Lecture 6 Fracture strength and testing methods

Transcription

1 ADVANCED DESIGN OF GLASS STRUCTURES Lecture 6 Fracture strength and testing methods Viorel Ungureanu CZ-ERA MUNDUS-EMMC

2 Glass is not able to yield plastically (no stress redistribution) thus its fracture strength is very sensitive to stress concentrations. Since surface flaws cause high stress concentrations, the characterization of the fracture strength of glass must incorporate the behaviour of such flaws. Fracture BEAM Edge L6 Fracture flaw caused strength by and grounding testing methods Surface flaw on an accessible glazing 2

3 Glass fracture mechanics Fracture The initial acceleration of a flaw starts on a relatively smooth surface known as the mirror zone. As the flaw continues to accelerate, the higher stress and greater energy release produce some form of micro mechanical activity close to the crack tip, producing severe surface roughening that finally causes the crack to bifurcate or branch along its front. An elevation of the crack surface will reveal a progressive increase in the roughness of the fracture surface from mirror to mist to hackle. The mirror radius R is approximately 8 to 16 times larger than the initial flaw depth a R a 3

4 Glass fracture mechanics Fracture one (critical) flaw flaw population INERT STRENGTH LEFM (short-term) LEFM + PROB AMBIENT STRENGTH LEFM + (long-term) LEFM + + PROB LEFM : Fracture : Subcritical Crack Growth PROB: Theory of Probability 4

5 Linear elastic fracture mechanics Fracture STEEL or CONCRETE: homogenous TIMBER or GLASS: not homogenous: defects test strength = strength of the material test strength << strength of material σ n < critical stress σ n.y.(π.a) 0.5 < critical value σ n : uniform stress σ n : uniform stress Y: correction factor (defects) a: depth of defects (flaws) critical value: material constant 5

6 Linear elastic fracture mechanics Fracture There is stress magnification near the tip of a crack. The stress intensity factor K I : elastic stress intensity near a crack tip. Provides a means to characterize the material in terms of its fracture toughness. K = Y. σ. π a I n. K I : stress intensity factor [MPa.m 0.5 ] Y : geometry factor [-] σ n : stress normal to the flaw s plane [MPa] a : flaw depth [m] Instantaneous failure of glass occurs when the elastic stress intensity K I, due to tensile stress at the tip of a crack, reaches or exceeds a critical value. This critical value is a material constant known as the fracture toughness or the critical stress intensity factor K IC. 6

7 Linear elastic fracture mechanics Fracture K = Y. σ. π a I n. K I : stress intensity factor [MPa.m 0.5 ] Y : geometry factor [-] σ n : stress normal to the flaw s plane [MPa] a : flaw depth [m] a Y = 1.12 Ground edge flaw Surface flaw σ n = σe σ = σ + σ n E r Annealed glass Tempered glass Y=0.80 a Cut edge flaw The fracture toughness or the critical stress intensity factor K IC can be considered to be a material constant known with a high level of precision. Its value for SLSG is around 0.75 MPam

8 Linear elastic fracture mechanics Fracture Inert conditions Failure when: K I = K Ic Y. f inert. π. a c = Y f K I : stress intensity factor K IC : critical stress intensity factor = inert MPa.m 0.75 f inert. π. a c 0.75 = Y. π. a c MPa Stress causing failure of a crack of depth a c (a c : critical flaw depth) Resistance of a crack to instantaneous failure (not triggered by sub critical crack growth) 8

9 Sub critical crack growth Fracture Instantaneous failure of glass occurs when the elastic stress intensity K I due to tensile stress at the tip of a crack reaches or exceeds or the critical stress intensity factor K IC. In vacuum (inert conditions), the strength of glass is time independent. In the presence of humidity, however, stress corrosion causes flaws to grow slowly when they are exposed to a positive crack opening stress. This happens for values of stress intensity at the crack tip lower than K IC (sub critical crack growth). Stress corrosion is the chemical reaction of a water molecule with silica at the crack tip. Water H O H Si-O-Si+H 2 O Si-OH+HO-Si Glass Si O H H Si O O H Si O H Si Si O Si Stress corrosion - chemical phenomenon Sub-critical crack growth - consequence of stress corrosion

10 Sub critical crack growth Fracture The growth of a surface flaw depends on the properties of the flaw and the glass, the stress history and the relationship between crack velocity and stress intensity. The crack velocity scales with the kinetics of the chemical equation for the stress corrosion (region I). Used for lifetime predictions In region III, close to K IC ν is independent of the environment and approaches a characteristic propagation speed very rapidly ( 1500m/s). In region II the kinetics of the chemical reaction at the crack tip are no longer controlled by the activation of the chemical process but by the supply rate of water (water rate can t keep up when the crack speed increases very fast) For usual conditions, only region I (extremely slow sub-critical crack growth) is relevant for determining the design life of a glass. Stress intensity factor, K I n, v 0 - crack velocity parameters for structural design n=16 is reasonable and v 0 =6mm*/s should be conservative K IC Fracture toughness (material constant = 0.75 Mpa m 0.5 for SLSG)). K th Threshold below which no crack growth occurs 0.55 Mpa m 0.5 for SLSG. Parameters affecting the relation between ν and stress intensity facroe K I : Humidity, temperature, PH value. Loading rate (if it is too fast the water supply suffer a shortage and the stress corrosion is slow down). Chemical composition of glass (affects all the parameters in sub critical crack growth). 10

11 Fracture single flaw Integration of the crack growth law, considering a constant stress history, constant n and yields: f ct = t f ( n - 2 ). v.( Y. π / K ). 2 n 0 Ic. a da dt ( n-2)/ 2 i Risk integral or Brown s integral (to characterize damage accumulation in glass) K = v0.( K 1/ n I Ic ) n K I = Y. σ ( t). π. a( t) n Given a stress story enables the calculation of: the lifetime of a crack given its initial depth or the allowable initial crack depth given its required lifetime This is asymptotic to inert strength, i.e.(t f t r ) 0 as a i a f, and asymptotic to the threshold strength, i.e. (t f t r ) as a i a TH

12 Fracture random surface flaw population (RSFP) The single flaw model is adequate when the critical flaw is known and it is sure that it will lead to failure. In situations other than that a random surface flaw population has to be considered. If the physical characteristics of the surface cracks are unknown, the characteristic tensile strength of glass is evaluated statistically, from the 2-parameter Weibull distribution of test specimens. P f σ = 1 exp θ β P f - Cumulative probability of failure σ Failure stress of specimens which the surface area A is exposed to tensile stress. θ Scale parameter (depends on A) β Shape parameter of the Weibull distribution

13 Glass fracture mechanics Fracture one(critical) flaw flaw population INERT STRENGTH LEFM (short-term) LEFM + PROB AMBIENT STRENGTH LEFM + (long-term) LEFM + + PROB LEFM : Fracture : Subcritical Crack Growth PROB: Theory of Probability 13

14 Glass fracture mechanics Fracture Flaw characteristics known Flaw and environment characteristics known One flaw Flaw population One flaw Flaw population f inert P f,inert (Weibull) f ambient P f,ambient (Weibull) Flaw characteristics known and environment characteristics not known Testing Testing Testing Inert + micr. ambient Inert or ambient Y, a: flaw parameters n, ν 0 : crack velocity parameters Test results (fitting Weibull) P f,inert Test results (fitting Weibull) P f,ambient 14

15 Fracture The (characteristic) strength of glass can be estimated experimentally with the coaxial double ring (CDR) or the four point bending (4PB) test setup. glass specimen reaction ring load loading ring reaction Coaxial double ring test load glass specimen reaction reaction Four point bending test 15

16 Fracture Three point bending test: one flaw is tested Four point bending test: a flaw population is tested 16

17 Fracture Coaxial double ring test standardized in EN large (EN : mm²) or small (EN : 254 mm²) test surface area stress rate: 2 MPa/s ± 0,4 MPa/s rel. humidity: 40 % to 70 % equibiaxial stress field (σ 1 = σ 2 ) the failure strength is influenced by the surface conditions only Technische Universität Darmstadt, Germany 1 Load ring 2 Specimen 3 Supporting ring 17

18 Four point bending test standardized in EN Fracture size of the specimens: 1100 x 360 mm stress rate: 2 MPa/s ± 0,4 MPa/s rel. humidity: 40 % to 70 % uniaxial stress field the failure strength is influenced by the edge and the surface conditions the failure can occur from the edge or from the surface the test results outside the load span are excluded MPA Darmstadt, Germany L s = 1000 mm L b = 200 mm 18

19 Fracture The characteristic value corresponds to a fractile of 5%, and can be determined according to EN 1990, EN and the relevant product standards. Glass type bending strength f g;k i [N/mm 2 ] Annealed glass 45 Heat strengthened glass 70 Fully tempered glass 120 Chemical strengthened glass 150 *) *) depends on the surface conditions Glasbau /Wörner et al. 19

20 Fracture Example: Determination of a the characteristic value in the drilled area of a flat glass. Parameters (specimens): Glass type: Annealed glass Nominal size: 250 mm x 250 mm Diameter of the central borehole: 50 mm Nominal glass thickness: 6 mm Parameters (testing): Coaxial double ring test Stress rate: 2 MPa/s ± 0,4 MPa/s Rel. humidity: 50 % 20

21 From the coaxial double ring test, the following values were obtained: Fracture N Measured failure stress [N/mm 2 ] N Measured failure stress [N/mm 2 ] N Measured failure stress [N/mm 2 ] MPA Darmstadt, Germany measured failure stress 21

22 Fracture Histogram and 2p-Weibull fitting: For brittle materials, the Weibull distribution is measured the most failure appropriate statistical strength distribution. In Europe, stress the standard EN specifies procedures on evaluation of test results with the 2p-Weibull distribution. 22

23 Fracture For brittle materials, the Weibull distribution is the most appropriate statistical strength distribution. In Europe, the standard EN specifies procedures on evaluation of test results with the 2p-Weibull distribution. The cumulative distribution function of the 2p-Weibull distribution is given by: x F(x) =1-exp(-( ) θ β ) The experimental results were fitted to the 2p-Weibull distribution, using the Maximum Likelihood Estimation method. (The method according to EN can be used alternatively) Using Matlab, the Weibull parameters were estimated: θ = MPa and β =

24 A probality plot shows the failure probability P f of the measured data, which were fitted to a 2p-Weibull distribution (large deviations at the lower bound!!!) Fracture 24

25 Fracture A probality plot shows the failure probability P f of the measured data, which are fitted to a 2p-Weibull, Normal and 2p- Lognormal distribution: for these data the tail fits best to the Weibull distribution 25

26 Fracture The probability of failure for the fitted 2p-Weibull distribution: P f fg =1-exp(-( θ ) β ) The characteristic strength f g;k corresponds to P f = 0.05: 005=1. -exp(-( f g; k β θ ) ) θ = MPa and β = are the estimated Weibull parameters. Under the assumption, that the count of specimens is high, the characteristic strength f g;k can be calculated approximately: f g;k =7218MPa. *(-ln( )) 1/ =5806MPa. 26

27 References Anderson, T.L., Fracture mechanics, Fundamentals and Applications, Taylor & Francis Group, Evans A.G., A method for evaluating the time-dependent failure characteristics of brittle materials and its application to polycrystalline alumina. Journal of materials science 7: , Fink A., Dissertation D17: Ein Beitrag zum Einsatz von Floatglas als Dauerhaft tragender Konstruktionswerkstoff im Bauwesen. Technische Universität Darmstadt, Institut für Statik, Bericht Nr. 21, Griffith A. A., The Phenomena of Rupture and Flow in Solids. Philosophical Transactions, Series A, 1920, 221: Haldimann M., Thèse n 3671: Fracture strength of structural glass s analytical and numerical modelling, testing and design. EPFL, Lausanne, Haldimann M, Luible A, Overend M., Structural Engineering Document 10: Structural use of glass. IABSE / ETH Zürich, Zürich, Irwin G., Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied, 1957, 24: Irwin, G.R., Crack-extension force for a part-through crack in a place. Journal of Applied, 1962, pp Porter M., Thesis: Aspects of Structural Design with Glass. Trinity, Oxford, Schneider, J., Schula, S., Weinhold, W.P. (2010) Characterisation of the scratch resistance of annealed and tempered architectural glass. Thin Solid Films - article in press, doi: /j.tsf Schneider, J., Schula, S., Burmeister, A. (2011) Two mechanical design concepts for simulating the soft body impact at glazings Part 1: Numerical, transient Finite Element simulation and simplified concept with equivalent static loads. Stahlbau Spezial 2011 Glasbau/Glass in Building 80 (1) pp Veer F.A., Rodichev Y.M., The structural strength of glass: hidden damage. Strength of materials, May 2011, Vol. 43, nr. 3. Weller B., Nicklisch F., Thieme S., Weimar T., Glasbau-Praxis: Konstruktion und Bemessung. 2 Aufl. Berlin: Bauwerk, Wiederhorn S.M., Bolz L.H., Stress corrosion and static fatigue of glass. Journal of the American Ceramic Society, 1970, Vol. 53, p Wörner, J.-D., Schneider, J., Fink, A. (2001) Glasbau: Grundlagen, Berechnung, Konstruktion. Springer, Berlin. 27

28 References EN Glass in building - Determination of the bending strength of glass - Part 1: Fundamentals of testing glass EN Glass in building - Determination of the bending strength of glass - Part 3: Test with specimen supported at two points (four point bending) EN Glass in building - Determination of the bending strength of glass - Part 5: Coaxial double ring test on flat specimens with small test surface areas EN 1990 Eurocode: Basis of structural design EN Glass in building - Pendulum tests - Impact test method and classification for flat glass EN Glass in building ó Procedures for goodness of fit and confidence intervals for Weibull distributed glass strenght data DIN Glass in Building - Design and construction rules - Part 1: Terms and general bases DIN Glass in Building - Design and construction rules - Part 2: Linearly supported glazings DIN Glass in Building - Design and construction rules - Part 3:Point fixed glazing DIN Glass in Building - Design and construction rules - Part 4: Additional requirements for anti-drop device DIN Glass in Building - Design and construction rules - Part 5: Accessible glazing 28

29 This lecture was prepared for the 1st Edition of SUSCOS (2012/14) by Prof. Sandra Jordão (UC). Adaptations brought by Prof. Viorel Ungureanu (UPT) for 2 nd Edition of SUSCOS 29