Fracture toughness of a material depends on its microstructure and the imposed loading condition s. Intuitively , the resultant fracture surfaces must contain the information about the interlacing of these intrinsic (microstructure) and extrinsic (imposed loading) characteristics. Mandelbrot’s revelation that fracture surfaces are fractals, excited both the scientifi c and engineering communities , spurr ing a series of works focused at correlatin g the fracture toughness and the fracture surface roughness. Unfortunately , these studies remained inconclusive and later on it was shown that the fractal dimension of the fracture surface roughness is in fact universal . Here , we show that by going beyond th e universality, a definite correlation between the fracture toughness and indices of the fracture surface roughness is obtained. To this end, f racture experiments on an aluminum alloy were carried over a wide range of loading rates (10-2 – 106 1MPa ms− ), and the resulting fracture surface w ere reconstructed using stereography. The correlation lengths, extracted from the reconstructed surfaces, were found to be linearly correlated with the measured fracture toughness. The correlation unraveled in our work , along with the proposed mechanistic int erpretation, revives the hope of correlating fracture toughness and fracture surface roughness , allowing quantitative failure analysis and a potential reconstructive approach es to designing fracture resistant material s. Keywords: Dynamic fracture; Fracture mechanisms; Fracture toughness; Fracture surface roughness ; Mechanical testing 1. Introduction The fracture process in ductile materials , in general involves nucleation, growth and coalescence of micron -scale voids , resulting in relatively greater crack growth resistance. To put this in to perspective , the atomic work of separation for any material is 1 – 10 J/m2 (Hayes et al., 2004; Van der Ven and Ceder, 2004) whereas the work of crack advance in ductile m aterials is 104 – 105 J/m2 (Lautridou and Pineau, 1981) . The difference comes from the micron and larger scale dissipation s associated with the ductile fracture process and lead to highly irregular fracture surfaces . These 2 fracture surfaces are expected to contain the evidence of the influence of both the intrinsic (microstructure) and extrinsic (imposed loading ) characteristics of the fracture process. Thus, i t is no wonder , that the study of fracture surfaces dates back to the sixteenth century (ASM Metals handbook, 1987) . However, in the absence of a quantitative geometrical method for characterizing the irregular surfaces , these studies were strictly qualitative. Nevertheless these qualitative studies helped unravel – micromechanism of ductile fracture, origin of striations in fatigue fracture and cleavage pattern in brittle fracture , to name a few (Pineau et al., 2016, 2015) . The quest for correlating the post mortem analysis of fracture surfaces with the energy invested in their creation gained attention with the technological advancements in the characterization tools and in the description of complex geometries in the twentieth century. In particular, Mandelbrot et al. (1984) were the first to quantitatively characterize the scale invariant properties or fractal nature of the fracture surfaces (Mandelbrot et al., 1984) . Mandelbrot et al. (1984) also introduced the fractal dimension of the fracture surfaces as a quantifying parameter that can potentially be correlate d with the apparent toughness of the material. This spurred a series of works focused at correlating the fracture toughness and the fracture surface roughness via the fractal dimension (Carney and Mecholsky, 2013; Charkaluk et al., 1998; Davids on, 1989; Mecholsky et al., 1989; Wang et al., 1988) , resulting in contradictory observations ranging from negative, positive or no correlation . This discrepancy is largely due to relatively small variation in the value s of the fractal dimension (Bouchaud, 1990) and the different methods used to estimate the fractal dimension (Charkaluk et al ., 1998) . Later on, it was shown that the fractal dimension or the exponent characterizing the scale invariance of the fracture surface roughness is universal, i.e. independent of the material and i ts toughness (Bonamy and Bouchaud, 2011; Bouchaud, 1990) as long as the overall fracture mechanism remains the same . Indeed, t he scale invariant properties of fracture surfaces have been observed in a variety of materials , ranging from engineering alloys to composites to geomaterials , and under a variety of loading conditions ranging from quasi -static to fatigue t o dynamic fracture (Bouchaud, 1997; Charkaluk et al., 1998; Cherepanov et al., 1995; Dauskardt et al., 1990; Underwood and Banerji, 1986) . The universality of the scaling exponent, and the various theoretical models used to explain it (Bonamy and Bouchaud, 2011; Bouchbinder et al., 2004; Ramanathan et al., 1997) , do not coincide with the fact that the fracture toughne ss depends on both the material and the loading 3 conditions, and can vary over orders of magnitude even if the overall fracture mechanism remains the same. The universality of the scaling exponent thus limits the applicability of statistical analyses of the fracture surfaces for the characterization of the f racture process and fracture toughness. Recently , we (Osovski et al., 2015; Srivastava et al., 2014) carried out three dimensional calculatio ns of ductile crack growth under mode I small scale yielding conditions with the aim of predicting the variation of crack growth resistance and fracture surface roughness with material microstructure and imposed loading rate . The calculations were carried o ut using an elasto – viscoplastic constitutive relation for a progressively cavitating solid with two populations of void nucleating second phase particles : small homogeneously distributed particles that nucleate voids at large plastic strains, and discret e large inclusions that result in void nucleation at an early stage of the deformation history. The extent of crack growth in the se calculations were sufficient to extract the crack growth resistance parameters and quantify the statistical features of the fracture surface roughness . From the calculations, a quantitative correlation between the correlation length scales beyond which the scale invariant properties of the fracture surface roughness break down , and the material microstructure and loading ra te were unraveled. Similar, dependence of correlation length scales on material microstructure were also observed in some experimental works (Hinojosa et al., 2000, 1998; Hinojosa and Aldaco, 2002) . Interestingly, in Srivastava et al. (2014) it was found that the variations in the fracture toughness arising from changing the density of the void nucleating large inclusions in the calculations correlate linearly with the correlation length scales extracted from the calculated fracture surface roughness. A si milar correlation between the fracture toughness and fracture surface roughness was found in Osovski et al. (2015), for a material with fixed density of large inclusions subjected to varying loading rates . Here , inspired by our previous computational results, we carry out an experimental study to show that indeed by going beyond the universality of the scale invariant properties , a definite correlation between the fracture toughness and the fracture surface roughness can be obtained. In particular, we show that the lateral and vertical correlation lengths, beyond which the scale invariant properties of the fracture surface roughness break down, scale linearly with the fracture toughness. 4 2. Material microstructure and mechanical property 2.1 Microstructural characterization The material examined in this work is an Aluminum alloy (AA) 6061 -T6. The AA6061 -T6 is a precipitation -hardened alloy with a high magnesium and silicon content. Commercial AA6061 -T6 rolled sheets with 1.6 mm thickness were acquired from McMaster -Carr Supply Company . Extensive metallographic studies were performed on the as -received material to determine the material’s intrinsic length scales , such as, mean inclusion spacing and grain size. Metallographic specimens of the as -received material were mounted using cold acrylic resin , and then grinded using Si C papers (320 & 1000 grit) with water as lubricant. Next, the specimens were mechanically polished on lapping cloth with 3 m and 1 m diamond suspensions . Special care was given throughout all grinding and polishing stages to avoid applying excessive force that may embed the polishing particles into the specimen surface. Finally, the specimens were electropolished using an ethanol -perch loric acid solution. Some of the mechanically polished specimens (prior to electropolishing) were lightly etched using Kroll’s reagent. Metallographic analyses were performed using a Tescan Mira -3 FEG -SEM. The surfaces were image d at a resolution of 4096 4096 pixels over a view field of 500 m resulting in a pixel size of 122 nm. A representative SEM – BSE image of the microstructure of as -received AA 6061 -T6 showing the distribution of inclusions is shown in Figure 1. 5 Figure 1: SEM – BSE image showing the distribut ion of inclusions in the as -received AA6061 - T6 sheet metal . Large variation in inclusion size and inclusion -inclusion distances are noted . Mean inclusion spacing The rolling process leaves behind inclusions shattered into tiny particles along with larger inclusions, Figure 1. To measure the inclusion size and me an inclusion spacing, we first apply a grayscale binary image thresholding to the SEM – BSE images to separate the inclusions from the background. The size of the tiny particles were found be ~ 20.122 m and that of larger inclusions were found to be in the range 2~ 7 12 m− with a few very large inclusions reaching up to 250m . A large amount of these sub -micron particles, clustered together, ha s a significant influence on t he mean inclusion spacing measurements . Hence, we must somehow balance the weight of the small and large inclusions when quantifying the mean inclusion spacing. To this end , several algorithms were considered (k -means, user defined size thresholding and nearest neighbors analysis to name a few ) to extract the mean inclusion spacing . All of the considered algorithms, were found to be extremely sensitive to the user’s input such as number of clusters, threshold inclusion size etc. and failed to successfully balance the weight of large versus small particles in the measurement of the mean inclusion spacing. Following this we relied on the dimensionless geometrical parameter: ()()ˆi i iiA N tMt Ln=   (1) where iA is the area of inclusion i , ˆ iL is the average distance between centroids of inclusion i and each of its near est neighbors, in is the number of near est neighbors of inclusion i , and finally, ()Nt is the total number of inclusions for the given threshold t . The use of the dimensionless geometric parameter in Eq. (1) allows us to assign more weight to larger inclusions without determining a specific threshold. By plotting th e variation of ()Mt , we can classify the particles into 5 size groups. Considering the threshold size as the value corresponding to the peak after which there is no trend in the variation of ()Mt , the mean spacing between the inclusions was quantified to be 030lm= . 6 Grain size Metallographic specimens were extracted along all three sheet directions (R olling , Transverse and Normal directions ). The specimens were polished following the standard techniques and subsequently electropolished before being subjected to EBSD analysis using an Oxford EBSD detector . The EBSD pattern was acquired at 20 kV beam voltage and 0.7 m step size. A representative inverse poll figure map of the as received AA6061 -T6 sheet metal in shown in Figure 2. T he line interception method was then used to extract the average grain size along all the three sheet directions . The average grain size through the thic kness of the sheet was found to be 15 m which is roughly half of the mean inclusion spacing. Figure 2: EBSD inverse poll figure map of as-received AA6061 -T6 sheet metal . The normal direction of the sheet metal is along the viewing direction. 2.2 Mechanical characterization The uniaxial stress -strain behavior of AA6061 -T6 sheet metal along the rolling direction was examined over a range of imposed strain rates varying from 0.001 s-1 to 5500 s-1. Quasi static tests were carried out using an Instron servo -hydraulic machine while the high rate experiments were carried out using a split Hopkinson tension /compression bar (SH TB/SHPB ) utilizing C300 harden ed Maraging steel bars. Both compression and tension experiments were carried out at high strain rates to better characterize the rate dependent flow response of the material . The tensile specimens were connected to the SH TB using a designated cylindrical specimen clamp fitted to the sheet specimen geometry as shown in the insert in Figure 3. Two strain gages were glued on each bar and sampled at 2 MHz using a Nicolet 450 oscilloscope . The measured signals were used 7 to obtain the force -displacement curves , which were then converted to nominal stress-strain curves shown in Figure 3. On an average, i n the quasi -static regime the observed yield stress 270s y MPa while in the dynamic regime the observed yield stress 300d y MPa . Furthermore, the strain rate sensitivity of the material is found to vary over the deformation history with the strain rate sensitivity exponent bein g ~0.014 for flow stress taken at the onset of plastic deformation whereas the exponent is ~0.025 for flow stress taken at a strain of 0.1. Due to extensive ringing in the signals obtained for tension at strain rates , 11500 s− , only compression curves are presented for higher strain rates. Figure 3: The e xperimentally obtained quasi -static and dynamic uniaxial nominal stress strain curves. The curves labelled , Tens, are obtained from tension tests while the other curves are obtained from compression tests. The inse rt shows the cylindrical specimen clamp fitted to the sheet specimen geometry for tension tests . 3. Fracture toughness Double edge notched tensile (DENT) specimens with the axis aligned parallel to the rolling direction of the as -received sheets of AA6061 -T6 were machined using electro -discharged machining. The DENT geometry is shown i n Figure 4 a. The fracture tests using DENT specimens 0 0.2 0.4 0.6 0.8 1 1.20100200300400500  [MPa] 5500 [1/s] 3900[1/s] 3400 [1/s] 2200 [1/s] 0.01 [1/s] 0.001 [1/s] Tens 1500[1/s] Tens 0.001[1/s] 8 were carried out at velocities ranging from 74 10 ms− to 16ms resulting in imposed rate of the mode I stress intensity factor ranging from 210IK MPa m s− to 610IK MPa m s . To cover this large range of loading rates, we used two experimental setups: (i ) Standard Instron load frame to characterize the fracture toughness in the quasi -static regime; and (ii) A split Hopkinson tensile bar to characterize the fracture toughness in the dynamic regime. Prior to conducting the experiments, the specimens were co ated with a random speckle pattern consisting of black and white paint (Figure 4b) . Special care was taken to ensure that the white layer of coating was not continuous as in early attempts a continuous layer of white color resulted in deboning of the coati ng from the specimen’s surface under high loading rates. The quasi -static tests were imaged using an allied vision camera recording at 30 fps, while t he high rate experiments were continuously imaged using a Kirana high -speed camera capturing 180 frames with ~1 megapixel resolution at 1 million fps (Crooks et al., 2013) . The digital image correlation (DIC) technique was used to obtain full field displacements/strains throughout the experiment using the “Match ID” software with a pixel size of 0.01 -0.03 mm. The displacement field obtained from the DIC measurement s were used to measure the crack tip opening displacement (CTOD ), . Fracture toughness was then measured based on the relation between  , K and J (Rice, 1968) : 1/2 1 2Iy IyK JE m E K m E   −== = (2) The value of the CTOD at crack growth initiation, c , was then used to estimate the fracture toughness (Shih, 1981) , IC y cJm= , of the material, where m is taken to be 1 due to the plane stress conditions and y is the yield strength of the material corresponding to the respective strain rate regime . 9 (a) (b) Figure 4: (a) Geometry and d imensions of the double edge notch tension specimen used for fracture experiments . (b) Random speckle pattern on a double edge notch specimen . To determine the critical value of  (i.e. the value at crack growth initiation - c) one needs to identify very precisely the exact moment of crack growth initiation. To avoid large errors in the values of c, arising from mis -identification of the crack growth initiation time, three separate methods were applied. Method 1 takes advanta ge of the derivative of δ with respect to time throughout the experiment. It is assumed that once a crack starts propagating, the value of δ will increase rapidly. Therefore, the frame where the derivative increases rapidly is considered as the frame of in itiation and δc is the value of δ measured in this frame , Figure 5a, b. Methods 2 and 3 are both based on the assumption that the measured mechanical response can be divided into three regimes, the first is linear elastic, followed by a regime which is governed by plasticity without any crack growth and finally crack growth takes place. In method 2, the experiment s were simulated using a commercial finite element software (Abaqus \Explicit) using the experimentally measured material properties shown in Fig ure 3 . The numerical simulations w ere carried out within the framework of J2 isotropic elasto -visco plastic formulation using the full kinematics as to account for the large deformation near the notch tip. A power law material model was used: 10 ()()0 0m n flow y p k  =+   (3) with flow being the flow stress and 0270y MPa= being the yield stress at the reference strain rate, 1 00.001 s−= . The hardening coefficient 358k MPa= , hardening exponent 0.054n= and rate sensitivity exponent 0.014m= were calibrated to fit the experimental data. No damage model was used and therefore the simulations do not account for crack initiation and propagation . In the first two regimes (purely elastic deformation followed by the beginning of plastic flow) , the experiment and the simulation are expected to correlate quantitatively . However, once a crack will start to grow , a breakdown in the correlation is expected. Thus, wh ile comparing the experiment and the simulation it is expected that the curves will separate upon crack initiation. The value of δ at this separation is simply the δc. An example of the data analysi s usin g this method is given in Figure 5c for the case of quasi -static loading . Finally, method 3 used curve fitting. A linear fit was used for the elastic regime and two power law fits were used for the second (plastic flow) and the third regimes (the crack gro wth regime) . Using this method, initiation is considered a t the intersection of the second and third regime fits , see Fig ure 5d. The critical CTOD values extracted from all three methods were found to be in good agreement with each other. The final critical CTOD (δc) used to estimate the fracture toughness ( JIC) is the mean value of the δc obtained using the three methods. 11 (a) (b) (c) (d) Figure 5: (a) The value of δ as measured from the DIC analysis versus the frame number. (b) The time derivative of δ as measured from the DIC analysis versus the frame number. (c) A comparison between the evolution of δ as a function of the reaction force (F) measured from the experiment (dashed line) and extracted from the finite element simulation (dotted line). The separation between the curves is used to identify the value of δc (insert in (c) ). (d) The measured δ values (dotted line) and the three fitting regimes ( dashed lines ). The intersection of the second and thir d fitting curves is denoted as δc (insert in (d) ). The variation in the fracture toughness , ICJ, with imposed loading rate, IK , is shown in Figure 6. Here we note, that the reported values of IK are estimated using Eq . (2) and therefore are just an approximation at the late stages of deformation where extensive plasticity is present. As shown in the figure , the fracture toughness of the material in the quasi -static regime is not very sensitive to the changes in the applied loading rate whereas in the dynamic regime the fracture toughness of the material strongly depends on the applied loading rate . Similar observation w as made for A l- 2024 alloy by Owen et. al (1998) . 12 Figure 6: The variation of measured fracture toughness ( ICJ ) as a function of the imposed loading rate, IK . Here, both the fracture toughness ( ICJ ) and the rate of increase in the mode I stress intensity factor ( IK ) were extracted from the measured CTOD. 13 4. Fracture surfaces Figure 7: (a) Representative SEM -SE image of fracture surface of a double edge notch tension specimen deformed at a loading rate of 22.4 10IK MPa m s− . The SEM -BSE image of the region in (a) is shown in (b). (c) Representative SEM -SE image of fracture surface of a double edge notch tension specimen deformed at a loading rate of 56.5 10IK MPa m s . The SEM - BSE image of the region in (c) is shown in (d). The representative SEM -SE images of the fracture surfaces of AA6061 -T6 sheet metal presented in Figures 7a and c, show dimples on the fractu re surface, which indicates a ductile fracture mechanism governed by the nucleation, growth and coalescence of micron scale voids. In the SEM -BSE images, Figures 7b and d, inclusions can be observed within the dimples, highlighting their role in the void n ucleation process. The fracture surfaces shown in Figures 7a and b are representative for low loading rates ( 22.4 10IK MPa m s− ) while Figures 7c and d are 14 representative for high loading rate ( 56.5 10IK MPa m s ). Similar images taken from all of the reported specimens confirm that the fracture mechanism does not depend on the loading rate. To quantitatively characterize the highly irregular ductile fracture surface s shown in Figure 7, we first digitized the experimental fracture surfaces. To achieve this we captured several 2600 600 m regions of a fracture surface using a SE detector in the TESCAN MIRA3 FEG - SEM , at two tilt angles, 3o from each other , with a resolution of 0.1465 m per pixel. The 3D fracture surfaces and digital elevation maps were then recreated using stereographic reconstruction in the “Mountains -Map” software suite by “DigitalSurf”. A typical recreated fracture surface of dimension 2600 600 m corresponding to a specimen fractured at an applied loading rate of 56.5 10IK MPa m s is shown in Figure 8 a. Next, from each fracture surface we extract 1 00 equidistant line profiles of the variation in the out -of-plane height (h) of the fracture surface along the crack propagation direction (x) and calculate the height -height correlation function : ()()()()2 ,xnh x h x x h x x  = + −    (4) where, h is the height at a location x along the crack propagation direction,  is the scaling (Hurst) exponent and ,xn denotes averag e over x and n, with n being the number of line profile s. In Eq. (4), the quantity () hx is simply the typical difference of height s between two points separated by a distance x . The minimum value of x i.e. minx is determined by the resolution of the digital elevation map . The value of Hurst exponent,  , in Eq. (1), necessarily lies between 0 and 1 where 0= corresponds to a straight line with slope zero and 1= corresponds to a straight line with a nonzero slope. The value 0.5= correspond to a random walk suggesting no correlation between h and x . The value of 0.5 indicate persistence i.e. an increase (decrease) in h with increasing x will most likely be followed by an increase (decrease) with further increase in x . The value of 0.5 , on the other hand, indicate anti -persistence (or mean reverting) i.e. an increase (decrease) will be followed by a decrease (increase) or in other words h has the tendency to revert to a mean value with increasing x . 15 Figure 8: (a) A 3D stereographic reconstruction of a 2600 600 m region extracted from the fracture surface of a specimen loaded at 56.5 10 .IK MPa m s The pixel size is 0.1465 m . (b) The height -height correlation function of the surface in (a). The universal scaling exponent ( ), the lateral correlation length ()sx and its associated height ()sh are marked in the figure. A typical log-log plot of the correlation function obtained using min 0.1465 xm  = is shown in Figure 8b. The corre lation function in Figure 8b is for the fracture surface shown in Fig ure 8a. We note that the correlation function s obtained using min 0.1465 xm  = along the crack propagation direction for all the cases were found to be statistically invariant. In addition, irrespecti ve of the applied loading rate, IK , the correlation function s exhibit a power law scaling , ()() h x x    , with  varying within a narrow range of 0.76 to 0.82 across all tested specimens, regardless of the loading rate and measured value of the fracture toughness . The value of  or the Hurst exponent for all the cases was calculated as the slope of the line fitted to the range , 2.6 26 m x m  , on the log -log plot , as shown in Figure 8b. The insensitivity of the Hurst exponent for the range of applied loading rate considered , is in-line with the universality of the exponent characterizing the scale invariance of the fracture surface roughness . The value of 0.5 indicates persistence , which is also in -line with the fact that irrespective of the applied 16 loading rate a growing crack has the tendency to continue growing, at least for a while, in the same direction. Figure 9: (a) A 3D stereographic reconstruction of a 230 30 m region extracted from the fracture surface of a specimen loaded at 61.0 10 .IK MPa m s The pixel size is 0.0073 m . (b) The height -height correlation function of the surface in (a) and a second region of the same size take from a different location on the fracture surface . The scaling exponent extracted from both regions are marked in the figure. As can be seen from Figure 8b, if the value of  is calculated as the slope of the line fitted to small values of x i.e. 2.6xm , the value of  is greater than the value obtained as the slope of the line fitted to the range, 2.6 26 m x m  . In order to understand the scaling of the correlation function at small length scales , we analyzed the fracture surface at a magnification approximately 20 times greater than the one used in Figure 8a. A typical fracture surface of dimension 230 30 m ( )min 0.0073 xm  = extracted from a specimen fractured at a loading rate of 61.0 10IK MPa m s is shown in Fig ure 9a. The height -height correlation function along the crack propagation direction extracted from two 230 30 m regions of the same fracture surface are shown in Figure 9b . It was found that the correlation function obtained from 230 30 m regions are not stati stically invariant over the fracture surface . Both the correlation functions in Figure 9b do exhibit power law scaling but with different values of the Hurst exponent , 17 . In addition, s imilar to Fig ure 8b, here as well for small values of x the local slope of both the curves is still greater than their respective Hurst exponent s. First, t he fact that the correlation function obtained from the high magnification fracture surface using min 0.0073 xm  = is not statistically invariant is simply because of the non -uniform nucleation and growth of voids in the materi al within the limited field of view in high magnification images . Second, the fact that for small values of x , the local slope of the curve is still greater than the value of the Hurst exponent, is a result of the stereographic reconstruction algorithm, which uses an 88 pixel window to spatially correlate points between the tilted images which limits the applicability of this me thod when x approaches the pixel size. In Bonamy et al. (2006) it was predicted that the Hurs t exponent may obtain different values depending on the size of the region over which it is extracted . For regions that are smaller than the process zone, a value of ~0.5 is expected while a value of ~0.6-0.7 was predicted for a transition zone (i.e. when the size of the sampled region is comparable with the process zone size). Those predictions were later on observed experimentally for gl ass in (Hinojosa et al., 2008) . Here we note that the values of the Hurst exponent extracted from the 230 30 m regions were all in the range of 0.6 4-0.73, while the values of the Hurst exponent extracted from the 2600 600 m regions were 0.8±0.03 . Thus , the height -height correlation function at a scale comparable to the intrinsic length scale of the material i.e. mean inclusion spacing ( 030μm l= ) is very sensitive to the sampling region and hence we recommend to focus on a length scale which is larger than 0l for the analysis of the fracture surface roughness. Next, note that for large values of x, the local slope of the curve in Figure 8b is less than the Hurst exponent and it further decreases with increasing x . For large enough x the local slope eventually goes to zero. The value of the local slop e of the height -height correlation function below 0.5 indicates anti -persistence . Anti -persistence at large values of x is because, a t large length scales, the growing crack tend s to revert back to the mean fracture plane, despite the attraction it feels from the randomly distributed inclusions. Furthermore, the value of x at which the local slope of the correlation function goes to zero indicates the lateral correlation length ()sx at which the height -height correlation function is (nearly) constant and the surface features are no longer correlated. Our rigorous analysis of the scale invariant properties of the fracture surface roughness of the specimens tested over a range of applied loading rates clearly show that the Hurst exponent, 18 , cannot provide a quantitative correlation between the fracture toughness and fracture surface roughness as it was envisioned in the past. Nevertheless, a s evident from Fig ure 7, the observed variation in fracture toughness with the applied loading rate is not due to any variation in the micromechanism of d uctile fracture. Instead, the toughness increases due to the increase in the spread of damage (fracture process zone size) and plastic dissipation (plas tic zone size) ahead of the crack tip with increasing loading rate (Osovski et al., 2015) . As the loading rate increas es, the region in which the stress level is sufficient to trigger void nucleation spreads further away from the crack tip (due to the material’s rate sensitivity) , allowing for more damage sites to interact with one an other (Broberg, 1979) . This process may not affect the nature of crack meandering that dictates the universality of the scale invariant properties of the fracture surface roughness. B ut it must affect the extent of it that dictates the lateral correlation length ( sx ).We postulate that the distance over which voids interact will manifest itself on the fracture surface roughness in such a way that it will dictate the value of ,sx as was previously observed numerically (Osovski et al., 2015; Srivastava et al., 2014) . Following this, we go beyond the universality of the scale invariant properties of the fracture surface roughness and extract two characterizing parameters, sh and sx . The value of sh is the maximum value of the correlation function, () hx , and the value of sx is the value of x at the intersection of ()s h x h  = and the line fit to the linear portion of the log – log plot as illustrated in Figure 8b. The values of sx and sh quantify the extent of crack meandering. In a previous work, it was demonstrated that the correlation length extracted from a sinusoidal crack indeed return the sinus wave length and the corresponding height (denoted as sh ) reflects the sinus amplitude (Srivastava et al., 2017) . The variation s of sx and sh with fracture toughness , ICJ , are shown in Fig ure 10a and b. The values of sx and sh in Fig ure 10 are normalized with the microstructural length scale, 0l. For low loading rates, where variations in the fracture toughness were relatively small (Fig ure 6) the points are clustered together , however, with increasing loading rate, as the material becomes tougher, the values of sx and sh increase linear ly. The linear dependence of sx and sh on ICJ can be rationalized by assum ing that the extent of crack meandering depends on the size of the volume in which inclusions are activated to 19 play their role as void nucleation sites . The experimental results, summarized in Figure 10, are in direct agreement with our previously published roughness -toughness correlations obtained from numerical simulations (Osovski et al., 2015; Srivastava et al., 2014) . Figure 10: Correlating the fracture toughness with length scales extracted from the fracture surface roughness: A linear correlation between the two characterizing parameters of the fracture surface roughness and the fracture toughness is found . (a) The variation of the normalized lateral correlation length , /soxl - representing the in -plane roughness with the fracture toughness , ICJ. (b) The variation of the normalized corresponding height , /sohl - representing the out of plane roughness with the fracture toughness , ICJ. 5. Discussion The universality of the scaling exponents of fracture surfaces has attracted a lot of interest since its discovery . However, it has failed to provide a quantitative tool for the post mortem analysis of fracture surfaces . By going beyond this universality, and extracting the correlation length from the fracture surface, we were able to demonstrate a linear relation between the measured fracture toughness and the fracture surface roughness. Although the co rrelation length of fracture surfaces has been studied (Hinojosa et al., 2000, 1998; Hinojosa and Aldaco, 2002) and shown to yield the microstructural length scale associated with the fracture process, we are not aware of any experimental effort so far to correlate its variation with the variation in fracture toughness. Our 20 results indicate that this correlation, when studied over a varie ty of microstructures and loading scenarios, can manifest into a quantitative tool for failure analysis. The attractiveness of the method we propose lies in the fact that , unlike previous sought after correlations (Carney and Mecholsky, 2013; Mandelbrot et al., 1984; Wang et al., 1988) , it is not purely emp irical but rather originates from a mechanistic approach. The quantitative correlation between the fracture toughness with length scales extracted from the fracture surface roughness presented here are for a specific alloy . Nevertheless , we note that the continuum scale simulations carried out using a constitutive framework for a progressively cavitating isotropic elastic -viscoplastic material (Osovski et al., 2015; Srivastava et al., 2014) containing two populations of void nucleating particles , have demonstrated similar correlation s between the numerically obtained fracture surfaces and fracture toughness. A direct comparison between the previously published numerical results and the experimental data presented here is not straightforward, as the numerical model was not calibrated for a specific material. Nonetheless, several similarities are noted . In particular, the values of sx extracted from the numerical results of Srivastava et al. (2014) for low inclusion densities and under quasi -static loading conditions, scale lin early with 0l with a slope of ~2, and in our experiments, for quasi -static loading onditions, on an average 0/sxl is ~1.8 . Also, for both, the experiment s (current work ) and the numerical simulations (Osovski et al., 2015) , for a fixed 0l and varying imposed loading rates, 0/shl and ICJ exhibit a linear correlation. Following this, we can claim generality at least for ductile materials undergoing fracture by void nucleation, growth and coalescence. For ductile structural materials used in engineering applications, of course, the fracture processes can be more complex. For example, cleavage can occur in ligaments between two voids under extreme conditions. Even restricting attention to void nucleation , growth and coalescence induced crack growth, the fracture process in certain polycrystalline structural metals and alloys can be dominated by voids that nucleate and grow preferentially along grain boundaries. The fracture toughness of a material indeed depends on the material properties, active fracture mechanisms , microstructural length scales and loading conditions . Intuitively, these in turn will affect the fracture surface roughness and the analysis presented here can be extended to probe such effect s. 21 Whilst the work presented here aims primarily at elucidating t he correlation between fracture toughness and fracture surface roughness, th is has broader scientific implication s. For example, i t can serve in validating existing theories regarding the size and evolution of the process zone and its relation to the intrinsic and extrinsic characteristics of the fracture process. The understanding of the interlacing between the microstructure and the imposed loading and its effect on the fracture toughness and damage tolerance is essential for designing toughness oriented new materials. The modern data analysis tools (e.g. machine learning ) can be trained to preform quantitative fractography once a l arge enough data set is available covering different classes of materials and loading scenarios. These can then be used as a potential reconstructive approaches to designing fracture resistant materials. 6. Concluding remarks We have carried out fracture experiments using double edge notch tension specimens of an aluminum alloy over a wide range of loading rates, IK , varying from 2610 10 MPa m s−− . The fracture toughness of the material as a function of the imposed loading rate were quantified and the resulting fracture surfaces were reconstructed using stereography. The reconstructed fracture surfaces were then used to extract indices of the fract ure surface roughness. The key conclusions of this work are: 1. The value of the Hurst exponent , , is found to be constant over the entire range of the loading rate considered, in agreement with the universal value ( 0.8 ). 2. The value of the lateral correlation length scale, sx , and the maximum value of the correlation function, sh , extracted from the fracture surface roughness are found to increase with the increasing loading rate, in agreement with the hypothes is that the fracture process zone size increases with the increasing loading rate that in turn increases the crack tortuosity . 3. A linear correlation is found between the two indices of the fracture surface roughness, sx and sh , and the fracture toughness of the material over the entire range of the loading rate considered . 22 Acknowledgments We thank Dr. Alan Needleman of Texas A&M University and Dr. Daniel Rittel of Technion – Israel Institute of Technology for helpful and stimulating discussions. The financial support provided by the Pazy foundation Young Researchers Award (Grant no. 1176) and European Union’s Horizon2020 Program (Excellent Science, Marie -Sklodowska -Curie Actions, H2020 - MSCA - RISE - 2017) under REA grant agreement 777896 (Project QUANTIFY )is gratefully acknowledged .