Electron microscope examination (using chromium shadowed collodion replicas) showed that while the pearlitic cementite in the minimum stress region has a certain definite orientation in every grain (Fig. 2a), this is no longer so in the case of the material in the maximum stress region where the cementite lamellae tend to be oriented in the direction of ultrasonic waves (Fig. 2b); at the same time the cementite particles become broken up. The microhardness in the maximum stress region is slightly higher (I~ff = 206 kg/mm a) than that in the minimum stress region (H~~ = 170 kg/mm2). This means that a certain degree of hardening of steel under the influence of ultrasonic vibrations takes place in regions of maximum stress. It is known [3] that relaxation phenomena due to ultrasonic vibrations take place preferentially in structurally imperfect microvolumes which are tailed relaxation spheres. As a result of friction in the nodal point of a standing wave a certain quantity of heat is liberated in relaxation spheres of a viscous material (ferrite-pearlite) under the influence of each cycle of ultrasonic vibrations; this leads to overheating of the regions under consideration. This localized action of heat and deformation leads to an increase in the viscosity of the softer component due to the precipitation of carbon which forms eementite inclusions and to the fragmentation of cementite grains; it facilitates redistribution of eementite particles in a viscous matrix and changes their orientation in a given grain (Fig, 2b). The structure-refining effect of ultrasonic vibrations on metals was observed by other workers [4]. It appears that ultrasonic vibrations make it possible to redistribute structural constituents of steel wkhout a marked variation in its hardness: the ferrite constituent hardens, while the cementite lametlae in pearlite are broken up and slightly spherodized. REFERENCES 1. V. A. Kuz'menko, Sonic and Ultrasonic Vibrations in Dynamic Testing of Materials [in Russian], Izd. AN USSR, Kiev, 1963. 2. V. P. Severdenko, V. I. Elin, and E. I. Tochitskii, collection: Plasticity and Pressure Processing of Metals[in Russian], Izd. Nauka i tekhnika, Minsk, 1966. 3. Yu. F. Balalaev and S. Z. Bokshtein, collection: Diffusion Phenomena and Metal Structure and Properties [in Russian], Izd. Mashinostroenie, Moscow, 1964. 4. Hugo-Josef Seemann and Karl-Georg Pretoi, Z. Metallkunde, 57, no. 5, 347-349, 1966. 10 August 1967

Institute for Problems of Strength AS UkrSSR, Kiev

UDC 539.4.012 I N F L U E N C E OF U N I F O R M H E A T FLUX ON T H E C R I T I C A L LOAD FOR A PLATE WITH A CRACK G. S. Kit FizNo-Khimicheskaya Mekhanika Materialov, Vol. 5, No. 1, pp. 114-115, 1969 The stress state Of solids containing defects of the crack type under load has been studied by many workers both in the Soviet Union and abroad. However, the strength of many machine parts, structural elements, and engineering equipment is often determined not only by mechanical loads but also by thermal effects. The presence of sharp-ended, cracklike defects in a solid leads to a local increase in the temperature gradients in the vicinity of tips of such defects which, in turn, produces a substantial increase in thermal stresses, thereby affecting the material strength. This necessitates studies of the critical equilibrium Of brittle solids with cracks under a combined influence of mechanicai and thermal factors. This investigation was concerned with the effect of a uniform heat flux on the critical equilihrium of a plate with a rectilinear crack uniaxially strained by tensile loads applied at infinity. Let an infinite isotropic plate of unit thickness weakened by a through rectilinear crack of length 2l be strained in tension by monotonically increasing loads p appiied at infinitely distant points at an angle c~ to the crack piane.

88

i.

Ti i

f

I

i

/

#

/

/7

ta

1

t#

O.g

]

/

/ / 6'

7

1,#

'

0.6

3

02

I.

0

~/a

X/a

Fig. 1.

~/g

gJr/a

~i-/s

oL

Fig. 2.

In addition let there be (at a certain distance from the crack) a linear temperature field

(I)

t . = q ( x cos~ q- y sin~) + t o ,

where t o is a constant and q denotes the intensity of a uniform heat flux at infinity oriented at an angle 8 to the Ox-axis; the coordinate system xOy is chosen in the m i d d l e plane of the crack so that the crack is situated on the Ox-axis and is s y m m e t r i c a l in relation to O. tt is assumed that the crack walls and plate-base faces are heat insulated. The problem consists of determining the c r i t i c a l values of the external load p, and the heat flux q. at which one of the crack tips reaches the c r i t i c a l equilibrium state, as a result of which crack propagation occurs in a brittle p l a c e . * Using the basic concepts of the theory of equilibrium cracks under normal fracture conditions [1], we obtain the following equation for these values:

O, 1/'2K kl - - 3 k 2 tg-ff- = ~ cosSO~" 9 2

(2)

Here k 1 and kz are stress intensity coefficients which for the right crack tip (x = l) are given by [ 1 - 3 ] :

k I -----p l/'7-sin=a, k= = p V T ( sin ~ cos~ + "q),

(3)

where ~7= (atEql tsin 51)/4p; a t is the linear thermal expansion coefficient, E is modulus of elasticity, ~ . is the angle of the initial orientation of the crack measured anticlockwise from the positive half of the Ox-axis for the right tip of the crack and from the n e g a t i v e half of the Ox-axis for its left tip, and K is the cohesion modulus [4] which is a constant characteristic of the m a t e r i a l under g i v e n conditions. Since K is t e m p e r a t u r e - d e p e n d e n t , its value in (2) must correspond to the temperature of the crack tip region. Angle |

is given by O, = 2arc tg

1 -- V 1 +

4n

8n ~

k2 --. n -----kl

,

(4)

Thus, in accordance with Eqs. (2), (3), and ( 4 ) w e find

P~

=--

4k~t(

O,

= ] f Y s i n ~ cos a -2--(1 + 3 V 1 + 8n ~)

4,2p,~ '

q,

--

c~tEli sinai

.

(5)

*It should be noted that the stress state in the vicinity of the crack depends only on the temperature gradient and not on the general t e m p e r a t u r e l e v e l (if the temperature dependence of p h y s i c o m e c h a n i c a l properties of the m a t e r i a l is neglected). The temperature in the crack tip regions is assumed to be such that the fracture is quasi brittle in character.

89

In the special case of a plate under the influence of a temperature field only (i. e . , p = 0), we obtain the following formula for the c r i t i c a l heat flux:

9Vg X q~ = ~l~ %E Isin ~l

(6)

Curves representing the variation i n | for the right crack tip at ~ = 0, 0.5, 0.8, and 2.0 are reproduced in Fig. 1, while in Fig. 2, the value ~. = ~4/p,/4-2K for the same crack tip* is plotted against a(0 _< a _< 7r). It should be noted in conclusion that in the case of a plate under the influence of either uniaxial tension or a heat flux the crack will start propagating at both its tips simultaneously. In the case of a simultaneous action of m e c h a n i c a l and thermal factors, the conditions in each crack tip region are different (except when ct = 0, a = 7r/2, or a = It). At 0 < a < w/2 the crack will start propagating from its right tip and at 7r/2 < ct < ~r at its left tip (the line a = 7r/2 is the axis of symmetry). Increasing the heat flux produces a reduction in p., i . e . , the strength of a plate with a crack decreases. REF ERENC ES 1. V. V. Panasyuk, C r i t i c a l Equilibrinm of Brittle Solids with Cracks [in Russian], Izd. Naukova dumka, Kiev, 1968. 2. Ya. S. Podstrigach and G. S. Kit, c o l l e c t i o n : T h e r m a l Stresses in Structural Elements, vol. 7 [in Russian], Izd. Naukova dumka, Kiev, 1967. 3. G. S. Kit and Yu. S. Frenchko, Prikladnaya mekhanika [Soviet Applied Mechanics], no. 9, 1968. 4. G. I. Barenblatt, PMTF, no. 4, 1961. institute of Physics and lvlechanics AS UkrSSR, L'vov

17 September 1968

UDC 839.4.012.2 CRITICAL EQUILIBRIUM WITH AN ARBITRARILY

S T A T E OF A N I N F I N I T E BRITTLE ORIENTED ELLIPTICAL CRACK

BODY

V. V. Panasyuk and A. E. Andreikiv Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 5, No. 1, pp. 116-118, 1969 1. Let us consider an infinite brittle solid with an e l l i p t i c a l macroscopic crack in the plane z = 0 (Fig. 1). Let us select a rectangular system of coordinates Oxyz so that the Ox- and Oy-axes coincide, respectively, with the large (2a) and small (2b) axes of the ellipse. It is assumed that the crack surface is free from external stresses and that uniformly distributed and m o n o t o n i c a l l y increasing stresses (tensile or compressive) p, q, and g are applied to the solid at infinitely distant points at right angles to each other. Stresses g, q, and p are oriented, respectively, along axes O~, O~, and O~ of a rectangular system of coordinates Og~g whose orientation r e l a t i v e to the system Oxyz is definited by three Euler's angles q~, ~0, and | [1]. The problem is to determine the critical stress levels p = p., q = q., and g = g. at which the crack will start propagating (i. e . , at which l o c a l fracture of the solid will take place). To solve this problem, the following assumption is made: the local stress distribution in the i m m e d i a t e vicinity of the crack edge (in a plane normal to the edge) m a y be regarded as planar.** Starting from this assumption and using concepts of the theory of macrocracks [2, 3], we obtain the following relations for determining p., q., and g,:

. ~" - ]~cos ~-

3 k ,. .s F m~- =

K

,,

(1)

COS ~2-

=0

)

0kl ~~ 0k2 ~3. ~ - c o s ~- -- 3 ~ - s i n ~ ~=~* ---- O.

(2)

(3)

*Analogous results are obtained for the left crack tip if a is replaced by Or - a). '~The validity of this assumption in the case of an infinite solid with a circular d i s c - l i k e crack was proved in [4]. 90