This investigation deals with the fracture produced in a disc by simple rotation cycles as well as rotation cycles under an imposed hydrostatic pressure. In both the cases, relations are obtained between the critical angular speed omega and the number of cycles N, required to cause fracture

Muthuswamy, V. P.

Singh, R. Mohan.

English

Defence Science Journal

R~EGIONAL ENQINEERING COLLEGE, T I R U Q H I R A P ~  (Received 17 June 1972 ; revised 29 J a n w y  1974) This investigation deals with the fracture produced in a disc by simple rotatian cycles as  well ab rotation cyclae under an imposed hydrostatic pressure. In both the cases, relations are obtained between the critical angular speed u and the number of cycles I?, required to cause fracture. N O T A T I O N S  T ~ ,  7,-Major, intermediate, minor principal elastic stresses k, JZ -Slip moduli of the material a, w -Fatigue moduli of the material G, es , e,-Principal plastic strains a,, a#, 0,-Principal plastic stresws el, u2, 0,-Major, intermediate, minor principal plastic stressee a, So, 8,-Plastic stress deviators B A S I C  A S S U M P T I O N  The theory of fracture of brittle solids developed by T.Y. Thoma~l*~  is applied to a thin solid dim rotating about an axis through the centre perpendicular to its plane. Thc: fracture is produced in theidisc by simple rotation cycles as well as rotation cycles under an imposed hydrostatic pressure. (i) The material of the disc is brittle. It undergoes an abrupt chalrge from its elastic state to a state of plastic equilibrium, immediately prior to fracture. It fractures without the necessity of increasing the angular apeed beyond the point required to produce the initial plastic deformation. (ii) A simple rotation cycle is one in which the angular speed increases continually in numerical value from zero to a maximum value and then decrease8 continually to the initial value zero. The maximum numerical value of w, of a rotation vole, is called the peak load of the cycle. The peak load at which fracture occurs is called the critical load of the cycle. In  this paper only the critical values of w which depend on the number of cycles N have been used. It is a constant for a given va lu~  of N. (iii) A function is defined as a (7) = (T~-7,) - 2 G (1) (is) If fracture is produced during the Nth cyclie load, the equation ( 2/13. -j- 3.1) al - ( 4 1  + 7c2 - k) a, = 2 K - 2,a (N-1) f2 (7) (2) 611 be satisfied by the stress field in the plastic state precediog fracture. If N = 1, equation (2) reduce8 to ( 4 1  + k ) q - ( d l + @ - k )  u g = 2 K  (3) which gives the fracture problem for a single loading operation. (v) The moduli b, ii and K satisfy the inequality ( ~ / W + ~ ) G < K  where k is very small and li: is very large when measured in conventional units. (vi) The disc is assumed to be so thin that the condition of plane stress exists. DEB. SUI. J., VOL. 24, OOTOBER 1974 (vii) E n h n c e  Zhit : The magnitude L of the peak load satisfies the inequality L,<L <La for loads which are just sufficitnt to cause fracture d u g  the Nth loading operation. The upper limit L2 is the mag- nitude of the peak load when fracture is produced by a single loading operation ; the lower limit L, is r e fend  to as the asdurance limit. F O R M U L A T I O N .  The cylindrical coordinates r, 8, z are employed such that the z-axis lies along the axis of rotation of the disc. It is assumed that there are no external fcroes acting on the diao and that it is of constant thickness. In addition to the condition of fatigue fracture (2) , the following relations are satisfied by the non- vanishing strew and strain fields induced in the diso, prior to fracture : Equation of incompressibility e, + eg + e, = 0 ( 5 )  Hencky stress-strain relations f i r = $ &  ; So = $ a d ;  S z = $ e ,  (6) Equation of equilibrium where p is the density of the material. The proportionality factor $ is assumed to be a positive function and is regarded as one of the dependent variablas of the problem. - I The boundary condition of the problem is > ,  ur=O a t r = a  , (8) where r is the distance of any point of the disc from its centre and 'ay, its radius, ANALYSIS Following Hoffma and Sachs" the major, intermediate and minor principal' plsstio s t rmm are res- pectively Substituting these values of ol and age in (2), we have \ ( d l + ' + k ) ~ g  = 2 E - - 2 a ( N - - l ) D ( T )  The principal elastic stressed are Using these values in (I), we find th t  M U T ~ S W A M Y  & MOHAX Smcla : Study of Frapture-in a Rotating Diso and hence equation (9) becomes Prom plane strws and symmetry considerations a, and ae are functions of r only. Therefore equa- tion (7) reduces,to the form &a, 1 dr (a8 - a,) + pw2 r = 0 -r Usiq the valne of Oe ,from (11) in the above and solving, we get on taking the constant of integration to be zero to avoid a, becoming infinite a t  r = 0. On using the bo,udary condition (8) in (13), we find that whioh gives the critical angular speed o, prior to fracture as a function of N. Equation (14) c m  be written as where ill= oS. It represents a hyperbola in the variables M and N. The second factor on the left side of equation (16) ia always positive. Also the right hand member is positive because of (4). Hence "'h Theref~re the endurance limit Ll has the value I From the relation (14), we can deduce the value of o which will cause fraoture in a single loading opera- tion. On setting N =  1 in (14), we observe that M, --------------------------- ' - 6 K  ; N The nature of the graph of eqwtion (15) for N 3 1 Fig. I*--Graph of simple rotation ayoles. is shown in Pig. 1. The endurance limit L is given *HI 5 @ = Square of the endurance limit, Ll paa / A&* - Square of the magnitude of p a k  load, L ,  when fractwe ia Pmdoced W single loading pa3 (dl-+ +I 6  G by the asymptote M = -- 6; of the hyperbola. Also i t  ia seen from Fig. 1 that cyclea with M - wih P a* pa2 not cause fracture in the material. By setting N = 1, in eqktions ( 9 )  snd (13) and using the value of w2 from ( l 6 ) ,  we obtain the following stres field precedipg fracture ia the case of a single loading operation: - ' D I S C  R O T A T I N G  U N D E R  H Y D R O S T A T I C  P R E S S U R E  Denoting the hydrostatic pressure by p and the resulting stress field by o', we have o1 = olg = ag - p ,  0, = a: = 0, -p , o3 = ole = - P  where el '1 3 2  2 On substituting these values for and and for Sd ( 7 )  from equation (10) in (a),' we get after simplification t Equation of equilibrium (12)  now h o m e s  Uskg the value of a'g in the above equation and integrating, we get after omitthg the cons;tant of integration so as to avoid a', becoming infinite a t  r = 0. C On using the boundt~ry condition o', = - p a t  r = a  in (19), the critical speed preceding fraoture is given by W2 = - ( K + k p ) + 2 a & ( N - 1 )  ( - + k ) + 2 a ( N - 1 )  1 (20) Or, we have -- ( ' M a '  - 2 G  3 - -- * - I 11-; ( d ' l + l c ' + k )  f 2kp (21) .. - 14!2 M 1 where M = w2. The second factor on the left side , and the second term on the right side of equation (21) are always positive. Also the first t a m  on the right side is positive because of (4). Hence 1 -2 i ;> ( ,  3 .Therefore the endurance limit L, has the value 1 1 - ' a As before on setting N = 1, in equation (20), we get I N ' ,J2 = 3 6 ( K  + kp) pa2 ( d - w  +k) (22) Fig. 2*-(a) Graph pf simple rotation cycles (b) Graph of simple rotation cycles under hydrostatic prassure . which gives the angular speed necessary to cause fracture in a single loading operation. under hydrostatic pressure. The nature of the graphs of the hyperbolas (15) and (21) for N 3 1 are shown in-'FQ. (2) for the convenience of comparison. , It is clear from Fig. 2 that, for a, given number of cycles, the load requireat0 cause fracture under hp- / drostatic pressure must be larger than the loadrequired under atmospheric pressure. For a single loading operation, this result can be verified by comparing the two values of w 2  obtained in equations (16) and (22). On setting N = 1, in equations (17) and (19) and using the value of w2 from 1(22), we obtain the following stress field prior to fracture when the disc rotates under an imposed hydrostatic pressure, in the case of a single loading operation : r t 7 z  = - P  The above work has tremendous potentialities of practical application beoauseof its close relationship to the problem of ship propellers which become brittle due to passage of time under exposed and marine conditions. The propellers oan be rotated repeatedly without danger of collapse due to fracture provided the I 6 0  angular speed o is always such that o Q - ( p ) t .  Another p o ~ s i b l e ~ a p p l i o n  would be 6 the a design of rotor discs in turbo-prop aircrafts which rotate under high velocities and pressure. * M I ,  Ms have the same values and significance as i n  Fig. 1 MIa = B(K + k p )  Pas ( d y  +- k) =. Square of the magnitude of peak load when fracture is produced by single loading under hydrostatic pressure. 143 

A Study of Fracture in a Rotating Disc

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A Study of Fracture in a Rotating Disc

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