2 Influence of Water on the Strength of Limestone

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Influence of Water on the Strength of Limestoneby N. S. Parate

Detailed analysis is given of the influence of water on the str eng th of limestone.The mechanism involved in strength reduction is described. Such analysis is ofimportance in geological, civil, constructional, mining, and metallurgical engineeringand is closely associated with the rock mechanics involved in these areas. Minedumping, slope stability, excavations, sedimentation, etc., ar e very much affectedby the presence of water. Tensile stresses, which have an important role in thedevelopment and propagation of fissures, are produced by water in cracks. Thedevelopment of fissures can be responsible over a long period of time for an ultimate,catastrophic failure.

Knowledge of th e streng th and physical properties ofengineering materials aids in the design of safer andmore economic structures. Therefore, such materialshave been extensively investigated but rocks, eventhough they occupy a unique place as the base for allstructures, have been relatively neglected.

The first attempts to study the mechanical behaviorof rocks in a serious and scientific manner began alitt le more than 60 years ago. Study of the deformationof rocks under high pressure and temperature revealedthat rocks may fracture a s brittle materials or flow plas-tically, depending on the stress conditions.Robinson1 and others, while study ing the effects ofpore pressures on the strength of rocks, concluded tha tthe effective pressure theory is valid and is more pro-nounced in the case of porous rocks. Recently Skempton2examined th e influence of pore pressure on the stresscontrolling the shear strength and volume changes inporous materials.

Influence of WaterIt is well known tha t the presence of water modifiesth e behavior of various rock and rocklike materials be-

cause of its mechanical, physical, and chemical effects.These effects influence (1) deformation of rocks, (2 )resistance of rocks to fracture , and (3 ) stabili ty of rockmasses.

was carefully measured.Types of Tests: The tensile tests were

1) Uniaxial . . . . . . . . u1 0, ua = 0.3) Triaxial . . . . . . . UI > uz = @a# 0.

Detailed descriptions of the equipment and the testsare given in another publication by the author?

ResultsThe results of t he tests ar e given in Tables 1 and 2.

Table 1 presents the results of tests on t he specimensin the dry condition. Results for specimens testedin the saturated condition are given in Table 2.

ObservationsSeveral comments can be made regarding rocks inthe dry and saturated conditions.

Rock masses are exposed to the influence of w ater in 1 ) Figs. 1 and 2 are plots for triaxial test states invarying degrees during most of the ac tive life of mines, the form of Mohr's envelope and in a form of new cri-dams, and underground excavations. This influence of teria proposed by the author.'water is an important consideration for the inherentsafety of underground supports, mining or construc-tions works, deep foundation stability, large dams, orsurface excavations. Table 1. Limestone, Dr y Condition

Experimental StudyThe rock studied is a typical limestone of a compacttype (porosity of about 4%, very slightly fissured), paleyellow, and fine grained. Some rock samples were driedat temperatures up to 110C for two days in o rder toobtain samples in as completely a dry condition as pos-sible. Another set of rock samples were saturated invacuum for two days.The specimens were subjected to the load conditionsoutlined subsequently. In each case the rupture strengthN. S. PARATE, formerly Expert to the Government of Ni ger, is withWarnock Hersey Interna tiona l Ltee., Div. of Professional Services,Montreal, P.Q., Canada. TP 70AM286. Manuscript, Oct. 27, 1970.Discussion of this paper, submitted in duplicate prior to Sep. 15,1973, will appear in SME Transactions, December 1973, and in Al METransactions, 1973, Vol. 254.

a1 a1 U8size of Specimen Type of Teat (Bar) (Bar) (Bar)4 10mm x 90mm Simple tension 0 0 -50 and

8030 x 30mm x 30mm Double tension 0 -50 -50d 36 mm x 36mm Brazilian ROO 0 -100-. . ..-24 x 25mm x 25mm ensi ion and 200 0 -60compression 400 0 -40- --a 0 2 a~(Bar) (Bar) (Bar)---36mm X 72mm Simple compression 0 0 +2400

25 x 25mm x 20mm Biaxial compression 0 200 30800 400 34000 1200 38000 1600 35300 3500 43000 2500 4000

15 X 15mm x 12mm Double compression 0 4600 460036mm x 72mm Triaxial 250 250 3450classical 500 500 4500

1000 1000 55004 24mm x 24mm Brazilian with con- -260 500 2580finement -220 500 2580+ 115 1000 3572

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Fig. 1-Mohr's envelopes. Dot ted line, limestone,saturated condition; solid line, dry condition.

The envelopes for rocks in dry condition are parallelto those for saturated rocks. Colback and Wiid6 ob-served the same phenomenon in their studies of otherrocks.2) Fig. 3 gives the biaxial fracture surface curves.The fractured surface for the saturated condition iscompletely enveloped by the fracture surface for thedry condition. The fracture surface seems to have beendisplaced parallel to the bisector (u2 = us) axis to asignificant amount.

3) Figs. 4 and 5 indicate the increase in strength withincreasing confining pressure.

DiscussionFurthe r study of Fig. 3 shows that the point X' on thecurve for dry rock estimates the amount of force created

by the presence of liquid in the sock. This phenomenoncan be verified by a simple technique based on theswelling experiment.Tables 1 and 2 show clearly the reduction of s trengthdue to the presence of water. The decrease in strengthindicates the fictitious hydrostatic tensile forces createdby the presence of water or any other liquid. In thiscase, ug = -75 bar. From Fig. 4 and 5 this value agreeswell with a stress value necessary to cancel completely

Table 2. Limestone, Saturated Condition

68 UZ 61Size of Speclmen Type of Test (Ba r) (Bar) (Bar)

@ 36 mm x 72 mm Simple tension -20 0 0 036 mm x 36 mm Frazilian -75 0 25036 mm x 72 mm Simple compression 0 0 + 150025 x 25 mm x 20 mm Biaxial compression 0 400 2400

0 200 2340

15 x 15 mm x 12 mmdiam. 36 mm x 72 mmdiam. 24 mm x 24 mm

Double compressionTriaxialclassicalBrazilian with con-finement

latera l dilation due to saturation of the rock with theliquid (Morliere). The following tabulation gives valuesof u, for a typical limestone when saturated with dif-ferent liquids:

CompressiveType of Llquid uu (b ar) Strength

Oi lAlcoholWaterDr y

550 ba r200 ba r150550

The presense of these stresses reduces the shear s trengthwithout disturbing the frictional properties. This fact issupported by the parallel nature of two envelopes inFigs. 1 and 2 and can be described by the followingequations (see Table 3 for Nomenclature) :

TI= Cl+ u tan @ (dry condition)and C1 > C2

T, = C2+ u tan @ (saturated condition).Various results expressed by intrinsic curves based on

Table 3. Nomenclaturea15 25 s P rincipal stressesa1< 0 Tensile stressa1> 0 Compressive stre ssuo Hydrostatic tensile stresses at microscopic levelI shear stressesCohesionAngle of internal frictionVolume change due to swellingThe maximum valuePressure of gaseous phasePressure of water filmSaturatio n co efficient lying between 0-1Relative humidityGas constantAbsolute te m~e ratu reTensile strengthEffective normal stressesMolecular mass of liquidSpecific volumeVapor pressureSaturated vapor pressureFree surface energy

Interatomic spaceModulus of elasticityRh Compressive stren ah in dry stateR . Compression strength In saturated state

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Mohr's theory and discussed previously predict the factthat the presence of water diminishes the cohesivestrength.

The shear strength of rock is influenced by (1) co-hesion, (2) internal friction, and (3) dilatancy.Cohesion is the resistance of the mate rial t o separa-tion and depends on (1) internal molecular and intra-granular forces; (2) absorption and intersurface phe-nomenon; (3) the hydrostatic tension of the liquid,which can be calculated according to Kelvin's equation;and ( 4 ) elementary surface stresess of solids whichexist because of surface tension and its modification bythe variation of free energy of the surface.

Fig. 6, a typical representation of a fissure, gives anidea of the forces and phases in rock which often ar e

Fig. 2-Plot of trioxiol tests according to ou-thor's proposed criteria.' Dashed line, saturatedslate; solid line, dry stote. Star, simple com-pression; triangle, bioxiol compression; circle,Brazilian; X, triaxiol compression.

characterized by the presence of fissures, micropres-sures, grains, discontinuities, etc.According to Kelvin's equation, the presence of con-densed liquid depends on the degree of humidity, ten-sion due to liquid = RT/MV log P/P., and the tensileresistance of water can be atta ined for log P/P, = 0.4.Hence, the mechanism takes place in two zones and thestress responsible for swelling for fissured solids couldbe expressed, according to Kelvin

AV - I RT P- --- log-K MV P.and

Fig. 3-Biaxiol compression. Dashed line, satu-ra te d condition; solid line, dry condition.

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Experimentally it is seen that there is a clear transitionpoint in the relative rate of reduction of strength aroundthe critical ratio of P/P,= 0.4 (unpublished data).Superficial Tension: The superficial tension exists at thelimit of two different phases between a ir or vapor andwater, and water and solids or solids and vapor. In thiszone, water is in a state of tensions (p,) and the effec-tive stresses due to the presence of the water film canbe calculated using Bishop's equation developed for par-tially saturated soils (Powers7).

Ueit = a - pa - x(p, - Iwhere p, is pressure of the gaseous phase, and x is thesaturation coefficient that lies between 0-1.

Fig. &Limestone under tria xia l compression.Dashed line, saturated condition; solid line, drycondition.

Absorption: The presence of various phases in a satu-rated rock-rock being heterogeneous in nature--givesrise to this phenomenon which results in internal ten-sile stresses.Free Surface Energy: Absorption of the gases or vaporon the internal surfaces of th e rocks (surfaces causedby fissures, voids, Griffith cracks) diminishes the freesurface energy.In general, it is assumed that the value of E and a inthe Orawan are constant for the tensile strength ofsolids. The Orawan formula is

Fig. %Brazilian tests on limestone. Dashed line,saturated condition; solid line, dry condition.

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The reduction in cohesive strength as the free surfaceenergy diminishes because of th e superficial tension andabsorption of vapors or gases is evident and supportsthe experimental observations.

ConclusionFrom this study, it is concluded that the presence ofwater in rocks (1 ) reduces compressive, tensile, andshear strengths; (2 ) affects the cr iteria of br itt le frac-ture developed by the author,& n which cri teria t he

mean stress is of significant importance; and ( 3 ) theangle of in ternal friction remains constant.

1 L Film

I-~ondenaed li q u i dFig. &Typi cal representation of a fissure (elementary solid),after Barrenblatt.

The foregoing conclusions are valid for all types oftests to which rocks are subjected.

The author believes that it would be of considerableinterest to extend and further develop this study to in-clude the basic fundamental physicomechanical prop-erties such as porosity, permeability, swelling, etc.

References1 Robinson, L.H., The Mechanics of Rock Failure," Quarterly ofthe Colora do School of Min es, Vol. 52, No. 3, July 1959, pp. 177-198.ZSkempton. A.W., Effective Stress in Soils. Concrete and Rocks,"Civil Engineering Dept. Imperia l College of Science & Technology,London, 1964.

Parat e. N.S.. "La Critere des Rup tures des Roches Fragiles." Ph.D.thesis, paris. 1968.4 Parat e, N.S., "Failure Criterion of Brittle Rock near th e PeakStrength," International Symposium on Rock Mechanics, Madrid,1968. Theme 11-2.Colback. P.S.B.. and Wiid, B.L., "The In!uence of Moisture Con-ten t on the Compressive Stren gth of Rocks. South African Councilof Scientific and Indu strial Research. 1965.Morlier, P., Pri vat e commu nication, 1968.7 Powers,

BibliographyBrace, W., "Dependence of Fractu re Stren gth of Rocks on GrainSize, Bulletin No. 76, 1961, Min eral Indu stries E xpe rimen t Sta tion,The Pennsylvania State University.Chenvert, M.E.. "Absorptive Pore Press ures of Argillaceous Rocks."Rock Mechanics-Theory and Pract ice, W.H. Somerton, ed., AIME,New York. 1970, pp. 599-627.Nadai, A., Theory of Flow an d Fr ac tur e of Solids, Vol. 1. 1950, p.244-.Para te, N.S., "La C ritere des Rupture s des Roches Fragiles," in-ternal report, Laboratoire de Mecanique des Solids, Ecole Polytech-

nique, Paris, July 1967.Terzaghi K., and Peck. R., Soil Mechanics in Engineering, JohnWiley, New York, 1967.Willard, R.J., et al., "Microstructural Techniqu e in the S tudy ofPhysical Properties of Rocks." Intern ation al Jour nal of Rock Me-chanics and Mining Sciences, Vol. 6, 1969, pp. 1-12.

TECHNICAL NOTE

Transfer Function for a Continuous Mechanical FrothFlotatio~iCell with a Distributed Rate Colistant

by Leon Y. Sadler, Ill,and E. K. Landis

Fro th flotation has been described by several authors'"as being analogous to a first-order ra te process. Althougha few investigator^'^^" have found orders other than onefit their data best, the rate is described most often bythe first-order equation,.

where r is rat e of disappearance of floatable mineral byflotation, C ( t ) is concentration of floatable mineral inthe cell, t is time, and k is first-order flotation rat e con-stant.

One of the difficulties in using this r ate equation todevelop a mathematical description of an ac tual flotation* This equation has been used in both "single" an d "two-phase"flotation models. For simplicity the "single-phase" model is used heresince the primary purpose of this note is to illustrate the develop-ment of a transfer function for continuous froth flotation with a dis-tributed rate constant.

Both authors ar e associated with U.S. Bureau of Mines, TuscaloosaMetallurgy Research Laboratory, Tuscaloosa, Ala.L. Y. SADLER, Ill, is Graduate Student in Chemical Engineering, andE. K. LANDIS is Professor, Chemical Engineering, respectively, Uni-versity of Alabama, University, A la. T N 71822 1. ~ a-n usc ript , Aug.5,1971.

process is the determinat ion of a sui table value of theconstant k. The value of k, which reflects the floatabilityof the floatable mineral, is dependent on the hydro-dynamic conditions in the cell, type and concentrationof flotation reagents used, pulp density, and pH, methodof f roth removal, aeration rate, particle characteristics,and many other factors. Therefore, it is reasonable toexpect that each particle of floatable material has adistinct value of k and that a unique distribution of kvalues exists for given feed and cell conditions. Thishas been observed and discussed by several investiga-tors, and mathematical descriptions of the batch celland steady-state continuous cell with distributed flota-tion rate constant have been developed."" The mathe-matical description in the form of a transfer function ofthe continuous or flow cell under unsteady-state condi-tions with a distributed flotation rate constant has notbeen presented in the literature. The purpose of thisnote is to make such a presentation.

The basic features of the perfectly mixed? continuousi odels assuming that the entire cell is perfectly mixed cannot be

applied to the whole cell in which the entire volume is not perfectlymixed. However, several inv est iga tor ss. l~a ve ound that perfect mix-ing is approxim ated in the m echanical flotation cell. If the e ntire cellis not perfectly mixed, then th e perfect mixing model may be appliedto the portion of the cell which i s perfectly mixed.TRANSACTIONS- OL. 254 Society of Mining Engineers, AIM JUNE 1973- 31

2 Influence of Water on the Strength of Limestone

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