thermal tempering simulation of glass plates: inner and edge residual stresses

THERMAL TEMPERING SIMULATION OF GLASS PLATES:INNER AND EDGE RESIDUAL STRESSES

Laurent DaudevilleLaboratoire de Mecanique et Technologie

Ecole Nonnale Superieure de CachanCNRS, Uniu. Paris 6

Cachan, France

Helene CarreCentre Scientifique et Technique du Biitiment

Champs-sur-Mame, France

Narayanaswamy's model is used to describe the thennomechanical behavior ofglass. Itincludes both stress relaxation (to take into account the viscous aspect of glass) andstructural relaxation (to take into account the structure state of glass). The necessarythermal and mechanical characteristics are given for the float soda-lime silicate glass.

The thermal tempering of thin glass plates is simulated. Transient and residualstresses are given for the inner part of the plate. Computational results are comparedwith experimental results ofprevious works. This comparison validates Naranaswamy'smodel associated with material characteristicsgiven previously.

The edge effect (variation ofstresses close to the edges) is described for thin plates.The thermal tempering of thick plates is simulated, and computational results arevalidated with optical measurements and a fractographic analysis.

The objective of this work is to provide a reliable computational method for thesimulation of the thermal tempering of soda-lime silicate glass plates. Quenchingconsists of heating glass to a temperature above 600°C and cooling it brutally by aircasts. Such thermal treatment induces residual stresses in the plate at the end ofthe cooling, thereby increasing the mechanical strength of glass parts.

In previous works, the simulation of an infinite glass plate has been carried outexperimentally and numerically. Both inner and edge effects are analyzed in thisarticle. The knowledge of residual stresses in the vicinity of edges is very importantin the case of elements loaded in their plane. The main application of these glasscomponents concerns building structures (beams or columns).

Glass is described as a viscoelastic material. Its behavior varies with time andtemperature. Narayanaswamy's model, which includes both stress and structuralrelaxation (influence of the state of the glass structure), is used in this work.

The tridimensional residual stress state is computed using the finite elementmethod (FEM). First, the model is checked by comparing the residual stresses inthe inner part of thin plates from simulations and literature. The authors found it

Received 30 October 1996; accepted 27 October 1997.Address correspondence to Laurent Daudeville, Laboratoire de Mecanique et Technologie, Ecole

Norrnale Superieure de Cachan, CNRS, Univ. Paris 6, Cachan, France. E-mail: [email protected]cachan.fr

Journal of Thermal Stresses, 21:667-689, 1998Copyright © 1998 Taylor & Francis

0149-5739/98 $12.00 + .00

667

668 L. DAUDEVILLE AND H. CARRE

difficult to obtain reliable data for the concerned material. No material datum wasidentified, and a sensitivity analysis to material parameters was conducted.

Then, the analysis of transient and residual stresses in the neighboring edgesof tempered glass plates is presented. Inner and edge residual stresses weremeasured optically on several tempered glass plates. The heat transfer coefficientsare temper process parameters. A comparison of two optical measurements withsimulation results facilitates the identification of a priori unknown heat transfercoefficients.

The tempered glass plates were tested up to failure. A fractographic analysisgives the residual stress on the plate edge that cannot be measured optically.

The edge effect analysis is validated by a comparison of computed stresses withfractographic and optical information.

THERMOMECHANICAL MODELING OF GLASS

Generalized Maxwell's Model

For a tridimensional solicitation, the stress response (a(t) of a viscoelasticmaterial can be separated into its volume and deviatoric parts:

lSi/ x , t) = f' G(t-t')~ejj(x,t')dt'

-0> at

f' au(x,t)= K(t-t')-,fi(x,t')dt'_0> at

(1)

where aij = sij + u8 jj, Eij = eij + fi8ij, G = relaxation shear modulus, and K =relaxation bulk modulus. The generalized Maxwell's model is a rheological discretemodel. It is an association of several Maxwell's models (spring and dashpot inseries). The proposed rheological model of glass is illustrated in Figures 1 and 2 for

Figure I. Generalized Maxwell's model for the deviatoric part.

THERMAL TEMPERING OF GLASS 669

"

Figure 2. Generalized Maxwell's model for the volume part.

the deviatoric and volume parts, respectively [1, 2]. Relaxation shear and bulkmoduli are described with instantaneous and deferred moduli and Prony's series.

For the deviatoric part

(2)

where

Gg = instantaneous shear modulus, and G, = deferred shear modulus = O.For the volume part

(3)

where

Kg = instantaneous bulk modulus, and K, = deferred bulk modulus.

Stabilized Glass

When its structure is stabilized, glass can be considered a thermorheologicallysimple material. It means that the mechanical behavior is the same for differenttemperatures; only the rate of microstructural mechanisms changes. The relaxationfunction can be determined at any temperature if it is known at the referencetemperature (Tref ) . It consists of a shift of the relaxation function in a timelogarithm scale [3].


The influence of the actual temperature T can be introduced by means ofreduced time (~), which is defined as

i = 1,2 (4)

Weights W l i and W 2i, defined in G(t} and Kii), are constant with temperature. AnArrhenius relation allows a correct description of the times of relaxation changeswith temperature [4, 5]:

it Tre! it~(t, T) = ( ') dt' = <I>(T(t» dt'

oTT, t 0

where

H (1 1)In(<I>(T» = - - - --R Tre! T

H = energy of activation, and R = perfect gas constant.

(5)

Structural Relaxation

Quenching is a brutal thermal treatment. During cooling, the glass structure maynot be stabilized. Since glass is sensible to the thermal loading rate, the structurestate must be taken into account. That is Narayanaswamy's model [4, 6, 7].

The influence of the thermal loading rate can be shown by varying the specificvolume with temperature (Figure 3). That phenomenon essentially concerns temperatures close to the transition range between the "glass" and "liquid" states.

Specific volume

T2~ T1f ~ Tl Temperature" IItransition range

Figure 3. Variations of specific volume with temperature for two cooling rates.


When glass is heated to a temperature Tz (point A on Figure 3) below thetransition range, the structure of glass evolves to a stable state characteristic of theliquid glass (point B). This evolution is the direct structural relaxation. If thetemperature is far below the transformation range, this evolution is too slow to bevisible.

If glass is heated brutally to T, above Tg , the glass state goes to a point belowthe "liquid" straight line (point C). This unstable configuration changes to a stableone; the "liquid" glass state (point D). This process is called reverse structuralrelaxation.

There are several possible structural glass states for the same temperaturedepending on the cooling rate.

In I946, Tool [8) introduced the fictive temperature (Tf ) to take into accountthe structure of glass. Variations of the specific volume with temperature are usedto describe variations of Tf •

• T, = T if T is above the transition range.• T, = the intersection between the "liquid" straight line and the parallel to the

"glass" straight line in the transition range.• T, = the intersection between the "liquid" straight line and the "glass" straight

line below the transition range.

A response function (M) is defined by varying the specific volume with temperature as

V(O - I-:, zMv(O = V. - v.: .

0,2 al,2(6)

where T[ - Tz = step of temperature, T, = fictive temperature at Tz, V =

instantaneous specific volume, Vo,z = volume just after the temperature change,and I-:,.z = equilibrium volume at Tz.

The dependence of the response function on temperature is given by reducedtime g. Mv(O can be considered the volume relaxation function and can definevariations of the fictive temperature

By analogy with viscous relaxation, the response function can be described witha Prony's series as

(8)


The fictive temperature variations are calculated with Markovsky and Soules'algorithm [9, 10]

t:l.tTr,(1 - Ill) + T(t}>:

Tr,(l) = III '1+-

Ai

{ H[ 1 x I-X]}Ai = Airer exp - Ii T

rer- T(I) - Tr{t - Ill)

n

Tr(t} = L CiTr,{t)i= 1

(9)

Structural relaxation times (A) are considered proportional to shear relaxationtimes. A dilatometric curve facilitates obtaining their ratio, which has been foundto be very close to 9 [11].

The fictive temperature allows one to take into account the structure state ofglass. All characteristics vary with the actual temperature but also with the fictivetemperature. The fictive temperature is introduced in the reduced time expressionfor the variations of viscoelastic characteristics by modifying Eq. 5

where

( ) l' Trer '1' ,g I,T,Tr = ( ') dt = a(T(t}, Tr(l» dto T T,I 0

H( 1 x I-X)In(a(T, r;» = - - - - - - --R Trer T t,

(10)

x = constant (0 < x < 1), H = energy of activation, and R = perfect gas constant.

Introduction of Fictive Temperature

The dependency of viscosity on the structure state induces very small variations inresidual stresses. But, the dependency of the density on the structure state has amuch more significant effect on residual stresses (Figure 4).

Variations of viscosity in relation to the fictive temperature are not consideredin our simulations. The influence of the structural state on the density is introduced by varying the thermal expansion coefficient in relation to temperature

E'h = f3/T{t) - Tr{t» + f31(Tr(t} - To)

= f3(I)(T(I) - To) (11)

THERMAL TEMPERING OF GLASS

Tension at the Inner part (nm I em)

673

- structural model- - viscoelastic model

-0- ,,:itho~tdependancy ofVISCOSIty I structure

-0- without dependancy ofdensity I structure

Figure 4. Influence of the fictive temperatureon residual stresses (5).

where f3g = thermal expansion coefficient of solid glass, f3 J = thermal expansioncoefficient of liquid glass, To = initial temperature, and f3 = equivalent thermalexpansion coefficient.

FINITE ELEMENT MODELING

The thermal tempering of glass consists of using air casts to cool very rapidly theplate that was previously heated.

The problem that must be solved is time dependent. At each moment in time,the temperature is computed; then the nonlinear mechanical problem is solved.The nonlinear aspect is due to the viscous feature of the glass behavior.

The temperature of the glass plate is assumed to be uniform at the beginningof the computation (at a temperature above 600°C). In fact, the surface temperature of the glass plate decreases between the time it is removed from the oven andthe beginning of cooling.

Heat Exchange Boundary Conditions

The cooling is modeled by a forced convection to each face of the plate. It ischaracterized by a heat transfer coefficient (h) and air temperature (TexJ For aglass plate, three planes of symmetry and three forced convections define thecooling (Figure 5).

The exchange boundary condition by forced convection is written as

where q = heat flow, h = thermal transfer coefficient, Ts = surface temperature,and Text = room temperature.

B,.. II~~UDX:,VFI I"[-, -,AM) a t , C A R R ~

Radiiaiion is ~ ~ c c ~ ~ u n t c ~ $ for i fi ~ ~ B R C C L ! c_~g331$plecti0n a I 1 hmmui trms fc g- ccxfficien~ The n:rcii;s$iajn bour1~3aq cvn~iition can bc written as (&4' eqkxiv~ai~aace with c<3n\cctiofi)

e r e = 2 f'h.3~4, uSw = Stef:jr~-B01txrn;ann"s cc3effieie,*nt, c3 = emf~sivity, 'T% =

iurfdcc tcmperaturc, q,, = morn icmpcratore, and ir,,(rr,,,, e. T,. 7,,, 1 -- equivalent therx12ai zr;~nsfegP c i c t Thermal r e e f t ;ire ; I 3 he cwstkant during te,.n2pcring,

Ir~iercs%iz~g results arc %he stress stiate i~ the rxxidcile c3f the plltte md i-i~ the carre~f r t h s 3 prsshlt;m th~it needs t s ~ be soived d t - 3 ~ -,not ~lepei~d _"em s, hec,.;nase the ph%e is cp"~nsidere~i infinite in tixis ~lis-cetio~~ This is ;I piitae i3rc3131ex11 since it depends uniy ual ,Y and p., Nevcrt!aciess- tritiime,.nasic>~~:tl eienaen$s a - - e uhed xo ixct>xxnt for the$-nxal cicfarmi$titj~ :tlong z7

'rhc nmc~deling of infksite strip i~~pl ies quite complicated hnundi$~y condi- tic~as, Then, the temp~ring of an eighth of plate is sixm3al;ited with the finite clement code & A R C e m e is refi~ed i zones where t e e present high v:~riz- tions-it4 the thickness of the piate (itl~s~l$ .Y) ared clash: to the edge (along 3;).

The edge effect BS not st~gdled along x bec;arss: it is similar to &h& ;doaxg J, The tlispiitce~xxe~2t ht>~laadaq cganditig~f3s this t are inimdarced arc sy~>~nctxy cmdi titaaas: xbt is, there is nn dispiaaccn~ent perperadicuh the pkane of s y m ~ ~ e t ~ y BFiyure 51,


composition and are difficult to measure experimentally (in particular, for theviscoelastic characteristics and the thermal expansion coefficient of the liquidglass).

The six mechanical characteristics of Narayanaswamy's model that are used insimulations are as follows.

• Elastic characteristics at room temperature [1]

Young modulusPoisson ratio

E = 7.1010 Pa

v= 0.22

• VIScoelastic characteristics [1] (Table 1),• Structural characteristics for the calculation of T, [12] (Table 2).

The energy of activation is not a well-known parameter. The dependence ofglass viscosity on temperature was determined in a previous work [1]. It is anempirical relation of Vogel-Fulcher-Tammann

BLogfn =A +-

T-To

4003.5= -1.993 + for T> Tg where 1) = (Tl)G (14)

T-542

The value of the ratio H/ R was identified by comparing the previous relation withthe function <I>(T) of Narayanaswamy's model

H/R = 55,000 K

• Thermal expansion coefficient [12]

solid glass (room temperature)

liquid glass (high temperature)

fJg = 9.1O- 6°C- 1

fJl = 25.1O- 6°C- 1

Table 1 Characteristics of shear and volume moduli

Deviatoric part (T", = 869 K)

a, (10' Pal Tli (8)

Volume part (K,/K. ~ 0.18) (T,et = 869 K)

s; (10' Pal T2j (8)

1.5845 6.65810- 5 0.7588 5.00910- 5

2.3539 1.197 10- 3 0.7650 9.94510-'3.4857 1.51410-2 0.9806 2.02210- 3

6.5582 1.672 10- 1 7.301 1.925 10- 2

8.2049 7.49710- 1 13.47 1.19910- 1

6.4980 3.292 10.896 2.033

676

.',

L. DAUDEV[LLE AND H. CARRE

Table 2 Characteristics of the response functionfor the structural part

Structural part

c,552310- 2

8.20510- 2

1.2[510- 1

2.28610- 1

2.86010- 1

2.26510- 1

Ai(s)

5.965 10-'1.07710- 2

1.36210- 1

1.5056.747

29.63

The thermal conductivity and the specific heat are the thermal characteristics.They vary with temperature [12].

• Thermal conductivity (W/m.K)

A = 0,975 + 8,58.10-4 T where T in ·C

• Specific heat (Jykg.K) [11] (with T in K)

liquid glass (T> Tg = 850 K)

solid glass (T < Tg )

INNER EFFECT

Cp,l = 1433 + 6.5 10-3 T

Cp,s = 893 + OAT - 1.8 1O-7/T 2

Experimental studies give the residual stress state in the inner part of thin plates(0.61-cm thick) [4, 6]. The tempering simulation of these glass plates is presentedand compared to previous works for the validation of the model [13, 14].

Results

During tempering, the temperature decreases more rapidly at the surface than inthe inner part (Figure 6). A thermal gradient appears in the thickness (which canbe greater than 150·C); it implies transient and residual stresses.

At the beginning of cooling, the surface contracts more rapidly than the innerpart, which leads to a compression state at the core and a tensile state at thesurface by equilibrium. The transient tensile stress on the surface can inducefailure at the onset of cooling.

When the surface temperature is below the transition range, the surfacecongeals. The core, still liquid, continues to cool and contract. The surface is thenin compression and the core in tension. After several tens of seconds, stresses arestabilized (Figure 7).

In the middle of the plate, the residual tractions are equal for directions y andz. Stresses are parabolic in the thickness. The surface compressive stress is twice


Temperature ('C)

700

z

- node I (center of the plate)- node 2 (surface of die plate)

TIme(s)

400300200100

T. =6S0'Ch, = 280.1 W/m'.K

100

200

300

400

500

600

Figure 6. Variations of temperature with time.

higher than the tensile midplane stress (Figure 8). This stress state gives plates abetter mechanical strength.

The characteristic parameter values of the model were obtained from differentworks. A large discrepancy in the values can be observed in the literature, possiblydue to the difficulty involved in measuring some characteristics and to sensibility tothe glass chemical composition.

In order to study the influence of characteristic parameter values, a sensibilityanalysis is carried out by varying the mechanical and thermal parameters: Young'smodulus (E), the Poisson ratio (IJ), the solid and liquid glass thermal expansioncoefficients ({3g and (3,), reference temperature (Trof ) ' specific heat (C p), andthermal conductivity (A). The parameters are modified (+ 20%, + 10%,-10%, - 20%) from a current case: To = 650°C and hi = 280.1 W jm2.K.

Variations of residual stresses in the surface and in the midplane are observed.The Poisson ratio, the solid glass thermal expansion coefficient, the specific heat,and the thermal conductivity are parameters of less importance than the others.The variations of the residual stresses are quasi-proportional to the variations ofYoung's modulus and of the liquid glass expansion thermal coefficient. Thereference temperature is a parameter of great importance because it controls theviscoelastic behavior (Figures 9 and 10).

Comparisons with Experimental Works

Experimental results issued from [4, 6] are compared with FEM results obtainedwith the previous glass characteristics.

Figure T7, \'lthtukic.%n% c j f hlrc.gs with time

Tl ie variation of thc tr:ictiiin i q,) in the ihicknehs i3 compared with uptic;tl measu~cmentz fronz if,] in ;I pitriicuiitr case trf quenching (T,, = 73XC<' aiiil k , = 211 .R % V ~ Q ~ ' ~ K I (Figure 1 11-

In Figpure i2, csperiment:il f51 ainJ ~zumerical results are givcrl, T~Ic): ~ h i - i ~ ~ the infiue~cr: o%' the cs~oling raiia, (83 1 and the: irnitial tempcraiu~e /'/;,I on iht: re.;itl%a;tl tensile stress in the core uf the plotc- 'rhc higher h , t9-w Esigher this residua! tractic38-9, They also reveal ;i dey>enrlcnce c~ (he iniiial tea~per;i$um.c: the z-esid~arti

Speifi"ne heat

Liquid g % w &ermd expmsion coeff jcie~t

Residu~L stress in sarfaee

Di @ere@%@ ( % 1

Conductivity \

Liquid g i a ~ thermal expmsion coefficieat

Saiid giass them& expsnsisn caeffjcient

Poisson ratio 7 Yauag modui

I--! , , , , , 4

680 L. DAUDEV[LLE AND H. CARRE

Stress (MPa)

100 ..---------------.......,

so

-so

-[00

To =738°Ch =221.5W/mJ.K

-ISO '------------------'Figure 11. Variation of stress in thethickness.

stress increases with the initial temperature until 680°C, where it becomes quasiconstant with To.

This satisfying comparison between computational and experimental resultsvalidates Narayanaswamy's model given the chosen data.

EDGE EFFECT

The goal is the determination of tridimensional transient and residual stresses onthe edge of glass plates where fracture may occur. The inner effect of temperingwas presented for thin plates. The edge effect is now studied, on one hand, to thesame thin plates and, on another hand, to thick plates (l9-mm thickness) for acomparison with optical measurements.

The tempering parameters are the initial temperature (easily determinedduring the induslrial process) and the thermal transfer coefficients (unknown).Thermal transfer coefficients can be calculated by an inverse analysis.

Results

As for the inner effect, the stress sign changes during tempering on the edges. Thetransient traction (uyy) is very high at the comer (point 4). At the end of thecooling phase, the three points on the surface (points 2,3,4) are in compressioneven though the core is in tension (point I) (Figure 13).

In the middle of the plate, the stresses are parabolic along the thickness(Figure 14). The stress profile in the thickness on the edge depends on the ratio


Residual tenslon stress (MPa)

100

80

60

~ h, = 376.2 W/m' .K

~~ .==-=::::::::, h,=280.1 W/m'.K

40

~__--===-. hI = 188.1 W/m'.K~

~::::~::'~=~==.~. h, =92 W/m'.K

20

0600 650

experimentscalculations

700

IDldal temperature (0C)

750 800

Figure 12. Variations of residual tension stress with quenching parameters.

hl/hz• where h, is the thermal transfer coefficient of convection on the plane andh z is the thermal transfer coefficient on the edge (Figure 5). If hz is below hI' thetraction along z (uzz ) is more important at the corner than in the middle of theedge. If hz is far above hI' it is the contrary.

Time(s)

500

-node I- node 2--- node 3---node4

400300

------------------------------------

200

T o=6S0·ChI =280.1 WIm'.Khz =100 WIm'.K

-" ----

100

•\\

\ "\ \

\ '\ ", "

"">'"~----------------------------------_.

-50

·100

50

Stress zz (MPa)

100

·150

Figure 13. Variations of stress (Tn with time in the inner part and at the edge.

Figx~re $4, Vurzatww 4 ~ $ : xifess (5 ;an rhe thicknc3% i ia Ihc middtc n ~ d ;$I: the edge &sr 4evcral e.aBiac.a of

Influenee af the G h a ~ ~ f e r

The chara~fer introduced for the cs:~mputat&r~s caf thin glass pla~.es is bi,fi-~lm wide, "I'hc thermal transfer caefficient is assumed to hc the sa144e in the broke11 eorgncr as in the edge, The charx~fcr has XIO Infiaxenee on the residual stresses (Figure 15) lmt cva decrease t hc valare of tlxc maximum trafisicnt surface tension stress (Figure I h)-

Case of a Thiek Plate

Optical measuren$esrtga 'fkmpered glass is birefringent. The phc~tc)ei;$stici~xefq tectmnoit)@ attaws stress mcasurc~f~ents 1151- TWO kinds of these measuremefits are rcaiizcd an thick tempered glass pkates,

First, thc Epihiascrspe 116, 17: is an zipparatus that is used &B obtoirs the surhtce stress. I r is massred in sevcsal points i w the innca part of thc plate tr;.,, = $;I

mean = 130,3 MPa standard deviation = 4,9 MPa

wi$h prccrsic3n of measurement = & 6 MPa

The B2ibinct compefisator measures the difference of the apticaf path in the thickness of the plate, which is gsroporhioaai ta, the i~tcgral in the thicknless of the difference of principal stresses (ii,, - 3;) 1181. This integral is equal to zero in the middle of the plate, Thc ificasuregnents are done close to the edge (1.5 nmm from

n ehc m-8sddlc of the pi P

a--- -)"a5 -2a t

at &e edge of &e plate ,,e^"r :z z" Z"G*"

,A"&-

7-

i - watheur chamfer

- 1 2 ~ # : - wath cha

1 , ,--- i.- -~l_hcha - Figare ISfJx Strcs., $ J - _ 1-$1 ~ h c thibxsbnc$~ is $he mi~kllc at.stl ;nB lt4c ctlgc t a f the plate ~ i i t f m . and wlthew.r ch;fnnfcr,

the edge), at the limit of thc chamfer:

mean = 42,O MPa standard dcvii3lion = 3-3 MPa

with precision id $Ileasure = f 1.2 MP3

Figam 16, I4;irratssjna ot $$re%% ~ ' 7 ~ ; ~vilih tame En the mrdslie and at the edge nilh 2nd ~8vi fh~wf ch;i~~~ter.


Several measures at different distances from the edge were carried out. Thevariations of the integral in the thickness of the difference of principal stresses withthe distance from the edge are given (Figure 17).

Results.

Determination ofthermal transfer coefficients Uniform tempering of a glass plate ischaracterized by four parameters:

To = initial temperature

h , = thermal transfer coefficient of convection on the plane

h 2 = thermal transfer coefficient of convection on the edge

Text = temperature of blown air

The initial temperature and the temperature of blown air can be determined quiteeasily during the industrial process. For studied thick plates, their values are 620°Cand 20°C, respectively. hI and h 2 are very difficult to measure. These coefficientswill be identified by a comparison of optical measurements with calculation results.Since the surface residual stress values in the inner part of the plate are onlygoverned by hI' the Epibiascope measurements allow one to determine h ,

The edge stresses are controlled by both h ; and h 2 • h 2 can be identified with theBabinet compensator measurement at the limit of the chamfer (point A on Figure17)

Validation of simulations.

Validation by optical measurements In Figure 17, only one point was used todetermine the thermal transfer coefficient h 2 • Figure 18 shows a comparison ofoptical measurements and -computational results from the edge to the inner part.

Validation by fractographic analysis Thick tempered glass plates were tested tofailure under four-point bending. The local failure stress is obtained with theultimate load using beam theory.

A semicircular shiny zone (mirror of failure) can be observed in the failurepattern of annealed glass. The mirror radius, corresponding to a crack branching

THERMAL TEMPERING OF GLASS

Integral of the dilTerence of principal stresses (MPa)

685

80

60

40

20

-20

o 5 10 15

x

20 25

d =Distance to the edge (rom)

Figure 17. Variations of the integral of the difference of principal stresses with the distance from theedge.

Integral of the dilTerence of principal stresses (MPa)

120

100

80

60

40

20

-20

h I = 135 WI m'.K

h 1 =115 WI m'.K

5

x

..... Experimental points_ Calculations

d =Distance from the edge (mm)

Figure 18. Variations of the integral of the difference of principal stresses with the distance from theedge.

{C'Bl, ca1-z bc linked to $he tensile htilurc stress by 6191

where r-,,, - mirmr radius and M = 1.85 MPa.rn':< The prcbtious reliltion is oh- tziined u.;isfg iinex elastic f i " ; l~$~re nxeehrrnics theory,

For tempered glass, the failure mirror is also ohscnrcd (I;igurc i91. 'l'hc residual strespa st;ak mtst he tbrken ir-ittl axarrant in the frad.lugaagixic analysis, Thc stress intensity factor eirgl f ~ e 8~.rii%m [;?I%, 211 if3

whcrc ~d - cmck radius, Y, ( 81 = sh;:lgc factt.rr assoeiatcd with hcmading, grnd Y;,( r l ) - shapc fiict6ra assoei;itcd with residki~~l S ~ ~ C C S = l a The i \ p ~ a ~ e ~ l i failarc stress ( u, 1, $ 1 ~ rc&iiJ%~;~I ct~mprcal"air'vii7 stress f vJt), rlix! the xnirro~ radius <r,,B car?i he linked I K 3 - Kt and n --as4,,) via

wt~erc Q,, - 2.3 % MI3a,n3 l j.2 lind Y, ( r l b - K, ,$/M 2.33/ 1-85 - I,2Er, Mcilsurc- meats were taken fix two pl;t%e~:

TIlERMAL TEMPERING OF GLASS 687

-9.5

These values of residual stresses must be compared to the mean value of thecompression stress on the edge obtained with calculations: 101.1 MPa (maximum =108.8 MPa and minimum = 93.8 MPa). The deviation is only about 1%.

Residual stresses. The residual stresses in the inner part of the plate are parabolicalong the thickness, as seen earlier for thin plates. The residual stress value on theedge of the plate is quasi-constant and slightly inferior to the surface residualstress in the inner part of the plate (Figure 20).

CONCLUSION

The FEM of modeling glass tempering was presented. The calculation of thetridimensional residual stress state in the edge regions is necessary for a possiblefailure analysis. The use of Narayanaswamy's model to describe the glass thermomechanical behavior is validated thanks to comparisons between our calculatedresults and the experimental results of previous works. The sensibility analysis hasshown the foreseeable importance of Young's modulus and the liquid glass thermalexpansion coefficient values on the residual stress state. Among the other materialparameters of our model, the reference temperature is the more sensitive. The

Stress zz (MPa)

o

-50

at the ed e of the plate

Figure 20. Variations of u" in the middle and at the edge of the plate in the thickness.

GBB L. DAUDEVILLE AND H. CARRE

glass behavior depends on the chemical glass composition, and a slight change inthe reference temperature implies great changes in the residual stresses.

The edge effect can be analyzed using the variations of stresses close to theedges. The residual stress shape in the edge depends on the thermal transfercoefficient of forced convection applied in the edge. Two kinds of experiments aredone to validate residual stress values close to the edge: optical measurements andfractographic analysis.

An experimental campaign was carried out to identify the fracture features ofglass [22]. The lifetime of glass components in building structures can be analyzedwith the presented approach.

REFERENCES

I. L. Duffrene, Comportement viscoelastique d'un verre silico-sodocalcique dans Ie dornaine destemperatures interrnediaires, Thesis, Ecole Nationale Superieure des Mines de Paris, Paris, 1994.

2. R. Gy, L. Duffrene, and M. Labrot, New Insights into the Viscoelasticity of Glass, J. Non-CrystallineSolids, vol. 175, pp. 103-117, 1994.

3. F. Schwarzl and A. J. Staverman, Time-Temperature Dependence of Linear Viscoelastic Behavior,J. Appl. Phys., vol. 23, no. 8, 1952.

4. O. S. Narayanaswamy, Stress and Structural Relaxation in Tempering Glass, J. Amer. Ceramic Soc.,vol. 61, no. 3-4, pp. 146-152, 1978.

5. R. Gardon, Thermal Tempering of Glass, in D. R. Uhlmann and N. J. Kreidl (eds.), Glass: Scienceand Technology. Academic Press, New York, 1980.

6. O. S. Narayanaswamy and R. Gardon, Calculation of Residual Stresses in Glass, J. Amer. CeramicSoc., vol. 52, no. 10, pp. 554-558, 1969.

7. O. S. Narayanaswamy, A Model of Structural Relaxation in Glass, J. Amer. Ceramic Soc., vol. 54,no. 10, pp. 491-498, 1971.

8. A. Q. Tool, Relation Between Inelastic Deformability and Thermal Expansion of Glass in ItsAnnealing Range, J. Amer. Ceramic Soc., vol. 29, no. 9, pp. 240-253, 1946.

9. A. Markovsky and T. F. Soules, An Efficient and Stable Algorithm for Calculating FictiveTemperatures, J. Am. Ceram. Soc., vol. 67, no. 4, C-S6-C-57, 1984.

10. M. A. Burke, T. F. Soules, R. F. Busbey, and S. M. Kheson, Finite-Element Calculation of Stressesin Glass Parts Undergoing Viscous Relaxation, J. Amer. Ceramic Soc., vol .70, no. 2, pp. 90-95,1987.

11. C. Guillemet, R. Gy, and M. Labrot, Viscosity, Configurational Entropy and Structural Relaxationof Silica-Soda-Lime Glass, Proc. XVII Int. Congress in Glass, Madrid, 1992.

12. Saint-Gobain Recherche, Private communication.13. H. Carre and L. Daudeville, Finite Element Calculation of Stresses in Tempered Glass, Proc. 1st

Int. Sympos. Thermal Stresses and Related Topics, Hamamatsu, Japan, 1995.14. H. Carre and L. Daudeville, Numerical Simulation of Soda-Lime Silicate Glass Tempering, Proc. J.

Physique IV, Colloque C1, Suppl. J. Physique lll, vol. 6, pp. 175-185, 1996.15. V. Novotny, Relation entre les contraintes au voisinage de la surface des verres trernpes, deterrninees

aI'aide d'un epibiascopc, leur degre de trempe etleur destruction apres Casse, VerTes et refractaires.vol. 35, no. 5, 1981.

16. H. Aben and C. Guillemet. Photoelasticity of Glass. Springer-Verlag, Berlin, Germany, 1993.17. A. S. Redner, How to Measure Stress in Glass, Glass Digest, 1994.18. N. K. Sinha, Stress State in Tempered Glass Plate and Determination of Heat-Transfer Rate,

Experimental Mechanics, vol. 18, no. 1, pp. 25-34, 1978.19. J. C. Conway and J. J. Mecholsky, Use of Crack Branching Data for Measuring Near Surface

Residual Stresses in Tempered Glass, J. Amer, Ceramic Soc., vol. 72, no. 9, pp. 1584-1587, 1989.


20. E. B. Shand, Breaking Stress of Glass Determined from Dimensions of Fracture of Mirrors, J.Amer. Ceramic Soc., vol. 42, no. 10, pp. 474-477, 1959.

21. M. J. Kerper and T. G. Scuderi, Relation of Strength of Thermally Tempered Glass to FractureMirrors Size, Amer. Ceramic Soc. Bull., vol. 44, no. 12, pp. 953-955, 1965.

22. H. Carre, Determination de la resistance du verre trempe pour son utilisation en genie civil, Proc.14eme Rencontres Unioersitaires de Genie Civil, Clermont-Ferrand, 1996.

thermal tempering simulation of glass plates: inner and edge residual stresses

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