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Scaling of strength and lifetime probability distributions of quasibrittle structures based on atomistic fracture mechanics

Contributed by Zdenêk P. Baẑant, April 30, 2009 (received for review January 27, 2009)
Abstract
The failure probability of engineering structures such as aircraft, bridges, dams, nuclear structures, and ships, as well as microelectronic components and medical implants, must be kept extremely low, typically <10^{−6}. The safety factors needed to ensure it have so far been assessed empirically. For perfectly ductile and perfectly brittle structures, the empirical approach is sufficient because the cumulative distribution function (cdf) of random material strength is known and fixed. However, such an approach is insufficient for structures consisting of quasibrittle materials, which are brittle materials with inhomogeneities that are not negligible compared with the structure size. The reason is that the strength cdf of quasibrittle structure varies from Gaussian to Weibullian as the structure size increases. In this article, a recently proposed theory for the strength cdf of quasibrittle structure is refined by deriving it from fracture mechanics of nanocracks propagating by small, activationenergycontrolled, random jumps through the atomic lattice. This refinement also provides a plausible physical justification of the power law for subcritical creep crack growth, hitherto considered empirical. The theory is further extended to predict the cdf of structural lifetime at constant load, which is shown to be size and geometrydependent. The size effects on structure strength and lifetime are shown to be related and the latter to be much stronger. The theory fits previously unexplained deviations of experimental strength and lifetime histograms from the Weibull distribution. Finally, a boundary layer method for numerical calculation of the cdf of structural strength and lifetime is outlined.
A comprehensive theory of the statistical strength distribution exists only for perfectly brittle structures failing at macrocrack initiation from a negligibly small representative volume element (RVE) of material. The objective of the present article is to extend recent studies (1–3) by developing a comprehensive and atomistically based theory for strength and lifetime distributions of structures consisting of quasibrittle materials. These materials, which include fiber composites, concretes, rocks, stiff soils, foams, sea ice, consolidated snow, bone, tough industrial and dental ceramics, and many other materials on approach to nanoscale, are characterized by a RVE that is not negligible compared with structure size D. The space does not allow commenting on previous valuable contributions made by Coleman (4), Zhurkov (5, 6), Freudenthal (7), Phoenix (8, 9) and others (10, 11) (for such comments, see, e.g., refs. 2, 3, 12, and 13).
Fracture Kinetics on Atomic Scale.
Consider a nanoscale size atomic lattice block undergoing fracture (Fig. 1 A). The separation δ between the opposite atoms across the crack gradually increases by small increments as the distance from the crack front grows. The work of the force F _{b} transmitted across each pair of opposite atoms on their relative displacement δ defines a certain local potential Π_{1}(δ), which is a part of the overall potential function Π (or free energy) of the atomic lattice block. The equilibrium states of these atomic pairs (bonds) are marked on the local potential curves Π_{1}(δ) by circles. As the fracture separation grows from one atomic pair to the next, the state marked by the circle moves on the curve Π_{1}(δ) up and right (Fig. 1 B). These states are also shown on the corresponding curves of bond force F _{b}(δ) = ∂Π_{1}/∂δ between the opposite atoms in each pair (Fig. 1 C). The local bond failure begins when the peak point of the curve F _{b}(δ) is reached. This point corresponds to the point of maximum slope of the curve Π_{1}(δ) (state 3), and represents the end of the cohesive crack [an old idea attributed to Barenblatt (14)]. The true crack ends at the pair where the bond force is reduced to 0 (state 1).
The fracture process zone (FPZ) in the lattice spans approximately from state 1 to state 3, which lies many atoms apart. As the fracture propagates, the diagram P(u) of load P = ∂Π/∂u applied on the atomic lattice block versus the associated displacement u caused by elasticity of lattice and by fracture growth would have the usual shape shown in Fig. 1 D if the lattice were treated as a continuum (the curvature of the rising portion is caused by finiteness of the length of growing FPZ). On the nanoscale, however, the lattice is not a continuum. The fracture advances by random jumps over the activation energy barriers Q on the surface of the state potential Π (free energy) of the atomic lattice block. These barriers cause the diagram P(u) to be wavy; see Fig. 1 E, where the radial lines marked as a _{1},a _{2},… correspond to the unloading lines for subsequent crack lengths ending at different atomic pairs marked in Fig. 1 A.
The interatomic crack propagates by jumps equal to the atomic spacing δ_{a}. During each jump, one barrier on the potential Π as a function of u must be overcome (see the wavy potential profile in Fig. 1 G). After each jump, at each new crack length, there is a small decrease (Fig. 1 F and G) of the overall potential Π of the atomic lattice block, corresponding to a small advance along the loaddeflection curve P(u) (Fig. 1 E). An important point is that the separation of opposite atoms (in their equilibrium positions) increases during each jump by only a small fraction of their initial distance δ_{a}.
Because of thermal activation, the states of the atomic lattice block fluctuate and can jump over the activation energy barrier in either direction (forward of backward, Fig. 1 F), although not with the same frequency. When crack length a (defined by the location of state 3 in Fig. 1 A–C) jumps by one atomic spacing, i.e, from a _{i} to a _{i} + δ_{a}, i = 1,2,3,…), the activation energy barrier Q changes by a small amount, ΔQ, corresponding to the energy release by fracture (Fig. 1 F and G) that is associated with the equilibrium load drop P caused by fracture (Fig. 1 E).
To calculate ΔQ, consider planar 3dimensional cracks that grow in an affine, or selfsimilar, manner (e.g., expanding concentric circles or squares). The FPZ of the lattice crack cannot be expected to be negligible compared with the size l _{a} of the atomic lattice block, and so we will use the approximation of equivalent linear elastic fracture mechanics (LEFM) in which the tip of an equivalent sharp LEFM crack is considered to lie approximately in the middle of the FPZ. The LEFM stress intensity factor (of mode I, II, or III) is in general expressed as
The energy release rate of the crack (with respect to crack length rather than time), per unit length of crack perimeter, is G_{a}(α) = K _{a} ^{2}/E _{1} = k _{a} ^{2}(α) l _{a}τ^{2}/E _{1}. Here E _{1} = Young's (elastic) modulus for a continuum approximation of the lattice (which is larger than the macroscopic Young's modulus E); k _{a} ^{2}(α) represents the dimensionless energy release rate function of LEFM for continuous bodies, characterizing the geometry of fracture and of the atomic lattice block (15) (if the block boundaries are distant, k _{a} ^{2}(α) ∝ γα). Let γ_{1} = geometry constant such that γ_{1}a = γ_{1}αl _{a} = crack perimeter, the crack being assumed to grow radially in an affine manner (γ_{1} = 2π for a circular crack of radius a = αl _{a}, and γ_{1} = 8 for a square perimeter crack). Similar to the expression in ref. 3 (derived by assuming 2 rather than 3dimensional cracks), the increment of energy that is released when the crack advances by δ_{a} along its entire perimeter of length γ_{1}αl _{a} is: Here, Π* = complementary energy potential (Gibbs' free energy) of the atomic lattice block, and V _{a}(α) = δ_{a}(γ_{1}αl _{a} ^{2})k _{a} ^{2}(α) = activation volume. [Note that if the stress tensor is written as τ s, where τ = stress parameter, one may write V _{a} = s: v _{a}, where v _{a} = activation volume tensor, as in the atomistic theories of phase transformations in crystals (16).]
Because the cohesive crack is much longer than δ_{a}, the separation δ changes very little during each crack jump by one atomic spacing δ_{a}, and so the activation energy barrier for a forward jump, Q _{0} − ΔQ/2, differs very little from the activation energy barrier for a backward jump, Q _{0} + ΔQ/2 (Q _{0} = activation energy at no stress) (Fig. 1 F). Note that multiple activation energy barriers Q _{0} = Q _{1},Q _{2},… are always present; however, the lowest one always dominates. The reason is that the factor e ^{−Q1/kT} is very small, typically 10^{−12} (e.g., if Q _{2}/Q _{1} = 1.2 or 2, then e ^{−Q2/kT} = 0.0043e ^{−Q1/kT} or 10^{−23} e ^{−Q1/kT} and thus makes a negligible contribution; and if Q _{2}/Q _{1} = 1.02, then Q _{1} and Q _{2} can be replaced by a single activation energy Q _{0} = 1.01Q _{1}). Consequently, the jumps from one metastable state to the next on the surface of the atomic lattice block potential Π* must be happening in both forward and backward directions, although at slightly different frequencies. According to the transition rate theory (17, 18), the firstpassage time for each transition (in the limit of a large freeenergy barrier, Q _{0} ≫ kT) is given by Kramer's formula (19). Thus, the net frequency of crack length jumps is (3): where V _{T} = 2E _{1}kT/τ^{2}; ν_{T} is the characteristic attempt frequency for the reversible transition, ν_{T} = kT/h, where h = 6.626 · 10^{−34} Js = Planck constant = (energy of a photon)/(frequency of its electromagnetic wave); T = absolute temperature; and k = 1.381 · 10^{−23} J/K = Boltzmann constant. Since δ_{a} is of the order of 0.1 nm, V _{a} ∼ 10^{−26} m^{3}. Volume V _{T} depends on τ = cσ where c is expected to be > 10. For example, in the nanostructure of hardened Portland cement gel, the nanoscale remote stress may perhaps be τ ≈20 MPa, which gives V _{T} ∼ 10^{−25} m^{3}, and so V _{a}/V _{T} ≪ 1. Therefore, Eq. 3 becomes f _{1} ≈e ^{−Q0/kT}[ν_{T}V _{a}(α)/kT]τ^{2}/E _{1}. Denoting H _{T} = e ^{−Q0/kT}(δ_{a}l _{a} ^{2}/E _{1}h) and F _{T} = γ_{1}αk _{a} ^{2}(α)H _{T}, we thus have:
The fact that frequency f _{1} is a power law of stress τ with a zero threshold is essential for the forthcoming arguments. Important for the foregoing derivation is that ΔQ ≪ kT ≪ Q _{0} or τ ≪ (E _{1}kT/V _{a})^{1/2}. Another noteworthy point is that stressindependent crack front diffusion (as a random walk) should be negligible because the Péclet number Pe ≫ 1 [Pe = 2f _{1}l _{a}/δ_{a}f = 4(l _{a}/δ_{a})(V _{a}/V _{T}), where f = ν_{T}e ^{−Q0/kT}].
In classical macroscale continuum fracture, by contrast, the jumps representing fracture reversal play no role because the crack face separations of interest equal many atomic spacings, rather than a small fraction of one atomic spacing; ΔQ is thus entirely negligible and the freeenergy change is smooth (as shown in Fig. 1 G) (this point was made in ref. 20, figure 1.18).
Failure Frequency and Probability at Nanoscale.
The atomic lattice fails when the nanocrack propagates from its original length a _{0} to a certain critical length a _{c} at which the crack looses stability and propagates dynamically [this occurs at the point of the P(u) curve in Fig. 1 E where the downward tangent slope becomes equal in magnitude to the stiffness of embedment of the lattice block in the surrounding microstructure (21)]. To get to a _{c}, the crack experiences many length jumps, say n jumps, where the frequency of each jump is given by Eq. 4. At the atomic lattice scale, it may be assumed, in general, that each jump is independent (i.e., the frequency of the jump is independent of the particular history of breaking and restoration sequences that brought the nanocrack to the current length) (20). The failure probability of the atomic lattice on sudden application of stress can thus be written as
NanoMacro Transition of Failure Probability.
We consider the broad class structures that fail as soon as a macrocrack initiates from one RVE (15, 22, 23) (they are called the positive geometry structures, characterized by ∂K/∂a > 0; K = stress intensity factor). Their strength is statistically modeled by a chain of RVEs, i.e., by the weakestlink model. However, in contrast to Weibull theory, the number of RVEs (or links in the chain) is not infinite because the RVE size is not negligible. For softening damage and failure, and in contrast to the homogenization theory, the RVE must be defined as the smallest material volume whose failure triggers the failure of the whole structure (1). Typically, the RVE size l _{0} is ≈2 to 3 inhomogeneity sizes (2, 25).
The transition from the nanoscale of atomic lattice crack to the macroscale of RVE can be represented, for statistical purposes, by a hierarchy of series and parallel couplings (i.e., chains of links and bundles of fibers) (see ref. 1, figure 1E). In such couplings, a power law tail of strength cumulative distribution function (cdf) is indestructible (2). The series couplings preserve the tail exponent and extend the reach of the powerlaw tail. In parallel couplings, the tail exponents of the elements are additive [as proven for both brittle (2, 24) and plastic or softening elements (2)], and the reach of the powerlaw tail shortens drastically. The cdf of strength of one RVE can be approximated by a Gaussian core onto which a farleft Weibull tail is grafted at failure probability P _{gr} ≈ 10^{−4} − 10^{−3} (ref. 1, equations 6 and 7) [in more detail (2)]. The meaning of Weibull modulus m, which is typically 10 to 50, is the minimum of the sum of the Weibull tail powers among all possible cuts separating the hierarchical model into two halves.
The elements of the hierarchical model may be imagined to represent atomic lattice cracks, for which the tail exponent of strength cdf is 2 (Eq. 5). Isn't that in conflict with the preceding simpler model (1), in which the elements of the hierarchical model had the tail exponent of 1? No, because the elements in ref. 1 represented the break of a bond between a pair of atoms, which occurs one scale lower than the break of atomic lattice block. According to the hierarchical model, the passage from that scale to the scale of atomic lattice should raise the exponent, and the present analysis shows it should raise it to 2. Although refs. 1 and 2 led to equivalent results on the RVE level, the treatment of interatomic pair breaks in refs. 1 and 2 was oversimplified because the present nearsymmetry of forward and backward activation energy barriers Q _{0} ± ΔQ does not exist for isolated atomic pairs. Fracture mechanics of the atomic lattice (Eq. 4), as introduced here and in ref. 3, is essential.
Size Effect on Strength Distribution.
The chain of N RVEs survives if all its elements survive. So, according to the joint probability theorem, and under the hypothesis of independent random variables the failure probability of the structure, P _{f}, follows the equation: Here, σ_{N} = c _{g}P _{max}/bD = nominal strength of structure (22, 23, 25), b = structure width, and c _{g} = arbitrarily chosen dimensionless geometry parameter (for a suitable choice of c _{g}, σ_{N} = maximum normal stress in the structure); σ_{i}( x _{i}) = σ_{N}s( x _{i}) = maximum principal macroscale stress at the center x _{i} of the i th RVE; s( x) = dimensionless stress field; P _{1}(σ) = grafted Gauss–Weibull cdf of the strength of a single RVE (ref. 1, equations 6 and 7).
As N increases, the Weibull tail gradually spreads and the Gaussian portion of cdf shrinks until virtually the entire cdf becomes Weibullian, which means that the structure becomes brittle and occurs when the equivalent number of RVEs N _{eq} ≥ N _{b}, where N _{b} = 5/P _{gr} (typically 5 × 10^{3}). For N _{eq} < N _{b}, P _{f} must be computed from Eq. 6 (N _{eq} results by scaling N according to the stress field (1, 2); for a homogeneous stress field, N _{eq} = N).
Macrocrack Growth Law as Consequence of Subcritical Nanocrack Growth Rate.
To predict the lifetime of structures, we need to know the frequency of crack jumps that governs the rate of growth, ȧ, of the atomic crack. According to Eq. 4,
Eqs. 7 and 8 have different exponents but the same form, except that, unlike A, factor ν_{1} depends on the length of nanocrack. However, the FPZ does not change significantly as it travels through the structure (which is a central tenet behind the constancy of G _{f}). Therefore, all the different relative crack lengths α in the nanostructure of a FPZ must average out to give a constant A, which is independent of the length of macrocrack. Note that parameter A depends on structure size [just like the size effect on Paris' law (15)].
To explain the difference in exponents, we may use the condition that the energy dissipation power of the macroscale crack a must be equal to the combined energy dissipation power of all the active nanocracks a _{i} (i = 1,…N) in the FPZ of the macroscale crack. So,
On each material scale, the number of activated cracks q within the FPZ of the next higher scale μ must be a function of the relative stress intensity factor K/K _{μ}, i.e., q = q(K/K _{μ}), where K _{μ} = critical value of K for cracks of scale μ. It may be expected that the function q(K/K _{μ}) increases rapidly with increasing K, whereas the ratio v _{i}ω_{i} ^{4} E/E _{i} varies far less. Therefore, one may replace E _{i}, ω_{i}, and ν_{i} by some effective mean values E _{a}, ω_{a}, and
Since there appears to be no characteristic value of K at which the behavior of q(K/K _{μ}) would qualitatively change, function q(K/K _{μ}) should be selfsimilar, i.e., a power law, q(K/K _{μ}) = (K/K _{μ})^{r}. Consequently, function ϕ(K) should be a power law as well: Setting rs + 2 = n and substituting ϕ(K) back to Eq. 10, one thus finally obtains the power law for macrocrack growth.
According to the foregoing arguments, the main reason why the powerlaw exponent of crack growth rate increases from 2 at the atomic scale to n ≈10 to 30 at the macroscale is that the number N of energy dissipating nanocracks steeply increases with the applied macrostress. Nevertheless, these arguments merely constitute a plausible explanation but not proof. The fact that they deliver agreement with experiments is essential.
Distribution of Structural Lifetime.
Consider load histories in which applied stress σ is first raised rapidly to a value σ_{0} less than the shorttime strength, then is held constant for a certain time period Δt, and after that is raised rapidly up to failure, occurring at stress σ*. When Δt is increased from 0 to arbitrarily large periods of time, there is a continuous transition of failure stress σ* from the shorttime failure stress σ_{N}, obtained in a typical laboratory strength test, to failure stress σ* = σ_{0} obtained in a test of lifetime λ at constant stress σ_{0}. Therefore, the same failure criterion, based on the same theory, must apply to the shorttime strength and to the lifetime.
Now consider a dominant subcritical macrocrack of length a _{R} growing within the RVE. Its stress intensity factor may be expressed as
Eliminating I from Eqs. 13 and 14, one gets the following simple relationship between σ_{N} and λ: where β = [r(n + 1)]^{1/(n+1)} = constant.
Eq. 15 makes it possible to calculate the cdf of lifetime of one RVE from the cdf of strength of the RVE, defined as Gauss–Weibull grafted distribution in equations 6 and 7 of ref. 1. The result is where λ_{gr} = β^{−(n+1)}σ_{0} ^{−n}σ_{N,gr} ^{n+1}, λ_{1} = γλ_{gr} ^{1/(n+1)}, λ_{2} = γλ^{1/(n+1)}, γ = βσ_{0} ^{n/(n+1)}, s _{λ} = s _{0} ^{n+1}β^{−(n+1)}σ_{0} ^{−n}, and μ_{G},δ_{G},s _{0} are statistical parameters for the strength cdf as defined in equations 6 and 7 of ref. 1. For one RVE, the cdfs of both strength and lifetime have Weibullian tails, but with very different exponents. The Weibull modulus for the lifetime cdf is which is significantly lower than the Weibull modulus m for the strength cdf. However, the grafting probability P _{gr} is the same for both. In contrast to the core of strength cdf, the core of lifetime cdf is not even approximately Gaussian.
Based on the finite chain model, the lifetime distribution of a structure can be calculated from the joint probability theorem: Similar to the size effect on the cdf of strength, the lifetime distribution of large structures depends only on the tail of the lifetime cdf of one RVE. As the structure size is increased, the cdf of lifetime of the whole structure will eventually converge to the Weibull cdf, i.e., P _{f}(λ) = 1 − exp[−(λ/S)^{m/(n+1)}].
Optimum Fitting of Strength and Lifetime Histograms.
The observed strength histograms for ceramics, concrete, and fiber composites (10, 30, 31) typically exhibit in Weibull scale a kink that separates two segments. The lower segment is a Weibull straight line and the upper one deviates from this line to the right. These histograms have often been fitted by the 2parameter Weibull distribution, which has a zero threshold. However, systematic deviations always occurred unless the data were too few and the structure size ≫ RVE.
Fig. 1 H–K documents further close fits of histograms of flexural strength of various industrial and dental ceramics (32–34). Previous investigators fitted them closely by the 3parameter Weibull distribution with a nonzero threshold, though with unreasonably small Weibull moduli m. It is now clear that close fits were possible because only a few hundred tests were made. Weibull' s data for concrete (30) (with about 5,000 tests still being the most extensive to date) cannot be fitted closely by the 3parameter Weibull distribution (ref. 2, figure 3d). Further similar histograms of ceramics have also been fitted closely by the present theory (12, 13).
In logarithmic plots of size effect on the mean structural strength, the Weibull theory gives a straight line, whereas tests show a large upward deviation for smaller sizes; see figure 1q in ref. 23 for concrete and figure 2b in ref. 35 for laminates. As for ceramics, unfortunately, no size effect tests accompanied the histogram testing, although it would have been the easiest way to detect the inadequacy of Weibull distribution.
Fig. 1 L–O documents that the present theory fits closely the lifetime histograms of organic fiber (Kevlar 49) composites (36), whereas the 2parameter Weibull cdf does not. The test temperatures were 100– 110° C. The specimens were bars under uniform uniaxial tensile stress equal to ≈70% of the mean strength of the specimen.
Similar to the strength histograms, the lifetime histograms in Weibull plots are again found to have two segments separated by a kink, which cannot be fitted by the 2parameter Weibull distribution. The present theory fits the entire histograms well. The lower segment is a straight line, whose slope represents the Weibull modulus m _{λ}. The fits show that m _{λ} ≈2.3–3, which is significantly lower than Weibull modulus m of the strength distribution [for Kevlar 49 fiber composites, typically m = 40 –50 (8)]. From Eq. 18, the crack growth rate exponent n ≈ 16.
Effective Experimental Determination of Weibull Modulus of Lifetime Distribution.
Determining the Weibull modulus of lifetime distribution by histogram testing is time consuming and costly and is rendered unnecessary by Eq. 18. One merely needs the Weibull modulus m of the strength distribution, which is best determined from mean size effect tests (13). The crack growth rate exponent n can then be obtained by standard tests that measure the subcritical crack growth velocity.
Size Effect on Mean Structural Strength and Lifetime.
The mean of strength as well as lifetime for a structure with any number of RVEs may be calculated as
The mean strength and lifetime must be related by Eq. 15. Therefore, the mean size effect on lifetime may be written as: where m = Weibull modulus of strength cdf, and n = crack growth rate exponent. Parameters C _{a}, C _{b}, and r can again be determined by three asymptotic matching conditions. Clearly, the size effect on the mean structural lifetime is far stronger than the size effect on the mean structural strength. This phenomenon is intuitively plausible.
Generalization for Boundary Layer and Nonlocal Interior.
Direct use of Eq. 6 to compute P _{f} is possible if the structure can be subdivided into elements of equal size. Such a subdivision is impossible for general bodies (Fig. 1 P). One might think of replacing the finite sum corresponding to the weakestlink model with a nonlocal integral over the structure volume V, similar to the nonlocal models for structures with softening distributed damage (39). However, these models lead to troublesome, still unresolved, problems with the treatment of boundaries when the nonlocal integration domain protrudes beyond the body surface.
Therefore, we need a different idea. We separate from the interior volume V _{I} a boundary layer of thickness h _{0} ≈ l _{0} = size of the RVE. In V _{I}, a nonlocal continuum treatment is free of trouble since, for the points in V _{I}, the nonlocal integral domain cannot protrude outside the body surface if it chosen to extend no further than distance h _{0} from the center point (Fig. 1 Q). In the boundary layer, one can introduce nonlocal integration domains for the points of the middle surface, which also cause no trouble with the boundary condition since this surface is closed (i.e., has no boundaries). Eq. 6 thus yields:
where P _{1}(σ) is the grafted Gauss–Weibull cdf of RVE strength. The first integral runs over all the points x _{M} of the middle surface Ω_{M} of the boundary layer, and the second integral runs over all the points x in the interior V _{I}; and
Concluding Remarks.
A detailed development of numerical implementation, including the boundary layer theory for quasibrittle failure probability, is beyond the scope of this article and is relegated to a separate article. The present results are bound to impact the safety and lifetime assessments of large loadbearing fiber composite parts for modern fuelefficient aircraft and ships, and large concrete structures and rock masses. They will also affect the reliability and lifetime assessments of microelectronic devices. If the buildup of residual stresses in the microstructure can be successfully introduced into the formulation, a comprehensive unified theory encompassing also fatigue under cyclic loading is in sight. Finally, note that a mathematically analogous formulation can describe the lifetime statistics of electrical breakdown of gate dielectrics (41).
Acknowledgments
This work was supported in part by National Science Foundation Grant CMS0556323 and Boeing, Inc. Grant N007613.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: zbazant{at}northwestern.edu

Author contributions: Z.P.B. designed research; Z.P.B., J.L.L., and M.Z.B. performed research; J.L.L. analyzed data; and Z.P.B. and J.L.L. wrote the paper.

The authors declare no conflict of interest.
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