seeing the unseen uncover the bulk heterogeneous deformation processes in metallic glasses through surface temperature decoding

Materials Today  Volume 20, Number  January/February 2017 RESEARCH RESEARCH: Original Research Seeing the unseen: uncover the bulk heterogeneous deformation processes in metallic glasses through surface temperature decoding Yang Song1,4, Xie Xie2,4, Jiajia Luo1, Peter K Liaw2,*, Hairong Qi1,* and Yanfei Gao2,3,* Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, TN 37831, USA Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Deformation processes in various materials are inhomogeneous in space and jerky in time, with the shear banding in bulk metallic glasses (BMGs) as a quintessential example, but there is a lack of in situ, nondestructive observations of such processes on the appropriate spatio-temporal scales This work solves this long-lasting difficulty by the integration of in situ infrared (IR) measurements and innovative signal processing algorithms A spatio-temporal unmixing method is developed to identify the discrete surface ‘hot-spots’ that are responsible for the initiation and propagation of macroscopic shear bands during the serrated flow The use of a thermal-electric analogy further identifies depths of these hotspots, whose magnitudes and locations evolve as the successive shearing process repeats on the major shear band Seeing the previously ‘unseen’ localized heat sources and their 3D evolution patterns, both in situ and inside the bulk, reveals for the first time how the coupled structural/thermal softening mechanisms govern the heterogeneous deformation processes in BMGs Introduction Unlike crystalline materials, atoms in bulk metallic glasses (BMGs) pack in a glassy and amorphous state, and thus there are no grains, dislocations, or other crystalline defects that are responsible for plastic deformation as in typical metals and alloys In contrast, BMGs deform essentially through the formation and propagation of shear bands, which are localized plastic deformations concentrated in narrow bands (about 10–50 nm) [1–9] From the atomistic point of view, shear bands originate from local fluctuations of atomic structures, such as shear transformation zones, which eventually coalesce and form microscopic bands with shear strain localization Our understanding of the above processes, however, is mostly limited to atomistic simulations since in situ microscopy investigations are limited in both temporal and spatial resolutions On the other hand, one can redistribute the applied macroscopic strain field to one or multiple *Corresponding author: Liaw, P.K (pliaw@utk.edu), Qi, H (hqi@utk.edu), Gao, Y (ygao7@utk.edu) These two authors contributed equally to this work shear bands If designed appropriately, the shear band arrangements may reduce the strain in individual shear bands and thus prevent the catastrophic failure The above two asymptotic viewpoints on the atomistic and macroscopic length scales clearly require a knowledge or a microscopic observation window, with which we can develop a structure-property relationship that controls the initiation and evolution of shear bands, as well as their evolution into failure The primary challenge in the BMG study is the lack of an in situ, nondestructive experimental investigation of the shear banding dynamics on the commensurate spatial and temporal scales The shear banding process has been found to be jerky in time and inhomogeneous in space, which poses great challenges in the shear band characterizations as summarized in Fig Inhomogeneity in space indicates the localization of the strain field into narrow bands; the nanometer nature of these shear bands makes it difficult to characterize Serrations on the stress–strain curves (an example will be given in Fig 2) indicate the jerky nature in time [2–4]; a slow acquisition rate will miss detailed correspondence between temporal and spatial information Studies 1369-7021/ß 2016 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) http://dx.doi.org/10.1016/ j.mattod.2016.12.002 RESEARCH Materials Today  Volume 20, Number  January/February 2017 RESEARCH: Original Research FIGURE Schematic illustration of the spatio-temporal visibility of shear bands in BMGs The unmixing method in this paper provides a critical link on microscopic length scales, with an additional dimension in depth by the thermal-electric analogy FIGURE using strain gauges [2], high-speed cameras [3], or digital image correlations (DICs) [4] found that as the macroscopic deformation transitions from the elastic to plastic stages, the strain fields fluctuate in space, and these variations evolve in time until a small number of major shear bands emerge Synchrotron X-ray diffraction studies [5] and surface-based measurements (e.g., the damping characteristics using atomic force microscopy) [6,7] suggest the important roles played by the atomistic structural heterogeneities on the early stage of deformation inhomogeneity Nanoindentation tests on BMGs often show sudden displacement excursions on the load-displacement curves, denoted as pop-ins [8,9] The corresponding pop-in stress can be used as a structural probe to characterize the structural heterogeneity on the microscopic scale For example, Li et al [9] suggested that a seemingly amorphous structure actually consists of a perfect glassy state and weak/soft zones which serve as the shear band nucleation sites Although these characterization methods are either surface based or indirect, their findings suggest that the precursor of the macroscopically observed shear bands is a complicated evolution of strain fields associated with microscopic structural heterogeneities Referring back to Fig 1, shear-band arrangements can be determined using electron microscopy, but these measurements are usually ex situ and post-mortem and thus leave a large gap in the temporal resolution The history of shear-band arrangements has been measured by the in situ transmission electron microscopy experiment (limited to ultra-small samples), DIC method (limited to surface measurements), and the infrared (IR) thermography technique For the last technique, highly concentrated plastic deformation within the shear bands in BMGs generates a great amount of heat and makes the shear bands hot and visible on the thermograph [10–15] The thermography technique requires no contact with the specimen, and can reach a thermal resolution up 10 Stick-slip or serration behavior on the stress–time plot The compression test was performed on a Zr52.5Cu17.9Ni14.6Al10.0Ti5.0 (at.%) BMG cylindrical sample with f2 mm  mm The corresponding thermal evolutions (which are fontal views of the cylindrical samples as measured by an IR camera at the acquisition rate of 300 Hz) in two representative serrations are illustrated: (a–e) correspond to the early stage where multiple small shear bands are emerging, and (f–j) are at the later stage when one major shear band dominates The temperature scale (8C) is indicated by the color legend to 10À3 8C and a micrometer spatial resolution We have previously reported in situ thermography observations of the Zr52.5Cu17.9Ni14.6Al10.0Ti5.0 (Vitreloy-105, in atomic percent, at%) BMG specimens under tensile and compressive loading with various strain rates, typically showing a hot band with the sub-millimeter width and temperature rise of about 0.25 8C [12,15] In spite of the advantages of IR imaging, it has its own unique challenges and drawbacks First, the infrared camera gives limited spatial and temporal resolutions (Fig 1) The font plane of this three-dimensional plot shows the temporal scale and the gauge length of the surface view The shear band propagating velocity can be as high as 4200 mm/s [3], and the shear band width is in the range of 10–50 nm The existing commercial IR cameras can only observe the hot band after the heat generated in the shear band is conducted to its nearby area Due to these hardware limitations in the IR image acquisition, it remains challenging to extract the hidden information about shear bands From the image processing viewpoint, the surface temperature field can be considered as a ‘mixture’ caused by many heat sources inside the BMG with each one having its own unique temperature evolution pattern (or called footprint) From a spatio-temporal unmixing algorithm, the evolution process (in the temporal domain) and location (in the spatial domain) of each individual heat source can be extracted and identified Second, IR images can only capture the surface RESEARCH temperature on the material In theory, given the surface temperature fields, we can obtain the heat pattern of various internal elements by solving the inverse heat transfer problem, but finding such a solution is a formidable task Here we develop a new methodology from the thermal-electric analogy, which would allow us to identify the depths (i.e., the corresponding 3D locations) of the localized heat sources without having to solve the inverse problem In Fig 1, this line of study means that the scope of the unmixing method on the surface can be extended to the depth dimension, as shown by the gauge length in the depth direction This provides the key information on the true nature of the heterogeneous process, both spatially and temporally, associated with the serrated flow of BMGs in Fig Unmixing the spatio-temporal evolution of surface temperature fields A compression test on the Vitreloy-105 BMG shows the serrated flow on the stress–time plot in Fig Two sets of representative IR images are also presented, one corresponding to a small serration near the yield point in the early stage of the deformation, and the other to a large serration in the later stage See Methods for experimental details Strain fluctuations appear when the applied stress is below or near the macroscopic yield stress, as indicated by many small ‘immature’ shear bands in Serration Strain fields eventually evolve into one or a small number of major shear bands Also in the later stage of the deformation, most serrations (but not all) correspond to the appearance and disappearance of one hot band at the same location, as indicated by the example in Serration 21 This trend suggests repeated shearing processes in a macroscopic shear band More importantly, the surface temperature in this major hot band is not uniform, and such a nonuniform distribution varies in each serration These observations suggest that the strain and strain-rate fields are not uniform even in a single shear band, likely due to the heterogeneities on this shear plane in the depth dimension Consequently, a relationship among temperature rises, serrations on the stress–strain curves, and shear banding dynamics will not be feasible with only the surface temperature field mapping when the bulk information is missing The surface temperature field obtained from IR images arises from a number of heat sources Such ‘mixed’ measurements are frequently encountered in real world applications Due to the resolution issue associated with discrete sampling and the effect of unknown sources, the broad existence of mixed measurements has brought the decomposition (or unmixing) technique into a variety of applications [16,17] A general problem formulation is given by a linear mixing model (LMM): X ẳ AS ỵ B; subject to 1T s ¼ 1; s ! 0; (1) where X is the N  M observation matrix (or mixture) with each column recording one observation vector, A is an N  c source matrix with each column representing a pure source signature, c represents the number of sources, S is the c  M abundance matrix showing the amount of contributions or proportions of each source signature in forming the mixture (i.e., X), s is an arbitrary column vector of S, and 1T is a  c vector of ones The last term, B, takes possible errors and noises into account The constraints given in Eq (1) ensure that elements in each column of the abundance matrix S should sum to one and be no less than zero Due to these two constraints, the mixture becomes a convex combination of source signatures; that is, all the mixtures are within a (c À 1)-simplex, whose c vertices correspond to the source signatures [18,19] We can take advantage of the linear mixing model in Eq (1) to unmix the surface temperature field and construct the observation matrix, X, by stacking a sequence of IR images taken at different times In this way, we can obtain the temperature-variation curves over time (i.e., the temporal information) of all pixels across the entire image plane (i.e., the spatial information) That is, in Eq (1), X RNÂM (N is the number of IR frames, and M is the number of pixels of each frame) denotes the observation matrix with each column recording the surface-temperature change profile of the corresponding pixel over time, and A RNÂc denotes the source matrix with each column representing the temperature-change profile of each heat source A total of 3238 IR frames were obtained for the stress–time curve in Fig 2, which consists of 27 serrations If using the GDME unmixing method on all these IR images (i.e., using these images as X), the identified source matrix A will miss some heat sources that are short-lived and only elapse in the early stage or in the small tiny serrations (e.g., Serrations and 11) A Gradient Descent Maximum Entropy (GDME) method is employed to determine the abundance matrix for these 27 serrations (see Methods for details) The unmixing results are given by the abundance maps in Fig In these abundance maps, each pixel value indicates the fraction (ranging from to 1, as shown by the color scale) of the corresponding source signature in forming the mixture at the same pixel location as the IR image They are not temperature or heat content fields In the first seven serrations, there are many places where the heat sources are located, such as the lower right corner and the upper left part That is, in the early stage of plastic deformation, the fluctuations of the strain fields and, correspondingly, the temperature fields have small amplitudes and emerge everywhere on the surface Note that this is a cylindrical compression test, so that shear bands and their planes can be seen with various inclination angles These fluctuations gradually evolve into one or a few major shear bands which have significantly larger plastic strains and, correspondingly, much higher temperature fields After the 6th serration, a primary shear band begins to extend across the entire specimen, which later is responsible for most serrations afterwards However, at the 14th serration, another shear band shows up, and this secondary shear band is slightly below the primary one identified in the 6th serration These two shear bands are so adjacent that a visual inspection of the surface IR images will fail to separate their evolutions In these results, the improved resolution reaches mm in the spatial domain and 0.1 ms in the temporal domain, both of which are about one order-of-magnitude higher than the IR instrument resolution Surprisingly, even for the major and the secondary shear bands that run across the entire cylinder, the identified heat sources from the unmixing method are discrete For example, a total of six localized heat sources are found for Serration in the Supplementary Information (Fig S2) In this regard, the unmixing method enables the study of shear band evolution to go beyond what are immediately ‘seen’ from the IR surface temperature map and reveal the ‘internal’ heat sources, which would otherwise be ‘unseen’ from the surface temperature map 11 RESEARCH: Original Research Materials Today  Volume 20, Number  January/February 2017 RESEARCH Materials Today  Volume 20, Number  January/February 2017 RESEARCH: Original Research FIGURE Abundance maps by employing the spatio-temporal unmixing algorithm in individual serrations Multiple incipient shear bands can be observed in the early stage, while one major shear band dominates in the later stage The abundance field is dimensionless and varies from to The size corresponds to that of the sample, that is, diameter of mm and height of mm Bulk knowledge of the localized heat sources on one major shear band Even at the stage of deformation that is governed by one or a few shear bands, the strain field on the seemingly uniformly-shearing plane might involve further spatial localizations, indicating the existence of generation and evolution processes (i.e., the number of heat sources, their locations on the shear plane, and the heat content of each heat source vary with time) in one macroscopic shear band Results in Fig have unraveled the hidden information of the surface temperature map, that is, any shear band is illustrated with a small number of localized heat sources The depth information of these localized heat sources can be derived from a thermal-electric analogy, which does not require a direct solution to the inverse heat transfer problem but reveals the relationship between the surface temperature and internal heat sources See Methods and Supplementary Information for details The material parameters for Vitreloy-105 include thermal diffusivity a, material density r, and specific heat, CP, being  10À6 m2 sÀ1, 6730 kg mÀ3 and 330 J kgÀ1 KÀ1, respectively As previously noted in Fig 3, even in the late stage of deformation, occasionally some serrations result from the shearing process on secondary shear bands Therefore, only serrations 12 corresponding to the major shear-band plane will be analyzed (Fig S4, Supplementary Information) Results in Fig present the 3D construction of the heat sources on the inclined plane of the major shear band in three successive serrations The direct results from the thermal-electrical analogy give the heat contents of these heat sources (see Table S1 and Fig S4), which are converted to the temperature fields whose magnitudes clearly rely on the pixel size and the acquisition rate That is, the temperature rise is about 100 8C for our study (i.e., 300 Hz) as opposed to the estimate of thousands of degrees in [20] which used a much smaller time scale We also note that our analysis starts from the temperature field mapping on the front surface, thus leading to heat sources mostly located in the left half of the inclined ellipses in Figs and S4 The most striking finding is that the total number, locations, and heat contents of these heat sources differ in successive serrations For example, the numbers of heat sources in the 19th, 20th, and 21st serrations were determined as 6, 6, and 4, respectively As shown by the stress–time plot in Fig 2, the heat generation in one serration is instantaneous and corresponds to the sudden load drop What these localized heat sources represent when successive serrations take place by repeatedly shearing on an individual shear RESEARCH RESEARCH: Original Research Materials Today  Volume 20, Number  January/February 2017 FIGURE Using the thermal-electric analogy, the spatio-temporal evolution of the internal localized heat sources on the major shear band is identified The left figure shows the perspective view on how to determine the depth information of the heat source from the surface temperature field Images on the right show these heat sources on the major shear band with the viewing direction given in the left figure This approach clearly proves that the shearing process in a macroscopic shear band is actually realized by the heterogeneous processes that operate on the 1–100 mm scale band/plane? The heat generation rate is proportional to the plastic p p work rate, given by s ij e_ ij , with stress s ij and plastic strain rate e_ ij The thermal transport equation becomes p rCP T_ ẳ kr2 T ỵ aTQ s ij e_ ij ; (2) where k is the thermal conductivity a ẳ k=rCP ị, and aTQ is the TaylorQuinney coefficient that describes the fraction of the plastic work that is converted to heat (e.g., 70%) Because plastic deformation leads to local heating and temperature increase, the appearance of localized heat sources is a consequence of an inhop mogeneous plastic strain rate field, e_ ij , on the shear plane Note this is a 2D inhomogeneous field on the major shear-band plane (i.e., the ellipses in Figs and S4) that differs from the 3D inhomogeneous deformation field in the early stage of deformation as shown in Fig Upon ‘seeing the unseen’ localized heat sources inside the shear band, what governs the above inhomogeneity and the spacing between these localized heat sources? Before answering this question, it should be reminded that from the continuum mechanics point of view, strain localization is a type of instability arising from the loss of the ellipticity in the constitutive law – one example being strain softening Regardless of different views on the atomistic mechanisms, the shear banding process can be viewed as a stress-driven structural-change process, which leads to viscosity decrease, strain softening, and thus subsequent strain localization Various models differ on the exact processes of stress-driven atomistic structural evolution, such as the free volume and shear transformation zone models, all of which are nevertheless thermally activated and depend on temperature and loading rate Thus a deformation mechanism map can be constructed in this way to delineate the homogeneous versus heterogeneous deformation modes However, none of these models provides a length scale and thus cannot explain the 1–100 mm spacing among the localized heat sources that we have discovered in Figs and S4 Consider a characteristic timescale, t0, on which the shear banding process operates, and we naturally obtain a heat conducpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion length, LT ¼ kt =rCP from Eq (2) The relevance of this length scale to the spacing of localized heat sources in Fig can be explained as follows In addition to the atomistic mechanisms, the thermal softening can be introduced by @h=@T < 0, where h is the viscosity In the incipient stage of the shear band, the strain field tends to localize, so that Eq (2) leads to an instantaneous temperature rise that results into a viscosity drop, which further promotes the strain localization in a positive feedback manner In other words, the relationship, p @_eij @T ¼À p e_ ij @h > 0; h @T (3) naturally provides the necessary condition for the thermal softening and thermal transport to couple with the strain localization process [14] A linear stability analysis thus must be conducted to examine the conditions for the unstable growth of the coupled temperature and plastic strain rate fields From the thermal transport equation in Eq (2), regardless of the atomistic mechanisms that trigger the strain softening and incipient shear band, shortwavelength fluctuations in strain, strain rate, and temperature fields will be stable; that is, their amplitudes will decay because of the rapid heat conduction when the perturbation wavelength 13 RESEARCH RESEARCH: Original Research falls below LT [14] Perturbations with long wavelengths will be unstable, and the perturbation of plastic strain field will amplify, as the perturbation of temperature amplifies, eventually leading to the localized heat sources Since t0 varies within 10À7–10À5 s, the conduction length LT is about 1–10 mm Consequently, the ‘seen’ localized heat sources and strain rate heterogeneity on the shear plane results from a thermal-softening-induced instability, and the spacing among these localized heat sources correspond to the wavelength of the fastest-growing perturbation that is characterized by LT With the spacing among localized heat sources explained above, we now address another critical question: can the locations of these heat sources tell us anything about structural heterogeneity? The incipient shear bands are indeed prompted by structural heterogeneities However, our unmixing method (still having limited resolutions of mm and 0.1 ms) is unable to provide such information including the characteristic lengths associated with the atomistic structural heterogeneities, as well as their growth spectra as the strain localization proceeds However, in the sequence of images in Figs and S4, one can readily find that the locations of heat sources are changing as deformation proceeds on the major shear-band plane, suggesting a complex evolution of structural heterogeneities in successive serrations Imagine that in one serration, the heterogeneous deformation and the localized heat sources on the shear plane may have disrupted the atomic structures, so the locations of these localized heat sources are unlikely prone to activation in the next serration For example, the strain localization process could correspond to the evolution of free volume or STZ densities, which are certain measures of the atomic structure Recent experimental works [5–7] have shown that mechanical or thermal history can rejuvenate the glass structure; in other words, prior thermomechanical loading may lead to a new atomic structure that is less prone to further strain localization The lack of the correlation of these heat sources is the key in understanding the serrated flow That is, shear bands are triggered by structural heterogeneities, which leads to severe plastic deformation and thus generates transient heating (as identified by the hot spots in Figs and S4) The raised temperature field results into thermal softening that synergistically facilities the strain localization, and also governs the spacing these localized spots in Fig However, the rapid conduction is equivalent to a quenching process that changes the atomic structure in the shear band and potentially ‘hardens’ these hot spots Thus additional stress is required to start a new strain localization process (i.e., the next serration) but at different hot spots from those noted in Figs and S4 Further insights on the deformation mechanisms will be reported in a future work Materials Today  Volume 20, Number  January/February 2017 evolution of the localized heat sources This truly interdisciplinary example clearly demonstrates the capability of ‘seeing the unseen’, that is, how structural heterogeneities evolve when coupled with strain, strain rate, and temperature inhomogeneities The methodology developed in this work can certainly be extended to many other material problems Methods Experimental A ladle-hearth-type arc-melt tilt-casting technique has been employed to manufacture the Zr52.5Cu17.9Ni14.6Al10.0Ti5.0 (atomic percent, at.%) Vitreloy-105 BMG samples The BMG ingot was cut into cylindrical rods of 2:1 length-to-diameter ratio, for example, f2 mm  mm The two end sides of the sample were polished to a 1200 SiC grit-surface finish perpendicular to the longitudinal axis of the specimens, using a polishing fixture to ensure that these two sides were parallel to each one The cylindrical sample was loaded between the upper and lower tungsten carbide platens for the compression test A computer-controlled Material Test System (MTS) servohydraulic testing machine was employed for compressive experiments at a strain rate of  10À3 sÀ1 Thermography detections were conducted using a FLIR SC5000 Infrared Imaging System with a cooled Indium Antimonide (InSb) detector The temperature sensitivity is 20 mK at 23 8C, while the spatial resolution can be as small as 15 mm [12,15] The IR camera was used to monitor the temperature evolution during the above test at a frame rate of 300 Hz Unmixing method The primary challenge in developing an effective unmixing algorithm to solve a particular problem is to find the appropriate constraint(s) to confine the solution space in that problem domain Given the model in Eq (1) and assuming that the source matrix, A, is known a priori, the problem of unmixing becomes a constrained linear-regression problem However, in our problem, we have no prior knowledge about the temperature signatures of internal heat sources (i.e., A is unknown) Hence, an unsupervised unmixing algorithm is needed to solve the problem We apply the Gradient Descent Maximum Entropy (GDME) learning method [16], as it has the inherent capacity of providing more accurate estimates when the observation is corrupted by the strong noise or when the sources are quite similar to each other This method progressively identifies the sources, A, using an iterative process and derives the abundance, S, based on the maximization of the Shannon entropy That is, minimize f sị ẳ c X sj lnsj jẳ1 Summary Our results present a comprehensive 3D evolution pattern for all internal heat sources with their heat content, and the nature of in situ and nondestructive characterization from the unmixing algorithm extends far beyond all the present techniques Interestingly, we find that regardless of the atomic structures in metallic glasses, eventually the heterogeneous deformation process on a shear-band plane is governed by a length scale that relates to the thermal transport The serrated flow corresponds to the repeated shearing processes on a major shear band/plane with a complex 14 subject to h0 sị ẳ 1T s1 ẳ 0; hi sị ¼ c X Aij sj Àxi ¼ 0; i ¼ 1; ; N j¼1 (4) where the form of f0 takes the negative entropy function of an ideal solution, and xi is the ith component in the observation column vector, x X The first constraint, h0(s), ensures that the elements within the abundance vector (i.e., each column in S) sum to unity, and the second constraint, hi(s), reflects the sensor measurement model (i.e., the reconstruction error using the linear mixing model) To find the optimal solution, the method of Lagrangian Materials Today  Volume 20, Number  January/February 2017 RESEARCH T Ls; l; l0 ị ẳ f sị ỵ l Hsị ỵ l0 1ịh0 sị; (5) where l and l0 are the Lagrangian multipliers, and H is a column vector assembled by hi ðsÞ Note that l is a vector in order to provide component-wise optimization of the abundances Our previous work [16] has proved that the maximum entropy formulation yields a sigmoid-like analytical solution to the abundance distribution in terms of the Lagrangian multipliers, which is guaranteed to satisfy both the non-negative and the sum-to-one constrains as used in Eq (1) during the learning process The unmixing process has the great potential to improve the resolution in both the spatial and temporal domains and to provide unrevealing details in a non-destructive way Acknowledgements The authors acknowledge the financial supports from NSF IIS 1239478 (YS and HQ) and NSF CMMI 1300223 (YG) PKL acknowledges the Department of Energy (DOE), Office of Fossil Energy, National Energy Technology Laboratory (DE-FE-0008855 and DE-FE-0024054), with Mr V Cedro and Mr R Dunst as program managers, and NSF CMMI 1100080 with the program director, Dr C Cooper XX and PKL appreciate the support from the DOE, Office of Fossil Energy (DE-FE-0011194) with the program manager, Dr J Mullen, and from the U.S Army Research Office project (W911NF-13-1-0438) with the program manager, Dr D.M Stepp Appendix A Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.mattod.2016.12.002 Depth estimation First, a thermal-electric analogy is proposed to study the relationship between the surface temperature (as measured by IR imaging) and ‘emissions’ from the heat source inside The heat source inside the material can be simulated as a battery with voltage, and the heat loss inside the heat source can be simulated as the heat loss on a resistor The temperature of the heat source can then correspond to the voltage of the battery, and the heat flux to the circuit current The surface temperature corresponds to the output voltage, which can be calculated easily from the circuit theory Second, once the temperature field is identified, the location of the heat source can be determined by finding the half power point of a Gaussian distribution Further details can be found in the Supplementary Information, where an example with six heat sources shows the procedure to identify their locations and heat contents Author contributions All authors discussed the results and made critical contributions to the work References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] C.A Schuh, T.C Hufnagel, U Ramamurty, Acta Mater 55 (2007) 4067–4109 W.J Wright, et al Acta Mater 57 (2009) 4639–4648 S.X Song, T.G Nieh, Intermetallics 17 (2009) 762–767 Y Wu, et al Int J Plast 71 (2015) 136–145 W Dmowski, et al Phys Rev Lett 105 (2010) 205502 Y.H Liu, et al Phys Rev Lett 106 (2011) 125504 Y Yang, et al Sci Rep (2014) 6699 C.A Schuh, T.G Nieh, Acta Mater 51 (2013) 87–99 W.D Li, et al Appl Phys Lett 103 (2013) 171910 C.T Liu, et al Metall Mater Trans A 29 (1998) 1811–1820 C.J Gilbert, et al Appl Phys Lett 74 (1999) 3809–3811 B Yang, et al Appl Phys Lett 86 (2005) 141904 J.J Lewandowski, A.L Greer, Nat Mater (2006) 15–18 Y.F Gao, B Yang, T.G Nieh, Acta Mater 55 (2007) 2319–2327 W.H Jiang, et al Int J Plast 24 (2008) 1–16 L Miao, H Qi, IEEE Trans Image Process 16 (2007) 1008–1021 L Miao, H Qi, IEEE Trans Geosci Remote Sens 45 (2007) 765–777 J Luo, et al Intermetallics 29 (2012) 1–13 H Qi, P.T Kuruganti, Z Liu, IEEE Int Symp Biomed Imaging: Macro to Nano, Washington, DC, July, (2002), pp 309–312 [20] J.J Lewandowski, A.L Greer, Nat Mater (2006) 15–18 15 RESEARCH: Original Research multipliers is used to convert the original constrained optimization to an unconstrained problem,