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Chapter 2: Mechanical Behav i our and PropertiesChapter 2.1MODELLING WEATHERING EFFECTS ON THE MECHANICAL BEHAVIOUR OF NATURAL BUILDING STONESRoberto NovaMilan University of Technology (Politecnico), P.za L. da Vinci 32, 20133 Milan, Italy Abstract:The main features of the mechanical behaviour of natural building stones are first recalled. It is shown that a convenient constitutive model can be conceivedin the framework of the theory of strain-hardening plasticity, successfullyapplied to soils and soft rocks. Extension of the theory to chemo-mechanicalconditions is presented and the main results of the numerical analyses of twogeotechnical problems are shown.Key words:soft rocks; plasticity; constitutive modelling; weathering; chemo-mechanics. 1.INTRODUCTIONSoft rocks were extensively used in the past as natural building stones. Limestones, tuffs, calcarenites and similar rocks were in fact employed as construction materials for temples, churches and palaces in all times, all over the world. Their mechanical properties are similar to those of concrete, the principal building stone of our age. They are therefore strong enough to bear high loading but at the same time weak enough to be easily quarried. Lime-stone, for instance, has a tensile strength that is one order of magnitude less than that of granite. They are also widespread so that their cost is limited: an ideal material to work with.The low tensile strength is due to the weakness of the chemical bonds link-ing together the mineral grains constituting the rock skeleton. These bonds are attacked by environmental agents such as water, wind, bacteria, so that their strength is progressively reduced with time of exposure to the atmosphere.The surface degradation of statues and monuments in the polluted air of our metropoleis is an apparent clue of the effect of weathering on natural55S.K. Kourkoulis (ed.), Fracture and Failure of Natural Building Stones, 55–70.© 2006 Springer.56 RobertoNovabuilding stones. The decay of the mechanical properties is not onlysuperficial but extends to the core of the stone, so that even its structuralefficiency can be jeopardised by the action of atmospheric agents.It is therefore of some interest also for engineers to know more about theweathering effects on the mechanical behaviour of natural building stones.Furthermore, it is important to clarify on which lines a constitutive model forsuch materials should be developed in order to cope with the observedphenomena.In this paper, the fundamental properties of soft rocks are recalled first.The way in which phenomena as yielding and subsequent hardening, brittleductile transition, effects of chemical attack can be dealt with is sketchednext. Comparisons with experimental data in laboratory tests and descriptionof the progressive degradation of structures in engineering problems are thenpresented. Finally, a simplified model for structural elements is discussed inelementary cases. It is shown how it is possible to calculate the variation ofthe bearing capacity of a structural element with time.2.GENERALITIES ABOUT THE MECHANICALBEHA VIOUR OF SOFT ROCKSTraditionally, engineers divided natural geomaterials in two classes,depending on the degree of bonding between the mineral grains that consti-tute the geomaterial skeleton: soils and rocks. The former are characterizedby little or no bond strength. Permanent strains take place as soon as loadingis applied and failure is controlled mainly by friction. The latter are char-acterized by large bond strength. Strains are more or less reversible untilfractures occur in the rock mass. Even though, for the sake of simplicity, forhard rocks engineers use a plastic failure criterion, characterized by frictionand cohesion, it is more the fracture energy that controls the strength and thedeformability of such materials.In the last fifteen years, it was recognized that there exists a wide transi-tional class of geomaterials that behave in a way intermediate between theformer two. At low stress levels their stress strain response is qualitativelysimilar to that of a hard rock. On the contrary, at high stress levels theirbehaviour in more similar to that of an unbonded soil. This class of materialswas named hard soils-soft rocks. Many natural building stones, especiallythose characterized by high porosity, belong to it.Typical examples of behaviour are given by Figures 1 and 2 (after Lagioiaand Nova, 1995). Both refer to constant cell pressure triaxial tests on speci-mens of Gravina calcarenite (Southern Italy). In Figure 1 the behaviour ofthree specimens under relatively low confining pressures is shown. The in-itially linear stress - strain behaviour as well as the occurrence of a peak fora certain value of the deviator stress and the following loss of strength andModelling Weathering Effects on the Behaviour of Building Stones 57 strain uniformity of the specimen can be noticed. The peak is in fact associ-ated to the occurrence of a planar fracture across the specimen. From that point onwards, the specimen deformation is essentially due to the sliding of one rigid block over another and strains, calculated as if they were uniform, are meaningless. In Figure 2 the behaviour of a specimen made of the same material but with a much higher confining pressure is shown. Also in this case the initial stress-strain relationship is linear, characterized by a stiffness that is more or less equal to that of the tests at low confining pressure.Yielding occurs in a different way, however. After a while in which the displacement controlled specimen deforms at more or less constant load, hardening takes place up to a failure value characterized by large uniform strains. The clearly observed saw-tooth shape of the constant load step is apparently a clear sign that some sort of instability is occurring. Figure 3 shows that the stiffness in unloading-reloading is not affected by the large permanent strains experienced. In unloading-reloading the behaviour can be therefore assumed to be linear elastic.A convenient conceptual framework to model this behaviour can be constructed, starting from the elasto-plastic strain-hardening model proposed first by the School of Cambridge (Schofield and Wroth, 1968). The fund-amental idea on which such models are based is that it is possible to define a domain in the stress space, as shown in Figure 4, in which the behaviour of the rock can be considered linear elastic.Figure 1.Drained constant cell pressure tests on specimens of Gravina calcarenite con-solidated at low values of confining pressure.58 RobertoNovaFigure 2.Drained constant cell pressure test (ıǯ3 = 900 kPa) on Gravina calcarenite.Figure 3.Constant cell pressure compression test (ıǯ3 = 600 kPa): unloading-reloading cyclesperformed in the elastic phase, the destructuration phase and the hardening phase.Yielding occurs when the stress point touches the frontier of the domain.The occurrence of plastic strains induces either strain-hardening or strain-softening depending on the sign of plastic volumetric strain rates, which inturn is governed by the so-called normality rule. It is apparent from FigureModelling Weathering Effects on the Behaviour of Building Stones 59 4b that such a model can qualitatively describe at least some of the observed features of soft rock behaviour. For instance, it is possible to model the strength increase with the increase of confining pressure, the transition from brittle to ductile behaviour, the occurrence of dilating plastic volumetric strains at small confining pressures, the occurrence of yielding at lower deviator stress levels for higher confining pressures (point H in Figure 4).To obtain quantitative agreement with experimental tests, many things have to be modified. First of all the shape of the yield locus must be as in Figure 5 (Nova, 1992). The normality rule is no more valid and a plastic potential different from the yield function is introduced. Hardening is con-trolled by one parameter linked to the grain structure, p s, as in soil, and two parameters p m and p t related to the existence of bonds, (Nova, 1999; Gens and Nova, 1993). Possibly these latter parameters could be considered proportional to each other. In general, hardening depends on both volumetric and deviatoric plastic strains. With these modifications and appropriate choice of constitutive functions, Lagioia and Nova (1995) were able to reproduce the behaviour of Gravina calcarenite with remarkable accuracy in drained tests with low and high confining pressure, in pǯ constant and undrained tests, in isotropic and oedometric compression. The behaviour of other materials such as tuff, marl, natural and artificial calcarenite, limestone was also simulated (Nova, 1992; Lagioia and Nova; 1993, Nova and Lagioia, 1996).Figure 4.Schematic picture of the Cam Clay model: (a) stress paths in constant cell pressure tests with different overconsolidation ratios; (b) stress-strain behaviour; (c) volumetric strains;(d), (e) normality rule for different stress states.60 Roberto Nova Figure 5.Initial yield surface of cemented (continuous line) and uncemented (dotted line) soil and evolution of the former during destructuration.3.MODELLING OF WEATHERING EFFECTSAs previously discussed, the action of the atmospheric agents alters the intimate structure of geomaterials and influences their mechanical behaviour. It is mainly the plastic part that is affected, while the elastic properties can be considered constant with time, as a first approximation at least. The results presented in Figure 3 show in fact that the mechanical breaking of the bonds associated to plastic strains does not modify the elastic stiffness. Castellanza and Nova (2004) have shown that the elastic properties of an artificial cemented silica sand are not altered by the chemical aggression of an acid. They showed instead that such a chemical attack transforms a soft rock in an assembly of grains without any bonding.To model such a behaviour, the following way of reasoning is possible (Nova, 2000): it is assumed that there exists an elastic domain governed by the following expression:(,,)0ij k f p s p d(1)where p is the isotropic pressure and s ij is the deviator stress: 13ij ij p V G { (2) ij ij ij s p V G { (3) In addition p k is a set of hardening parameters. To make things simple it is assumed here, making reference to Figure 5, that:m t p p D(4)where Į is a constant and *1c c s m s t p p p p p p D { (5)Modelling Weathering Effects on the Behaviour of Building Stones 61The hardening parameter p s depends on the plastic strains experienced p s s ij p p H (6) while p t is assumed to depend on a scalar measure of the degradation, x d , that is linked to the particular process considered. It can be a function of the quartz content in a granite specimen, as in the original definition of Lumb (1962) who studied the weathering of granite. Or it can be a function of the exposure time to an acid attack or to the temperature at which a specimen is tested, and so on. To make things simple it is assumed that:201t t d p p x (7) Several experimental data (Park, 1996; Baynes and Dearman, 1978; Cas-tellanza and Nova, 2004) support this hypothesis for different geomaterials. Clearly, bonds brake also for mechanical reasons, as shown in Figure 2. The value of p t should also depend on plastic strains (Nova et al., 2003):20,1p p t t ij d t t ij d p p x p P x H H (8)For space reasons, it is assumed that P t is a constant equal to 1. Only the non-mechanical agents are considered to be responsible for the degradation of the mechanical properties. Even with this limitation, it is possible to des-cribe interesting phenomena, as it will be seen in the following.Since when plastic strains occur f must be equal to zero before and after the “load” increment (an increment in x d is also considered as a generalized load), the total differential of f must be zero. Therefore0p s t ij rs d p ij s rs t dp p f f f f df dp ds d dx p s p p x H H w w w w w ww w w w w w (9) Since p rs rs g d H V w /w (10)where g(ırs ) is the plastic potential and ȁ is a scalar, called plastic multiplier. Taking account of Eqs.(5) and (7) and solving for ȁ one gets:021ij t d d ij t s p s rs rsf f f dp ds p x dx p s p p f gp H V w w w w w w / w w w w w w (11)62 RobertoNovaFigure 6.Schematic picture of the weathering test device (a) and photograph of the apparatus(b); after Castellanza (2002).From Eq.(11) it is apparent that plastic strains are generated both fromstress increments and by a variation of x d. In particular plastic strains canoccur even under constant stresses.Consider as a special case an oedometric test on an artificial specimen ofsoft rock composed of a mixture of quartz grains and cement constituted byhydraulic lime and water. The test is conducted in the so-called WTD(weathering test device, Figure 6 after Castellanza, 2002). The apparatusconsists of a thin metal ring that laterally constrains the geomaterial speci-men. Un upper and a lower platen are used to load the specimen in thevertical direction. The two platens are permeable so that a fluid (water oracid) can seep through the specimen. Horizontal strains are almost preventedas in a normal oedometer but not completely, since the ring is slightly de-formable. The test is composed of two phases. In the first one the specimenis subject to an increasing vertical loading as in a normal oedometer. In thesecond one, the load is kept constant to the level achieved in the first phaseand an acid is forced to seep into the specimen. With time, the acid dissolvesthe calcareous bonds so that further vertical strains take place.A very sensible strain gauge is attached along the circumference of theright section of the ring. This can record the circumferential strains and fromthose it is possible to determine the radial stresses by means of the Mariotteformula. Therefore the entire stress path can be determined.Figures 7-10 (after Nova et al., 2003) show the comparisons of the expe-rimental and calculated stress path, radial stresses, vertical settlements andthe variation of the calculated internal variables for a specimen of cementedsilica sand. The constitutive model used is presented in detail in Nova et al.(2003), where also more complex tests are considered.It is apparent that the model proposed can describe with sufficient ac-curacy the effects of weathering on specimens of soft rock.Modelling Weathering Effects on the Behaviour of Building Stones 63Figure 7.Experimental versus calculated stress paths in a weathering test on cemented silicasand, after Nova et al. (2003).Figure 10.Evolution of the calculated internal variables for the stress path of Figure 7 (after Nova et al. (2003). xxFigure 8.Experimental versus calculatedradial stress for the stress path shown inFigure 7, after Nova et al. (2003).Figure 9.Experimental versus calculated vertical strains for the stress path shown inFigure 7, after Nova et al. (2003).4.APPLICATIONSBy employing the model of Nova et al. (2003), Castellanza et al. (2002) showed that it is possible to simulate the progressive failure of geostructures, subjected to chemically induced bond degradation. In Lorraine, for instance, several collapses were caused by the progressive degradation of the pillars of abandoned iron mines, attacked by bacteria, which live in the water that flooded the mine chambers after the end of the mining activities.The damage progression was simulated simply by imposing a degradation law to the pil-lars. The closer the rock element is to the flooded chamber, the higher is the value of x d (up to one that corresponds to full degradation). The value of x d increases with time at a conveniently established rate. Furthermore the zone of the pillar affected by degradation also increases with time at a given rate.The results obtained are shown in Figure 11. The main effect of the de-gradation is that of reducing the bearing capacity of the layers that are more exposed to the chemical attack. There is therefore a progressive stress migra-tion towards the centre of the pillar that can eventually collapse under the weight of the overburden that cannot be sustained anymore. Figure 11b shows how relevant can be the weathering induced displacements at the surface with respect to the original ones due solely to mine excavation.(a)(b)Figure 11.Subsidence induced by the degradation of the pillars of an abandoned iron mine;(a) Stress redistribution within the pillar; (b) Settlements, after Castellanza et al. (2002).Recently, Parma (2004) considered the effect of a pollutant migrating with-in a bonded soil mass on the settlement of a shallow foundation. To solve this problem, the contaminant diffusion equations were treated by means of a finite difference computer code combined with GeHoMadrid, the finite ele-ment code (Fernandez Merodo et al., 1999) in which the aforementioned Nova et al. model is implemented.Figure 12 shows the evolution of the contaminant, the associated plastic strains induced by chemical degradation and the progression of the found-ation settlements with time. It is apparent that the calculated results look reasonable. A series of experimental tests are presently performed in Milan on a model foundation resting on an artificially cemented soil, to have a benchmark against which checking the validity of the proposed theory.(a) (b) (c) Figure 12.Effects of the diffusion of a contaminant under a shallow foundation; (a) Variation of x d; (b) Deviatoric plastic strains; (c) Vertical displacements for successive time steps (Parma, 2004).5.SIMPLIFIED MODEL FOR STRUCTURALBEHA VIOUR OF NATURAL BUILDING STONES The conceptual structure of the constitutive model presented so far can be applied to structural problems involving the weathering of natural building stones. To achieve this goal, a simplified model is formulated in this section. At variance with foundation soils, structural elements are not laterally confined. Therefore the range of interest is that of low confining pressures, where the natural building stones behaviour is qualitatively similar to that of hard rocks. The yield condition is therefore also a failure condition, since yielding is associated to negative hardening values (softening) and a sharp peak occurs in the deviatoric stress-strain law. It is apparent from Figure 13 that such a failure condition can be approximated by a straight line of equationm q mp q c (12) at least for positive p ǯ. It is also shown in Figure 13 that as the degradation proceeds, m remains more or less constant, while q m decreases more or less in proportion to p m . One can therefore assume that the unconfined compressive strength of a natural building stone q C varies with the weathering degree as201C C d q q x (13) or even more simply as01C C d q q x (14) The plastic potential rules the ratio of plastic strains. A constant dilation rate can be assumed again for the sake of simplicityp p v d H \H (15) where p v Hand p d H are the volumetric and deviatoric plastic strain rates, res-pectively. Furthermore, the value of q C can be made to depend on plasticstrain experienced, for instance as:C C C p d q q U H w w (16)If ȡc is taken to be zero, a ductile behaviour is assumed after yielding. If ȡc is taken to be infinite, a perfectly brittle behaviour is considered. This latter possibility is considered here. In practice, this means that when a ma-terial element reaches the yielding condition it suddenly looses the ability of bearing stress. The existing stress must be therefore instantaneously trans-ferred to the parts that have not yet yielded to guarantee the fulfilment of equilibrium. Finally, in accordance with many experimental observations, the elastic moduli are considered to be constant, affected neither by weather-ing nor by the occurrence of plastic strains.6.AN ANALYSIS OF THE STABILITY OF A CIRCUL-AR PILLAR SUBJECTED TO WEATHERINGAs an application, consider a cylindrical pillar with circular cross section subjected to a constant load Q (consider negligible the pillar own weight). When the pillar is intact at time zero the pillar contraction is given by020QL E R G G S { (17)where L is the height of the pillar, R 0 its radius and E the Young’s modulus of the material. Assume now that weathering affects the lateral surface of the pillar and progresses towards the centre of the pillar linearly with time. The front of intact rock is therefore at a distance from centre equal to0R R t D (18) Assuming that the behaviour is perfectly brittle, the rock strength in a point at distance R falls to zero as soon as the front reaches it. The pillar therefore collapses abruptly when:200c Q R t S D V (19) where ıc0 is the compressive strength of the stone. This occurs for a time t fafter weathering begins equal to:01f R t D ­° ®°¯ (20)The contraction increases with time as:022200011QLt t E R R R G G D D S §·§· ¨¸¨¸©¹©¹ (21)until the time of failure is achieved and then sudden collapse occurs. The displacement corresponding to t f according to Eq.(20) is equal to:00f Q Q G G (22) where Q 0 is the force the intact pillar can bear. The results are summarized in Figure 14. The values of the parameters of Eqs.(20) and (21) can be de-termined either by recording the contraction variation with time or by per-forming appropriate tests in the laboratory. To give an order of magnitude, by assuming that the degradation front progresses at 1 mm/year for a column with a diameter 1 m wide loaded at 0.1 of its bearing capacity in non-weathered conditions, the failure time is 342 years. This time reduces to 146 years if the load is one half of the bearing capacity.7.CONCLUSIONSOften old constructions are made of natural materials belonging to the class of soft rocks. Their behaviour is intermediate between that of a hard rock and that of a soil and depends on the confining pressure and the degreeFigure 14.of weathering. At high confining pressures or high weathering degree the rock behaves as a soil, which is as an unbonded material (no cohesion).By exploiting the conceptual framework of soil models, with appropriate modifications to take the role of bonding into account, it is possible to model the behaviour of rock specimens under mechanical loading and/or chemical attack, with reasonable agreement between experimental data and numerical results. The constitutive model developed can be implemented in a finite element code and, in principle, any type of boundary value problem can be tackled, taking into account the effect of progressive chemical weathering. Two geotechnical problems were discussed.Also structural problems could be analysed in the same way. However, since in this case the load condition is simple, simplified methods may be used. Such methods can be derived by drawing inspiration from the more realistic 3D model described here. A simple model for evaluating the life-time of a column under centrally loaded subject to environmental degra-dation was finally proposed.REFERENCESBaynes F. J., and Dearman, W. R., 1978, The relationship between the microfabric and the engineering properties of weathered granite, Bulletin of the International Association of Eng. Geol.18:191-197.Castellanza R., 2002, Weathering effects on the mechanical behaviour of bonded geoma-terials: an experimental, theoretical and numerical study, PhD Thesis, Politecnico di Milano.Castellanza R., Nova R., 2004, Oedometric Tests on Artificially Weathered Carbonatic Soft Rocks,J. Geotechnical and Geoenvironmental Engineering ASCE 130(7):728-739. Castellanza R., Nova R., Tamagnini C., 2002, Mechanical effects of chemical degradation of bonded geomaterials in boundary value problems, Rev. Fran. Génie Civil6(6):1169-1192.Gens A., Nova R., 1993, Conceptual bases for a constitutive model for bonded soils and weak rocks, in Proceedings of the International Symposium on Hard soils-soft rocks, Athens, 485-494.Lagioia R., Nova R., 1993, A constitutive model for soft rocks, in Proceedings of the International Symposium on Hard soils-soft rocks, Athens, pp. 625-632.Lagioia R, Nova R., 1995, An experimental and theoretical study of the behaviour of a cal-carenite in triaxial compression. Géotechnique45(4):633–648.Lumb P., 1962, The properties of decomposed granite, Géotechnique12:226-243.Nova R., 1992, Mathematical modelling of natural and engineered geomaterials. General lecture 1st E.C.S.M. Munchen, European J. Mech. A/Solids11(Special issue):135–154. Nova R., 2000, Modelling weathering effects on the mechanical behaviour of granite, in Con-stitutive Modelling of Granular Materials, D. Kolymbas editor, Springer, Berlin, 397-411. Nova R., Castellanza R, Tamagnini C., 2003, A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation. Int. J. Num. Anal. Meth. Geomech.27(9):705–732.Nova R., Lagioia R., 1996, Soft Rocks: Behaviour and Modelling” Keynote lecture, Proc.Eurock ’96, Turin, Balkema 2000, 3, 1521-1540.Parma M., 2004, Modellazione dell’accoppiamento chemo-meccanico nei materiali cementati, Tesi di Laurea (in Italian).Park H. D., 1996, Assessment of the geotechnical properties of the weathered rocks at historical monuments in Korea, in Proceedings of Eurock ’96, edited by G. Barla, Balkema, Rotterdam, 1413-1416.Schofield A. N., Wroth C. P., 1968, Critical State Soil Mechanics, McGraw Hill, New York.。