International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Nanomechanics based theory of size effect on strength, lifetime and residual strength distributions of… Salviato, Marco; Kirane, Kedar; Bažant, Zdeněk P. 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Nanomechanics Based Theory of Size Effect on Strength, Lifetimeand Residual Strength Distributions of Quasibrittle Failure: A ReviewMarco SalviatoPost-doctoral associate, Dept. of Civil and Env. Engineering, Northwestern Univ,Evanston, IL, USAKedar KiraneGraduate research assistant, Dept. of Mechanical Engineering, Northwestern Univ,Evanston, IL, USAZdeneˇk P. BažantProfessor, Dept. of Civil and Env. Engineering, Mechanical Engineering, and MaterialScience and Engineering, Northwestern Univ, Evanston, IL, USAABSTRACT: The paper reviews a series of studies at Northwestern University which led to the establish-ment of a theory of probability distributions of short-time strength, residual strength after static preloadand lifetime of structures made of quasibrittle materials such as concrete, fiber composites and toughceramics. The theory is based on the frequency of probability of interatomic bond breaks on the atomicscale and on the multi-scale transition of power-law probability tail. The conclusion is that if the failure isnot perfectly brittle, the probability distribution of strength and lifetime is a graft of Gaussian and Weibulldistributions and varies from nearly Gaussian at the scale of one RVE to Weibullian for very large struc-tures consisting of many RVEs. As a consequence, the safety factors should depend on structure size.Numerous experimental comparisons and computational simulations are given.1. INTRODUCTIONIn most engineering applications such as bridges,dams, ships, aircraft and microelectronic compo-nents, it is essential for the design to ensure a verylow failure probability such as 10−6 throughoutthe lifetime (Bazant and Pang (2006)). Therefore,the cdf of the structure must be known up to thevery tail region, which must be established theo-retically since such small probabilities are beyonddirect experimental verification. The type of cdfof strength for perfectly ductile structures must beGaussian (from the central limit theorem), whereasfor perfectly brittle structures, it must be Weibullian(from the weakest link model with infinite links).However this is more complicated for quasibrittlematerials, which represent heterogeneous materi-als characterized by brittle constituents that are notnegligible compared to structural dimensions e.g.concrete, fiber composites, tough ceramics, rocks,and many more (Bazant and Planas (1998)). Thesebehave as ductile when small and brittle when large,thus making the type of cdf a function of the struc-ture size.The type of cdf of strength and of static life-time for quasibrittle structures, was mathemati-cally established from atomistic scale argumentsbased on nano-scale cracks propagating by manysmall, activation energy-controlled, random breaksof atomic bonds in the nanostructure (Bazant andPang (2007); J.-L. Le and Bazant (2011)). It wasshown that a quasibrittle structure (of positive ge-ometry) must be modeled by a finite (rather than in-finite) weakest-link model, and that the cdf of struc-tural strength as well as lifetime varies from nearlyGaussian to Weibullian as a function of structuresize and shape. Excellent agreement with experi-mentally obtained distributions was demonstrated.In this paper, the theory is briefly reviewed and112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015010 1Normalized time tR/lNormalized stresss/sNStrength testResidual strength testsLifetime testOABCDEFigure 1: Schematic of various load historiesextended to the probabilistic distributions of resid-ual strength after a period of sustained load. Know-ing the statistics of residual strength is importantfor meaningful estimates of safety factors by takinginto account the strength degradation of the struc-ture depending on the load history and duration. Itis also important to better estimate the remainingservice life of structures, for which maintenance de-sign is a primary concern such as modern large air-craft made of load bearing quasibrittle composites.2. THEORETICAL FORMULATIONThe nano-mechanical derivation of the cdf ofRVE strength as well as lifetime under static andcyclic loads is based on the fact that failure proba-bility can be exactly predicted only on the atomicscale because the bond breakage process is quasi-stationary, which means that the probability mustbe exactly equal to the frequency (Kramer’s rule).To derive the statistics of residual strength of anRVE, it is first noted that the crack growth rate onthe atomic scale must follow a power law of ap-plied stress with the exponent of 2. Equating thetime rates of energy disssipations on the RVE andon the atomic level explains why Evans’ law forsub-critical macrocrack growth has a much higherexponent, typically about 10 for concrete and 30 fortough ceramics (Evans (1972); M. D. Thouless andEvans (1983); Evans and Fu (1984)).Using Evans’ law to integrate the failure proba-bility contributions over time yielded a simple rela-tion be-tween the strength and static lifetime statis-tics (J.-L. Le and Bazant (2011)) – assuming themechanisms of crack growth in a strength test anda static lifetime test are the same. The argument isextended here to the statistics of residual strength.2.1. Relation between structural strength, staticlifetime and static residual strengthEvans’s law for subcritical crack growth undersus-tained load readsa˙ = Ae−Q0/kT K1n (1)Where a is the crack length, a˙ = da/dt, t=time,A=material constant, Q0 = activation energy,k=Boltzmann constant and T = absolute tempera-ture. The stress intensity factor is denoted as K1where the subscript 1 indicates the RVE level. So,we have K1 = σ√l0k1(α) where σ = F/l20 = nomi-nal stress, l0 = RVE size, α = a1/l0 = relative cracklength and k1 = dimensionless stress intensity fac-tor. Accordingly, the above equation becomes:a˙ = Ae−Q0/kTσnln/20 k1n(α) (2)Consider now the different load histories illustratedin Fig. 1. The load history O-A corresponds to thestrength test, O-B-C to a static lifetime test and O-B-D-E to a residual strength test. Integration overload history O-B-D-E provides:1r∫ σ00σndσ +∫ tRt0σ0ndt +1r∫ σRσ0σndσ =eQ0/kT∫ αcα01Aln−220 k1n(α)dα (3)By a similar integration of load histories O-A andO-B-C and appropriate substitution, one gets a verysimple correlation betweenσN ,λ , and σR asσR = [σNn+1−σn0 (n+1)(rtR−σ0)]1n+1 (4)This is the equation for the degradation of the resid-ual strength as a function of two independent (de-terministic) variables, applied load and time of sus-tained load application. This equation also repre-sents a link between the short-time strength and theresidual strength.212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150. 0.2 0.4 0.6 0.8 1n=6  20  12  26  Normalized time tR/lNormalized residual strength s R/s NFigure 2: Predicted degradation curves for variousvalues of static crack growth exponent n2.2. Analysis of residual strength degradation forone RVEWe now proceed to analyze the effect of the ex-ponent of the crack propagation law (Eq. 1) onthe residual strength degradation. Fig. 2 showsthe degradation in strength of one RVE under staticload for various values of n for applied load σ0 =0.5σN . The time of load application, normalizedwith respect to the lifetime, is shown on the hor-izontal axis. σN is assumed to be unity and theloading rate is taken as 0.5 MPa/second. It is seenthat the rate of strength degradation is negligibleinitially but progressively increases and the mostrapid degradation is seen in the end. This effectis seen to be more pronounced for higher values ofn. Based on this observation, the degradation curvecould be roughly divided in two regions, one of rel-atively slow degradation and one of rapid degrada-tion – the distinction being more pronounced forhigher values of n. This study reveals the useful-ness of Eq. 4 since for given load parameters andcrack growth exponent, one may determine the por-tion of lifetime for which the strength degradationis negligible.2.3. Formulation of statistics of residual strengthfor one RVEThe analysis of inter-atomic bond breaks andmultiscale transitions to the RVE has shown thatthe strength of one RVE must have a Gaussian dis-tribution transitioning to a power law in the tail ofprobability within the range of 10−4 to 10−3 (J.-L. Le and Bazant (2011)). Starting from the cdf ofstrength, it is now possible to determine the cdf ofresidual strength for one RVE by means of Eq. 4.This yields (M. Salviato and Bazant (2014)) :P(σR) = 1− exp[−(〈σn+1R +σA〉/sR)m] (5)for σ0 ≤ σR < σR,gr, andP1,R = Pgr +r f√2piδG∫ (σn+1R +σA)1n+1σR,gre−(σ′−µG)2/2δ 2Gdσ ′ (6)for σR ≥ σR,gr > σ0Note that in above eqs, σA =σn0 (n+1)(rtR−σ0),σR,gr = (σn+1N,gr−σA)1/(n+1), while for the parame-ters sR = sn+10 , m = m/(n+ 1); P1,R represents theprobability of failure of one RVE under an over-load, and P1,R(σ0) represents the probability of fail-ure of one RVE before the overload is applied. Notethat only the part of the cdf where the residualstrength is defined, i.e. where σR ≥ σ0 is consid-ered.Unlike the strength distribution, the residualstrength cdf of one RVE does not have a pureWeibull tail. It is noteworthy that Eq. 5 describesa three parameter Weibull distribution in the vari-able σn+1R , which has a finite threshold. Althoughit was proved that there can be no finite thresholdin the distribution of strength (J.-L. Le and Bazant(2011)), the same does not hold true for the resid-ual strength. The existence of a threshold value, σAin the cdf stems from the fact that some specimenscould fail already during the period of sustainedpreload, which excludes them from the statistics ofthe overload. These are the specimens for whichλ < tR or σN < σ0 .2.4. Formulation of residual strength cdf forstructures of any sizeOnce the cdf of residual strength related to oneRVE is found, the cdf of failure of a structure ofany size and geometry can be determined by meansof the weakest link theory. The general applicabil-ity of this theory for brittle, ductile or quasi-brittle312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015structures is guaranteed by the definition of RVE it-self and the fact that failure is considered to occur atmacro-crack initiation. One RVE is defined as thesmallest part of the structure whose failure causesthe failure of the entire structure. Thus, the RVEstatistically represents a link (the failing RVE is theweakest link) and the structure can be statisticallytreated as a chain.Similar to the definition of nominal strength, wedefine the nominal applied stress, σ0 = cnP/bDor cnP/D2 for two- or three-dimensional scaling,where P = applied load. Then, by applying thejoint probability theorem to the survival probabil-ities, the residual strength distribution of the struc-ture can be expressed as:Pf ,R = 1−N∏i=1{1−P1,R[〈σ0s(xi)〉, tR,σR]} (7)where s(x) = dimensionless stress field and x isthe position vector. Similar to the chain model forthe cdf of structural strength, the residual strengthof the ith RVE is here assumed to be governedby the maximum average principal stress σ0s(xi)within the RVE, which is valid provided that theother principal stresses are fully statistically corre-lated.3. RESULTS AND DISCUSSION3.1. Optimum fits of strength and residualstrength histograms of borosilicate glassIn this section, we determine the parameters ofthe distribution by fitting strength histograms andthen we use them to predict the cdf of residualstrength of borosilicate glasses. The predictionsare compared to experiments by (Sglavo and Renzi(1999)). Figure 3a to 3d show the experimentallyobserved strength and residual strength histogramsplotted in the Weibull scale. All the data consideredwere determined by conducting, in deionized water,four-point bend tests of borosilicate glass rods witha nominal diameter of 3 mm and length of 100 mm.The loading rate was set to about 60 MPa/s and dif-ferent sustained load durations were used. Sinceglass is a brittle material and its RVE size is verysmall compared to the tested specimen size, the dis-tribution of strength is virtually indistinguishablefrom the Weibull distribution, as can be seen in Fig3. By the optimum fitting of strength and resid-ual strength, a Weibull modulus m of about 6 and avalue of n of about 30 have been estimated. The fitpredicted by the statistical formulation, shown bythe solid line curves, is seen to be in good agree-ment with the experimental results. Except for theone hour case, all the other plots show the devia-tion of the residual strength distribution from thestrength distribution to reach the probability value.It should be emphasized that, despite the scatter anda low number of data, all the residual strength distri-butions are predicted using the same set of parame-ters.3.2. Optimum fit of strength histograms and pre-diction of lifetime and mean residual strengthfor unidirectional glass/epoxy compositesThe methodology of the previous section is nowpursued for the strength, lifetime and residualstrength data on unidirectional glass-epoxy com-posites re-ported by (Hahn and Kim (1975)). Eachspecimen analyzed consisted of 8 unidirectionalplies. 71 specimens were tested to obtain thestrength and lifetime distributions. A constant sus-tained load 758 MPa was applied for all the lifetimetests. Fig. 4a shows the fit of strength histogramsby means of the grafted Gauss-Weibull distributionin the Weibull scale. This fit shows a kink in thecurve corresponding to the transition from Weibullto Gaussian distribution. A value of m equal to56 and a value of n equal to 27 are estimated byleast-square optimum fitting. Now that the requiredparameters of the distribution have been identified,the theory is applied to predict the mean residualstrength and compare it to the experimental data.The comparison is made only for the mean since thenumber of available data is not sufficient to studythe entire cdf. The resulting cdf of residual strengthis then used to compute the mean values. The re-sults are shown in Figure 4b for the different initialoverloads and durations considered. Note that thepredictions agree with the experiments, the differ-ence being always less than 7%. The agreementprovides another support for the present theory.412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015- 4- 3- 2-10124 4.5 5 5 .5 6Strength dataResidual strength dataPredicted distributiont = 1 hourσ0= 97  4 MPaBorosilicate glass, Sglavo 1999a)  Failure probability (Weibull scale) ln[-ln(1-P f)]ln sR  - 4- 3- 2-10124 4.5 5 5.5Strength dataResidual Strength dataPredicted distributiont = 1 dayσ0= 82  8 MPaBorosilicate glass, Sglavo, 1999b)  Failure probability (Weibull scale) ln[-ln(1-P f)]ln sR  - 4- 3- 2-10124 4.5 5 5.5Strength dataResidual strength dataPredicted distributiont = 10 aysσ0= 81  9 MPaBorosilicate glass, Sglavo, 1999c )  Failure probability (Weibull scale) ln[-ln(1-P f)]ln sR  - 4- 3- 2-10124 4.5 5 5.5Str ngth ataResidual strength dataPredicted distributiont = 20 daysσ0= 77  3 MPaBorosilicate glass, Sglavo, 1999d)  Failure probability (Weibull scale) ln[-ln(1-P f)]ln sR  Figure 3: Optimum fits of residual strength histogramsfor borosilicate glass Hold times : (a) 1 hour (b) 1 day(c) 10 days and (d) 20 days- 6- 5- 4- 3- 2-101236.6 6.7 6.8 6.9 7Strength dataFit55.941Unidirectional glass/epoxy compositeHahn and Kim 1975a)  Failure probability  (Weibull scale) ln[-ln(1-P f)]ln sN Hold time [min]  05001000150010 15 20 40Experimental data Predicted'=6.6% '=1.9% '=0.9%'=4.6%c )  Mean s R [MPa]Figure 4: (a) Optimum Gauss Weibull fit of strengthhistogram (b) Comparison of predicted mean resid-ual strength for unidirectional glass/epoxy composite(Hahn and Kim, 1975)3.3. Size effect on mean residual strengthA more severe check on the theory would be totest the size effect on the mean lifetime and resid-ual strength. However, no such test data seem to beavailable in the literature. It is nevertheless interest-ing to predict the size effect on the mean residualstrength integrating Eq. 7. Figure 5 shows the cal-culated size effect on the mean residual strength of99.6% Al2O3 . The set of parameters of the distri-bution is determined from the strength and lifetimehistograms reported in (Fett and Munz (1991)). Anapplied load σ0 = 0.78σN is considered. Differenttimes of load application are used, as reported in thefigure, depending on the mean strength, i.e., rtR =βσN . Note that, for a given tR, the mean residualstrength shows a similar trend as the strength andlifetime for the large size limit. In fact, the meanstend to a straight line with the same slope as themean strength.It is impossible to obtain closed-form analyticalexpressions for the mean residual strength. How-ever, sufficiently accurate analytical formulas canbe derived by asymptotic matching. The size effect512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015-0.4- 2 4 6StrengthtR=20 s0tR=50 s0tR=10 5 s0Size effect fits99.6% Al2O3Rel. residual strength log(s R/s 0)Rel. size log(D/D 0 )  Figure 5: Calculated size effect curves on the meanstrength residual strength at different hold times for99.6% Al2O3can reasonably be approximated by the equation:σR =[MaD+(MbD)η/m]1/η(8)where m is the Weibull modulus of the cdf ofstrength and Ma,Mb and η can be derived by match-ing three asymptotic conditions:1. [σR]D→l02. [dσR/dD]D→l0 and3. [σRD1/m]D→∞As can be noted from Figure 5, the approxima-tion given by Eq. 8 is rather good for all the differ-ent times of load application. In deriving the fore-going result, the two ratios, i.e., the applied loadto strength and the hold time to lifetime, were keptconstant across the sizes. It is trivial to note how-ever that if the absolute value of the applied load orthe hold time, or both, are kept constant, the sizeeffect will of course be much stronger. However,in this case, the mean residual strength does not re-semble the strength curve and it cannot be describedby Eq. 8.4. CONCLUSIONS• A theory for predicting the probabilistic dis-tributions of residual strength after a period ofstatic load has been developed and validatedagainst test data. An important practical meritof the present theory combined with predeces-sors (Bazant and Pang (2006, 2007); J.-L. Leand Bazant (2011)) is that it provides a way todetermine the strength, residual strength andlifetime distributions without any histogramtesting.• The rate of degradation of strength under aconstant static load is not constant. Initially itis very slow and in the end very rapid. This ef-fect is more pronounced for higher static crackgrowth exponents.• The cdf of residual strength of quasibrittle ma-terials may be closely approximated by a graftof Gaussian and Weibull distributions. In theleft tail, the distribution is a three parameterWeibull distribution in the variable . Unlikethe cdf’s of strength and lifetime, the cdf ofresidual strength has a finite threshold, albeitoften very small.• The finiteness of the threshold is explained bythe fact some specimens may fail during thesustained static preload and are thus excludedfrom the statistics of overload.• An expression for the size effect on theresidual strength is derived using asymptoticmatching. It is shown that the size effect onthe residual strength is as strong as the size ef-fect on strength.• Good agreement with the existing test dataon glass-epoxy composites and on borosilicateand soda-lime silicate glasses is demonstrated.5. REFERENCESBazant, Z. P. and Pang, S. D. (2006). “Mechanics basedstatistics of failure risk of quasibrittle structures andsize effect on safety factors.” Proc. Natl. Acad. SciUSA, 103(25), 9434–9439.Bazant, Z. P. and Pang, S. D. (2007). “Activation energybased extreme value statistics and size effect in brittleand quasibrittle fracture.” J. Mech. Phys. Solids, 55,91–134.Bazant, Z. P. and Planas, J. (1998). “Fracture and size ef-fect in concrete and other quasibrittle materials.” CRCPress.Evans, A. G. (1972). “A method for evaluating thetime-dependent failure characteristics of brittle mate-rials and its application to polycrystalline alumina.” J.Mater. Sci., 7, 1146–1173.Evans, A. G. and Fu, Y. (1984). “The mechanical be-haviour of alumina : in fracture in ceramic materials.”Noyes Publications, Park Ridge, NJ, 56–88.Fett, T. and Munz, D. (1991). “Static and cyclic fatigue612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015of ceramic materials. in: Vincenzini, p. (ed.), ceram-ics today—-tomorrow”s ceramics.” Elsevier SciencePublisher B.V., 1827–1835.Hahn, H. and Kim, R. Y. (1975). “Proof testing of com-posite materials.” J. Compos. Mater., 9, 297–311.J.-L. Le, Z. B. and Bazant, M. (2011). “Unified nano-mechanics based probabilistic theory of quasibrittleand brittle structures: I. strength, static crack growth,lifetime and scaling.” J. Mech Phys Solids, 59, 1291–1321.M. D. Thouless, C. H. H. and Evans, A. G. (1983). “Adamage model of creep crack growth in polycrystals.”Acta Metal., 31(10), 1675–1687.M. Salviato, K. K. and Bazant, Z. P. (2014). “Statisticaldistribution and size effect of residual strength aftera period of constant load.” J. Mech Phys Solids, 64,440–454.Sglavo, V. M. and Renzi, S. (1999). “Fatigue limit inborosilicate glasses by interrupted static fatigue test.”Phys. Chem. Glasses, 40(2), 79–84.7


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