Strength of porous materials

It is shown that Ryshkewitch's exponential and Schiller's logarithmic formulae for the strength of porous mate

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Strength of porous materials

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CEMENT and CONCRETERESEARCH. Vol. I, pp. 419-422, 1971. Pergamon Press, Inc Printed in the United States.

STRENGTH OF POROUSMATERIALS K. K. Schiller, Dr. P h i l . , F.Inst. P. Senior Scientific Project Advisor BPB Industries (Research & Development) Ltd. East Leake, Loughborough, Leics.

(Communicated by R. W. Nurse)

ABSTRACT I t is shown that Ryshkewitch's exponential and Schiller's logarithmic formulae for the strength of porous materials are numerically indistinguishable except in the neighbourhood of the extremes of 0% and 100% porosity.

Es wird gezeigt, dass Ryshkewitch's exponentielle Schiller's logarithmische

und

Formel fuer die Festigkeit

poroeser Stoffe numerisch ununterscheidbar sind,ausser in der Nachbarschaft der Extreme von 0% und 100% Porositaet.

419

420

Vol. I , No. 4 HARDNESS, ELASTICITY MODULUS, POROSITY

In t h e i r paper on the " I n t e r r e l a t i o n of Hardness, Modulus of E l a s t i c i t y , and Porosity in Various Gypsum Systems,"Soroka and Sereda (I) base t h e i r presentation of the results of measurements of hardness and e l a s t i c i t y

on the

exponential r e l a t i o n s h i p which Ryshkewitch o r i g i n a l l y obtain e m p i r i c a l l y ; also applies to strength measurements.

it

In t h e i r bibliography the authors also

r e f e r to my publication of 1958 (2) which both t h e o r e t i c a l l y and experimenta l l y leads to a logarithmic expression.

In spite of the at f i r s t

sight rather

radical difference between the two formulae i t may be of i n t e r e s t to show that w i t h i n the range of practical

p o r o s i t i e s they lead to almost the same r e s u l t s .

Distinguishing the expressions for strength by the i n i t i a l s

of the authors

as subscripts we have:

and

SR = Soe-bP

(I)

SS = q In Pcr P

(2)

where So is the strength of the non-porous m a t e r i a l , b and q are constants and Pcr the porosity at which the strength p r a c t i c a l l y vanishes. that neither formula can be quite true.

I t is evident

According to the f i r s t ,

even a body

with 100% porosity would have some strength l e f t w h i l s t , according to the second, a non-porous body would have i n f i n i t e qualitatively

both show a monotonic f a l l

strength.

We note f u r t h e r that

of strength at a decreasing rate with

increase in porosity in accordance with experimental facts. We show in the following that there are r e l a t i o n s between the parameters of the two equations which render them numerically i n d i s t i n g u i s h a b l e except in the neighbourhood of p = 0 and p = Pcr" I f the strengths worked out according to these two expressions are to agree over a s i g n i f i c a n t

range, t h e i r differences must be almost independent

of p, i . e . d (SR - SS) = -b SO e -bp + ~p ~p or This w i l l

p e -bp

= & bSo

=

Constant

= 0

(3) (4)

be the case in the neighbourhood of the maximum of the expression

on the lefthand side of equation (4), namely at 1 Pm = b

(5)

We now r e p l a c e p by" 1

P =F

+'rr

(6)

VoI. I , No. 4

421 HARDNESS, ELASTICITY MODULUS, POROSITY

,1

.2

.3

.4

.5

.6

,?

.8

.9

1.0

FIG. 1 Comparison of Ryshkewitch's and S c h i l l e r ' s formulae in standardised co-ordinates 1

and assume Inserting

~T