 | 0.12327
18 | 61.56953 | 0.07985 | 78 | 2201.42923 | 0.12732
19 | 67.49483 | 0.08892 | 79 | 2298.89449 | 0.12484
20 | 73.44082 | 0.09063 | 80 | 2394.89766 | 0.11944
21 | 79.09578 | 0.09768 | 81 | 2495.77787 | 0.12956
22 | 86.77363 | 0.07735 | 82 | 2605.74169 | 0.13207
23 | 92.97928 | 0.08211 | 83 | 2711.33234 | 0.12492
24 | 102.04513 | 0.09434 | 84 | 2824.85956 | 0.12661
25 | 110.29707 | 0.11099 | 85 | 2941.44724 | 0.11752
26 | 117.91848 | 0.09071 | 86 | 3061.39759 | 0.11652
27 | 127.6346 | 0.09414 | 87 | 3182.59563 | 0.12828
28 | 137.35956 | 0.10321 | 88 | 3315.34195 | 0.11445
29 | 148.41189 | 0.09947 | 89 | 3443.04218 | 0.12567
30 | 158.94798 | 0.09437 | 90 | 3582.2955 | 0.12406
31 | 170.66804 | 0.08901 | 91 | 3718.89783 | 0.12712
32 | 184.3826 | 0.10115 | 92 | 3869.18493 | 0.13725
33 | 196.7869 | 0.08587 | 93 | 4017.25116 | 0.12485
34 | 210.43677 | 0.08737 | 94 | 4168.54847 | 0.12696
35 | 225.56907 | 0.09862 | 95 | 4331.43286 | 0.12761
36 | 240.88302 | 0.09293 | 96 | 4493.13283 | 0.13195
37 | 256.87634 | 0.07142 | 97 | 4660.26291 | 0.14771
38 | 275.35135 | 0.0887 | 98 | 4837.76617 | 0.13474
39 | 293.73678 | 0.08964 | 99 | 5017.62166 | 0.13879
40 | 311.97289 | 0.08487 | 100 | 5189.481 | 0.13698
41 | 331.23519 | 0.08369 | 101 | 5380.76496 | 0.14203
42 | 352.6805 | 0.09294 | 102 | 5583.12942 | 0.13622
43 | 376.23075 | 0.08993 | 103 | 5783.47385 | 0.14555
44 | 398.6735 | 0.09228 | 104 | 5975.85655 | 0.14192
45 | 423.94248 | 0.08565 | 105 | 6191.0202 | 0.13435
46 | 449.21152 | 0.08802 | 106 | 6417.74146 | 0.14723
47 | 475.82454 | 0.09796 | 107 | 6639.16311 | 0.16445
48 | 504.91543 | 0.08404 | 108 | 6874.66909 | 0.15783
49 | 533.0923 | 0.0842 | 109 | 7102.18887 | 0.16578
50 | 564.68611 | 0.08471 | 110 | 7347.30777 | 0.15691
51 | 596.11318 | 0.08891 | 111 | 7594.78445 | 0.13629
52 | 629.66989 | 0.08598 | 112 | 7854.36082 | 0.1576
53 | 665.20734 | 0.09349 | 113 | 8109.97729 | 0.16195
54 | 702.49227 | 0.0911 | 114 | 8382.98127 | 0.16341
55 | 740.72401 | 0.09057 | 115 | 8649.03082 | 0.16926
56 | 780.88126 | 0.09311 | 116 | 8945.83198 | 0.18487
57 | 822.83684 | 0.10031 | 117 | 9235.93133 | 0.17883
58 | 865.98754 | 0.10064 | 118 | 9540.39072 | 0.16098
59 | 911.66121 | 0.09852 | 119 | 9841.1845 | 0.19399
60 | 959.20555 | 0.09646 |  |  |
Table 6: Standard deviation of Dynamic hardness for peak-aged Z5 sample. S.No | Strain Rate (1/s) | Dynamic Hardness_STD | S.No | Strain Rate (1/s) | Dynamic Hardness_STD
---|---|---|---|---|---
1 | 11.4317 | 0 | 60 | 9847.84811 | 0.13541
2 | 13.6059 | 0.01506 | 61 | 959.09597 | 0.07525
3 | 15.34068 | 0.07662 | 62 | 1006.28321 | 0.07614
4 | 17.09732 | 0.08133 | 63 | 1057.7589 | 0.07772
5 | 18.93417 | 0.04807 | 64 | 1111.99088 | 0.07028
6 | 21.22634 | 0.02831 | 65 | 1169.34094 | 0.07438
7 | 24.0556 | 0.07375 | 66 | 1224.38848 | 0.07151
8 | 26.33735 | 0.0614 | 67 | 1284.33602 | 0.07956
9 | 29.12948 | 0.04923 | 68 | 1346.84544 | 0.07384
10 | 32.66838 | 0.06954 | 69 | 1413.14168 | 0.07135
11 | 34.83625 | 0.07593 | 70 | 1480.21105 | 0.07279
12 | 39.06283 | 0.07651 | 71 | 1547.48288 | 0.07005
13 | 42.68435 | 0.06671 | 72 | 1619.76735 | 0.06744
14 | 46.66146 | 0.06548 | 73 | 1697.59205 | 0.07034
15 | 51.6477 | 0.061 | 74 | 1771.03775 | 0.07
16 | 56.48525 | 0.07657 | 75 | 1850.10272 | 0.06852
17 | 61.35494 | 0.07169 | 76 | 1936.11215 | 0.06958
18 | 66.7823 | 0.06858 | 77 | 2021.17417 | 0.07083
19 | 72.75592 | 0.07499 | 78 | 2108.78121 | 0.07107
20 | 79.22108 | 0.07385 | 79 | 2199.38027 | 0.06533
21 | 86.62227 | 0.07211 | 80 | 2296.34924 | 0.07429
22 | 93.20207 | 0.06887 | 81 | 2398.49568 | 0.07144
23 | 101.08401 | 0.07093 | 82 | 2494.59703 | 0.06693
24 | 110.4806 | 0.08604 | 83 | 2603.17046 | 0.07155
25 | 118.78563 | 0.07363 | 84 | 2711.5216 | 0.0709
26 | 126.10944 | 0.08027 | 85 | 2822.9624 | 0.07366
27 | 137.75045 | 0.08467 | 86 | 2941.70274 | 0.0729
28 | 148.36915 | 0.06979 | 87 | 3058.78717 | 0.0659
29 | 159.36976 | 0.08922 | 88 | 3184.64014 | 0.07176
30 | 170.63119 | 0.08195 | 89 | 3312.5738 | 0.07171
31 | 184.15915 | 0.06833 | 90 | 3443.66464 | 0.06909
32 | 196.82152 | 0.06965 | 91 | 3578.73639 | 0.07519
33 | 210.05017 | 0.0644 | 92 | 3722.1544 | 0.07387
34 | 224.44571 | 0.07862 | 93 | 3864.67708 | 0.07607
35 | 241.10177 | 0.05486 | 94 | 4013.35726 | 0.07265
36 | 258.33593 | 0.07631 | 95 | 4174.19533 | 0.07723
37 | 274.58269 | 0.06617 | 96 | 4323.27745 | 0.07867
38 | 292.54554 | 0.07678 | 97 | 4490.96335 | 0.07297
39 | 312.2643 | 0.06738 | 98 | 4661.79432 | 0.07392
40 | 331.84418 | 0.0753 | 99 | 4826.88121 | 0.0823
41 | 352.08983 | 0.06947 | 100 | 5002.43394 | 0.08408
42 | 374.76696 | 0.07932 | 101 | 5190.53005 | 0.08859
43 | 398.12235 | 0.06503 | 102 | 5389.91886 | 0.08471
44 | 422.10922 | 0.06256 | 103 | 5577.82261 | 0.07853
45 | 448.38299 | 0.08049 | 104 | 5782.94352 | 0.0954
46 | 476.39404 | 0.07129 | 105 | 5984.84442 | 0.09328
47 | 503.35696 | 0.07023 | 106 | 6191.55544 | 0.09271
48 | 533.48216 | 0.07662 | 107 | 6410.68876 | 0.09653
49 | 565.34942 | 0.07533 | 108 | 6645.10721 | 0.09031
50 | 596.27527 | 0.07716 | 109 | 6868.25354 | 0.10244
51 | 630.23991 | 0.07234 | 110 | 7099.88638 | 0.09946
52 | 664.73547 | 0.07376 | 111 | 7352.04756 | 0.09845
53 | 703.19504 | 0.07361 | 112 | 7596.21588 | 0.09558
54 | 741.68297 | 0.07765 | 113 | 7850.56626 | 0.09473
55 | 779.67713 | 0.07314 | 114 | 8104.09223 | 0.1263
56 | 822.55709 | 0.08055 | 115 | 8385.46575 | 0.09898
57 | 868.25771 | 0.07562 | 116 | 8665.08226 | 0.10278
58 | 910.25145 | 0.06809 | 117 | 8942.38358 | 0.10007
59 | 9530.94562 | 0.11103 | 118 | 9245.37519 | 0.10506

##  7 Terminology Table

Terminology
---
$\theta$ | Angle | $\tau_{s}$ | Resolved shear stress
$A$ | Area | $G_{m}$ | Shear modulus
$C_{0}$ | Bulk wave-speed | $\gamma$ | Shear strain
$b$ | Burger’s vector | $\tau$ | Shear stress
$\dot{H}$ | Change in hardness over time | $\Sigma^{*}$ | Spall strength
$\tau_{r}$ | Critical resolved shear stress | $\epsilon$ | Strain
$N_{1}$ | Density of grain boundary | $n$ | Strain hardening exponent
$N_{2}$ | Density of particle nucleation sites | $\dot{\epsilon}$ | Strain rate
$N$ | Density of Potential Nucleation sites | $B$ | Strain rate hardening modulus
$r_{0}$ | Dislocation core radius | $\tau_{ss}$ | Strengthening via distortion of lattice
$\rho$ | Dislocation density | $M_{S}$ | Taylor factors for random solutionized Z5
$\delta$ | Elastic-Plastic correction factor | $M_{P-A}$ | Taylor factors for weakly textured peak-aged Z5
$G$ | Formation energy | $T$ | Temperature
$d_{s}$ | Glide plane spacing of precipitates | $\beta_{0}$ | Thermal-softening coefficient
$l$ | Grain size | $U_{fs}$ | Velocity drop
$k$ | Hall-petch slope | $V$ | Volume
$H$ | Hardness | $\sigma_{Y}$ | Yield sterngth
$h$ | Indentation depth | $E$ | Young’s modulus
$\dot{h}$ | Indentation depth over time | Alloy Designations (All at.%)
$\dot{\epsilon}_{i}$ | Indentation strain rate |  |
$w$ | Interaction energy | Z$n$ |  $n$% Zn, rest Mg
$\lambda_{e}$ | Inter-particle spacing | Abbreviations
$P$ | Load |  |
$\dot{P}$ | Loading rate | ADF | Annular dark-field
$c$ | Nominal concentration | CSM | Continous Stiffness Measurement
$I$ | Nucleation rate | CRSS | Critical Resolved Shear Stress
$r_{p}$ | Planar radius | DOE | Diffractive Optical Element
$\nu$ | Poisson’s ratio | EFL | Effective Focal Length
$d_{t}$ | Precipitate diameter ($=2r_{p}$) | EBSD | Electron Backscatter Diffraction
$t_{t}$ | Precipitate thickness | GP zone | Guinier–Preston zone
$f$ | Precipitate volume Fraction | HCP | Hexagonal close-packed
$\mathcal{R}_{y}$ | Probability distribution function | LDMF | Laser Driven Micro-Flyer
$\sigma_{G}$ | Quasi-static strength | LIPIT | Laser Induced Particle Impact Test
$r$ | Radial distance | PDV | Photonic Doppler Velocimetry
$R$ | Radius | RMS | Root Mean Square
$\beta_{1}$ | Rate-sensitivity parameter | SEM | Scanning Electron Microscope
$\rho_{0}$ | Reference density | STEM | Scanning Transmission Electron Microscopy
$l_{0}$ | Reference grain size | TEM | Transmission Electron Microscopy
$d^{0}_{s}$ | Reference mean particle spacing | XRD | X-ray Diffraction
$\dot{\epsilon_{0}}$ | Reference strain rate |  |

##  8 Supplementary References

  1. 1.

Gordon R Johnson. A constitutive model and data for materials subjected to
large strains, high strain rates, and high temperatures. Proc. 7th Inf. Sympo.
Ballistics, pages 541–547, 1983.

  2. 2.

Hoon Huh and WJ Kang. Crash-worthiness assessment of thin-walled structures
with the high-strength steel sheet. International Journal of Vehicle Design,
30(1-2):1–21, 2002.

  3. 3.

G Rr Cowper and Paul Southworth Symonds. Strain-hardening and strain-rate
effects in the impact loading of cantilever beams. Technical report, Brown
Univ Providence Ri, 1957.

  *[$\dot{P}/P$]: ratio of loading rate to load
  *[$\dot{\varepsilon}_{i}$]: indentation strain rate
  *[$\dot{h}$]: velocity
  *[$h$]: indentation depth
  *[$\dot{P}$]: loading rate
  *[$P$]: load
  *[$\dot{H}$]: change in hardness over time
  *[$H$]: hardness
  *[$\dot{P}/P$s]: ratio of loading rate to load
  *[CSM]: continuous stiffness measurements
  *[RMS]: root means square
  *[$h_{c}/h$]: ratio of contact depth to depth
  *[$A_{c}$]: contact area