Ultrasonic measurements are widely used as NDT method in historical structures for evaluation of stone material strength properties. Actual stone degradation in situ is estimated based on various laboratory tests which allow researchers to correlate the number of artificial ageing cycles of stone specimens with ultrasonic wave velocity measured on these specimens. Paper presents the results obtained for granite, marble, limestone, travertine and sandstone which undergone various cyclic ageing tests including freezing and thawing, high temperature and salt crystallisation. Analysis of the obtained results shows that independent of the stone type tested and independent of the ageing test applied a rate of change of the stone elastic properties is described by an ordinary differential equation which solution is an exponential law analogues to the Newton’s law of cooling. The degradation function model can be used for further research on expected residual strength and dynamics of the heritage materials degradation processes. Keywords stone deterioration, decay function model, monuments, ultrasonic measurements, cyclic ageing Introduction Stone material ageing is a process of natural degradation of mechanical and decorative properties of stones due to long lasting environmental processes (ICOMOS ISCS, 2008)[1]. In geology these processes are related to a weathering of stone material (Irfan and Dearman, 1978) [2]. Susceptibility of a stone to weathering depends on its morphological and physical- mechanical properties, environmental conditions and exposure time. The ageing process dynamics can hardly be observed in natural conditions because of its slowness. An important role of engineering geology is the assessment of actual and future dynamics of stone material degradation and its susceptibility to environmental loadings. The deterioration processes adversely change structural and aesthetic properties of stone material. This problem is economically important also in Poland which has a long tradition of using stone structural and decorative elements in heritage buildings. Nowadays civil engineering and aesthetic use of stones also pursue the research on the dynamics and modelling of deterioration processes. Engineering geology developed geomechanical research methods for non-invasive diagnostics of stone susceptibility to environmental risks. These methods can be studied in laboratories [3,4,5] and advantageously forecast stone integrity based on geomechanical tests of accelerated ageing. Measured values of e.g. ultrasonic wave velocity, reboundness hardness, strength etc. can be plotted against number of cycles of the ageing and observed trend extrapolated to predict further decrease of stone elastic properties [6], There are also papers presenting deeper insight in stone deterioration mechanisms however they publish fresh and degraded stone material properties after one time period or one number of cycles without intermediate stages which does not inform about a dynamics of deterioration processes [7,8,9] or valuable works showing deterioration trends in graphical or mathematical form however with an insufficient information about measurement data to allow a reader to analyse the results by himself [10]. Comparing an impact of various deterioration processes it is possible to determine dominating and less important environmental factors for various building materials. To this purpose more interesting than simple mathematical extrapolation of the test results is having a physical model of a stone deterioration process and of degradation of its properties. The interesting idea was presented by Mutlutürk et al. in 2004 [11] of adopting to this purpose well known radioactive decay function. The same decay function was used to describe a decrease of the compressive strength of natural hydraulic lime mortar due to varying leaching time [12]. In 2005 the Mutlutürk model was considered to describe stone durability under various ageing processes within EC Project MCDUR and generalised version of the decay function called fatigue law there, with non-zero asymptotic stone integrity level (residual integrity), was proposed by Exadaktylos and analysed by the Project Partners. According to the authors best knowledge, it has not been published except in MCDUR Final Report [13]. 3 In the paper a resilience of stones from Mediterranean Basin and Poland to deterioration processes and influence of moderate and Mediterranean climates environmental conditions on these processes is analysed. The process is modelled by the exponential law proposed in MCDUR being the general exponential decay function analogues to the Newton’s law of cooling. Tested stone materials Tested stone materials from Mediterranean Basin are represented by Gioia Marble (MG), Italy and travertine from Pamukkale (PA), Turkey. These stones have been used as structural and decorative materials both in ancient times and nowadays. The stones from Poland represented by sandstone from Szydłowiec (SZ), granite from Strzegom (ST) and limestone from Raciszyn (RA) known under technical name “Polish travertine” are the stones used in today constructions for internal and outside cladding. Travertine, marble, sandstone and limestone were fresh from quarry while granite was displaced from XIX c. grave tombs during conservation works. The petrographic characteristics of the tested stone materials are presented in Table 1. Table 1. General petrographic characteristics of tested stone materials Stone Quarry / Country (material) Geological period Petrographic characteristics MG Carrara / Italy (marble) Lower Jurassic marble with typical granoblastic polygonal microstructure compact texture, non-directional (calcite 99%, dolomite traces) PA Pamukkale / Turkey (travertine) Pleistocene crystalline limestone composed of carbonate sediment, consisting mainly of calcite, sparite, porous sedimentary stone , there are remains of plants in the stone SZ Szydłowiec / Poland (sandstone) Lower Jurassic fine-grained and medium-grained sandstone with siliceous binder ST Strzegom / Poland (granite) Carboniferous medium-grained granite, light grey colour, full-crystalline, non-directional texture; it consists mainly of plagioclase, quartz, biotite, rarely hornblende RA Raciszyn / Poland (limestone) Upper Jurassic limestone with ooid, with a biomorphic structure, macroporous with a stoney background composed of organic fragments, pseudooolites combined with fine-grained calcite Physical and mechanical properties of the tested stone materials are given in Table 2. The selected stones represent sedimentary, metamorphic and igneous stones and cover a broad range of these properties. For example uniaxial compressive strength varies from16.8 MPa to 109.1 MPa for Mediterranean stones and from 42.8 MPa to 120.8 MPa for stones from Poland. Similar observation can be done in the case of other stone properties. These allowed to analyse dynamics of degradation processes for relatively large variety of stone materials and present conclusions regarding possibility of efficient modelling of stone integrity loss being an interesting and practically important geomechanical problem. Table 2. Physical and-mechanical properties of tested stone materials Stone Real density [kg/m3] Bulk density [kg/m3] Total porosity [%] Water absorption [%] Longitudinal wave velocity [m/s] Uniaxial compressive strength [MPa] MG 2716 2708 0,76 0,28 1900 109,1 PA 2010 1790 23,7 5,75 3950 16,8 SZ 2670 1940 27.38 9.60 2980 46.0 ST 2670 2590 2,88 0,52 2680 120,8 RA 2680 2300 9,89 2,72 4050 42,8 Laboratory methods of deterioration testing Non-destructive testing (NDT) of stones has a main advantage of using the same specimens throughout the whole research programs – from origin stone state and after each cycle of laboratory accelerated ageing. Thus measurement results in each cycle are performed on the same material and not influenced by possible differences between various specimens. Ultrasonic measurements are among the most often used NDT methods because of their reliability, ease of use and direct relation of the sound propagation velocity V to modulus of elasticity E of the material ( V2~E). Assuming the Continuum Damage Mechanics damage model of the material [20] its integrity IN after N deterioration cycles is equal to EN/E0 and hence 𝐼ே=ாಿ ாబ=ቀ௏ಿ ௏బቁଶ (1) where EN, VN are the elasticity modulus and wave propagation velocity after N cycles and E0, V0 represents the elasticity modulus and the ultrasonic wave velocity recorded at the beginning of ageing experiments. Non-destructive ultrasonic measurements of VP and VR velocities were made to tests weathering behaviour of five stones with known physical and mechanical properties. Longitudinal waves VP were measured in transition mode on granite, sandstone, travertine and limestone along the height of the cylindrical specimens having 50 mm in diameter and the slenderness equal to 1. Ultrasonic VP probes had frequency of 1 MHz and 0.5 MHz. Rayleigh surface wave VR [14] was measured on side walls of the prismatic marble specimen using edge probes emitting 2 MHz ultrasonic wave [15]. The time of flight of ultrasonic waves was recorded and sound velocity calculated for known travel distance. In each measurement case several recordings were done and the average ultrasound velocity 5 values were calculated. The ultrasound velocity values before laboratory weathering are presented in Table 2. The following geomechanical accelerated ageing tests were applied: (A) freezing and thawing (F/T) of the marble, granite, limestone and travertine in the temperature range (-25 oC ÷ 20 oC) according to Standard (PN-EN 12371, 2010) [16]; (B) artificial insolation of the sandstone, granite, limestone and travertine, at 40 oC to simulate sun operation at Mediterranean and moderate climatic zones according to originally developed procedure by the second author (AB) [17]; (C) salt crystallisation for the limestone and travertine according to Standard (PN-EN 12370, 2001) [18]. Theoretical model of stone deterioration Mathematical modelling of stone deterioration processes can be done using thermo- physical models [19] requiring many material parameters and extensive computer calculation time. Opposite to rigorous theoretical analysis it is reasonable to look for phenomenological description of stone deterioration. The main reason is operational suitability to geotechnical analysis, ease of use and neglecting the local variations of material parameters influencing the complicated thermo-physical-mechanical modelling results. A good phenomenological model can describe material deterioration with statistically acceptable accuracy at relatively low experimental and calculation costs. However it should not be simply a function approximating experimental data. For a model it is required to describe the law governing observed deterioration phenomena, otherwise deterioration processes of various stones under various environmental loadings could not be compared among each other. An interesting and simple phenomenological model was suggested by Mutlutürk and co- author [11] and verified for 10 various stone types (limestones, marbles, travertines and diabase) using Shore hardness SH index (scleroscope method) as a measure of stone integrity. The authors observed that decay law known e.g. for describing a decay of radioactive elements can well describe their experimental observations. In this case the rate of loss of a stone integrity I is proportional to its integrity at the beginning of the each ageing cycle N −ௗூ ௗே=𝜆𝐼 (2) where -dI/dN is a disintegration rate (integrity loss rate) and λ a positive constant number called decay constant. Integrity I equals to 1 for non-deteriorated material and equals to zero in the case of the complete degradation. The solution of this differential equation is 𝐼ே= eି஛୒ (3) where λ is a decay constant, shows that stone integration IN after N ageing cycles diminishes exponentially with the number of cycles. The law assumes that a stone completely deteriorates and losses its full integrity which is not always the case and depends on a stone properties and environmental degradation factors. The mathematical model of the generalized deterioration law [13] is based on experimental observations that in some cases the final residual integrity is not zero and a continuation of the stone ageing process does not result in its further deterioration. Such a generalized model is analogous to the Newton’s law of cooling and is expressed as −ௗூ ௗே=𝜆(𝐼−𝐴) (4) which means that in general the disintegration rate -dI/dN is proportional to the difference between the actual integrity I and its asymptotic value A≥0 which is the residual integrity. Obviously when there is no residual integrity and a stone completely deteriorates then A = 0 and equation (3) becomes the same as (2). The solution of equation (4) 𝐼ே=𝐴+(1−𝐴)eିఒே (5) allows to describe and compare degradation of stones with zero and non-zero residual asymptotic integrity A. In the present research integrity model was calculated based on measurements of elastic waves propagation velocity using equation (5) and experimentally measured stone integrity is calculated as Material Elasticity Index (MEI) 𝑀𝐸𝐼=ቀ௏ಿ ௏బቁଶ (6) The wave propagation velocity measurements were performed for longitudinal ultrasonic wave for every tested stone except of Gioia Marble for which ultrasonic surface Rayleigh wave was used. 7 Results and discussion Measured ultrasonic wave velocities before and after deterioration cycles for the tested stones and various applied degradation tests performed in the laboratory of Institute of Hydrogeology and Engineering Geology of Warsaw University are presented in the below tables. MEI values calculated according to (6) are rounded to 4 meaningful digits. Table 3a. Ultrasound velocity (Rayleigh surface wave) in F/T tests for Gioia Marble Cycle 0 40 60 80 100 120 VN [m/s] 1900 1600 1500 1450 1300 1500 (VN/V0)2 1 0,7091 0,6233 0,5824 0,4681 0,6233 Model parameters Decay constant λ = 0.0251 | Residual asymptotic Integrity A = 0.53 | Correlation coefficient Rsq = 0.9099 Table 3b. Ultrasound velocity in F/T test for Strzegom Granite Cycle 0 5 10 15 20 VN [m/s] 4359 3792 3613 3391 3072 (VN/V0)2 1 0,7568 0,6870 0,6052 0,4967 Model parameters Decay constant λ = 0.087 | Residual asymptotic Integrity A = 0.42 | Correlation coefficient Rsq = 0.9765 Table 3c. Ultrasound velocity in F/T tests for Raciszyn Limestone Cycle 0 5 10 15 20 25 VN [m/s] 4942 4895 4816 4836 4805 4788 (VN/V0)2 1 0,9811 0,9497 0,9576 0,9453 0,9386 Model parameters Decay constant λ = 0.0814 | Residual asymptotic Integrity A = 0.93 | Correlation coefficient Rsq = 0.9289 Table 3d. Ultrasound velocity in F/T tests for Pamukkale Travertine Cycle 0 5 10 15 20 25 VN [m/s] 3882 3821 3781 3662 3644 3662 (VN/V0)2 1 0,9688 0,9486 0,8897 0,8811 0,8897 Model parameters Decay constant λ = 0.0508 | Residual asymptotic Integrity A = 0.83 | Correlation coefficient Rsq = 0.9250 Table 4a. Ultrasound velocity in Insolation tests for Syzdłowiec Sandstone Cycle 0 7 14 21 28 35 42 49 56 VN [m/s] 3406 3368 3368 3339 3333 3333 3327 3339 (VN/V0)2 1 0.9778 0.9778 0.9610 0.9576 0.9576 0.9541 0.9610 Model parameters Decay constant λ = 0.0983 | Residual asymptotic Integrity A = 0.96 | Correlation coefficient Rsq = 0,9300 Table 4b. Ultrasound velocity in Insolation tests for Strzegom Granite Cycle 0 7 14 21 28 35 42 49 56 63 70 VN [m/s] 4132 4063 3987 3755 3583 3488 3258 3222 3250 3190 3210 (VN/V0)2 1 0.9669 0.9310 0.8258 0.7519 0.7126 0.6217 0.6080 0.6187 0.5960 0.6035 Model parameters Decay constant λ = 0.0178 | Residual asymptotic Integrity A = 0.39 | Correlation coefficient Rsq = 0.9565 Table 4c. Ultrasound velocity in Insolation tests for Raciszyn Limestone Cycle 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 VN [m/s] 4703 4627 4568 4457 4461 4500 4526 4525 4512 4475 4445 4399 4390 4390 4371 (VN/V0)2 1 0.9679 0.9434 0.8981 0.8997 0.9155 0.9261 0.9257 0.9204 0.9054 0.8933 0.8749 0.8713 0.8713 0.8638 Model parameters Decay constant λ = 0.0395 | Residual asymptotic Integrity A = 0.88 | Correlation coefficient Rsq = 0,7669 Table 4d. Ultrasound velocity in Insolation tests for Pamukkale Travertine Cycle 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 VN [m/s] 450 5 4481 4482 4499 4474 4468 4462 4448 4453 4436 4468 4425 4413 4400 4398 4391 (VN/V0) 2 1 0,989 4 0,989 8 0,997 3 0986 3 0,983 6 0,981 0 0,974 9 0,977 0 0,969 6 0,986 4 0,984 8 0,959 6 0,953 9 0,953 1 0,950 0 Model parameters Decay constant λ = 0,0005 | Residual asymptotic Integrity A = 0,07 | Correlation coefficient Rsq = 0,8893 Table 5a. Ultrasound velocity in salt crystalization tests for Raciszyn Limestone Cycle 0 3 6 9 12 15 VN [m/s] 5021 5009 5004 5001 4999 4997 (VN/V0)2 1 0.9952 0,9932 0,9920 0,9913 0,9905 Model parameters Decay constant λ = 0.1906 | Residual asymptotic Integrity A = 0.99 | Correlation coefficient Rsq = 0.9956 Table 5b. Ultrasound velocity in salt crystalization tests for Pamukkale Travertine Cycle 0 5 10 15 20 25 VN [m/s] 3627 3607 3599 3598 3594 3589 (VN/V0)2 1 0,9890 0,9846 0,9841 0,9819 0,9792 Model parameters Decay constant λ = 0.2352 | Residual asymptotic Integrity A = 0.98 | Correlation coefficient Rsq = 0.9778 The presented figures (at the end of the manuscript) are graphical representations of data shown in the Tables 3, 4 and 5. In the Figure 1 and Figure 5 a lack of one measurement data point can be seen. There is no experimental point for 20th F/T cycle of Gioia Marble and for 28th insolation cycle of Szydłowiec Sandstone. In both cases the measured wave velocities fallen outside the 95% error confidence bounds and were rejected. Altogether 85 experimental data recorded for five stones were analysed. The results are presented in Figures 1-10 with continuous lines representing integrity model function calculated according to equation (5) and discrete points show experimental MEI indices calculated using equation (6). Asymptotic integrity values A predicted by the model and correlation coefficient Rsq between the data and the model are also given in each figure. In nine cases out of 10 the residual integrity A of the tested stones is moderate (0.39÷0.53) or high (0.88÷0.99). Only Pamukkale Travertine undergoing ageing due to artificial insolation (test type B) shows the residual integrity as low as A = 0.07 but degrades very slowly – only about 5% of integrity loss after 105 artificial insolation cycles. On the other hand Pamukkale Travertine sustains in-situ tens of thousands of real insolation cycles and still has the integrity greater than zero (monuments still exist). This suggests that developed artificial insolation test is probably much more aggressive than natural environmental loads. 9 Correlation of MEI with the model is very high in the most cases. The lowest value of Rsq = 0.7669 is for Raciszyn Limestone under insolation ageing. Calculated model integrity functions allowed us to compare the dynamics of degradation processes under F/T for marble, travertine, granite and limestone. The comparison is presented in Figure 11. Without a well correlated model such a comparison would not be possible. Conclusions Laboratory experiments of artificial ageing of granite, marble, limestone, travertine and sandstone using freezing and thawing, insolation and salt crystallization tests combined with measurement of ultrasonic wave velocities allowed to observe deterioration of elastic properties of stones. The measured ultrasound velocities were used to define Material Elasticity Index (MEI) which describes experimentally detected loss of stone material integrity. The mathematical model of degradation of stone elastic properties analogues to the Newton’s law of cooling shows good correlation with experimental MEIs. Calculated model degradation functions can be used for further research on expected residual strength and dynamics of the stone cultural heritage buildings, sculptures and artifacts degradation processes. As an example the comparison of dynamics of degradation of four different stones under cyclic freezing and thawing is presented. It can be seen that both a susceptibility of the stone to the environmental loading and the expected residual degradation level can be qualitatively described by the model. Experimental data recorded and analysed for Pamukkale Travertine suggest that developed artificial insolation test is probably much more aggressive than natural environmental loads. FIGURES Fig. 1 Gioia Marble (IT); Freeze and Thaw test. Fig. 2 Strzegom Granite (PL); Freeze and Thaw test. Fig. 3 Raciszyn Limestone (PL); Freeze and Thaw test. Fig. 4 Pamukkale Travertine (TR); Freeze and Thaw test . Fig. 5 Szydłowiec Sandstone (PL); Insolation test. Fig. 6 Strzegom Granite (PL); Insolation test Fig. 5 Szydłowiec Sandstone (PL); Insolation test. Fig. 6 Strzegom Granite (PL); Insolation test 11 Fig. 7 Raciszyn Limestone (PL); Insolation test Fig. 8 Pamukkale Travertine (TR); Insolation test. Fig. 11 Comparison of generalized decay functions of four stones under freeze and thaw action based on ultrasound R-wave (Marble) and P-wave (Granite, Travertine) velocity measurements