Revista Latinoamericana de Metalurgia y Moteriolee, Vol. 9, NOB. 1-2, 1989 29

Obtention of the specitic-risk-of-fracture function of a square prismatic bar subjectedto torsion

P. Kittl, G. Díaz

Departamento de Ciencia e Ingeniería de Materiales, IDIEM, Facultad de Ciencias Físicas y Matemáticas,Universidad de Chile, Casilla 1420, Santiago, Chile.

SurnrnaryThe cumulative probabilities offracture and the specific-risk-of-fracture functions are obtained forvolume and surface brittlenesses of .a square prismatic bar, In orderto solvethe problem analytically the square prismatic bar is assumed to be included w ithin a cylindricalmatrix made ofthe same material as the bar. When the specific-risk-of-fracture function is unknown it is obtained using the method ofintegral equations, and when the material ofthe bar is simultaneously exhíbitingvolume and surface brittlenesses. both can he separa·;:ec1.'Sesides, the impossioility 01 reducing the integral equacions to one having an exact analytical solution is shown,inasmuch as thekern 05 such equations 18 infinite. The disoersion or the parameters ofthe specific-risk-of-fracture functions is determined usíng the Fis-her informatio« matrix.

1. INTRODUCTIONThe problem of the torsion of prismatic bars

having a cross-section that is not circular is compli-cated even when pure states of torsion are beingconsidered, i.e. when the cross-sections of the barsare exhibiting only a torsional moment and, for ins-tance, there is no flexure in the bars. Díaz and Mo-rales have determined in a recent work [1]the func-tions of cumulative probability of fracture in cylin-drical glass bars subjected to a state of pure torsion[2-6].In that case the problem was less complicatedbecause the hypothesis of the invariability of planecross-sections was allowing to simplify the same.However, strictly specaking such hypothesis is notvalid in the torsion of prismatic bars whose cross-section is not circular; the cross-sections of the barbecome warped and hence the stress-field in such asection is different. In view of the complexity of tor-sion problem, the same is to be dealt with using thetheory ofelasticity, which solves this problem by re-sorting to indirect methods such as the method ofmembrane analogy, for instance [7]. In the methodof the membrane, independently of the cross-sectionof the bar, the torsion problem thereof leads us tothe same differential equation corresponding to theproblem of the equilibrium of some membrane stret-ched over a contour that is exhibiting the same geo-metry as the cross-section of the bar and that isbeing subjected to a uniformly distributed pressu-re. Up to now the problem of determining torsionstresses of some bar whose cross-section is arbi-trary has been solved, in a first approximation, re-placing cross-section contour by the inscribed circlecontour, thus leaving aside those cross-sectionalzones located in the vecinity of the vertices and

whose stresses are very reduced and of little in-fluence on bar strength.

The purpose of this work is, first, obtaining thespecific-risk-of-torsional-fracture function of asquare prismatic bar considering therefor volumeand surface brittlenesses, and, secondly, determi-ning the disprsion of bar parameters. In order to beable to solve the problem in an analytical way weare proposing to include the square prismatic barwithin a cylinder, i.e. to circunscribe a circle in thesquare section. As concerns experimentation it isnecessary that the square bar and the circunscribedcylinder must be of the same material and, as far aspossible, be bonded using some epoxy resin. Ofcourse the stress field thus obtained is not the sameas that ofthe bar considered without the cylindricalmatrix; however, this is a method allowing to getthe volume and surface functions of the specific riskof racture, without resorting to the numerical met-hods and using therefor the stress field that is ae-ting within some prismatic bar of circular cross-section, solving the problem in a purely analyticalway. In this proposed approximation we shall dis-regard the interphase generated between the squa-re bar and the circunscribed cylinder. Two extremesituations are to be distinguished in the experien-ces carried out, namely: glassy material s and opa-que materials subjected to the torsion test. For theglassy materials the cylindrical matrix is to bemade of the same material as the square bar andthrough sintering sol as to get a transparent matrixand then control the experiences using sorne opticalmethod and consider only the fractures that arestarting in the volume and/or on the surface of thesquare prismatic bar. The other extreme situation

30 LatinAmerican Journoi of Metallurgy and Materials, Vol. 9, Nos. 1-2, 1989

namely that of opaque materials, obliges to controlthe experiences using therefor some acoustic emis-sion apparatus to the end of detecting the facturesthat are starting in the cylindrical matrix and in thesquare bar; and this detection will be possible owingto the reflection of the acoustic wave on the interp-hase between the square bar and the circunscribedmatrix.

2. VOLUME BRITTLENESS

According to the probabilistic strength of ma-terials, the cumulative probability of fracture ofsorne hornogeneous and isotropic material exhibi-ting volume brittleness and subjected to a varyinguniaxial stress fields, is given by the following ex-pression [8-12]:

F(r) = 1 - exp {- ~ h[r(r)] dV} (1)'6vwhere V is the volume of the material, V is the unitvolume, r is the position vector, r is the hfghest frac-ture stress, T(r)<r is the stress field, and <j>(r)is thespecific-risk-of-fracture function, also known as Wei-bull's function. In general this function is exhibitingthe following analytical expression:

{(r - r\m

cp(r)= OTo'Jr~r ,

r<r ,

where rr is the stress below which there is no fractu-re, Le. the stress forwihich the probability offractu-re is null, m and T are parameters that depend onthe manufacturing process of the material.

The stress field of some bar of round section,having one end restrained and whose free end issubjected to torsion moment M is given by

r(p)=.J!.. r ~r = 2 Mr 7Tr8

where r is the radius and Lis the length of the bar.Let us consider a square prismatic bar, of side a,inscribed within a cylindrical matrix of the samematerial as this bar; then, in accordance with theproposed model the radius of this circunscribed cy-linder is r = al V16. Hence the expression of thestress field within thesquare kern ofthe cylindricalbar is:

r(p)= V2 P t:a

0<OSrr/4 cosO

Equation (1) can be rewritten as follows [13]:

(5)

Using the method of the defined functions, i.e.the method that is postulating a potential form ofthe specific-risk-of-fracture function, we can deter-mine the curnulative probability ofvolume fracturethrough torsion. If the specific-risk-of-fracturefunction cp(r) is expressed by means of equation (2),the consideration of equations (4) and (5) yields:

';(T) = 4La2

( ~LO)m (~L )2Vo(m+ 1)(rn+2). •

71'/4

k[1+(m+1) ;/rLJP cosOO

] (r/rL )m+dO

v2 cosO - 1 (6)

71'/421-m/2 r dO

rn+2 J c~sm+20O

(7)

(2)

Weibull's parameters m. r and n can be obtainedofrorn above equation (6) inasmuch as ';(r) is knownthrough thetests. If r, =0 a Weibull diagrarn can beplotted, and then the parameters m and r and T can

obe obtained using equation (7).

If a known analytical form is not postulated forthe specific-risk-of-fracture function, then equa-tions (4) and (5) allow to get the following integralequation wherein cp (r) is the unknownfunction [14-16]:

(3)

71'/4 r/v2 cosO

4La21ff~(!')=- - dO <j>(1J)1Jd1JV r2o

O O

(8)

This integral equation 98) can be rewritten asfollows:

(4)

(9b)

(9a)

Revista Latinoamericana de Metalurgia y Materiales, Vol. 9, Nos. 1-2, 1989 31

Equation (9a) is a new integral equation whe-rein the unknown function is cp , which once solvedand introduced into equation (9b), allows solvingtheproblem, i.e, getting the function cp(r). Butequation (9a) has no known analytical solution, itskern is infinite and hence equation (8) too has noknown analytical solution. By directly operating onequation (8) itis possible to obtain its solution bymeans of Taylor series expansions. Therefore, thesolution for cp(r) is:

00

cp(r)= Vo ~ (n+2)~/2-1C; (n)'(O) ~ (10)La2 rr/4"I. dOn=O

. cosm+20()

3. SURF ACE BRITTLENESSIn the case of materials exhibiting surface

brittleness, the cumulative probability offracture isgiven by the following expression;

F«) =Tvexp {- ~ ~ [«r)]dS } (11)

where S is the surface of the material, and S is theunit surface. Similarly to equation (5), above equa-tion (11) can be rewritten as follows:

{(TI~ In 1 _ ~«) ~ ~ ~ [«r)] dS (12)

In accordance with the model that has beenproposed, namely a square prismatic bar inscribedwithin a cylindrical bar, the stress field, consideringall the sides of the square bar, is given by:

VI + (~)2 vf O~Z<L; O~X~a/2 ; y = a/2

(p)= VI + (~)2 h- O<Z<L; O<y~a/2; x = a/2

.y'2 p rO<0<rr/4 ; O<p~a/2 cosO; Z=L (13)

. a

The torsion moment M is acting in the (X,Y)planeo Hence, ifcp(r) is expressed by means of equa-tion (2), the consideration of equations (12) and(13) yields: r/r

c;(r)= 4y2 aL (rL)m rL ¡;(r¡-l)m r¡dr¡

So ro r rV r¡2 - (r/v'2 rL 'fy2rL

+ 4_a2

__ ( ~o )m (~L)2So(m+ 1)(m+2) e e

rr/4

k[1+(m+1) . ;/rLJL1v~ cosO

O

(14).

](.s»:_1 )m+ 1dOy2 cosO

If rl = U we get: rr/4

{(r)- ~-m/2La (_r )m[l + a/L ] r_dO~- So ro 2(m+2) -!:Osm+20

(15)Weibull's parameters m, ro and ri can be obtainedfrom above equation (14) inasmuch as {(r) is knownthrough the tests.If r = O a Weibull diagram can beobtained using equation (15).

If a known analytical form is not postulated forthe specific-risk-of-fracture function, in the samefashion as for the case ofvolume brittleness there isobtained the following equation wherein cp(r) is theunknown function for the present instance of surfa-ce brittleness:

1

~(r)= 4Laf (VI +t¡2 ~) dt¡Vo D r

4a2frr/4+-So dO

O

v2 cosO

p(n)ndnO

(16)

This integral equation (16) can be rewrittenas follows:

(17a)

11. a 1f

cp(r¡)= r¡cp(r¡) + 2L ¡:- cp(';Rd'; (17b)

Above equation (17a) is of the same type asequation (9a), so that here applies the same corn-ments as for that equation. Therefore, by directlyoperating on equation (16) and by means of someTaylor series expansions, the solution forcp(r) is:

00 2 n/2 - 2 .; (n) (O)cp(r)= ~ ~. rr/4

La ¿;",,¡ n! [ a/L] fi dOn=O 1+ 2(n+2) cosn+20

0(18)

4. JOINT VOLUME AND SURFACEBRITTLENESSES

For square prismatic bars subjected to torsionand exhibiting joint volume and surface brittlenes-ses, the respective specific-risk-of-fracture func-

32 LatinAmencan Journai o/ Metallurgy and Materials, Vol. 9, Nos. 1-2, 1989

tians can be separately obtained. This can be achie-ved using the method of integral equations. Theconsideration of equatians (9a), (9b), (17a) and(17b) yields:

{(r) = {v(r) + {s (r) (19a)

(19b)

If we take two groups of samples similarlymade using the same material, and having the di-mensions L. and a. , where i = 1,2, then we get a sys-tem of line~r equations having a non-trivial solu-tion, because of a,. ~ ~ the associated determinantis not null and the system if linearly independent.There is obtained the following pair of integralequations wherein the unknown functions are 1>vand 1>5:

rVo [{l(r) ';2(r)] ~ r c!>s«()d{

2y2(al- a2) Ll3.¡ - ~~ =V V {2 - (r/y'2)2

r/vIl (20a)

4y2S¡a1

_ a2) [L¿~ ';2 (r) - ~~ ';1 (r) ]=

r1 f c!>s(.;)d.;r V ';2 _ (r/v2 )2

r!vIl(20b)

These two equations (20a) and (20b) canstitutethe solution to the prablem of the separation of thevolume and surface functions of the specific risk offracture. Solving integral equatian (9b) we obtainthe specifie-risk-of-volume-fracture function, namely

1>v(r) = r1

d~ [r1>v (r) ] (21)

where 1>v(r) is obtained from equation (20a). Inorder to get the specific-risk-of-surface-fracturefunction it is necessary to solve the integral equa-tion given by (17b) and his solution is:

r

cjJs('r)= 21+.!.fdd [1]cjJs(1])]1]Ed1] (22)r 2L 1]

Owhere 1>8(1])is known from equation (20b).

Hence, far equations (20a) and (20b) the onlyalternative left to us is resarting to the numericalmethads in arder to thus separate the two specific-risk-of-fracture functions.

5. PARAMETERS DISPERSIONFisher's information matrix [17] is determined

using the folIowing expression:

_ ( a2Inf(r;m,ro»r..- -n E 08 00IJ • •, ,

(23)

{O}= {m, rol

where the aperator E is the expected value, n is thenumber of samples tested, and f(r) is the densityfunction given by

f(r)= df(r) = k m (-L)m-1 exp {_ k (r: )m}dr ro ro

(24)

where K has the following expressions for the res-pective volume and surface brittlenesses:

1T/421-m/2 r dOm+2 J c~sm+20

O

(25)

1T/4

[ a/L] f dO1 + 2(m+2) cosm+20

O

The elements oí the Fisher matrix are:

r =n (1 ak)2 + 2n ak 10.42277-lnk)11 Tom Kmom \

= ::.r (1.82379- 0.84555 Ink + In-k )

r _.J1. (- ~ ak -lnk + 0.42277)12 - ro K am

(26)

r22 = n (~tTherefore, is we consider only the cases n =0, oíequations (7) and (15), forvalume and surface britt-lenesses respectively, then the matrix oí variancesand covariances is easily obtained through the in-version afthe Fisher matrix, with the condition thatrqqz.O. FinalIy the variances and covariances aredetermined using the follawing expressions:

(27)

Revista Latinoamericana de Metalurgia y Materiales, Vol. 9, Nos. 1-2, 1989

ACKNOWLEDGEMENTSThe authors wish to thank to the Fondo N acio-

nal de Desarrollo Científico y Tecnológico FON-DECYT for its grant N° 516/88, to the ExclusivelyDedication Program (1988), to professor ErnestoGómez for his enthusiastic suppoert concerning theprogress of Materials Science, and Raymond Tole-do for his help with the readying of the manus-cript.

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