T H E S I M I L A R I T Y T H E O R Y A P P L I E D TO THE T H E R M A L S H O C K R E S I S T A N C E OF B R I T T L E CERMETS G. N. T r e t ' y a c h e n k o Institute of Metal Physics Problems, Academy of Sciences of the UkSSR Translated from Poroshkovaya Metallurgiya, No. 2(26) pp. 83- 89, February, 1965 Original article submitted January 8, 1964 Investigating the thermal-shock resistance of engineering components is a complex problem. The complexity results from the thermal-shock resistance being a function of a large number of values such as strength, linear expansion, modulus of elasticity, thermal conductivity, etc. Since the temperature field is not stationary, complex relationships exist for all these values with temperature. It is difficult to trace the influence of a particular value and it beoomes difficult to arrange experiments to study this value. In view of the above, we can use the similarity theory which states how it is possible to group separate values into dimensionless complexes and simplexes and how to reduce the number of these complexes and simplexes to a minimum. The similarity theory also indicates the mathematical structure of relative equations for the connection between complexes and simplexes, and also forms the conditions for similarity of classes and groups of similar phenomena for which the relative equations obtained from experiments remain true. Thus, the theory is a base for experiments and makes it possible to collate data from investigations of the properties of many materials. The similarity theory has been developed in many sections of science, chiefly in heating technology and in gas-dynamics. It has been developed mainly by Soviet scientists M. V. Kirpichev, A. A. Gukhman, P. K. Konakov, et al. As regards the investigation of the strength of materials the theory has scarcely been developed, despite the fact that in our country the foundation of the theory's development was based on mechanics by M. V. Kirpichev and his work dealing with investigations of similarity in elastic phenomena. Subsequently, the theory was used in the work of N. N. Davidenkov, G. P. Zaitsev, and some others. However, since the theory has frequently been used without taking into account the similarity of the properties of the material itself but only the similarity of equilibria, elasticity, and geometry, some investigators have rejected it for dealing with questions of strength and confirmed that the theory contradicts the results of experiments made on the strength of materials. The similarity theory began to be opposed by the so-called "scale factor" although there was no basis for this opposition. The similarity theory and the "scale factor" in no way exclude each other since the first is a theory of processing experimental results and descriptions of phenomena, whereas the "scale factor" is a phenomenon depending on the description. Let us show by example for brittle cermets how the theory can be used to study the strength properties, in particular, thermal-shock resistance. The equation connecting the deformation and strains in a body in the presence of a temperature field (physical equation) can be shown in the form of the general equation for Hooke's law [1].

1 ~. = -~- [~x - - ~ (~v + ~z)l+aT; 1 % = ~- [ % - - ~ (% + % ) l + a T ; 1 % = ~- [%- - Vt(~x + ay)l+ aT; "~xy .

~ x y = -G- '

T.~z. YY"-

O "l~zx

7 ,x - - - 6 - .

154

'

(1)

The equation connecting the forces, displacement, and time (equilibrium equation)takesthe form

O~x O'c~u O~x~ 02u Ox + -gy- + - ~ - + QX = Q ot2 ; a'%x 0% a~u~ Ox + - @ - + - ~ + Q Y = Q

a~y

O~

a~

O~v . at ~

(2)

@w

ax + - O ~ + - - & - + Qz = ~

at~

9

The geometric equations connecting the displacement with deformation take the form

Ou

Ov

Ow

(3) au

Ov

~x,=3~+~;

Ov

Ow.

am

au

v,,=~/+h-y -, ~.x=-0~+~.

The conditions of similarity for the problem in the absence of a temperature field will be

Px~ = %l + ~ / n + ~,n;

p . = ~u,l + %m + ~un;

(4)

p~ = ~xl + %urn+ %n. where

Pv =

P~ T-

"

Iv

(5)

In Eqs. (1)-(3) in nonstationary temperature-field conditions, the following equations should be connected:

z = f(r);

(6)

a

(7)

=

fCT),

and also the thermal-conductivity equation

OT -Oi- = aver"

(s)

The conditions of synonymity of theseequations will be: a) initial condition

To = To(x, Y, z);

(9)

in the frequent case

To = const; b) limiting condition

a* [T= (t) -- Tel ----0. ~0T)=,+_ -

(lo)

This condition corresponds to the so-called limit (boundary) conditions of the third order or to Newton's law. In the frequent case, it is possible to assume T c = const. For the sake of simplicity, other forms of limiting conditions in this paper are not examined. A11 the equations given above take into account the working schedule for the component, but,except the physical Eq. (1),do not take into account the properties of the real material. If we examine these equations from the

155

point of view of the theory of similarity without the connection with the properties of the material being studied, it is possible to conclude that experimental data contradict the similarity theory. The regularities of the mechanical properties of brittle materials thoroughly explain the statistical theory of strength and the theory embracing the effect on strength of microconcentrates of strains, existing in the material in the form of microcracks distributed over the volume, pores,and other internal defects [2]. As an example, we can give the following equations connecting the strength of the material with other characteristics of the material and component. According to the theory of A. A. Griffiths, the strain o k developing a fracture in the planar strained state of a component containing a microcrack of elliptical form with a line bigger than the semiaxes of the ellipse C k equals

2UE

]os (11)

O"K ~

The theory of W. Weibulltakes the following form for the magnitude of average strength:

a~v = Ira(l* ~ L.

(12)

Reducing all the values used in these equations by employing the theory of similarity we note similarities of the form u 8 = u~,Us.

ulh;~ o.

=

(13)

And we shall have ~ a~8,~ e8t~ = eSfi8 ~ El~E

08~ ~8~Y8,~ Ix8~

a~o Ixl3~a*8a -- F80

%oE80 Es~

T~0

88fi8o Es~

+

aooTo. ~

eSo asaTST;

TxYsz .

~x,8~ = ~8oG~o GSo ' 9

.

,

9

9

9

,

,

tgtrxao Ir8o01rxya, ~ ~8o O'~uzS, _t Qs~176176 . . . azs, " .~ % xsx

88088e --~ 9

9

9

0"i3ol~o O~uSt ~----~0.13 "8o % ~ ~% ;

(14)

OllSl . OXSl '

9

Ou~l

9

,

Or8 z

~oVSv-ays~ + Oxsf "

Equations (14) will be formally identical for 8-similar phenomena, if the following conditions hold ~$' --

aBTs' ---- 1;

Qs~

---- 1;

QSo l~______j, = 1' (15)

~80 ~13,G8~ = 1;

156

880 = I ;

~/80= I;

x8_2, = 1. a8o

Hence, we get the following invariants of similarity:

a~

= idem;

*~

%% - -

= idem;

idern; (16)

~x13= idem;

Q~X~l~=

idem; ~

= idem.

(113

The first of these equations suggest that to provide mechanical similarity in the geometrically similar specimens made of one and the same material, it is necessary that they have the same strains at the same points, the same mechanical deformation, and the same temperature deformation. When we equate conditions of similarity for different materials the signs of similarity will be: At the same points of geometrically similar specimens the ratios of the magnitudes of mechanical strain to the modulus of elasticity, multiplied by the magnitude of relative deformation, will also be the same. With the same deformation at the same points of the bodies being compared the strains will be located in the same ratio as the modulus of elasticity of the materials being compared; deduction of coefficients of linear expansion for temperature at the point should be identical. The conditions of synonymity (4), (5), (9), and (10) with the same conversions give the similarity criteria

Pv~

Bi=--

-- idem;

(17)

a*l 3,

=idem.

(18)

Condition (17) in the literature is known as the Barba-Kik law, and this condition especially has been examined by many investigators when comparing the results of tests for brittle and plastic bodies. However, this is inadequate for studying the properties of real materials. Thus (11), if it is true for the material class in question, gives the following extra similarity condition:

o~c~_ U~E~ Equation (12) gives

V

((1

av . (1*lraL

idem.

(19)

= idem.

(20)

As a result, for thermai-shock tests on components, according to the second similarity tiaeorem (g-theorem), it is possible to get the following equation: a) if the Griffiths equation is used as a base

(is

[' (1K.

(1~o

\ ~~

(12

aoTo

~o' U o E o '

so

p ' (1~ol~

c~ U ) '

Bi,

CKo~ U 0

(21)

"

b) if the Weibult theory is used aav

.~

[ (ran

(

(lay

Vo (1; oL ~

) % T O P Vme*] m~

Bi,

'

lo

,

, ---;

Vo mo (1o

.

(22)

These equations assume the absence of forces of inertia and mass forces. Equations (21) and (22) state in what form we should find the strength formula and what values to study experimentally. Thus, in the second case the complexes will be P/omean0l~ and Bi, if the temperature ranges of the test are unchanged; tile simplexes will be V/V0, m/m0, and o*/o~. All the values with index "0" are called parametric. T h e y are subject to the conditions

157

of the problem. Omean/Omean is a simplex function. Equations (21) and (2) are correct for any group of similar phenomena. If other strength theories are used as a base, it is possible to get equations identical with (21) or (23). Connotations not given in the text are ex, ey, Sz, ?xy, ?yz, and ?zx and are components of relative deformation; o x, Oy, o z, rxy, ry z, and rzx are components of strain, E the elasticity modulus, p the Poisson ratio, c~ the coef-linear expansion, T the temperature, p the density, X, Y, and Z the mass-force components, u, v, and w the displacement components, t the time, Pxv, Pyv, and Pzv the strain components on the surface area of the body with a normal v, l, m, and n the directional cosines with the normal, v the normal of the surface, Pv the force acting on the normal, ~ the area of action of the force, k the coefficient of thermal conductivity, a* the coefficient of heat exchange, T c the temperature of the surroundings, Ts the surface temperature of the body, To the initial temperature of the body, t5 the order index of the similar phenomena being studied, U the surface energy of the material, I m the parameter which is a function of parameter m, m the coefficient of the heterogeneity of the material, o* the constant for the material equal to the strain which for unit volume of material gives the probability of failure, equal to 0.63, V the volume of the specimen, L the function of the form of the strained state, l the typical geometric dimension, and Bi the Bio criterion. LITERATURE

1, 2.

158

CITED

N. N. Bezukhov, Theory of elasticity and plasticity, Gostekhizdat ,Moscow (1953). G. S. Pisarenko, V. T. Troshchenko, et al., The strength of cermets and alloys at normal and high temperatures; published by the Acad. Sci. UkSSR, Kiev (1963).