TNEMEC dna ETERCNOC .HCRAESER Vol. I, .pp 419-422, 1971. nomagreP Press, Inc Printed in the United States. HTGNERTS FO SUOROP SLAIRETAM .K .K Schiller, .rD Phil., F.Inst. .P Senior Scientific Project Advisor BPB Industries hcraeseR( & Development) Ltd. East Leake, Loughborough, Leics. detacinummoC( yb .R .W Nurse) TCARTSBA It is nwohs that Ryshkewitch's exponential dna Schiller's logarithmic formulae for the strength of porous materials are numerically indistinguishable except in the neighbour- dooh of the extremes of %0 dna %001 porosity. sE wird gezeigt, dass Ryshkewitch's exponentielle dnu Schiller's logarithmische Formel fuer die Festigkeit poroeser Stoffe numerisch ununterscheidbar sind,ausser in der Nachbarschaft der Extreme nov %0 dnu %001 Porositaet. 914 024 Vol. I, .oN 4 ,SSENDRAH ELASTICITY ,SULUDOM YTISOROP In their paper no the "Interrelation of Hardness, Modulus of Elasticity, dna Porosity in Various muspyG Systems,"Soroka and Sereda (I) base their pre- sentation of the results of measurements of hardness dna elasticity no the exponential relationship which Ryshkewitch originally obtain empirically; it also applies to strength measurements. In their bibliography the authors also refer to ym publication of 1958 (2) which both theoretically dna experiment- ally leads to a logarithmic expression. In spite of the at first sight rather radical difference between the two formulae it yam eb of interest to show that within the range of practical porosities they lead to almost the same results. Distinguishing the expressions for strength by the initials of the authors sa subscripts ew have: S R = Soe-bP (I) dna S S = q In Pcr (2) P where S o is the strength of the non-porous material, b dna q are constants dna Pcr the porosity at which the strength practically vanishes. It is evident that neither formula can eb quite true. According to the first, even a body with %001 porosity would have some strength left whilst, according to the sec- ond, a non-porous body would have infinite strength. eW note further that qualitatively both show a monotonic fall of strength at a decreasing rate with increase in porosity in accordance with experimental facts. eW show in the following that there are relations between the parameters of the two equations which render them numerically indistinguishable except in the neighbourhood of p = 0 dna p = Pcr" If the strengths worked out according to these two expressions are to agree over a significant range, their differences must eb almost independent of p, i.e. d e -bp + ~ = 0 (3) ~p S( R - S )S = -b S O p or p e -bp = & = Constant (4) Sb o This will eb the case in the neighbourhood of the mumixam of the expression no the lefthand side of equation (4), namely at 1 mP = b (5) eW won replace p by" 1 P =F +'rr (6) VoI. I, No. 4 124 ,SSENDRAH ELASTICITY ,SULUDOM YTISOROP ,1 .2 .3 .4 .5 .6 ,? .8 .9 1.0 FIG. 1 Comparison of Ryshkewitch's dna Schiller's formulae in standardised co-ordinates 1 and assume T~ <<b- (7) Inserting (6) in (I) and (2) and making use of (7) gives: -b(~ + )7 S o (I - bn) (8) S R = Soe = e and : q In Pcr Pcr In (I + b~) S S 1 = q In ~ = q In b Pcr + ~ (I + b~) : q In b Pcr b~ (9) If equations (8) and (9) are to agree sa long as ~ remains small ew must have S o e - q In b Pcr (I0) S and (II) °b = qb e 224 Vol. I, No. 4 ,SSENDRAH ELASTICITY ,SULUDOM YTISOROP S o ecneH q - e (of the order of S o ) )21( and b : e (an inverse representative porosity (13) Pcr The equations (I) and (2) won take the form e Pcr p S R = eoS (14) °S nI Pcr )51( and SS e P or, calling for convenience x : (0 < x < I) (16) Pcr m __ this becomes RS -ex S - e (17) o dna SS__S = 1 In _I (18) S e x 0 These two expressions are shown no the graph to eb almost identical except at the extremes x = O, I. References I. Soroka I., and .P J. Sereda. J. .mA Cer. Soc. ,I__5 No. 6 (June 1968) 337. 2. Schiller, .K Mechanical Properties of Non-Metallic Brittle Materials. Edited by .W .H Walton (Butterworth, 1958) p. 35.