1 /18 M.Chrzanowski: Strength of Materials SM2-12: Fracture Introduction to FRACTURE MECHANICS.
Lecture 6 – Fracture strength and testing methods
Transcript of 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
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Sustainable Constructions under Natural Hazards
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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
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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
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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
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Linear elastic fracture mechanics
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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
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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Linear elastic fracture mechanics
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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.
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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Linear elastic fracture mechanics
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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.
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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Linear elastic fracture mechanics
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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)
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
KI : stress intensity factor
KIC : critical stress intensity factor
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Stress corrosion & Sub critical crack growth
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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
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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Stress corrosion & Sub critical crack growth
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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
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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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
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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
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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
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Glass fracture mechanics
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
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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
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Testing methods
Three point bending test:
one flaw is tested
Four point bending test:
a flaw population is tested
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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
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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
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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.
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Characteristic values in design
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 %
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Characteristic values in design
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
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods Characteristic values in design
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Characteristic values in design
Histogram and 2p-Weibull fitting:
measured failure stress
Introduction Glass Fracture Mechanics Linear Elastic Fracture Mechanics Stress corrosion & SCG Lifetime of a glass element Testing methods 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.
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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=
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Characteristic values in design
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!!!)
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Characteristic values in design
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
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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
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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.
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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
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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