Metals strengthen with increasing temperature at extreme strain rates

The strength of materials depends on the rate at which they are tested, as defects, for example dislocations, that move in response to applied strains have intrinsic kinetic limitations1–4. As the deformation strain rate increases, more strengthening mechanisms become active and increase the strength4–7. However, the regime in which this transition happens has been difficult to access with traditional micromechanical strength measurements. Here, with microballistic impact testing at strain rates greater than 106 s−1, and without shock conflation, we show that the strength of copper increases by about 30% for a 157 °C increase in temperature, an effect also observed in pure titanium and gold. This effect is counterintuitive, as almost all materials soften when heated under normal conditions. This anomalous thermal strengthening across several pure metals is the result of a change in the controlling deformation mechanism from thermally activated strengthening to ballistic transport of dislocations, which experience drag through phonon interactions1,8–10. These results point to a pathway to better model and predict materials properties under various extreme strain rate conditions, from high-speed manufacturing operations11 to hypersonic transport12.


Metals strengthen with increasing temperature at extreme strain rates
The strength of materials depends on the rate at which they are tested, as defects, for example dislocations, that move in response to applied strains have intrinsic kinetic limitations [1][2][3][4] .As the deformation strain rate increases, more strengthening mechanisms become active and increase the strength [4][5][6][7] .However, the regime in which this transition happens has been difficult to access with traditional micromechanical strength measurements.Here, with microballistic impact testing at strain rates greater than 10 6 s −1 , and without shock conflation, we show that the strength of copper increases by about 30% for a 157 °C increase in temperature, an effect also observed in pure titanium and gold.This effect is counterintuitive, as almost all materials soften when heated under normal conditions.This anomalous thermal strengthening across several pure metals is the result of a change in the controlling deformation mechanism from thermally activated strengthening to ballistic transport of dislocations, which experience drag through phonon interactions 1,[8][9][10] .These results point to a pathway to better model and predict materials properties under various extreme strain rate conditions, from high-speed manufacturing operations 11 to hypersonic transport 12 .
Metals usually soften at high temperatures, because the motion of defects that control strength-dislocations-is thermally activated: temperature helps these carriers of plasticity move across the crystal lattice and bypass any obstacles there 6,10,13,14 .The softening of metals with temperature is extremely broadly observed, being the norm over more than ten orders of magnitude in strain rates, spanning 10 −6 -10 4 s −1 .However, as strain rates increase beyond 10 4 s −1 , more deformation mechanisms can become active; these contribute to a marked increase in strength through dislocation-drag-controlled plasticity 2,4,15,16 .Understanding material properties at such high strain rates is critical for designing and engineering new materials for use in extreme conditions.Deformation rates greater than 10 4 s −1 are commonly seen in hypervelocity meteorite impacts 12 , high-speed metal machining 11 , sandblasting and erosion 17,18 and in metal additive manufacturing processes such as cold spray 19 .The drag mechanisms that prevail under high rates are interesting because they are not rate limited in the same manner as the thermally activated mechanisms described above; they are instead expected to be a mode of ballistic dislocation transport.It has been anticipated that, in this extreme range, metals indeed may not soften at high temperatures 4,16,20,21 , a behaviour that is counterintuitive and at odds with decades of studies in less extreme conditions.
Unfortunately, experimental data on extreme strain-rate deformation mechanisms are rather rare, owing to limitations of measurement tools that can access extreme strain rates.Kolsky bars 22 , flyer plates 23 , shock impacts 24 , high-power laser pulses 25 , shockless isentropic compression experiments 26 and gas-gun-driven projectiles 27 are examples of methods that are generally conducted on macroscale samples (ranging from a few millimetres to a few hundred millimetres) and require high impact velocities (hundreds of metres per second) to induce these large strain rates.In general, the quantitative measurements provided by these methods end around a strain rate of about 10 4 s −1 .At higher rates, these approaches are more typically aimed at studying material spall 7 and convolute the mechanical response with strong shock effects in the material during impact.We are not aware of data that are able to separate out shock effects and show the posited loss of thermally activated softening at high strain rates.
Advances in optically driven microballistic systems over the past decade have made it possible to accelerate 1-50-µm-sized impactors to velocities up to and beyond 1,000 m s −1 , producing extreme strain rates from 10 6 to 10 9 s −1 , strain rates on the same order of magnitude as, for example, discrete dislocation and molecular dynamics simulations 28,29 .Notably, because of the small scale of the test, these rates are achieved without necessarily passing into the strong shock regime that characterizes many other rapid experimental test methods, permitting a clean assessment of material strength under extreme rates.There are now at least two different means of using such tests to extract quantitative measures of strength from an impacted substrate: dynamic yield strengths can be extracted from the rebound behaviour of the impactor over a range of velocities [30][31][32] and dynamic hardness values can be measured on the basis of post-impact measurements of the impact crater sizes 33 .The availability of these strength measures at extreme rates opens the door to exploration of the drag mechanisms that are expected to dominate there, but are quantitatively underexplored.

Quantitative measurements of strength and hardness
High-strain-rate micromechanical impact experiments were carried out using laser-induced particle impact tests, detailed in Methods.Spherical alumina microparticles, roughly 12.5 ± 1 µm in size, were chosen as impactors onto copper substrates owing to their much higher hardness (static hardness 15.7 versus 0.56 GPa for annealed Cu) 34,35 , to isolate the plasticity to the substrate.Figure 1 shows some typical results, in which we track by high-speed camera the inbound and rebound trajectories of alumina particles as they impact the copper substrate.The three examples in Fig. 1 are chosen for the close similarity of the conditions: these three nominally identical particles are launched at almost the same velocity and thus follow the same inbound trajectories very closely.However, these three experiments were conducted at 20, 100 and 177 °C and, as the test temperature is increased, the particle rebounds more aggressively, as shown by the increasing rebound trajectories, as well as the differences in height in the inset black and white photographs of the particles 600 ns after impact.These results indicate a higher strength of copper at higher temperatures, as more kinetic energy is carried away in the rebound; this is a first signature of anomalous temperature dependence of strength at high rates.
The raw trajectory data in Fig. 1 are corroborated by an independent measurement of local plastic deformation.The upper insets in Fig. 1 show impact crater profile measurements using 3D laser scanning confocal microscopy for the same three impacts, in profile (top row) and in plan-view projection (middle row); the crater diameter and depth decrease with test temperature at the same impact velocity.Quantitatively, the crater volume decreases by about a factor of two between room temperature and 177 °C.Because crater volume is inversely proportional to strength 36 , the crater images in Fig. 1 independently confirm that copper is hardening at higher temperatures under these extreme impact conditions.
The two independent signatures of thermal hardening in Fig. 1the increased rebound and the decreased crater volume-are both amenable to quantitative analysis to extract strength measures and, in what follows, we take each of these in turn.We first look at a larger set of rebound data for many more impact experiments.The rebound response is quantified using the coefficient of restitution, CoR = v r /v i , the ratio of the rebound and impact velocities.Figure 2a plots the CoR as a function of the impact velocity in double-logarithmic format.For a given test temperature, higher impact velocities lead to lesser rebounds, as more energy is dissipated in plasticity of the substrate.In fact, the parabolic scaling in Fig. 2a is the expected form for plasticity, as developed by Wu et al. 30,31 , who showed that ideally plastic impacts follow the power-law relation: in which the coefficient α = 0.78 for an undeforming impactor on a deforming substrate, Y d is the dynamic yield strength of the substrate, E* is an effective elastic modulus of the materials involved and v y is a characteristic velocity at which plastic deformation is initiated (see Supplementary Information Section 1 for more details).Not only do the data in Fig. 2a conform to the scaling of equation ( 1) but the equation is fully predictive with the fitting of a single parameter, namely the dynamic yield strength Y d , which is an average strength measure describing the complex deformation of the substrate over the entire duration of an impact.At room temperature, fitting the CoR data in Fig. 2a yields a dynamic strength of 280 MPa for copper at the average strain rate of 10 7 s −1 for these tests.For comparison, under quasistatic conditions, annealed copper would have a typical strength on the order of 70 MPa (ref.37) and under extreme conditions involving shock (such as Richtmyer-Meshkov instability tests 38 ), strengths up to 1,000 MPa have been inferred.The present value between these limits represents substantial rate-hardening, but no superimposed effect of shock or high pressure.One advantage of this assessment of strength is that it averages out local heterogeneities in the microstructure (including grain boundaries, crystal orientation effects and defect structures) over the entire range of impact velocities; these effects contribute to the scatter in the data and provide a quantitative measurement uncertainty as a standard deviation of the data's departure from the trendline in Fig. 2a.
The more interesting trend in Fig. 2a is the effect of the test temperature on the CoR curves and the dynamic yield strengths they encapsulate.As seen previously in Fig. 1  One limitation of the CoR analysis above is that it averages the dynamic strength over a range of velocities because it requires the fitting of many data points together.A second strength measure, the dynamic hardness, permits a test-by-test analysis of the trends shown above.Dynamic hardness measurements also require the quantitative measurement of inbound and outbound velocities to assess the amount of energy deposited into the substrate and expended as plastic deformation, normalized by the crater volume, V (ref.33):  The impact and rebound trajectories of three alumina particles impacting a copper substrate at 20 °C (blue), 100 °C (orange) and 177 °C (red).As the temperature is increased, the rebound velocity of the impacting particle increases, as shown by both the plotted rebound trajectories as well as the grey insets of the particle 600 ns after rebound.Insets of plan and line-scan views of each impact crater, measured with 3D laser scanning microscopy, show that, as the temperature is increased, the impact crater depth and width decrease.Scale bar, 5 µm. in which m p is the mass of the particle.The crater volume can be accurately assessed with scanning confocal microscopy as in Fig. 1 and, combined with the associated velocity measurements, gives the dynamic hardness of copper for each individual experiment.The results of this analysis are plotted in Fig. 2b for the range of strain rates tested, with the net strain rate of a given experiment assigned as the ratio of the impact velocity to the particle diameter, v i /d.Clearly, beyond the intrinsic scatter of the experiments, these data confirm all of the observations above: at strain rates greater than about 10 6 s −1 , the hardness of copper increases with increasing test temperature.
The above results represent the first experimental data to show anomalous temperature dependence at strain rates greater than 10 6 s −1 , in experiments that are both quantitative and free of strong shock effects.This result is also not limited to pure copper; we have observed the same trends in pure Au and Ti as well, as shown in Extended Data Figs. 1 and 2. For all three metals, the two independent strength measures (dynamic yield strength and dynamic hardness) reflect a change in deformation mechanism owing to the synergy between strain rate and temperature; at extreme strain rates, all three metals harden when heated.

Deformation mechanisms
The flow strength in pure metals can be broken down into three additive components: thermal, athermal and drag strengthening 5,39,40 , each of which is plotted separately in Fig. 3 at our experimental strain rate of 10 7 s −1 : • The thermal strength component comes from dislocation interactions with short-range barriers (Peierls-Nabarro barriers, vacancies) that can be fully overcome by thermal fluctuations.This strength can be calculated as a function of temperature and strain rate following refs.8,41,42 as described in Supplementary Information Section 2.1 and follows a conventional thermally activated softening-with-temperature trend (red line in Fig. 3).• The athermal strength component arises from other barriers to dislocation motion that are not able to be overcome by thermal fluctuations alone, such as dislocation-grain boundary and dislocation-forest interactions 8 .As shown in Supplementary Information Section 2.2, the total athermal strength component of copper can be calculated with standard models for our strain rate of 10 7 s −1 , giving the blue line in Fig. 3, again a softening-with-temperature trend.The softening in this case is because of the lowering of the elastic modulus with temperature, which-in turn-lowers the energy of dislocationobstacle interactions.• The final strength component, drag strengthening, comes from the interaction of dislocations with phonons in the lattice 43 .This interaction gives rise to anomalous thermal strengthening, because higher temperatures increase the energy levels of phonons and thus increase the interaction of dislocations with them.Supplementary Information Section 2.3 presents the standard model for phonon drag, which, when evaluated with inputs for copper at our strain rate of 10 7 s −1 , gives the green line in Fig. 3. Fig. 2 | Copper dynamic strength and hardness.a, The impact velocity versus CoR, v r /v i , is plotted on a double-logarithmic scale for a range of impact velocities.Across the range of impact velocities, the impacts at higher temperatures led to a faster rebound.The solid lines represent the scaling law for ideally plastic impacts with the fitting parameter, Y d , increasing with temperature.Uncertainty in measured velocities is ±2%, which is on the order of the size of the data markers.b, Dynamic hardness measurements from each impact are plotted against deformation strain rate.The shaded regions serve as a guide for the eye to show where most of the hardness values fall for each temperature.As with strength, the hardness trends upward with increasing temperature and measurement uncertainties are on the order of the size of the data markers.The contribution of each strength term for pure copper, calculated for a range of temperatures at a fixed strain rate of 10 7 s −1 .The magnitude of the increase in dislocation drag strengthening is greater than the magnitude of the decrease in all other strength terms, leading to a net increase in the total strength with temperature at this extreme strain rate.The experimental data points, plotted in red, are in good agreement with the total strength resulting from the summation of each strength component.Error bars on the experimental data points are the standard deviation of the strength measurements.

Article
Although the first two contributions exhibit classical thermal softening, the third term has a large influence under high-rate conditions, which explains why the dynamic strength and hardness increase at our experimental strain rates.In fact, the increase predicted by the summation of the three terms are reasonably aligned in both absolute magnitude and the scale of the change with temperature, as shown by the black line in Fig. 3.
The unusual hotter-is-stronger trend in the present data are strictly an extreme-rate effect and, in fact, the experimental data in Fig. 2b allude to a crossover point at which the temperature dependence of strength inflects; at low enough rates, the data suggest a transition to classical thermally activated behaviour.This is elaborated more clearly in Fig. 4, in which we reproduce data from Fig. 2b, and now include literature data for copper of similar purity 20 , tested at the closest lower strain rates at which data are available (approximately 2 × 10 4 s −1 ).The data at those lower rates exhibit the opposite temperature dependence from the present data.A clear transition regime exists between strain rates at which those literature data end and our data begin, as seen in the crossover of the blue and red dotted schematic trendlines.The highlighted yellow band shows the range of strain rates at which the crossover must occur.The strength equations described above can be used to evaluate the putative position of this crossover as well, by assessing the apparent activation energy for plasticity, Q app : T ε app in which R is the gas constant.A theoretical value for the apparent activation energy for deformation can be determined from the strength equations in Supplementary Information Section 2 through the use of equation ( 3) and signals the tendency of the temperature dependence: positive values of Q app suggest thermal softening owing to thermal hopping of dislocations, whereas negative values are required to capture thermal hardening, as in our experiments, by virtue of a transition into a ballistic-like transport regime of dislocation motion 1,44 (more details are provided in Supplementary Information Section 3).The theoretical prediction for Q app is plotted for copper at a constant temperature of 20 °C on the secondary y axis in Fig. 4. Indeed, the model suggests a critical strain rate near 10 5 s −1 , at which Q app changes sign from positive to negative, signalling the change in controlling mechanism from thermally activated softening to dislocation drag strengthening in the ballistic regime.This result provides interesting new insight on previous work such as that in refs.45,46, in which thermal softening was observed in copper even at rates as high as about 7 × 10 4 s −1 ; the present results clarify that only at even higher rates above 10 5 s −1 is the crossover expected.

Implications and conclusions
Although models such as those used here have been in the literature for many years, microballistic testing provides a new opportunity to quantitatively measure the strength and hardness of materials at extreme strain rates, without complicating factors caused by strong shock physics.This not only provides a chance to test those models, as done here, but potentially to rapidly calibrate them for new materials and situations.The present data for three pure metals provides a starting point for focused study on those difficult-to-access mechanisms.The result that pure metals show marked thermal hardening at high strain rates also alludes to new strategies for materials design for extreme conditions; extrapolating conventional strength measurements to extreme conditions may lead not only to incorrect strength expectations but even incorrect directional dependencies of those strengths.For example, although pure copper is a soft metal at low strain rates and would normally be expected to soften at high temperatures, the strength of copper measured here at 10 7 s −1 and 177 °C is more than 300 MPa; this is comparable with the conventional strength of 304 steel at this temperature 43 .The present results thus bolster the call for new modes of thinking about materials optimization for extreme conditions 47,48 .

Online content
Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41586-024-07420-1.The apparent activation energy, Q app , for plasticity (solid black line) of copper is also plotted as a function of strain rate at a constant temperature of 20 °C based on the model in the Supplementary Information.At rates below roughly 10 5 s −1 , Q app changes from positive, indicating thermally activated dislocation motion, to negative, one of the necessary conditions for phonon drag to be substantial.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Methods
Spherical alumina powder particles, with nominal particle size of approximately 12.5 ± 1 µm, were purchased from Inframat Advanced Materials and used for microparticle impact experiments on copper and titanium, and nominal particle size of about 20 ± 1 µm for impact experiments on gold.Oxygen-free high-conductivity copper plates (OnlineMetals) and pure Ti rods (Beantown Chemical) were cut using a diamond saw and annealed at 600 °C for 10 h.Pure gold rods (Beantown Chemical) were cold rolled to an 85% reduction in thickness before being annealed at 125 °C for 3 h.The plates and rods were then polished using standard metallographic techniques to a 0.04-µm mirror finish.
We performed microballistic impacts using a laser-induced particle impact test, detailed elsewhere 28 .Launching pads for use at elevated temperatures consisted of a 210-µm glass slide with a 90-nm chromium layer sputter coated on top, a UV-curable adhesive and a second 210-µm glass slide, a construction specifically designed to attain high test temperatures without any materials degradation 28 .A monolayer of alumina particles was finely dispersed on top of the second glass slide using a drop of ethanol.A single particle was selected for impact based on predetermined parameters: size, shape and surface morphology.A laser pulse (Nd:YAG, 532 nm wavelength) was focused directly behind the particle, resulting in the ablation of the chromium layer and rapid acceleration of the particle towards the copper substrate placed 750 µm away.The velocity of the particle was predetermined by changing the laser pulse energy between 1 and 80 mJ.The velocities targeted in this study (below 250 m s −1 for copper and titanium and below 150 m s −1 for gold) are all slightly low compared with conventional high-rate testing; this is far below the point at which instabilities associated with hydrodynamic jetting/spall set on (approximately 650 m s −1 for the present geometry in ref. 32) and even farther from the regime of hydrodynamic penetration (about 900 m s −1 from ref. 49).Every impact falls within a regime in which no fracture-like events occur that could lead to fragmentation and ejection of the substrate material.The onset of such events would be identifiable by an excess loss of rebound velocity, seen as deviations from the power law in Fig. 2a (ref.32), as well as post-mortem analysis of the craters 50 ; nothing seen in our data suggests any material fragmentation or ejection, as expected for these conditions.Furthermore, all of the impact velocities for each material are also well above the elastic-plastic transition velocity (v y from equation (S2) in the Supplementary Information).Plastic deformation sets on at velocities below 1 m s −1 for each material in this study, meaning that each impact in this work leaves a plastic crater in the substrate.Our procedure includes particle-wise measurements of impact velocity and particle diameter, thus providing a net strain rate for every individual experiment as v i /d (ref.33); this quantity can be regarded as an average over the plastically deformed region over the duration of the impact event.
The flight, impact and rebound of the particle were captured using a high-speed camera (Specialised Imaging) able to capture 16 images with as small as 5 ns exposure times, with the event being illuminated by a second laser pulse (640 nm wavelength, 10 µs duration).The impact and rebound velocities of the particle were measured between frames.To conduct impacts at elevated temperatures, a custom resistive heating stage was fabricated using a machinable ceramic (Macor, McMaster-Carr) and nichrome wire (McMaster-Carr).Three thermocouples were placed around the substrate to ensure that a homogenous temperature distribution was reached.Before launch, the particles are heated by virtue of their proximity with the target substrate, with an offset distance of only 750 µm; the system was thermally equilibrated before launch, so particle, vapour and substrate are all at the same nominal temperature.Post-mortem analysis of the impact sites was conducted using 3D laser scanning confocal microscopy (VK-X200 Keyence) to measure the volume of the impact craters.Hardness measurements use this direct measurement of crater volume in equation (2).
Extended Data Fig. 1 | Gold dynamic strength and hardness.a, The impact velocity versus CoR is plotted on a double-logarithmic scale for alumina impacts on gold substrates at 20 °C (blue) and 97 °C (red).Using scaling for ideally plastic impacts, the dynamic yield strength for each impact temperature is calculated to be Y d,20°C = 297 MPa and Y d,97°C = 311 MPa.Uncertainty in measured velocities is ±2%, which is on the order of the size of the data markers.b, The dynamic hardness of titanium is plotted for both 20 °C (blue) and 97 °C (red).For impact at similar strain rates, the tests at 97 °C show a higher hardness, and measurement uncertainties are on the order of the size of the data markers.Uncertainty in measured velocities is ±2%, which is on the order of the size of the data markers.b, H d of titanium is plotted for both 20 °C (blue) and 200 °C (red).For impact at similar strain rates, the tests at 200 °C show a higher hardness, and measurement uncertainties are on the order of the size of the data markers.

1 Received
https://doi.org/10.1038/s41586-024-07420-Scienceand Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA.2Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA.✉ e-mail: schuh@northwestern.edu for several example cases, we now see very broadly that temperature promotes greater rebounds in copper, over the full range of tested velocities.The fitted solid lines reflect an increasing dynamic strength: Y d,20°C = 280 MPa, Y d,100°C = 310 MPa and Y d,177°C = 364 MPa.With increasing temperature, the dynamic yield strength of copper increases by roughly 30% over a range of about 150 °C.

Fig. 1 |
Fig.1| Impact trajectories and impact craters.The impact and rebound trajectories of three alumina particles impacting a copper substrate at 20 °C (blue), 100 °C (orange) and 177 °C (red).As the temperature is increased, the rebound velocity of the impacting particle increases, as shown by both the plotted rebound trajectories as well as the grey insets of the particle 600 ns after rebound.Insets of plan and line-scan views of each impact crater, measured with 3D laser scanning microscopy, show that, as the temperature is increased, the impact crater depth and width decrease.Scale bar, 5 µm.

Fig. 3 |
Fig.3| Contributions of each strengthening mechanism.The contribution of each strength term for pure copper, calculated for a range of temperatures at a fixed strain rate of 10 7 s −1 .The magnitude of the increase in dislocation drag strengthening is greater than the magnitude of the decrease in all other strength terms, leading to a net increase in the total strength with temperature at this extreme strain rate.The experimental data points, plotted in red, are in good agreement with the total strength resulting from the summation of each strength component.Error bars on the experimental data points are the standard deviation of the strength measurements.

Fig. 4 |
Fig. 4 | Apparent activation energy for plasticity.The blue and red dashed lines serve as guides for the eye to connect hardness values, converted from strength values from ref. 20 (open circles), at low rates and dynamic hardness values from this work (filled circles).The apparent activation energy, Q app , for plasticity (solid black line) of copper is also plotted as a function of strain rate at a constant temperature of 20 °C based on the model in the Supplementary Information.At rates below roughly 10 5 s −1 , Q app changes from positive, indicating thermally activated dislocation motion, to negative, one of the necessary conditions for phonon drag to be substantial.

Extended Data Fig. 2 |
Titanium dynamic strength and hardness.a, The impact velocity versus CoR is plotted on a double-logarithmic scale for alumina impacts on titanium substrates at 20 °C (blue) and 200 °C (red).Y d for each impact temperature is calculated to be Y d,20°C = 776 MPa and Y d,200°C = 890 MPa.