lecture 6 – fracture strength and testing methods

ADVANCED DESIGN OF GLASS STRUCTURES

Lecture 6 – Fracture strength and testing methods

Viorel Ungureanu

European Erasmus Mundus Master Course

Sustainable Constructions under Natural Hazards and Catastrophic Events

520121-1-2011-1-CZ-ERA MUNDUS-EMMC

L6 Fracture strength and testing methods


Sustainable Constructions under Natural Hazards

and Catastrophic Events

Introduction

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.

Edge flaw caused by grounding

Surface flaw on an accessible glazing

BEAM

Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design





3

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

Glass fracture mechanics Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design





4

Glass fracture mechanics

one (critical) flaw flaw population

INERT STRENGTH

LEFM (short-term) LEFM + PROB

AMBIENT STRENGTH

LEFM + SCG (long-term) LEFM + SCG + PROB

LEFM : Linear Elastic Fracture Mechanics

SCG: Subcritical Crack Growth

PROB: Theory of Probability






Linear elastic fracture mechanics

5

STEEL or CONCRETE: homogenous

test strength = strength of the material

σn < critical stress

σn: uniform stress

TIMBER or GLASS: not homogenous: defects

test strength << strength of material

σn .Y.(π.a)0.5 < critical value σn: uniform stress Y: correction factor (defects) a: depth of defects (flaws) critical value: material constant







6

The stress intensity factor KI: elastic stress intensity near a crack tip. Provides a means to characterize the material in terms of its fracture toughness.

KI : stress intensity factor [MPa.m0.5]

Y : geometry factor [-]

σn : stress normal to the flaw’s plane [MPa]

a : flaw depth [m]

aYK nI ... πσ=

Instantaneous failure of glass occurs when the elastic stress intensity KI, 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 KIC.

There is stress magnification near the tip of a crack.







7

KI : stress intensity factor [MPa.m0.5]

Y : geometry factor [-]

σn : stress normal to the flaw’s plane [MPa]

a : flaw depth [m]

aYK nI ... πσ=

a

Y = 1.12

Y=0.80

a Cut edge flaw

Surface flaw

Ground edge flaw En σσ =

rEn σσσ +=

Annealed glass

Tempered glass

The fracture toughness or the critical stress intensity factor KIC can be considered to be a material constant known with a high level of precision. Its value for SLSG is around 0.75 MPam0.5.







8

Inert conditions

IcI KK =

0.5MPa.m 75.0... =cinert afY π

Failure when:

MPa

c

inertaY

f..

75.0

π=

cinert afY

..

75.0

π=

Stress causing failure of a crack of depth ac (ac: critical flaw depth) Resistance of a crack to instantaneous failure (not triggered by sub critical crack growth)


KI : stress intensity factor

KIC : critical stress intensity factor





Stress corrosion & Sub critical crack growth

9

Instantaneous failure of glass occurs when the elastic stress intensity KI due to tensile stress at the tip of a crack reaches or exceeds or the critical stress intensity factor KIC.

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 KIC (sub critical crack growth).

Si-O-Si+H2O → Si-OH+HO-Si

Stress corrosion is the chemical reaction of a water molecule with silica at the crack tip.

Glass

Water

1

Si

O

Si

Si

O

H

HO

O

H

H

O

Si

Si

Si

2 3

H

H O

Stress corrosion - chemical phenomenon

Sub-critical crack growth - consequence of stress corrosion






Stress corrosion & Sub critical crack growth

10

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.

Stress intensity factor, KI

KIC – Fracture toughness (material constant = 0.75 Mpa m0.5 for SLSG)). Kth – Threshold below which no crack growth occurs ≈ 0.55 Mpa m0.5 for SLSG.

In region III, close to KIC 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 element.

Parameters affecting the relation between ν and stress intensity facroe KI :

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).

The crack velocity scales with the kinetics of the chemical equation for the stress corrosion (region I).

n, v0 - crack velocity parameters for structural design n=16 is reasonable and v0 =6mm*/s should be conservative

Used for lifetime predictions






Integration of the crack growth law,

considering a constant stress history, constant n and

yields:

)(.).(. tatYK nI πσ=

n

Ic

I

K

Kv

dt

da).(0=

( ) ( )n

ni

n

Icf

ctaKYvnt

f

/1

2/)2-(0 ./...2-.

2

=

π

Risk integral or Brown’s integral (to characterize damage accumulation in glass)

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

Lifetime of a glass element single flaw

This is asymptotic to inert strength, i.e.(tf –tr)→ 0 as ai → af , and asymptotic to the threshold strength, i.e. (tf –tr)→∞ as ai → aTH






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.

Lifetime of a glass element random surface flaw population (RSFP)

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.

β

θσ

−−= exp1fP

Pf - 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


one(critical) flaw flaw population

INERT STRENGTH

LEFM (short-term) LEFM + PROB

AMBIENT STRENGTH

LEFM + SCG (long-term) LEFM + SCG + PROB

LEFM : Linear Elastic Fracture Mechanics

SCG: Subcritical Crack Growth

PROB: Theory of Probability






14


Flaw characteristics known Flaw and environment characteristics known

One flaw Flaw population One flaw Flaw population

Flaw characteristics known and environment characteristics not known

finert Pf,inert

(Weibull) fambient

Pf,ambient (Weibull)

Testing Testing Testing

Inert + micr. ambient

Y, a: flaw parameters

n, ν0: crack velocity parameters

Inert or ambient

Test results (fitting Weibull)

Pf,inert

Test results (fitting Weibull)

Pf,ambient






15

Testing methods

The (characteristic) strength of glass can be estimated experimentally with the coaxial double ring (CDR) or the four point bending (4PB) test setup.

load

loading ring

reaction ring

glass specimen

reaction

Coaxial double ring test

Four point bending test

load

glass specimen

reaction reaction






16

Testing methods

Three point bending test:

one flaw is tested

Four point bending test:

a flaw population is tested






17

Testing methods

Coaxial double ring test

•standardized in EN 1288-1

•large (EN 1288-2: 240000 mm²) or small (EN 1288-5: 254 mm²) test surface area

•stress rate: 2 MPa/s ± 0,4 MPa/s

•rel. humidity: 40 % to 70 %

•equibiaxial stress field (σ1 = σ2)

•the failure strength is influenced by the surface conditions only

Technische Universität Darmstadt,

Germany

1 Load ring

2 Specimen

3 Supporting ring






18

Testing methods

Four point bending test

•standardized in EN 1288-3

•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

Dar

mst

adt,

Ger

man

y

Ls = 1000 mm Lb = 200 mm






19

Characteristic values in design

• The characteristic value corresponds to a fractile of 5%, and can be determined according to EN 1990, EN 12603 and the relevant product standards .

Glass type Characteristic bending strength fg;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.






20


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 %






21


From the coaxial double ring test, the following values were obtained:

measured failure stress

N° Measured failure stress

[N/mm 2]

N°

Measured failure stress

[N/mm 2]

N° Measured failure stress

[N/mm 2]

1 67.70 11 69.96 21 66.67

2 74.12 12 68.29 22 72.04

3 62.05 13 55.05 23 69.86

4 73.09 14 74.24 24 69.91

5 73.99 15 72.1 25 69.8

6 71.35 16 75.32 26 72.92

7 73.84 17 66.46 27 77.63

8 70.87 18 67.34 28 69.90

9 68.18 19 60.46 29 64.57

10 83.27 20 59.88 30 70.55

MPA Darmstadt, Germany






22


Histogram and 2p-Weibull fitting:

measured failure stress


For brittle materials, the Weibull distribution is the most appropriate statistical strength distribution. In Europe, the standard EN 12603 specifies procedures on evaluation of

test results with the 2p-Weibull distribution.





23

Characteristic values in design For brittle materials, the Weibull distribution is the most appropriate statistical strength

distribution. In Europe, the standard EN 12603 specifies procedures on evaluation of test results with the 2p-Weibull distribution.

The cumulative distribution function of the 2p-Weibull distribution is given by:

The experimental results were fitted to the 2p-Weibull distribution, using the Maximum Likelihood Estimation method.

(The method according to EN 12603 can be used alternatively)

Using Matlab, the Weibull parameters were estimated:

θ = 72.18 MPa and β = 13.64

))x

-(exp(-)x(Fβ

θ1=






24


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






25


A probality plot shows the failure probability Pf 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






26

Characteristic values in design The probability of failure for the fitted 2p-Weibull distribution:

The characteristic strength fg;k corresponds to Pf = 0.05:

θ = 72.18 MPa and β = 13.64 are the estimated Weibull parameters.

Under the assumption, that the count of specimens is high, the characteristic strength fg;k can be calculated approximately:

))f

-(exp(-Pg

fβ

θ1=

))f

-(exp(-.kg; β

θ1=050

MPa0658=050MPa1872=64131

.)).--ln(1(*.f./

k;g






27

References Anderson, T.L. , Fracture mechanics, Fundamentals and Applications, Taylor & Francis Group, 2005.

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: 1137-1146, 1972.

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, 2000.

Griffith A. A., The Phenomena of Rupture and Flow in Solids. Philosophical Transactions, Series A, 1920, 221: 163-198.

Haldimann M. , Thèse n°3671: Fracture strength of structural glass elements – analytical and numerical modelling, testing and design. EPFL, Lausanne, 2006.

Haldimann M, Luible A, Overend M., Structural Engineering Document 10: Structural use of glass. IABSE / ETH Zürich, Zürich, 2008.

Irwin G. , Analysis of Stresses and Strains near the End of a Crack Traversing a Plate. Journal of Applied Mechanics, 1957, 24: 361-364.

Irwin, G.R. , Crack-extension force for a part-through crack in a place. Journal of Applied Mechanics, 1962, pp. 651-654.

Porter M. , Thesis: Aspects of Structural Design with Glass. Trinity, Oxford, 2001.

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:10.1016/j.tsf.2011.04.104.

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. 81 – 87.

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, 2010.

Wiederhorn S.M., Bolz L.H. , Stress corrosion and static fatigue of glass. Journal of the American Ceramic Society, 1970, Vol. 53, p. 543 – 548.

Wörner, J.-D., Schneider, J., Fink, A. (2001) Glasbau: Grundlagen, Berechnung, Konstruktion. Springer, Berlin.





28

References EN 1288-1 Glass in building - Determination of the bending strength of glass - Part 1: Fundamentals of testing glass

EN 1288-3 Glass in building - Determination of the bending strength of glass - Part 3: Test with specimen supported at two points (four point bending)

EN 1288-5 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 12600 Glass in building - Pendulum tests - Impact test method and classification for flat glass EN 12603 Glass in building ó Procedures for goodness of fit and confidence intervals for Weibull distributed glass strenght data

DIN 18008-1 Glass in Building - Design and construction rules - Part 1: Terms and general bases

DIN 18008-2 Glass in Building - Design and construction rules - Part 2: Linearly supported glazings

DIN 18008-3 Glass in Building - Design and construction rules - Part 3:Point fixed glazing

DIN 18008-4 Glass in Building - Design and construction rules - Part 4: Additional requirements for anti-drop device DIN 18008-5 Glass in Building - Design and construction rules - Part 5: Accessible glazing





29

This lecture was prepared for the 1st Edition of SU SCOS (2012/14) by Prof. Sandra Jordão (UC).

Adaptations brought by Prof. Viorel Ungureanu (UPT) for 2nd Edition of SUSCOS

lecture 6 – fracture strength and testing methods

Documents