- Email: [email protected]
cladding: numerical simulation
Materials Science and Engineering B90 (2002) 254– 260 www.elsevier.com/locate/mseb
Fabrication of metal/sheathed high-Tc superconducting composites by explosive compaction/cladding: numerical simulation A.G. Mamalis *, I.N. Vottea, D.E. Manolakos Department of Mechanical Engineering, Manufacturing Technology Di6ision, National Technical Uni6ersity of Athens, 42, 28th October A6enue, 10682 Athens, Greece Received 18 May 2001; accepted 17 October 2001
Abstract Explosive compaction/cladding, usually followed by forming, is a technique used extensively for fabricating multilayer sandwich components of the same or different materials. In this paper, we report on experimental and numerical investigations into the explosive compaction/cladding for fabricating superconducting Y – Ba– Cu– O ceramic/metal composite grooved discs. The manufacturing process is numerically simulated by using the explicit finite element code LS-DYNA3D. The final dimensions of the compact and the pressure, temperature and density distributions during the entire cladding compaction process are predicted. The proposed model is validated, as the numerical results obtained were in good agreement with the experimental results. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Explosive compaction/cladding; Superconductors; Finite element
Nomenclature Cp D E J1
J2D k P R T V W X0
specific heat hardening law coefficient Young’s modulus first stress invariant second stress invariant thermal conductivity pressure surface axis ratio maximum hydrostatic tension volume hardening law coefficient hardening law coefficient
Greek letters h material parameter of the Drucker – Prager/cap model i material parameter of the Drucker – Prager/cap model k material parameter of the Drucker – Prager/cap model m plastic strain
* Corresponding author. Tel.: +30-1772-3688; fax: + 30-1772-3689. E-mail address: [email protected] (A.G. Mamalis). 0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 0 9 1 3 - 8
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
m pv q w z |
255
volumetric plastic strain material parameter of the Drucker–Prager/cap model Poisson’s ratio density stress
1. Introduction Explosive cladding is a process of joining a wide variety of both similar and dissimilar combinations of materials for fabricating multilayer sandwich components for application in the advanced materials, electrical and electronic industries. The main characteristic of this process is its ability to distribute the high energy, obtained from the explosive detonation, over the weld-
Fig. 1. (a) Experimental set-up of the explosive cladding process; (b) a cross-section along a diametral plane showing the initial FE mesh.
ing area in an economical fashion. During explosive cladding, the high shock pressure developed accelerates the flyer plate towards the parent plate and the shockwaves propagate along the explosive layer, resulting in the collapse of the metal plates impacting against each other with a high relative velocity [1–3]. The cladding process also constitutes a way of explosively consolidating metallic and ceramic powders. Explosive compaction is a low-cost and very rapid technique for fabricating components of a great variety of geometries. It is quite promising for processing difficult-to-consolidate materials, due to the final homogeneity of the compact and the interparticle bonding. In general, stronger compacts are fabricated by this technique compared to those processed by quasi-static compaction. In particular, in processing superconducting materials, the explosive compaction technology shows some advantages; the very short duration of the process and the development of high pressures up to peak shock values of 1–100 GPa, lead to a dramatic reduction in porosity, whilst fracture of the initial grains and the formation of new clean grain boundaries during the compaction process may result in increased inter-grain current transport in the superconducting state. Furthermore, the defect-structure, created during the shockwave passage through the porous media and its interaction with lattice distortions and dislocation substructures, can provide flux pinning centres in Type II superconductors and increase the critical current density. Finally, a high integrity metal–ceramic bond is typically formed at the metal sheath/ceramic and/or ceramic/embedded metal contact rod interface. In addition, the superconducting properties are maintained and oxygen losses are not induced [4,5]. Several industrial applications of bulk superconducting ceramics of the YBCO and BSCCO compounds have been developed by the author and his collaborators, using explosive and electromagnetic dynamic compaction techniques and subsequent net-shape manufacturing processes, e.g. rolling, cold and warm extrusion, wiredrawing, etc. These applications include HTS conductors for electrical machines, e.g. rotating synchronous generators, levitated bearings and current limiters [4,6,7]. In order to obtain sound compacts and to optimize the compaction process, the understanding of the me-
256
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
Fig. 2. XRD pattern of (a) the initial and (b) the explosively compacted YBCO powder.
chanical behaviour of powders under applied stress and of the shock-wave propagation in powders, is very important. Finite element techniques constitute a powerful tool towards this direction, as they can predict the deformation of the compact, the pressure, temperature and density distributions, revealing details of the dynamic process that can hardly be obtained experimentally. Note, that the finite element methods may minimise the extensive experimental ‘trial-and-error’ procedure, by determining the amount and type of the explosive material used and by using different experimental set-ups, allowing, therefore, for the processing near to the near net-shape conditions. It may also be mentioned that until now, most of the finite element models were associated with the static consolidation, while the modeling of the dynamic compaction and especially the explosive compaction, has not been widely investigated, since it is difficult to provide the dynamic properties of the powder and its related Hugoniot curves [8,9]. In the present work, explosive compaction/cladding was employed to fabricate superconducting ceramic YBCO/aluminium composite discs, for fabricating levitating bearings for advanced measuring equipment and devices of high numbers of resolutions. The whole process was simulated by using explicit finite-element techniques. Taking into account that the simulation of
ceramic powders is often based on constitutive models, originally developed for mineral materials, such as soils and clays, an assumption is made to describe the nonlinear behaviour of the superconducting powders by using the modified Drucker–Prager/cap elasto-plastic constitutive material model, which can reflect the stress state and the degree of densification. Note, also, that for the development of the numerical model, the macromechanical approach, which characterizes the overall behaviour of the powder mass by idealizing the powder as a homogeneous continuum material, is assumed. This approach, apparently, constitutes a valid Table 1 Material properties Mechanical properties
Thermal properties
Aluminium Young modulus (E) Poisson’s ratio (n)
Specific heat (Cp) Thermal conductivity (k)
70 GPa 0.33
YBCO superconductor Young modulus 20 GPa (E) Poisson’s ratio 0.25 (w)
Specific heat (Cp) Thermal conductivity (k)
0.9 J gK−1 231 W mK−1
0.43 J gK−1 1 W mK−1
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
257
in Fig. 1(a). The ignition point was located at the centre of the explosive layer. The dimensions of the flyer aluminium axisymmetric plate and the parent aluminium disc were Ø130 × 3 and Ø95 ×25 mm, respectively. A cylindrical groove of dimensions Ø50 × 12 mm was machined in the parent metal and filled with YBCO superconducting powder. The bulk density before explosive cladding was 50% of the theoretical density, considered : 6.28 g cm − 3. Explosion was carried out in sand by firing an electric detonator, resulting in the collapse of the flyer plate and the consolidation of the powder. After compaction, the dimensions of the final HTS disc were Ø50 × 9 mm. According to experimental measurements, after the explosive compaction, the Y123 phase was the dominant one, while some decomposition around the grains occurred, as can be observed from the X-ray pattern of Fig. 2(a,b), where the pattern of the powder before and after compaction are presented. After oxygen annealing, the critical temperature, TC was measured by a SQUID magnetometer; a TC onset close to 93 K was obtained, proving the correct reoxygenation of the sample and confirming the ability of the explosive compaction to preserve the good quality of the superconducting phase [6].
3. Numerical simulation Fig. 3. Final deformed disc (a) after explosion in sand, (b) at the end of the numerical simulation.
approximation and it is useful from an industrial point of view, since the global behaviour of the powder mass can be modeled at an industrial scale [9,10].
2. Experimental procedure Explosive compaction/cladding was employed for the fabrication of an aluminium sheathed YBCO superconducting disc. The YBCO powder used was produced after calcination at 950 °C with dwell time reaching 200 h; its critical temperature was 92.2 K. The configuration of the cladding process consisted of a flyer metallic axisymmetric plate, which was impacted, using high-explosives, on a grooved metallic parent plate filled with the high-Tc superconducting ceramic powder, resulting in the cladding of the metallic plates and the compaction of the ceramic powder. The experimental configuration, with an initial set-up angle of 0° and a stand-off distance of 3 mm, is shown schematically in Fig. 1(a). The explosive used was a P-24 high explosive, in granular form, with 1.05 g cm − 3 nominal density and 2400 m s − 1 detonation velocity. An explosive mass of 420 g was used, forming a layer of 35 mm thickness above the flyer plate, as shown
A thermomechanical numerical analysis of the explosive compaction/cladding process was performed, using the explicit finite element code LS-DYNA3D. The explosive layer, the metal block, the cladding plate and the superconducting powder were modelled by using 8-node brick elements, from the element library of the code; a cross-section of the initial mesh of the configuration is presented in Fig. 1(b), consisting of 2796 elements and 3796 nodes. Sliding interfaces were used to define the contact conditions between the ceramic powder and the metal, as well as between the metal and the explosive. The explosive material was described by the Jones– Wilkins–Lee (JWL) equation of state, which provides the detonation pressure, following the notation, as:
P= 3.5·105 1−
n
+ 0.11·105 1− +
1.2·105 V
0.343 − 7V e V 0.12 − 2V e V
1.12−
0.12V0 V
−1
(1)
where the constants are characteristic for the specific explosive material. The aluminium, with mechanical and thermal properties shown in Table 1, was modelled as elasto-plastic material, with isotropic strain hardening, where the yield (flow) stress dependence on plastic strain is described by the equation:
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
258
| =Km n
(2)
X(s) is the intersection of the cap surface with the J1 axis. The hardening parameter s is related to the plastic volume change, m pv = ln(V/V0) through the hardening law
The superconducting YBa2Cu3O7 − x powder followed the modified Drucker– Prager/cap elasto-plastic constitutive model [9], the yield surface of which in the
J2D − J1 space consists of The failure envelope surface, f1 is described by the equation f1 = J2D − min(Fe(J1), Tmises)
(3)
where Fe is given as Fe(J1) h −k exp( −iJ1) + qJ1
(4)
(5)
The materials parameters h, k, i, q constitute the required input for the application of the model. The cap surface, f2 is defined by f2 = J2D − Fc(J1, s)
(6)
where Fc is given as Fc(J1, s)
1
[X(s) − L(s)]2 −[J1 −L(s)]2 R
(7)
R is the ratio of the major to minor axes of the quarter ellipse, defining the shape of the cap surface,
(8)
The tension cut-off surface, f3 is given by f3 T− J1
and Tmises X(sn)−L(sn)
m pv = W{1− exp[− D(X(s)− X0)]}
(9)
where T is the input material parameter specifying the maximum hydrostatic tension sustainable by the material. The cap model contains a number of parameters, which must be defined in order to represent a particular material; in general, they are based on experimental data obtained by uniaxial and triaxial compression tests. For the YBCO material examined, the values of a= 0.0175 MPa, k =2·10 − 4 MPa, i= 0.0192 MPa − 1 and q= 0.286 were selected from Ref. [11]. The elastic properties of the superconductor, taken from Refs. [12,13], are presented in Table 1. The thermal properties of the YBCO compound were measured at the Laboratory of Crismat, Caen, France and they are also presented in Table 1.
Fig. 4. A cross-section along a diametral plane showing the deformation of the plate at different time steps of the simulation of the cladding process.
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
259
decreased from 3 to 2.5 mm, while the height of the metal block was reduced to 22 mm. The diameter of the superconductor was :49.8 mm and its height was 9.6 mm, which differs from the experimental measurements by 1 and 6%, respectively. The final deformation of the disc after the explosion and after the numerical simulation is shown in Fig. 3(a,b), respectively. The deformation of the plates and the powder at different time steps of the simulation, along a diametral plane, is presented in Fig. 4. As detonation proceeds along the explosive layer, the developed shock pressure accelerates the flyer plate towards the parent plate, resulting in the consolidation of the superconducting powder. Pressure distributions,
Fig. 6. (a) Density distribution inside the superconducting disc at the end of the simulation; (b) variation of density of the superconductor with time at the top and bottom of the ceramic disc. Fig. 5. (a) Distribution of pressure inside the cladding plate, the metal block and the superconductor at two different time steps of the simulation, development of pressure with time inside the ceramic and the metallic; (b) at the bottom; and (c) at the top of the disc.
4. Numerical results and discussion From the simulation results it is indicated that the flyer plate reaches the powder after 13.5 ms, impacting on the superconductor with a velocity of 1100 m s − 1 and it is cladded on the metal block after 25 ms. After compaction, the thickness of the cladding plate was
Fig. 7. Variation of temperature with radius, along a diametral plane, at the top of the superconducting disc, at the final step of the simulation.
260
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
at the 19th and 21st ms of the simulation, are shown in Fig. 5. High pressures are developed at the impacted surface of the metal sheathed disc with the metallic flyer plate, resulting in the formation of dense components. The pressure inside the ceramic is higher at the bottom, near the interface with the metal block, due to reflection of the shock waves, as can be seen from Fig. 5(b,c). The application of such high pressures results in the consolidation of the superconducting ceramic material. The initial density of the material z was 50% of the theoretical density, z0 considered :6.28 g cm − 3, whilst, after compaction, the density reached the value of 83%, of the theoretical one. The ceramic compact is slightly denser at the bottom of the disc, as can be seen from the density distribution of Fig. 6(a). In Fig. 6(b), a diagram showing the development of density versus time at the top and bottom of the superconducting disc, reaching the values of 80 and 83% of the theoretical density respectively, is presented. According to the numerical analysis presented, the maximum temperature of the explosive reached 6000 °C, while the maximum temperature of the aluminum and the superconductor was 120 and 435 °C, respectively. The temperature was higher at the top of the superconducting disc, decreasing towards the bottom, due to the fact that the thermal conductivity of the YBCO superconductor is very small. Based on the experimental measurements, the superconducting properties are maintained, indicating that the temperature inside the powder is much lower than 970 °C, since the YBCO superconductor decomposes above this temperature. Fig. 7 shows the developed temperature at various positions along a diametral plane radius at the top of the compacted superconducting disc.
5. Conclusions A superconducting ceramic YBCO/aluminium composite disc was fabricated employing the explosive compaction/cladding process, in order to be used in the construction of levitating bearings. The process was simulated by using the explicit finite element code LSDYNA3D. From the results reported the following concluding remarks may be drawn. The numerically predicted final deformed shape of the superconducting disc is in good agreement with that experimentally obtained; the predicted diameter and the height of the ceramic disc differ from the experimental diameter and height by 1 and 6%, respectively.
Distributions of pressure inside the metal discs and the superconductor throughout the whole process were numerically predicted. The maximum pressure obtained was in the range of 5 GPa. The density of the superconductor was determined, reaching the value of 83% of the theoretical density, mainly at the bottom of the ceramic disc. The predicted maximum temperature in the superconducting powder was 435 °C; this temperature is much lower than the temperature of 970 °C, where the decomposition of the ceramic superconducting material takes place. Experimental evidence was obtained from the observed crystal structure, where no changes occurred.
Acknowledgements This work was performed under the auspices of the INCO-COPERNICUS Project IC-CT98-0504. We are grateful to our partners: G. Desgardin, Crismat, Caen, France and A. Sza´ lay, Metalltech, Budapest, Hungary for providing experimental evidence.
References [1] A.G. Mamalis, N.M. Vaxevanidis, A. Sza´ lay, J. Proha´ szka´ , J. Mater. Proc. Technol. 44 (1994) 99. [2] A.G. Mamalis, A. Sza´ lay, D. Pantelis, G. Pantazopoulos, J. Mater. Proc. Technol. 57 (1996) 155. [3] M. Gerland, H.N. Presles, J.P. Guin, D. Bertheau, Mater. Sci. Eng. A280 (2000) 311. [4] A.G. Mamalis, A. Sza´ lay, D.E. Manolakos, G. Pantazopoulos, Processing of High-Tc Superconductors at High Strain-Rates, Technomic, USA, 2000, p. 268. [5] A.G. Mamalis, A. Sza´ lay, N. Go¨ bl, I. Vajda, B. Raveau, Mater. Sci. Eng. B53 (1998) 119. [6] A.G. Mamalis, J. Inorg. Mater. 2 (2000) 623. [7] A.G. Mamalis, J. Mater. Proc. Tech. 99 (2000) 1. [8] H. Zipse, J. Eur. Ceram. Soc. 17 (1997) 1707. [9] A.G. Mamalis, I.N. Vottea, D.E. Manolakos, A. Sza´ lay, G. Desgardin, Proceedings of the International Conference on Fundamental Issues and Applications of Shock-Wave and HighStrain-Rate Phenomena, EXPLOMET 2000, New Mexico, USA, 2000, p. 289. [10] A.R. Khoei, R.W. Lewis, Finite Elem. Anal. Des. 30 (1998) 335. [11] I. Chou, C. Hung, J. Chin. Soc. Mech. Eng. 20 (1999) 287. [12] F. Tancret, G. Desgardin, F. Osterstock, Philos. Mag. A75 (1997) 505. [13] J. Joo, S.-B. Jung, W. Nah, J.-Y. Kim, T.S. Kim, Cryogenics 39 (2) (1999) 107.
Fabrication of metal/sheathed high-Tc superconducting composites by explosive compaction/cladding: numerical simulation A.G. Mamalis *, I.N. Vottea, D.E. Manolakos Department of Mechanical Engineering, Manufacturing Technology Di6ision, National Technical Uni6ersity of Athens, 42, 28th October A6enue, 10682 Athens, Greece Received 18 May 2001; accepted 17 October 2001
Abstract Explosive compaction/cladding, usually followed by forming, is a technique used extensively for fabricating multilayer sandwich components of the same or different materials. In this paper, we report on experimental and numerical investigations into the explosive compaction/cladding for fabricating superconducting Y – Ba– Cu– O ceramic/metal composite grooved discs. The manufacturing process is numerically simulated by using the explicit finite element code LS-DYNA3D. The final dimensions of the compact and the pressure, temperature and density distributions during the entire cladding compaction process are predicted. The proposed model is validated, as the numerical results obtained were in good agreement with the experimental results. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Explosive compaction/cladding; Superconductors; Finite element
Nomenclature Cp D E J1
J2D k P R T V W X0
specific heat hardening law coefficient Young’s modulus first stress invariant second stress invariant thermal conductivity pressure surface axis ratio maximum hydrostatic tension volume hardening law coefficient hardening law coefficient
Greek letters h material parameter of the Drucker – Prager/cap model i material parameter of the Drucker – Prager/cap model k material parameter of the Drucker – Prager/cap model m plastic strain
* Corresponding author. Tel.: +30-1772-3688; fax: + 30-1772-3689. E-mail address: [email protected] (A.G. Mamalis). 0921-5107/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 1 0 7 ( 0 1 ) 0 0 9 1 3 - 8
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
m pv q w z |
255
volumetric plastic strain material parameter of the Drucker–Prager/cap model Poisson’s ratio density stress
1. Introduction Explosive cladding is a process of joining a wide variety of both similar and dissimilar combinations of materials for fabricating multilayer sandwich components for application in the advanced materials, electrical and electronic industries. The main characteristic of this process is its ability to distribute the high energy, obtained from the explosive detonation, over the weld-
Fig. 1. (a) Experimental set-up of the explosive cladding process; (b) a cross-section along a diametral plane showing the initial FE mesh.
ing area in an economical fashion. During explosive cladding, the high shock pressure developed accelerates the flyer plate towards the parent plate and the shockwaves propagate along the explosive layer, resulting in the collapse of the metal plates impacting against each other with a high relative velocity [1–3]. The cladding process also constitutes a way of explosively consolidating metallic and ceramic powders. Explosive compaction is a low-cost and very rapid technique for fabricating components of a great variety of geometries. It is quite promising for processing difficult-to-consolidate materials, due to the final homogeneity of the compact and the interparticle bonding. In general, stronger compacts are fabricated by this technique compared to those processed by quasi-static compaction. In particular, in processing superconducting materials, the explosive compaction technology shows some advantages; the very short duration of the process and the development of high pressures up to peak shock values of 1–100 GPa, lead to a dramatic reduction in porosity, whilst fracture of the initial grains and the formation of new clean grain boundaries during the compaction process may result in increased inter-grain current transport in the superconducting state. Furthermore, the defect-structure, created during the shockwave passage through the porous media and its interaction with lattice distortions and dislocation substructures, can provide flux pinning centres in Type II superconductors and increase the critical current density. Finally, a high integrity metal–ceramic bond is typically formed at the metal sheath/ceramic and/or ceramic/embedded metal contact rod interface. In addition, the superconducting properties are maintained and oxygen losses are not induced [4,5]. Several industrial applications of bulk superconducting ceramics of the YBCO and BSCCO compounds have been developed by the author and his collaborators, using explosive and electromagnetic dynamic compaction techniques and subsequent net-shape manufacturing processes, e.g. rolling, cold and warm extrusion, wiredrawing, etc. These applications include HTS conductors for electrical machines, e.g. rotating synchronous generators, levitated bearings and current limiters [4,6,7]. In order to obtain sound compacts and to optimize the compaction process, the understanding of the me-
256
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
Fig. 2. XRD pattern of (a) the initial and (b) the explosively compacted YBCO powder.
chanical behaviour of powders under applied stress and of the shock-wave propagation in powders, is very important. Finite element techniques constitute a powerful tool towards this direction, as they can predict the deformation of the compact, the pressure, temperature and density distributions, revealing details of the dynamic process that can hardly be obtained experimentally. Note, that the finite element methods may minimise the extensive experimental ‘trial-and-error’ procedure, by determining the amount and type of the explosive material used and by using different experimental set-ups, allowing, therefore, for the processing near to the near net-shape conditions. It may also be mentioned that until now, most of the finite element models were associated with the static consolidation, while the modeling of the dynamic compaction and especially the explosive compaction, has not been widely investigated, since it is difficult to provide the dynamic properties of the powder and its related Hugoniot curves [8,9]. In the present work, explosive compaction/cladding was employed to fabricate superconducting ceramic YBCO/aluminium composite discs, for fabricating levitating bearings for advanced measuring equipment and devices of high numbers of resolutions. The whole process was simulated by using explicit finite-element techniques. Taking into account that the simulation of
ceramic powders is often based on constitutive models, originally developed for mineral materials, such as soils and clays, an assumption is made to describe the nonlinear behaviour of the superconducting powders by using the modified Drucker–Prager/cap elasto-plastic constitutive material model, which can reflect the stress state and the degree of densification. Note, also, that for the development of the numerical model, the macromechanical approach, which characterizes the overall behaviour of the powder mass by idealizing the powder as a homogeneous continuum material, is assumed. This approach, apparently, constitutes a valid Table 1 Material properties Mechanical properties
Thermal properties
Aluminium Young modulus (E) Poisson’s ratio (n)
Specific heat (Cp) Thermal conductivity (k)
70 GPa 0.33
YBCO superconductor Young modulus 20 GPa (E) Poisson’s ratio 0.25 (w)
Specific heat (Cp) Thermal conductivity (k)
0.9 J gK−1 231 W mK−1
0.43 J gK−1 1 W mK−1
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
257
in Fig. 1(a). The ignition point was located at the centre of the explosive layer. The dimensions of the flyer aluminium axisymmetric plate and the parent aluminium disc were Ø130 × 3 and Ø95 ×25 mm, respectively. A cylindrical groove of dimensions Ø50 × 12 mm was machined in the parent metal and filled with YBCO superconducting powder. The bulk density before explosive cladding was 50% of the theoretical density, considered : 6.28 g cm − 3. Explosion was carried out in sand by firing an electric detonator, resulting in the collapse of the flyer plate and the consolidation of the powder. After compaction, the dimensions of the final HTS disc were Ø50 × 9 mm. According to experimental measurements, after the explosive compaction, the Y123 phase was the dominant one, while some decomposition around the grains occurred, as can be observed from the X-ray pattern of Fig. 2(a,b), where the pattern of the powder before and after compaction are presented. After oxygen annealing, the critical temperature, TC was measured by a SQUID magnetometer; a TC onset close to 93 K was obtained, proving the correct reoxygenation of the sample and confirming the ability of the explosive compaction to preserve the good quality of the superconducting phase [6].
3. Numerical simulation Fig. 3. Final deformed disc (a) after explosion in sand, (b) at the end of the numerical simulation.
approximation and it is useful from an industrial point of view, since the global behaviour of the powder mass can be modeled at an industrial scale [9,10].
2. Experimental procedure Explosive compaction/cladding was employed for the fabrication of an aluminium sheathed YBCO superconducting disc. The YBCO powder used was produced after calcination at 950 °C with dwell time reaching 200 h; its critical temperature was 92.2 K. The configuration of the cladding process consisted of a flyer metallic axisymmetric plate, which was impacted, using high-explosives, on a grooved metallic parent plate filled with the high-Tc superconducting ceramic powder, resulting in the cladding of the metallic plates and the compaction of the ceramic powder. The experimental configuration, with an initial set-up angle of 0° and a stand-off distance of 3 mm, is shown schematically in Fig. 1(a). The explosive used was a P-24 high explosive, in granular form, with 1.05 g cm − 3 nominal density and 2400 m s − 1 detonation velocity. An explosive mass of 420 g was used, forming a layer of 35 mm thickness above the flyer plate, as shown
A thermomechanical numerical analysis of the explosive compaction/cladding process was performed, using the explicit finite element code LS-DYNA3D. The explosive layer, the metal block, the cladding plate and the superconducting powder were modelled by using 8-node brick elements, from the element library of the code; a cross-section of the initial mesh of the configuration is presented in Fig. 1(b), consisting of 2796 elements and 3796 nodes. Sliding interfaces were used to define the contact conditions between the ceramic powder and the metal, as well as between the metal and the explosive. The explosive material was described by the Jones– Wilkins–Lee (JWL) equation of state, which provides the detonation pressure, following the notation, as:
P= 3.5·105 1−
n
+ 0.11·105 1− +
1.2·105 V
0.343 − 7V e V 0.12 − 2V e V
1.12−
0.12V0 V
−1
(1)
where the constants are characteristic for the specific explosive material. The aluminium, with mechanical and thermal properties shown in Table 1, was modelled as elasto-plastic material, with isotropic strain hardening, where the yield (flow) stress dependence on plastic strain is described by the equation:
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
258
| =Km n
(2)
X(s) is the intersection of the cap surface with the J1 axis. The hardening parameter s is related to the plastic volume change, m pv = ln(V/V0) through the hardening law
The superconducting YBa2Cu3O7 − x powder followed the modified Drucker– Prager/cap elasto-plastic constitutive model [9], the yield surface of which in the
J2D − J1 space consists of The failure envelope surface, f1 is described by the equation f1 = J2D − min(Fe(J1), Tmises)
(3)
where Fe is given as Fe(J1) h −k exp( −iJ1) + qJ1
(4)
(5)
The materials parameters h, k, i, q constitute the required input for the application of the model. The cap surface, f2 is defined by f2 = J2D − Fc(J1, s)
(6)
where Fc is given as Fc(J1, s)
1
[X(s) − L(s)]2 −[J1 −L(s)]2 R
(7)
R is the ratio of the major to minor axes of the quarter ellipse, defining the shape of the cap surface,
(8)
The tension cut-off surface, f3 is given by f3 T− J1
and Tmises X(sn)−L(sn)
m pv = W{1− exp[− D(X(s)− X0)]}
(9)
where T is the input material parameter specifying the maximum hydrostatic tension sustainable by the material. The cap model contains a number of parameters, which must be defined in order to represent a particular material; in general, they are based on experimental data obtained by uniaxial and triaxial compression tests. For the YBCO material examined, the values of a= 0.0175 MPa, k =2·10 − 4 MPa, i= 0.0192 MPa − 1 and q= 0.286 were selected from Ref. [11]. The elastic properties of the superconductor, taken from Refs. [12,13], are presented in Table 1. The thermal properties of the YBCO compound were measured at the Laboratory of Crismat, Caen, France and they are also presented in Table 1.
Fig. 4. A cross-section along a diametral plane showing the deformation of the plate at different time steps of the simulation of the cladding process.
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
259
decreased from 3 to 2.5 mm, while the height of the metal block was reduced to 22 mm. The diameter of the superconductor was :49.8 mm and its height was 9.6 mm, which differs from the experimental measurements by 1 and 6%, respectively. The final deformation of the disc after the explosion and after the numerical simulation is shown in Fig. 3(a,b), respectively. The deformation of the plates and the powder at different time steps of the simulation, along a diametral plane, is presented in Fig. 4. As detonation proceeds along the explosive layer, the developed shock pressure accelerates the flyer plate towards the parent plate, resulting in the consolidation of the superconducting powder. Pressure distributions,
Fig. 6. (a) Density distribution inside the superconducting disc at the end of the simulation; (b) variation of density of the superconductor with time at the top and bottom of the ceramic disc. Fig. 5. (a) Distribution of pressure inside the cladding plate, the metal block and the superconductor at two different time steps of the simulation, development of pressure with time inside the ceramic and the metallic; (b) at the bottom; and (c) at the top of the disc.
4. Numerical results and discussion From the simulation results it is indicated that the flyer plate reaches the powder after 13.5 ms, impacting on the superconductor with a velocity of 1100 m s − 1 and it is cladded on the metal block after 25 ms. After compaction, the thickness of the cladding plate was
Fig. 7. Variation of temperature with radius, along a diametral plane, at the top of the superconducting disc, at the final step of the simulation.
260
A.G. Mamalis et al. / Materials Science and Engineering B90 (2002) 254–260
at the 19th and 21st ms of the simulation, are shown in Fig. 5. High pressures are developed at the impacted surface of the metal sheathed disc with the metallic flyer plate, resulting in the formation of dense components. The pressure inside the ceramic is higher at the bottom, near the interface with the metal block, due to reflection of the shock waves, as can be seen from Fig. 5(b,c). The application of such high pressures results in the consolidation of the superconducting ceramic material. The initial density of the material z was 50% of the theoretical density, z0 considered :6.28 g cm − 3, whilst, after compaction, the density reached the value of 83%, of the theoretical one. The ceramic compact is slightly denser at the bottom of the disc, as can be seen from the density distribution of Fig. 6(a). In Fig. 6(b), a diagram showing the development of density versus time at the top and bottom of the superconducting disc, reaching the values of 80 and 83% of the theoretical density respectively, is presented. According to the numerical analysis presented, the maximum temperature of the explosive reached 6000 °C, while the maximum temperature of the aluminum and the superconductor was 120 and 435 °C, respectively. The temperature was higher at the top of the superconducting disc, decreasing towards the bottom, due to the fact that the thermal conductivity of the YBCO superconductor is very small. Based on the experimental measurements, the superconducting properties are maintained, indicating that the temperature inside the powder is much lower than 970 °C, since the YBCO superconductor decomposes above this temperature. Fig. 7 shows the developed temperature at various positions along a diametral plane radius at the top of the compacted superconducting disc.
5. Conclusions A superconducting ceramic YBCO/aluminium composite disc was fabricated employing the explosive compaction/cladding process, in order to be used in the construction of levitating bearings. The process was simulated by using the explicit finite element code LSDYNA3D. From the results reported the following concluding remarks may be drawn. The numerically predicted final deformed shape of the superconducting disc is in good agreement with that experimentally obtained; the predicted diameter and the height of the ceramic disc differ from the experimental diameter and height by 1 and 6%, respectively.
Distributions of pressure inside the metal discs and the superconductor throughout the whole process were numerically predicted. The maximum pressure obtained was in the range of 5 GPa. The density of the superconductor was determined, reaching the value of 83% of the theoretical density, mainly at the bottom of the ceramic disc. The predicted maximum temperature in the superconducting powder was 435 °C; this temperature is much lower than the temperature of 970 °C, where the decomposition of the ceramic superconducting material takes place. Experimental evidence was obtained from the observed crystal structure, where no changes occurred.
Acknowledgements This work was performed under the auspices of the INCO-COPERNICUS Project IC-CT98-0504. We are grateful to our partners: G. Desgardin, Crismat, Caen, France and A. Sza´ lay, Metalltech, Budapest, Hungary for providing experimental evidence.
References [1] A.G. Mamalis, N.M. Vaxevanidis, A. Sza´ lay, J. Proha´ szka´ , J. Mater. Proc. Technol. 44 (1994) 99. [2] A.G. Mamalis, A. Sza´ lay, D. Pantelis, G. Pantazopoulos, J. Mater. Proc. Technol. 57 (1996) 155. [3] M. Gerland, H.N. Presles, J.P. Guin, D. Bertheau, Mater. Sci. Eng. A280 (2000) 311. [4] A.G. Mamalis, A. Sza´ lay, D.E. Manolakos, G. Pantazopoulos, Processing of High-Tc Superconductors at High Strain-Rates, Technomic, USA, 2000, p. 268. [5] A.G. Mamalis, A. Sza´ lay, N. Go¨ bl, I. Vajda, B. Raveau, Mater. Sci. Eng. B53 (1998) 119. [6] A.G. Mamalis, J. Inorg. Mater. 2 (2000) 623. [7] A.G. Mamalis, J. Mater. Proc. Tech. 99 (2000) 1. [8] H. Zipse, J. Eur. Ceram. Soc. 17 (1997) 1707. [9] A.G. Mamalis, I.N. Vottea, D.E. Manolakos, A. Sza´ lay, G. Desgardin, Proceedings of the International Conference on Fundamental Issues and Applications of Shock-Wave and HighStrain-Rate Phenomena, EXPLOMET 2000, New Mexico, USA, 2000, p. 289. [10] A.R. Khoei, R.W. Lewis, Finite Elem. Anal. Des. 30 (1998) 335. [11] I. Chou, C. Hung, J. Chin. Soc. Mech. Eng. 20 (1999) 287. [12] F. Tancret, G. Desgardin, F. Osterstock, Philos. Mag. A75 (1997) 505. [13] J. Joo, S.-B. Jung, W. Nah, J.-Y. Kim, T.S. Kim, Cryogenics 39 (2) (1999) 107.