VOLTERRA PRINCIPLE IN CALCULATING SHEET GLASS DEFORMATION

A. I. Shutov andE. P. Sakulina UDC 666.151:539 37.001 ~"

Pressing and free shaping for sheet glass are becoming more important because of the increased proportion of bent items used in vehicles. There is also a tendency to reduce the glass thickness in automobiles, and to use glass items with complicated curvatures. This requires a careful approach to managing the shaping, including the use of automatic systems based on models.

Volterra's principle is used in one such approach. The specimen is a flat plate of silicate glass having a given temperature pattern.

Here we consider the strain characteristics for a given external load. We use Carte- sian coordinates in real time.

The main feature of the thermal deformation in glass even under constant load is the time dependence, i.e., the creep, which is best described by integral Boltzmann-Volterra inherited-creep equations: [i]:

t

n(O= E~:,{Ij--.E I Rit--Ej~(~jd ~ ; f~

I

o(t) t ! ~'ll~= --'U-: • ~ /<(t--.:.,)o(~)d(~),

( i )

in which o(t) is stress, E is the first-order elastic modulus, ~(t) the strain, R(t - 6) and K(t - $) influence functions, and $ a time preceding the observation instant t, which reflects the loading or strain history.

We assume that analytic expressions are known for the influence functions.

We extend that theory to states of strain other than uniaxial. The differential equa- tion for the deflection W of a rectangular plate is [2]

a ' w a'p;" a ' w g ( 2 ) - - ~ - + _ o~--~+--~u = T ,

in which g is the transverse load per unit area,

Eh ~ ( 3 ) D= 12(l--p,) '

in which h is the thickness and D Poisson's ratio.

Fig. i.

Y

Scheme for plate loading and

deformation.

,n~ ~i ~ute. G r i s h m a n o v B e l o r u s s i a n G l a s s M a t e r i a l s -r ~ - . , - T r a n s l a t e d f r o m S t e k l o i K e r a m i k a ~ ~,o. 8, pp. ,5-,6, August~ 1991o

1,070~-0348~.~ .50 1992 Plen~3 P u b l i s h i n g C o r p o r a t i o n 348 0 ~ 6 ~ - , ~ I 0 , . �9

One combines (2) with the dependence of the strain components on the stress ones. In the purely elastic case, the relationship is an extension of Hooke's law. With creep, one uses the second equation in (i).

The properties of the material that define the time operators are independent of the coordinates, so the time and spatial coordinates are interchangeable. One can perform the operations in any order. If one first performs all the operations with respect to the co- ordinates on the assumption that the time operators are constant and then uses the boun- dary conditions for the current instant, one gets the solution to the elastic problem. One replaces the elastic constants by the corresponding theological operators to get the solu- tion for inherited creep.

That method is known as Volterra's principle.

For example, the elastic solution for a rectangular plate (Fig. i) under load

with free support at the edges is

e = g o s i n ? sin my b '

II~'= ax . my //" sin -- Sin -- .

(4)

The unique parameter in (4) is the rigidity D, which is defined by (3).

We derive an analytic formula for the elastoviscous equivalent of I/D [3]:

6 m ( t ) = ~-p-[Brl (O+ t ~g, /2( t ) ] ,

in which B=3(I_2~0):s 2(t) is the compliance, gl/2(t) the ll'yushin [4] coupled creep, and

~0 the initial Poisson's ratio.

In turn,

t

r l l t i = _ 1__ l+ K(t)d ; ([- L;

0 t

0

A= I.~?'~.,P[" A; gt/2= -1"~12~/c, ;

~%= I --2p.,

Certain transformations for ~0 = 0.22 give

t J

in which Al/2 = 0.1867.

As we have assumed that the analytical formula for K(t) is known, one can calculate )if) without particular difficulty.

These calculations can be used to derive parameters for molding sheet glass by pressing or other means. On the other hand, we have a strain model, which can be used in a computer program to control the process.

3 4 9

i. 2.

3. 4.

LITERATURE CITED

Yu. N. Rabotnov, The Mechanics of Deformable Solids [in Russian], Nauka, Moscow (1979). S. P. Timoshenko and S. Wojnovsky-Krieger, Plates and Shells [Russian translation], Nauka, Moscow (1966). M. A. Koltunov, Creep and Relaxation [in Russian], Vysshaya Shkola, Moscow (1976). A. A. Ii'yushin, Mechanics of Continuous Media [in Russian], Nauka, Moscow (1978).

ELECTRICAL CONDUCTIVITIES OF LOW-ALKALI AND ALKALI-FREE GLASSES

AND GLASS CERAMICS

T. V. Dubovik, V. Ya. Sushcheva, G. N. Shmat'ko and L. I. Maikova

UDC 666. ~ ~ o. ~6~... 539.26

Glasses and glass ceramics that do not contain alkali oxides have low specific (bulk) electrical conductivities under normal conditions: 10-13-10 -12 S'm -I, which increase some- what with temperature. Such materials are used as high-temperature insulators in electrical and radio engineering. If the temperature is only slightly above the room value, one can use low-alkali glass ceramics having conductivities of i0 11-i0 -I~ S'm -!. The applications of glass ceramics as high-grade insulators are continuously extending, so it is important to make them with better electrical characteristics, which involves not only resolving technological problems but also new developments and research on the origins of the con- ductivity in glass ceramics, which should also be measured more accurately.

We have examined the conductivities of glasses and glass ceramics in the following systems: Li20-AI203-SiO2-TiO2, Li20-AI203-SiO2-P205 [sic] (Ag20), MgO-AI203-SiO2-TiO2-Ce02, MgO (MnO)-AI203-SiO2-TiO2, ZnO-AI203-SiO2-TiO 2, BaO(CaO, SrO)-AI20~-SiO2-TiO 2. The con- ductivities were determined by the two-electrode method by means of an E7-4 ac bridge operated at 102 Hz, temperature range 350-800 K. The electrodes were made by firing pastes

at 720 K, which were prepared from silver nitrate.

Table i gives the basic conduction parameters and the mineral compositions of the crystalline phases. We see that all the alkali-bearing materials have low T k - i00 (the temperature at which the resistivity is 108 ~'cm). A distinctive feature of the glasses is that the activation energy derived from o(t) varies from 0.60 to 1.17 eV in the range 540-620 K. The conductivity falls on going from a glass to a glass ceramic, which shows smooth increase in heating characterized by a constant activation energy and elevated T k - i00, so such ceramics are reasonably stable insulators.

Alkali-free materials have high T k - i00. Glass 370 had the lowest conductivity amongst the glasses. When an alkali-free glass crystallizes, the conduction activation energy and T k - i00 are reduced, although the value remains quite high by comparison with

T k - i00 for an alkali-bearing glass ceramic.

Figure 1 shows that o(t) varies for alkali-bearing glasses and glass ceramic having compositions 4~o,'1~ 023, and 37. Glass 418 contains ..... ~ n~ Li20 (here and subsequently, mass content), and shows only a slight change in activation energy on heating, which may mean that the carriers are of the same type in the range 350-380 K and are Li +. Glasses 023 and 37 have Li20 contents increased to ii . . . . . . =v and simultaneously contain o 5,o~ K20, which affects o(T) and increases the activation energy in the range 570-620 K. The highly mobile Li + are accompanied by the less mobile K + in the conduction.

The crystalline phases in ceramic 418 are mainly of $-spodumene solid solution (Li20" A1203-4Si02) and SiO 2 with small amounts of rutile TiO 2. The $-spodumene crystals contain channels, in which the Li + can move freely and participate in the conductivity. Here T k -

�9 ~ glass does not improve the electrical I00 is comparatively low The 1% BaO in the ~io characteristics of the ceramic�9

Automobile Glass Research Institute. 16-18, August, 1991.

Translated from Steklo i Keramika, No. 8, pp.

~' ~i,0,0u-0350~ ..... �9 1992 Plenum Publishing Corporation 350 0ju _7610/o I ~ o ~io ~n