Experimental and theoretical characterization of Li 2 TiO 3 and Li 4 SiO 4 s D. Aquaro 1 N. Zaccari ABSTRACT Dipartimento di Ingegneria Meccanica Nucleare e della Produzione University of Pisa (Italy) This paper deals with compression tests on one single of Li 2 TiO 3 and Li 4 SiO 4 ( the most promising candidates as breeder materials). Useful information have been obtained from these tests in order to assess a theoretical model developed by the authors. The experimental results have been used in order to implement a numerical model which simulates the mechanical behaviour of the. In the second part of the paper, numerical simulations of oedometer tests on bed, performed with FEM codes, are illustrated.. The experimental results are compared with numerical ones obtained using the Cam-Clay model. The issues derived from the use of the soil model for describing the bed behaviour are described. 1. INTRODUCTION Ceramic materials are investigated in order to develop breeding blanket modules which will be experimentally tested in the ITER reactor. Several research activities are being carried out on ceramic s as well as on beds with the aim to improve the technological processes, to determine the thermal and mechanical properties and to develop suitable models for simulating the behaviour. This paper reports the results of mechanical characterization tests on Lithium Metatitanate and Lithum Ortosilicate spheres. The test is a compression uniaxial test (without radial constraints) on single spheres made of Li 2 TiO 3 and Li 4 SiO 4. This test allows to determine the stiffness in a complete cycle of loading and unloading. The experimental results are compared with those obtained by FEM model which simulate the compression of the. These simulations demonstrate the importance to know very well the material characteristic of the which depend on the production technology. A numerical analysis of a bed in an oedometer test is performed using a model (Cam-clay) which describes the behaviour of the soil. Choosing in appropriate manner the model parameters it is possible to fit very well the experimental results, but the physical meaning of the parameters is not straightforward. 2. EXPERIMENTAL TESTS ON SINGLE PEBBLES Experimental compression tests were performed on Lithium-Metatitanate (Li 2 TiO 3 ) s, 1.1-1.3 mm of diameter, and on Lithium-Orthosilicate s (Li 4 SiO 4 ),.55-.6 mm of diameter. The first s were produced by CEA while the Lithium-Orthosilicate s were produced by Schott- FZK. The s were analyzed by SEM before the test in order to verify the actual shapes and dimensions. These examinations permitted to estimate the average diameter and to normalize the results to a nominal diameter equal to 1 mm for the Li 2 TiO 3 s and.5 mm for the Li 4 SiO 4 s. Moreover the s or their fragments (if the collapsed) were again examined by SEM for evaluating the failure mode and the fracture surfaces. 1 Corresponding author : E-mail aquaro@ing.unipi.it
T11 T12 T15 T17 T23 T22 Φ=1.1982mm Φ=1.19972mm Φ=1.131mm Φ=1.1761mm Φ=1.12 mm Φ=1.2696mm x=1.151mm x=1.1245mm x=1.188mm x=1.28mm x=1.1435mm x=1.2153mm y=1.231 mm y=1.275 mm y=1.74 mm y=1.144 mm y=1.966 mm y=1.324 mm T25 T27 T28 T31 T34 T37 -Φ=1.12851mm Φ=1.2362mm Φ=1.197 mm Φ=1.211mm Φ=1.1363mm Φ=1.15547mm x=1.1392mm x=1.218mm x=1.194mm x=1.2898mm x=1.1152mm x=1.152mm y=1.118mm y=1.236 mm y=1.2 mm y=1.1322 mm y=1.1573mm y=1.159 mm Figure 1 Average diameter and bounding box of the tested Li 2 TiO 3 s T42 T43 T44 T45 T48 T412 Φ=.5534mm Φ=.578 mm Φ=.58564 mm Φ=.59881 mm Φ=.5686 mm Φ=.5822mm x=.532 mm x=.5641 mm x=.5868 mm x=.59892 mm x=.5534 mm x=.59892 mm y=.5748 mm y=.5775 mm y=.5844 mm y=.5987 mm y=.5838 mm y=.5987 mm Figure 2 Average diameter and bounding box of the tested Li 4 SiO 4 s Figures 1 and 2 show the images of the s obtained by the SEM and reports their average diameters and the dimensions of their bounding box. The maximum variation of the average diameter of the tested s is 7.8% for the Li 2 TiO 3 s and 4% for Li 4 SiO 4 s. Some s were subjected to several loading cycles without reaching the failure while other s were compressed until the collapse. Figures 3 and 4 illustrate for the Li 2 TiO 3 s the curves of the load versus displacement obtained during the loading and unloading phases, respectively. Figures 5 and 6 show the correspondent curves of the Li 4 SiO 4 s. The s calculated with the least square method are expressed by the following equations: a) Lithium-Metatitanate - loading phase: P=687.55 s 1.257 r 2 =.9917 - unloading phase : P=-3.287 + 733.62 s r 2 =.9854 b)lithium-orthosilicate - loading phase: P=3613 s 1.927 r 2 =.927 - unloading phase : P=7657 s 1.8922 r 2 =.974
where P (N) is the applied load, s (mm) is the vertical displacement and r 2 is the correlation coefficient. 7 6 5 4 3 t37 t23 t22 t28 t11 t12 t15 t34 t27 6 5 4 3 t37 t22 t17 t27 t23 t28 t34 2 2 1 1.5.1.15 Fig. 3 Loading curve of Lithium-Metatitanate.2.4.6.8 Fig. 4 Unloading curve of Lithium-Metatitanate Before the regression analyses, the values of the experimental displacement were normalized at the nominal diameter assuming according the Hertz theory that the displacement are inversely proportional to the radius. 14 12 t412 t43 t45 t48 12 1 t44 t45 t43 t412 1 8 load( N) 8 6 4 2 6 4 2.2.4.6.1.2.3.4 Fig. 5 Loading curve of Lithium-Orthosilicate Fig. 6 Unloading curve of Lithium-Orthosilicate
Even though the number of the tested is not statistically significative, nevertheless the number of the collapsed s versus the maximum load could give an useful information on the ultimate strength of the s. Figures 7 and 8 show the fraction of the collapsed s versus the maximum load for the Li 2 TiO 3 and Li 4 SiO 4 s, respectively. No Li 2 TiO 3 s collapsed under 25 N while 1 N is the maximum load supported by the Li 4 SiO 4 s without any rupture (Indeed just only one collapsed at 1.6 N while all the other broke for load upper than 7 N..6.7 collapsed fraction.5.4.3.2.1 11 17 24 26 31 31 35 36 37 39 4 51 fracture collapsed fraction.6.5.4.3.2.1 1.6 6.97 7.29 7.67 9.77 1.16 Fig. 7 Collapsed fraction of Lithium-Metatitanate Fig. 8 Collapsed fraction of of Lithium- Orthosilicate 3. NUMERICAL SIMULATION OF THE PEBBLE COMPRESSION The compression tests on single s have been simulated numerically by means of the FEM code MSC- MARC [4]. The aims of the simulations were to implement a model which described correctly the behaviour of the comparing the results with the experimental ones as well as to understand the main phenomena which occur during the compression. 6 5 t22 sy=2mpa-e=3.5gpa-eu=.3-sc=1mpa sy=2mpa-e=5gpa-eu=.15-sc=1mpa 4 3 2 1.5.1.15 Fig. 9- Mesh of the Fig. 1 Compression test on Li 2 TiO 3 :comparison between numerical and experimental results
Figure 9 shows the implemented mesh. The mesh is made up of axial symmetric elements. The load is applied by means of a rigid upper plate and the is located on a rigid lower plate. The constraints between the and the plates have been simulated by means of unilateral contacts (reacting only to compression loads) without friction. Being the made of a ceramic material having a low tension resistance but a great compression strength, the implemented model assumes a linear elastic behaviour under tensile stress and an elastic perfectly plastic behaviour under compressive stress. The material fails if the tensile stress overcomes the cracking stress value (σ cr ) or if the plastic strain (under compressive stresses) reaches the crushing strain value (ε u ). Other parameters which characterize the material behaviour are: - the softening module ( Es), which represents the slope of the curve σ-ε after the cracking stress (that is, the manner in which the stress goes to zero), - the shear retention (τ), which means the capability of the material to transmits shear by friction if the crack closure occurs. The material data for the lithium metatitanate and Lithium Ortosilicate are reported in [1]- [3] and are summarize as follows: - Lithium-Metatitanate : E= 2.6 GPa - ν=.27 - σ u = 1113 MPa - Lithium-ortosilicate : E= 9 GPa - ν=.25 - σ u = 88 MPa where E is the Young module, ν, the Poisson coefficient and σ u the ultimate compression stress. 12 1 8 6 4 2 t44 sy=25mpa-e=5.5gpa-eu=.15- sc=139mpa vertical stress (- MPa) 45. 4. 35. 3. 25. 2. 15. 1. 5. d=.5 mm d=.1 mm d=.125 mm.2.4.6...5.1.15.2 strain Fig. 11- Compression test on Li 4 SiO 4 :comparison between numerical and experimental results Fig. 12 Stress versus strain in different points of the Li 2 TiO 3 Using these data for obtaining the material parameters, the numerical results do not fit the experimental ones. E and σ u seem too high values. In fact considering the average stresses and strains obtained in the single compression tests, the Young modules have to be two or three order of magnitude lower than the above mentioned values. The data reported in [1]- [3] are data collected from different sources and they could not match the material characteristics of the tested s. Indeed for the ceramic material, the thermal mechanical characteristics are strongly dependent on the production technology. This problem was overcame performing a parametric analyses varying the model parameters in manner that the load- displacement curves fitted the experimental ones. Figures 1 and 11 compare the experimental load-displacement curves of the lithium metatitanate and Lithium Ortosilicate with those obtained numerically. In Fig.1, an experimental cycle on a Lithium-
Metatitanate (sample T22) fits the numerical results with the following values of the material parameters : E= 5 GPa; σ y =2 MPa; ε u =.15; σ cr =1 MPa; Es=.2 MPa. In the same Figure a cycle built with the s fits the numerical curve assuming E= 5 GPa and ε u =.3. Figure 11 shows the comparison of three cycles of loading and unloading on a Lithium-ortosilicate (sample T44) and the correspondent curves obtained by means of the numerical simulation. A good agreement is obtained assuming the following values of the material parameters: E= 5.5 GPa; σ y =25 MPa; ε u =.15; σ cr =139 MPa; Es=.2 MPa. Figure 12 illustrates the diagrams of the vertical stress ( with the sign changed) versus the total equivalent strain for three points of the, located at different distance from the contact zone (.125mm,.1 mm and.5 mm). The simulation corresponds to the loading phase of the cycle in Fig.1 compared to the. On the centre the vertical stress and the strain are 4 times lower than near the contact zone. For the same case, Figure 13 shows the distribution of the plastic strains and Von Mises stresses on the Lithium Metatitanate. A great part of the is yielded for a load of 47 N. The maximum plastic strain is.19. Plastic strain Von Mises stress 9.5 N 19 N 28 N 47N Fig. 13 Distribution of the plastic strain and Von Mises stress on the Lithium Metatitanate 4 NUMERICAL SIMULATION OF A LITHIUM ORTOSILICATE PEBBLE BED WITH AN HOMOGENEOUS MODEL Several Authors simulate the bed behaviour by means FEM codes using the model developed for the soil ( Cap model of Drucker and Prager[5], Cam-clay model of Roscoe and Burland [6] and so on). These model depend on several soil parameters which have not a precise correspondence with the mechanical
characteristics of the s. In the following a numerical simulation of an oedometer test performed on Lithium-ortosilicate bed is illustrated. This simulation is performed using the MSC-Marc FEM code and assuming that the bed behaviour is described by means of the Cam-clay model. The numerical results ( vertical load versus vertical displacement) are compared with the experimental ones obtained by the Authors of this paper [7]. The Cam-clay model depends on several parameters. In order to find a best fit between the numerical and experimental results we have varied the main parameters in reasonable ranges. 6.E+4 M=.5 - k=2e-3 - pc=.9 MPa lbd=5e-3 6.E+4 M=.5 - k=2e-3 lbd=6e-3 load(n) 5.E+4 4.E+4 3.E+4 lbd=6e-3 exp.results lbd=6.5e-3 load(n) 5.E+4 4.E+4 3.E+4 pc=.9mpa exp.results pc=2 MPa pc=.7 MPa 2.E+4 2.E+4 1.E+4 1.E+4.E+..1.2.3.4 Fig.14 Influence of the parameter λ.e+..1.2.3.4 Fig.15 Influence of the parameter p c 6 5 M=.5 -lbd=6e-3-pc=.9 MPa 6 5 lambda=6e-3 - k=1e-3 - pc=.9 MPa 4 3 2 exp.results k=3e-3 k=2e-3 k=.5e-3 load(n) 4 3 2 exper. results M=.5 M=.8 M=1.5 1..1.2.3.4 Fig.16 Influence of the parameter k 1.1.2.3.4 Fig.17 Influence of the parameter M The main parameters of the Cam-clay model are the following: - M, the slope of the critical state line - p c, preconsolidation pressure ( a stress-like hardening variable) - K, the slope of the over consolidation lines - λ, the slope of the normal consolidation line. M and p c enter in the yield criteria while K and λ enter in the equation which describe the isotropic compression of the soil (K in the elastic range and λ in the elastic plastic range). The isotropic compression
constitutive relations give the specific volume ( ratio between the total volume of the soil and the volume of the solid part) as function of the logarithm of the pressure. No influence have in the Cam-clay model the Young module, the Poisson coefficient and the yielding stress. A good agreement between the experimental curve and numerical one was obtained assuming the following values for the four parameters : M=.5; K=.2; λ=.6; p c =.9 MPa. Figures 14-17 shows the influence of the different parameters on the solution assumed as the best. Obviously several other groups of variable could give a solution which agrees with the experimental curve. A big question is the possibility of the extrapolation of the simple model of the oedometer test at more complex stress states. The stresses in different part of the are very different (Fig.13) but the homogeneous model of the oedometer test gives everywhere a constant stress and a constant strain. It is not easy to relate this effective stress to the actual stress of the. 5. CONCLUSIONS This paper illustrated the experimental results of compression tests on single s. The tests permitted to estimate the fracture load of Lithium-Metatitanate s (1.1-1.3 mm of diameter) and of Lithium- Orthosilicate s (.55-.6 mm of diameter). The numerical simulations of the compression give results which agree very well with the experimental ones assuming values of the material characteristics different from those found in the technical literature for these materials. This fact could be explained considering that the mechanical characteristics of ceramic materials depend strongly on the production technology. An homogeneous model developed for soil has been applied to simulate an oedometer test of a Lithium- Orthosilicate bed. Choosing appropriate values for the several constants it is possible to fit the experimental curve, but the physical meaning of the parameter values is not immediately extrapolate to the bed. REFERENCES [1] W. Dienst, H. Zimmermann, Mechanical properties of ceramics breeder materials, J. of Nucl. Mat., 155-157 (1998) 476-479. [2] P. Gierszewski, Review of properties of lithium metatitanate, Fusion Engineering and Design 39 4 (1998) 739 743. [3] G. Schumacher at alii, Properties of Lithium Ortosilicate, J. of Nuclear Materials, 155-157 (1998) 451-454. [4] MSC-MARC Vol. A Theory and User information 23 MSC.Software Corporation [5] Drucker,D.C.,Gibson,R.E., Henkel,D.J. Soil Mechanics and work hardening theories of plasticity Trans. ASCE (1957) 122, 338-346 [6] Roscoe K.H., Burland J.B. On the generalized stress-strain behaviour of wet clay. Engineering Plasticity, (1968) pp.535-69. Cambridge Univ. Press, Cambridge [7] D. Aquaro, N. Zaccari, Experimental and numerical analyses on Li4SO4 and Li2TiO3 Pebble beds used in a ITER Test Module Blanket., Journal of Nuclear Materials, in press, 26