PHYSICAL REVIEW MATERIALS 3, 113602 (2019)

Failure of fracture toughness criterion at small scales

Yangyang Cheng,1 Long Yu,1 Lirong Chen,1 Wenbin Liu,1 Xin Yi,1,* and Huiling Duan1,2,†

1State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, BIC-ESAT,College of Engineering, Peking University, Beijing 100871, People’s Republic of China

2CAPT, HEDPS and IFSA Collaborative Innovation Center of MoE, Peking University, Beijing 100871, People’s Republic of China

(Received 7 April 2019; published 11 November 2019)

Fracture toughness testing at small scales is widely adopted for materials of limited size. Recent experimentalresults demonstrate that the fracture toughness of single-crystal tungsten decreases by half as the specimen sizedrops equally from the order of several dozen micrometers to 1 μm. To explore this size-dependent fracturefeature, we construct a fracture model by combining the Griffith strength theory and elastic-damage law. Itis shown that as the specimen scales down to a critical size, the fracture behavior changes from the fracturetoughness domination to the strength domination, leading to a seeming decrease of the fracture toughness.Reviewing mechanical fracture testing of different brittle materials at small scales, we found that their fracture isdominated by fracture strength instead of fracture toughness, which explains why the reported values of fracturetoughness obtained from the microscale testing are significantly less than the corresponding ones measured atthe macroscale. Our study has profound guiding significance for the characterization of the fracture toughness atsmall scales.

DOI: 10.1103/PhysRevMaterials.3.113602

I. INTRODUCTION

Fracture toughness is commonly defined by the criticalstress intensity factor Kc, which is used to characterize theresistance of materials against crack extension [1–3]. Conven-tional measurement approaches of Kc via bend specimen orcompact specimen configurations require a sufficiently largespecimen in which a crack with a sharp tip and well-definedshape can be introduced via cyclic loading [4,5]. However, it isdifficult to perform cyclic loading at a characterized specimensize below the centimeter scale, thus hardly characterizingKc in an appropriate manner based on conventional measure-ments [6,7].

This problem is mitigated by the fracture toughness testingat small scales with a combination of focused ion beam(FIB) milling and small-scale mechanical testing approaches[8–12]. In the toughness testing, the FIB milling is employedto fabricate a beam of nanoscale or microscale and makea sharp notch of a root radius around tens of nanometers,which can be viewed as a crack [13–16], and then the crackedbeam is subjected to a bending test in which the loadingforce is recorded to calculate the fracture toughness as thecrack propagation occurs [17–21] (as detailed in Part 1 ofthe Supplemental Material (SM) [22]). This method at smallscales is universal and not restricted by the material types andgeometrical sizes.

It is known that the stress intensity factor K scales with theapplied stress and the specimen size (referring to crack length,thickness, and width) [2,4]. Under a specific configuration, thefracture toughness Kc as an intrinsic property of brittle mate-

*Corresponding author: xyi@pku.edu.cn†Corresponding author: hlduan@pku.edu.cn

rials reveals a special relationship between the fracture stressand the crack length: The smaller the crack length, the largerthe fracture stress at which the crack propagates, and viceversa [23,24]. On this basis, from the macroscale testing of Kc

to the microscale, the fracture stress would increase around30-fold due to the size variation from around dozens of mil-limeters to around several micrometers, and probably exceedthe fracture strength. It is desirable to determine the validityof Kc in dominating the fracture behavior at small scales.

Recent experimental studies about the effect of specimensize on the fracture toughness of the crack system {100}〈100〉of single-crystal tungsten indicate that as the tungsten spec-imen scales down from several dozens of micrometers toless than 1 μm, the fracture toughness obtained decreasesby half [18]. Moreover, there exists an inflection point at thecrack length about 2 μm, above which the fracture toughnessvaries very little and is close to the value from the macroscaletesting, yet below which it drops dramatically [18]. Since thegeometries of different specimens in Ref. [18] are similar,the size dependence of the fracture toughness shall not resultfrom the variation of specimen thickness, i.e., the stress-statetransition (plane stress or plane strain) [25–27]. Here wedevelop a theoretical model to understand the underlyingmechanism of size-dependent fracture toughness. It is shownthat with the downscaling of specimen size, the fracturetoughness-controlled fracture changes to strength-controlledfracture, which leads to the seeming decrease of the fracturetoughness. Because the geometries of the specimens are sim-ilar, it is reasonable to select the variation of crack lengthto represent the change of specimen size. On this basis, thecritical crack length distinguishing the fracture mode canbe expressed quantitatively by our model. Our results shedlight on the validity of the fracture toughness testing at smallscales.

2475-9953/2019/3(11)/113602(6) 113602-1 ©2019 American Physical Society

CHENG, YU, CHEN, LIU, YI, AND DUAN PHYSICAL REVIEW MATERIALS 3, 113602 (2019)

TABLE I. Summary of crack length a(μm), width W (μm),ligament length (W − a)(μm), thickness B(μm) and loading spanL(μm), crack-geometry factor ψ , ratio α = (W − a)2/W 2, and frac-ture toughness KQ(MPa m1/2) for tungsten specimens with differentsizes [18].

a W (W − a) B L ψ α KQ

0.2 0.8 0.6 0.5 2.5 1.92 0.56 1.48 ± 0.10.5 1.5 1 1 4 2.05 0.44 1.86 ± 0.10.7 2.5 1.8 2 6.5 1.96 0.52 2.34 ± 0.12.1 5 2.9 4.5 8 2.3 0.34 3.02 ± 0.35.5 15 9.5 12 24 2.02 0.35 3.18 ± 0.316.5 45 28.5 25.5 72 2.13 0.4 3.43

II. RESULTS AND DISCUSSION

We first explore whether the fracture toughness measuredat small scales in Ref. [18] is valid. For the specimen at themacroscale, according to ASTM standard E399-17 [4], thecrack length a is specified to be not less than la = 2.5(KQ/σy)2

with KQ as the conditional fracture toughness and σy as theyielding strength of the material. This specification guaranteesthe applicability of linear-elastic fracture mechanics (LEFM).The standard is constructed based on the three-point bendspecimen configuration of a/W = 0.5 with W as the specimenwidth, while at small scales the widely used configurationis cantilever specimen. Our finite element simulations (seePart 2 in the SM [22]) reveal that la depends on the type ofspecimen configuration and the value of a/W . For the can-tilever specimen configuration of a/W = 0.5, our numericalresults indicate that la = 1.4(KQ/σy)2, different from that ofthe three-point bend specimen configuration. a/W in Ref. [18]ranges from 0.25 to 0.42, and la varies from 0.24(KQ/σy)2 to0.87(KQ/σy)2 accordingly. The detailed calculation on la isprovided in Part 2 of the SM [22]. In the present paper, forsimplicity, as to the cantilever configuration with a/W < 0.3,a � la = 0.25(KQ/σy)2 is taken, while as to 0.3 � a/W �0.45, a � la = (KQ/σy)2 is taken. To the best of our knowl-edge, almost all experimental reports [17,20,28] characteriz-ing the fracture toughness at small scales adopt the standard ofASTM mainly for the three-point bend and compact tensionspecimen configurations, which is not appropriate for thecantilever specimen configuration.

To ensure the reliability of the testing of KQ in a plane-strain condition, both the specimen thickness B and the differ-ence between the width W and crack length a are required asB � 2.5(KQ/σy)2 and (W − a) � 2.5(KQ/σy)2 derived fromnumerous experimental results [3,4,29]. However, actually theeffect of (W − a) on the measurement of fracture toughness islimited, and the requirement on it is not as strict as a and B,which is discussed in details in Ref. [29]. For the cantileverspecimen configuration considered here, we adopt the samespecification for B and W .

Table I presents the geometrical size and fracture toughnessof the microcantilever specimen of single-crystalline tungstenin Ref. [18]. It has been reported that under uniaxial compres-sion for 〈100〉-oriented tungsten pillars, the yielding strengthσy is around 2.3 GPa at specimen diameters of 0.5 μm,1 μm, and 2 μm, and around 1.6 GPa at a specimen diameter

FIG. 1. Comparison between the microcantilever size and thesize required by our proposed standard. The inset shows theschematic of the microcantilever with a crack.

of 5 μm [14]. Therefore, we take σy = 2.3 GPa for thefirst four specimens with (W − a) � 2.9 μm as listed inTable I and σy = 1.6 GPa for the rest specimens of (W − a) >

2.9 μm.For easy visualization, the specimen size [referring to

a, (W − a) and B] and the size required by our proposedstandard as discussed above are depicted in Fig. 1. It isshown that the crack length a of all six specimens is not lessthan the required size la, verifying that LEFM can exactlycharacterize the stress field ahead of the crack tip. Regardingthe thickness B, its value in the small specimens is slightly lessthan the corresponding required size 2.5(KQ/σy)2. Strictly,this means that the crack tip is not completely situated forplane strain. However, the fractographic observation indicatesthat the crack plane is almost flat, and there are no pronouncedshear lips and crack deflection around the edges [18], whichconfirms that the deformation around the crack tip is predom-inately under the plane-strain condition. The above analysiseliminates the possibility that the variation of KQ is resultedfrom the inapplicability of LEFM and the change of stressstate.

Based on the thin beam elasticity theory, the maximumnominal fracture stress σn in the crack plane is given by

σn = 6FL

BW 2, (1)

where F is the loading force when the fracture occurs and Lis the loading span. Considering that the crack has no load-carrying capacity, the net width of the plane should be (W −a), then the maximum net-section fracture stress σe ahead ofthe crack tip is given by

σe = 6FL

B(W − a)2. (2)

Although the cantilever specimen configuration in Ref. [18]is not an ideal thin beam, σn and σe obtained from the thinbeam theory are consistent with our finite element simulationresults (listed in Table SII in Part 3 of the SM [22]). For the

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FAILURE OF FRACTURE TOUGHNESS CRITERION … PHYSICAL REVIEW MATERIALS 3, 113602 (2019)

FIG. 2. Maximum bending stress versus crack length for speci-mens of different sizes.

small specimens with a of 0.2 μm, 0.5 μm, 0.7 μm, and2.1 μm, σe is insensitive to the crack length a and maintainsaround 2.9 GPa with a standard deviation of 0.17 (see Fig. 2).In contrast, regarding σn, the average value is 1.3 GPa witha standard deviation of 0.33. In this situation, we make apostulate that as the specimen is smaller than a critical size,its fracture is determined by the critical net-section stress, i.e.,fracture strength. In other words, there exists a transition fromfracture toughness-dominated fracture to fracture strength-dominated fracture with the decrease in the specimen size.

According to LEFM, the stress intensity factor KI for themode I crack is

KI = σψa1/2, (3)

where σ is the remote applied stress far away from the crackplane (equivalent to σn for cantilever specimen configuration),a characterizes the crack length (for an internal crack, a isone half of the length; for an edge crack, a is the length),and ψ is crack-geometry factor [2]. If the fracture of thesmall specimens is controlled by the fracture strength σf orequivalently σe, the fracture toughness of the small specimenscan be given as

KQ = σfαψa1/2, (4)

where α = σn/σe or equivalently α = (W − a)2/W 2. Eq. (4)indicates that KQ depends on the specimen size and is pro-portional to αψa1/2. As demonstrated in Fig. 3, for smallspecimens, KQ and αψa1/2 obey a linear correlation withthe fracture strength σf = 2.8 GPa obtained by linear fitting,consistent with the net-section fracture stress σe = 2.9 GPacalculated from experimental data [18].

To understand the transition from the fracture toughness-dominated fracture to fracture strength-dominated fracture asthe specimen scales down, an approach is developed com-bining the Griffith strength theory and damage mechanics.According to the Griffith theory, the fracture strength σf of

FIG. 3. Fracture toughness KQ as a function of αψa1/2 for spec-imens of different sizes.

a brittle plate with a through microcrack of length 2a is [2,5]

σf =√

EG

πa, (5)

where E is Young’s modulus and G is the fracture surfaceenergy. σf can then be rewritten as

σf = KQ√πa

(6)

by introducing the equivalency between fracture toughness KQ

and fracture surface energy G as G = K2Q/E .

Based on the experimental results in Ref. [18], KQ for thelarge specimen is around 3.2 MPa m1/2 as the inherent fracturetoughness of the model material, and the fracture strength σf

is 2.8 GPa as derived from linear fitting. Thus, via Eq. (6), thelength of the microcrack can be determined as 0.85 μm. Fromthe viewpoint of elastic-damage law [30,31], initially thematerial deforms elastically, but the damage fracture suddenlyoccurs once the fracture strength is reached. Incorporatingthis law with the Griffith theory, the brittle fracture can beinterpreted as follows. For a crack-free specimen subjected tothe tensile loading, as the applied stress reaches the criticalvalue (fracture strength σf ), there comes rapid damage, anda damage zone appears without load-carrying capacity [seeFig. 4(a) for illustration]. Here we assume that the size of thedamage zone is equal to that of the microcrack of the Griffiththeory (0.85 μm). As a result, the Griffith fracture conditionis satisfied and, simultaneously, catastrophic failure occurs.In other words, nucleating the damage zone correspondsidentically with the catastrophic failure.

For the specimens with crack length of 0.2 μm–2.1 μm,before the stress intensity factor K approaches the criticalvalue KQ, the net-section stress reaches the fracture strengthσf , and a damage zone rapidly appears with a length ofaround 0.85 μm. Therefore, the fracture of these specimensis controlled by the strength criterion. Because the ligamentsof two smallest specimens in Ref. [18] are 0.6 μm and 1 μmlong, respectively, less than or almost equal to the damagezone size, the two specimens fractured catastrophically within

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CHENG, YU, CHEN, LIU, YI, AND DUAN PHYSICAL REVIEW MATERIALS 3, 113602 (2019)

FIG. 4. Coupling the Griffith strength theory and elastic-damage law to illustrate the fracture behavior of brittle materials. (a) For a crack-free specimen under the tensile condition, as the loading stress increases to the fracture strength σf , a damage zone rapidly generates with a sizeequal to the Griffith flaw, and catastrophic fracture occurs. (b) For a microcantilever with a pre-existing crack, the fracture criterion is relatedto the crack length. For small specimens with a < ac, the net-section stress σe reaches the critical value σf first, and the fracture is controlledby the strength. In the case of extremely small specimens, the ligament ahead of the crack is less than the damage zone size, and catastrophicfracture happens. For large specimens with a > ac, the stress intensity factor K first reaches the critical value KQ, and the fracture is controlledby the fracture toughness.

the nearly linear-elastic region [Fig. 4(b)]. For the specimenswith ligaments 1.8 μm and 2.9 μm (significantly larger thanthe identified damage zone size 0.85 μm), the crack extendsbut does not cover the whole uncracked area [Fig. 4(b)]. Thisexplains the phenomenon in Ref. [18] that the fracture ofthese two smallest specimens occurs almost within the elasticregion while the larger ones exhibit stable fracture. For thespecimens with ligaments of 9.5 μm and 28.5 μm, K firstreaches the critical value KQ, and the failure is controlled bythe fracture toughness criterion.

As discussed above, the fracture of brittle materials isgoverned by the fracture strength or toughness, depending onwhich term first reaches the corresponding critical value. Asindicated by the dashed line in Fig. 4(b), the critical cracklength

ac =(

KQ

σfαψ

)2

(7)

deduced from Eq. (4) can be used to distinguish the fracturebehavior. At a < ac, the net-section stress first reaches σf , andthe fracture is dominated by the strength. In this case, thefracture stress is independent of the crack length. At a > ac,K reaches KQ first, and the fracture is governed by the fracturetoughness. In a log-log plot, the net-section fracture stress andthe crack length exhibit a linear correlation.

In Fig. 5, we summarize the critical crack length ac ofbrittle materials based on reported experimental data (seePart 4 in the SM for details [22]). Among these studies, onlyfor silicon compounds, the crack length a is significantlylarger than the critical crack length ac (illustrated in Fig. 5)and the fracture toughness characterized at the microscale isin agreement with that at the macroscale [32]. For metallicglasses and intermetallics, a is several orders of magnitudesmaller than ac. The corresponding net-section fracture stressis close to or significantly larger than the fracture strength.This indicates that the fracture could be controlled by the frac-ture strength instead of the fracture toughness, and explainswhy the fracture toughness of Zr-based metallic glasses,NiAl and TiAl intermetallics measured at the macroscale

is about 50 MPa m1/2, 8 MPa m1/2, and 20 MPa m1/2,respectively, significantly larger than the microscale valuesof 5 MPa m1/2, 5 MPa m1/2, and 3.7 MPa m1/2, respectively[9,33–39].

Our analysis also sheds light on the macroscale fractureof materials with high fracture toughness. For instance, it isreported that the measured fracture toughness of the precip-itation hardening high strength steel is about 110 MPa m1/2

when the specimen size satisfies the ASTM standard [40]. Asthe specimen size equally scales down, the fracture toughnessmeasured decreases. Based on our model, we find that thecrack length a of the small specimen is less than ac (il-lustrated in Fig. 5), and the decrease of the fracture tough-ness with the specimen size is mainly due to the changeof fracture mechanism (see Part 5 in the SM for detaileddiscussion [22]).

FIG. 5. Summary of crack length a (polygons) and critical cracklength ac (bars) in the mechanical fracture testing of different brittlematerials.

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III. SUMMARY

The present paper demonstrates that as the size of thebrittle materials scales down to a critical value, the fracturetoughness-controlled fracture changes to strength-controlledfracture. This indicates that the fracture behavior of the largespecimen is dominated by the stress level around the cracktip, i.e., stress intensity factor, while for the small specimenit is determined by the stress level on the net section. Theconcept can be used to explain the phenomenon reported onthe biocomposites whose mineral platelet is minimized tobelow a critical length so it is insensitive to cracklike flaw andpossesses a high strength [41].

To appropriately characterize the fracture toughness, thecrack length shall be sufficiently large not only to ensurethe exact description of the stress field via LEFM, but alsoto guarantee that the net-section stress is considerably less

than the fracture strength. Moreover, based on our model, thecritical crack length ac, regulating the fracture mechanisms, isproportional to the ratio between the fracture toughness andfracture strength. For the material of low fracture toughnessand high strength, ac is in a range from microscale down tonanoscale, while for the material of high toughness and highstrength, ac is on the order of millimeters.

ACKNOWLEDGMENTS

We thank Dr. Johannes Ast for providing the data of spec-imen size and the financial support by the National NaturalScience Foundation of China under Grants No. 11632001, No.11521202, and No. U1830121, the Science Challenge ProjectNo. TZ2018001, and the ENN Science and Technology De-velopment Co. Ltd. under Grant No. 9710L2018-001.

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Supplemental Material for “Failure of fracture toughness criterion at small scales”

Yangyang Cheng,1 Long Yu,1 Lirong Chen,1 Wenbin Liu,1 Xin Yi,1 and Huiling Duan1,2 1State Key Laboratory for Turbulence and Complex System, Department of Mechanics

and Engineering Science, BIC-ESAT, College of Engineering, Peking University, Beijing 100871, People’s Republic of China

2CAPT, HEDPS and IFSA, Collaborative Innovation Center of MoE, Peking University, Beijing 100871, People’s Republic of China

Part 1: Fracture Toughness Testing

At the macroscale, ASTM Standard E399-17 [1] specifies the method to measure the fracture toughness KIc of materials under the opening mode (Mode I). The most commonly used specimen configuration is singe edged bend specimen and compact tension specimen (Fig. S1). To introduce a sharp crack with a well-defined shape, a fatigue pre-cracking procedure is carried out under cyclic loading. It requires that the specimen possesses a large size of dozens of millimeters. The achieved data based on the linear-elastic fracture mechanics (LEFM) is referred to as conditional fracture toughness KQ which could be defined as KIc only if the specimen sizes satisfy the requirements of the ASTM standard E399-17 (illustrated in Fig. S1).

FIG. S1. Comparison of macroscale and microscale testing for fracture toughness. FIB is referred to as focus ion beam technology.

However, as to the materials of limited size such as films and wires, the overall sizes are on the nanometer or micrometer order. Therefore, it is impossible to characterize their fracture toughness via the macroscale approach. This dilemma is mitigated by the advent of focused ion beam (FIB) milling and small-scale mechanical testing approaches. The FIB milling is mainly utilized to fabricate nano/micro structures and has the capability to make a sharp notch with root radius as small as a few tens of nanometers that can be viewed as a crack. Furthermore, by combining with small-scale mechanical testing approaches, the fracture toughness of materials

on the order of several micrometers can be successfully measured. For instance, a micro-cantilever beam with a straight-through notch/crack or chevron notch/crack is subjected to bending tests, and then, when the specimen fractures, the load is recorded to calculate the fracture toughness (Fig. S1). Nevertheless, to the best of our knowledge, there is no standard for the micro-cantilever specimen configuration to specify the condition at which LEFM is still applicable. In this paper, via the finite element simulations, we construct the corresponding criterion, which is discussed in Part 2. Part 2: Specification on Crack Length

Note that the stress intensity factor K only can exactly describe the stress near the crack tip. The deviation between the true stress and the stress calculated by K increases with the distance r from the crack tip. Accordingly, the critical distance rc is defined within which the deviation is below the relative error δ required by the engineering. Furthermore, there exists a plastic zone ahead of the crack tip. To ensure the applicability of LEFM, the size of the plastic zone rp should be sufficiently small. The specification on the crack length a is derived from rp≤rc.

For the three-point bend specimen configuration, as r/a=0.02, δ≈6%-7%. Consequently, at rc=a/50, the error δ is below 10%. More detailed discussion is in the literature [2]. Next, considering that rp=(K/σy)2/ (4 2 )π ≤rc for plane strain condition, a≥2.8(KQ/σy)2≈2.5(KQ/σy)2 is achieved. This specification ensures the exact description of the stress field via LEFM and the small-scale yielding ahead of the crack tip. On this basis, we shall calculate rc and δ of the present cantilever specimen configuration to quantify the requirement on the crack length.

FIG. S2. The finite element model of the cantilever specimen with a/W=0.25 (a) and three-point bend specimen with a/W=0.5 (b).

We utilize finite element (FE) software Abaqus to calculate the stress intensity factor K and the stress filed ahead of the crack. The specimen is modeled as a two-dimensional plate in plane strain condition with CPE8 elements (shown in Fig. S2). The Young’s modulus E and Poisson ratio ν are taken as 410 GPa and 0.28, respectively. To ensure the accuracy of the simulations, the element mesh is refined in the vicinity of the crack tip, and collapsed singular elements are applied. An interaction integral method [3] is used to calculate K where the number of contour integral is set to be five, and the average value of the second to fifth integral is taken as K. In the

case that the crack is oriented along Y-axis (see Fig. S2), the normal stress σxx along X-axis can be calculated by = / 2xx K rσ π . Meanwhile, we recognize σxx directly from the simulative results as the true stress. By analyzing the deviation between them, rc and δ can be achieved.

For example, for an infinite plate with a central crack of length 2a=1 μm under an applied stress 4444 MPa, the stress intensity factor K is 5.57 MPa⸱m1/2 calculated by interaction integral, shown in Fig. S3(a). This value coincides with the value from the theoretical solution K a=σ π . Then, we calculate the normal stress along the Y-axis σyy by σyy= / 2K rπ , denoted by red dots in the Fig. S3(b). The stress directly extracted from the simulation is taken as the true stress denoted by black dots in the Fig. S3(b). The green squares represent the relative error δ between the true stress and the tress calculated by K.

FIG. S3. (a) Stress intensity factor K calculated by interaction integral for an infinite plate with a central crack of length 2a=1 μm. (b) Comparison of normal stress σyy ahead of the crack tip obtained from the simulative results (black dots) and the calculation of K (red dots); green squares represent the relative error δ.

Furthermore, simulation results are compared to theoretical results. According to Ref. [4], for an infinite plate with a central crack of length 2a, the relative error δ is given by

1 1 12r ra a

δ = − + +

.

It can be seen that as r/a=0.2, δ=12.6%; r/a=0.1, δ=6.8%; r/a=0.02, δ=1.5%. The corresponding simulative results from Fig. S3(b) are r/a=0.202, δ=11.6%; r/a=0.102, δ=6.6%; r/a=0.02, δ=2.8%. The good agreement of the results indicates the validity of our simulations.

Next, we simulate the value of δ for the three-point bend specimen with a/W=0.5. The dimension is listed in Table SⅠ, and the FE model is shown in Fig. S2(b). The δ versus r curve is depicted in Fig. S4(a). It is revealed that as r/a=0.02, δ=8.4% comparable to the theoretical value about 7%. It is noted that the specification on crack length a≥2.5(KQ/σy)2 in the ASTM E399-17 is determined based on the experimental results, which is explained in detail in Ref. [5]. Reversely, we can derive rc=a/44.4 from a≥2.5(KQ/σy)2, and the corresponding δ is 9.9%, about 10%. On this basis, we take the error of 10% to calculate rc of the cantilever specimen configuration. Additionally, it is noted that 10% is also selected to analyze the effect of geometry configuration on the accuracy of the calculation of K in Section 8.4 of Mechanical Behavior of Materials [1].

The representative FE model of the cantilever with a/W=0.25 is shown in Fig. S2(a). the

specimen dimension and stress intensity factor are listed in Table SⅠ. The stress ahead of the crack tip obtained from the simulative results and the calculation of K are depicted in Fig. S4(b). As δ=0.1, rc=a/4.2, apparently larger than that of three-point bend specimen configuration. Then, considering rp=(K/σy)2/ (4 2 )π ≤rc, we identify the crack length a≥0.24(KQ/σy)2 to ensure the appropriate description of the stress field via LEFM and the small-scale yielding ahead of the crack tip.

FIG. S4. Stress σxx varies with the distance r from the crack tip and the relative error δ for three-point bend specimen configuration with for a/W=0.5 (a) and for cantilever specimen configuration with a/W=0.25.

Analogically, we calculate rc of the other cantilever configurations with different a/W. The results are listed in Table SⅠ. It can be seen that a/rc increases as a/W increases, indicating the region in which K can be used to describe the stress field becomes smaller. Based on the value of a/rc, we quantify the corresponding specification on crack length: a≥0.58(KQ/σy)2 at a/W=0.33, a≥0.72(KQ/σy)2at a/W=0.37, a≥0.87(KQ/σy)2 at a/W=0.42 and a≥1.39(KQ/σy)2 at a/W=0.5. For simplicity, in the present paper, we take a≥0.25(KQ/σy)2 for the specimen with a/W<0.3; a≥ (KQ/σy)2 for 0.3≤a/W≤0.45.

TABLE SⅠ. Summary of specimen size (crack length a, width W, thickness B and loading span L) and the stress intensity factor K used to calculate rc where the relative error δ is 10%.

a, μm W, μm a/W B, μm L, μm K, MPa⸱m1/2 rc, μm a/rc

Cantilever

0.2 0.8 0.25 0.5 2.5 1.47 0.048 4.2 0.5 1.5 0.33 1 4 1.86 0.048 10.4 2.1 5 0.42 4.5 8 3.02 0.136 15.4 5.5 15 0.37 12 24 2.96 0.428 12.8 7.5 15 0.5 12 24 4.41 0.303 24.7

Three-point Bending

5 10 0.5 5 40 3.18 0.118 42.4

Part 3: Finite Element Simulations to Solve Bending Stress

We use the thin beam elasticity theory to calculate the maximum nominal fracture stress σn=6FL/(BW2), where F is the loading force prior to the failure of the specimen, but the cantilever specimen configuration used in Ref. [6] is not an ideal thin beam. To analyze whether the result is correct, the finite element simulations have been carried out. The specimen is modeled as a

plate in plane strain condition with CPE8 elements. The material parameters are the same with these in Part 2. Because the thin beam theory does not consider the effect of stress concentration, we utilize a crack-free beam and adjust the beam width to solve σn and the maximum net-section fracture stress σe=6FL/[B(W–a)2], respectively. Two representative FE models utilized to solve σn and σe of the cantilever with a crack of length 0.2 μm are shown in Fig. S5. The model in Fig. S5(a) has the sizes of width 0.8 μm, thickness 0.5 μm and loading span 2.5 μm to solve σn, and the model in Fig. S5(b) has the sizes of width 0.6 μm, thickness 0.5 μm and loading span 2.5 μm to solve σe. The same method is used in the other beams. All results are listed in Table SⅡ. It is indicated that the stresses σn and σe calculated from the thin beam theory coincide with the values from the simulations.

FIG. S5. Two models represent the cantilevers with crack of length 0.2 μm used to obtain the maximum nominal and net-section bending stresses. (a) A beam with width 0.8 μm for σn and (b) a beam with width 0.6 μm for σe.

TABLE SⅡ. FE simulations to solve the maximum bending stress

a,

μm W, μm

B, μm

L, μm

K, MPa⸱m1/2

σn from theory,

GPa

σe from theory,

GPa

σn from FE, GPa

σe from FE, GPa

Cantilever

0.2 0.8 0.5 2.5 1.48 1.72 3.1 1.87 3.28 0.5 1.5 1 4 1.86 1.28 2.9 1.28 2.88 0.7 2.5 2.0 6.5 2.34 1.42 2.8 1.42 2.76 2.1 5 4.5 8 3.02 0.91 2.7 0.89 2.62 5.5 15 12 24 3.18 0.63 1.6 0.62 1.53 16.5 45 25.5 72 3.43 0.4 1 0.39 0.96

It can be seen that σn=1.7, 1.3, 1.4, 0.9, 0.6, and 0.4 GPa and σe=3.1, 2.9, 2.8, 2.7, 1.6, and 0.99 GPa for the specimens with a of 0.2, 0.5, 0.7, 2.1, 5.5, and 16.5 μm, respectively. As these data indicate, at small length σe at a=0.2 μm, 0.5 μm, 0.7 μm and 2.1 μm reaches an average value of eσ =2.9 GPa with a standard deviation of 0.17 (5.8% of eσ ). Based on the data at a=0.2 μm, 0.5 μm, 0.7 μm and 2.1 μm, σn at small length seems to reach an average value nσ=1.3 GPa but with an error of 0.33 (25% of nσ ), significantly larger than that in the case of σe. Therefore, we prefer to use σe as the stress criterion to judge the fracture event.

Part 4: Calculation of Critical Crack Length Here we review a few studies in literature relevant to the fracture toughness testing of brittle

materials at small scales, and calculated the critical crack length via ac=[KQ/(σfψα)]2. The fracture toughness KQ is selected from the large-scale or macroscale measurement, while the fracture strength σf is selected at the same scale as far as possible due to the size effect. Meanwhile, the corresponding ψ and α under a given specimen dimension are also calculated. We collected seven materials listed in Table SⅢ. As to the NiAl intermetallics, NiAl-hard represents the case corresponding to the orientation of <100>, while NiAl-soft corresponds to the orientation of <110>.

TABLE SⅢ. Summary of data used to calculate the critical crack length

Materials Fracture

Toughness KQ, MPa⸱m1/2

Fracture Strength σf, GPa

Crack Length a, μm

ψ α Critical Crack Length ac, μm

Tungsten [6] 3.2 2.5

0.2 1.92 0.56 1.40 0.5 2.05 0.44 1.97 0.7 1.96 0.52 1.59 2.1 2.30 0.34 2.74

1.6 5.5

2.13 0.40 5.48 16.5

Zr-BMG [7-9]

20 1.7

0.5 1.9 0.56 121

4 22

0.75 2.59 0.25 59

100 1.7

0.5 1.9 0.56 3029

4 547

0.75 2.59 0.25 1487 CoTaB-BMG

[10, 11] 5

5 1.25 2.59 0.25

2.38 11 0.49

NiAl-hard [12] 8 1.8

0.75 2.12 0.41 26

NiAl-soft [12] 4 6

0.15 941

Silicon Compounds

[13]

1

4.9 0.38 1.96 0.67 0.024 5.2 0.34 1.78 0.70 0.024 7.52

0.38 1.96 0.67 0.010 7.32 7.7

1.5

9.1 0.3 1.81 0.73 0.015 9.6 0.34 1.78 0.70 0.016 9.1 0.35 1.76 0.69 0.018 9.2 0.29 1.84 0.74 0.014

TiAl [14, 15] 18 1.5

4.8 1.92 0.56 123

4.7 2.0 0.47 160 1.91 0.58 118

4.8 1.95 0.53 137

4.6 1.97 0.50 146 4.8 1.94 0.54 133 5 1.96 0.52 140

Steel [16] 100 1.5

26350 2.66 0.22 12429 11780 2.46 0.26 10922 11910 2.49 0.25 11106 7710 2.41 0.27 10541 7830 2.44 0.26 10737 6050 2.53 0.24 11458 6210 2.60 0.23 11976 4060 2.55 0.24 11599 4560 2.94 0.19 14381

Part 5: Applicability of Our Model

It is known that the high strength steel possesses large fracture toughness above 100 MPa·m1/2 and large fracture strength about 1.5 GPa. The critical crack length, distinguishing the fracture dominated by the fracture toughness or fracture strength, is 10 mm-15 mm as listed in Table SⅢ.

A recent study [16] on the fracture toughness of precipitation hardening high strength steels shows that the larger specimen satisfying the ASTM requirements exhibits larger fracture toughness 110 MPa·m1/2. However, as the specimen size equally scales down, the measured fracture toughness decreases [16]. The fractographic observation in Figure 2 of Ref. [16] reveals that with decreasing in specimen size, the proportion of the shear fracture zone apparently increases. Generally, the shear fracture zone dissipates more energy than the flat fracture zone [17]. This is the reason why under plane stress conditions the fracture toughness obtained is significantly larger. However, in Ref. [16], the fracture toughness does not increase with the shear fracture zone. We find that σe of the small specimens is around 1.7 GPa (see Fig. S6(a)), the fracture strength derived from the linear fitting of KQ-ψαa1/2 curve as 1.6 GPa (Fig. S6(b)) and the fracture strength measured under tension 1.5 GPa in Ref. [16] are almost equal, indicating that the fracture of the small specimens should probably be controlled by fracture strength, which leads to the seeming decrease of the fracture toughness.

FIG. S6. (a) Maximum bending stress versus crack length and (b) fracture toughness versus ψαa1/2 for high strength steels with different sizes. The experimental data are from Ref. [16].

On the other hand, in the field of fatigue, it is known that Kitagawa-Takahashi diagram reveals that the fatigue crack growth is controlled by the fatigue threshold for the specimen with a large crack while by the fatigue limit for the specimen with a small crack [18, 19]. The underlying mechanism is that fatigue threshold-dominated crack extension changes to fatigue limit-dominated crack extension with decrease in crack size. According to our model, for the specimen with a small crack, the stress-induced damage facilitates the crack extension, and the critical crack length can be solved quantitatively. References 1. ASTM E399-17, Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness KIc of Metallic Materials, West Conshohocken, USA (2017). 2. N. E. Dowling, Mechanical Behavior of Materials, Section 8.40, Pearson, Boston, USA (2006). 3. H.T. Hansen, Nonlinear Static and Dynamic Analysis of Slender Structures Subjected to Hydrodynamic Loading. Doctroral Disserration, The Norwegian Institute of Technology, Tronheim (1987). 4. H. Awaji, T. Matsunaga, and S.M. Choi, Mater. Trans., 47, 1532 (2006). 5. S.J. Brown WF, Plane Strain Crack Toughness Testing of High Strength Metallic Materials. 1966: American Society for Testing and Materials. 6. J. Ast, M. Göken, and K. Durst, Acta Mater., 138, 198 (2017). 7. V. Schnabel, B.N. Jaya, M. Kohler, D. Music, C. Kirchlechner, G. Dehm, D. Raabe, and J.M. Schneider, Sci. Rep., 6, 36556 (2016). 8. C.L. Li, J.P. Chu, and J.W. Lee, Mater. Sci. Eng. A, 698, 104 (2017). 9. Z.Q. Liu and Z.F. Zhang, J. Appl. Phys., 115, 16 (2014). 10. P. Kontis, M. Kohler, S. Evertz, Y.T. Chen, V. Schnabel, R. Soler, J. Bednarick, C. Kirchlechner, G. Dehm, D. Raabe, et al., Scr. Mater., 155, 73 (2018). 11. J. Li, Y.W. Wang, J. Yi, I. Hussain, R. Li, B. Zhang, and G. Wang, J. Alloy. Compd., 680, 43 (2016). 12. F. Iqbal, J. Ast, M. Göken, and K. Durst, Acta Mater., 60, 1193 (2012). 13. K. Matoy, H. Schonherr, T. Detzel, T. Schoberl, R. Pippan, C. Motz, and G. Dehm, Thin Solid Films, 518, 247 (2009). 14. T.P. Halford, K. Takashima, Y. Higo, and P. Bowen, Fatigue Fract. Eng. Mater. Struct., 28, 695 (2005). 15. T. Tetsui, K. Shindo, S. Kobayashi, and M. Takeyama, Intermetallics, 11, 299 (2003). 16. Q.Q. Duan, B. Wang, P. Zhang, R.T. Qu, and Z.F. Zhang, Chin. J. Mater. Res., 32, 561 (2018). 17. G.N. Haidemenopoulos, Physical Metallurgy: Principles and Design, CRC Press, p. 349, Boca Raton, USA (2018). 18. H. Kitagawa and S. Takahashi, Proc. of 2nd ICM, Cleveland (1976). 19. J. Maierhofer, H. P.Gänsera, R.Pippan, Int. J. Fatigue, 70, 503 (2015).