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Probabilistic bounds on effective elastic moduli for the superconducting coils
Computational Materials Science 11 (1998) 252±260
Probabilistic bounds on eective elastic moduli for the superconducting coils M. Kami nski Faculty of Civil Engineering, Arch. and Env. Engineering, Division of Mechanics of Materials, Technical University of è odz, Al.Politechniki 6, 93-590 è odz, Poland Received 11 December 1997; accepted 6 February 1998
Abstract The main purpose of the paper presented is probabilistic characterization of the eective elastic characteristics of a superconducting four-component composite. The probabilistic moments of these characteristics up to the fourth-order have been estimated on the basis of experimentally measured expected values and standard deviations of component materials elastic properties (Young moduli and Poisson coecients). The Monte-Carlo simulation technique has been used to carry out all computational experiments. The results computed have been discussed in details in order to verify the sensitivity of respective probabilistic moments to input random data and to total number of random trials used to estimate these values. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Superconductors; Eective properties; Homogenization; Monte-Carlo simulation
1. Introduction The main idea of the paper is to present the method and the results of probabilistic simulation of the eective mechanical properties of periodic composite superconducting microstructure. It is well known, considering manufacturing processes and experimentation, that physical and mechanical properties of both composites and their constituents have random, in fact stochastic, character [1,2]. There are many probabilistic computational methods enabling simulation of random elastic materials and composite structures, such as Monte-Carlo Simulation (MCS) method [2±4], the Stochastic Finite Element Method (SFEM) [5±7], Delaunay networks [8], Voronoi Cell Finite Element Method (VCFEM) [9] or wide range of different types of stochastic polynomial expansions [5]. The ®rst one seems to be the most eective, es0927-0256/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 0 0 6 - 8
pecially in very complicated models of microstructures or materials. Another problem of designing and numerical simulation of ®eld problems in microsystems and microstructures is the scale eect caused by the dierence between micro and macro scale. The homogenization method consisting in approximation of the eective (homogenized) behaviour appeared to be one of the most ecient method during the 80's and the 90's [3±5]. The fundamental idea is to obtain a new material equivalent to the real composite in the sense of energy principles (i.e. the total elastic strain and complementary energy for the eective elasticity tensor). This method found its signi®cant applications in the computational analyses of superconducting coils [10,11]. There are many dierent homogenization approaches which have been developed; the asymptotic approach [10] and the eective modules
M. Kami nski / Computational Materials Science 11 (1998) 252±260
method (direct approximation of the homogenized material) [3,4,12] with their further innovations as well as dierent types of the variational upper and lower bounds of the eective physical and mechanical composite properties [1,13,14]. Considering all these facts, the stochastic formulation of the homogenization problem seems to be very useful and applicable in the engineering of composite microstructures [3,4]. To obtain a mathematical model quite consistent with given experimental results, the Young moduli and Poisson coecients of the composite constituents have been assumed as uncorrelated Gaussian random variables with expected values and standard deviations speci®ed. The probabilistic version of respective variational upper and lower bounds [1] of the homogenized constitutive tensor are presented. The computational implementation into the MCCEFF [4±6] MCS and FEM-based system is used to test the example of the rectangular Representative Volume Element (RVE) of a superconducting coil cable. The probabilistic moments and coecients [15] of the eective elasticity tensor approximations are compared and discussed in detail. The probabilistic sensitivity problem of these tensor components to the input random variables is analyzed in the numerical analysis. The application of the probabilistic simulation approach to the computational analysis and designing of the superconducting coils cables [10,11] is proposed as well as to the stochastic structural interface defects in general composite materials [5,7]. 2. Eective characteristics of periodic composites Let us consider a composite structure Y built up with n linear elastic, isotropic and homogeneous components. The elastic properties of these components are de®ned deterministically by the Lame constants k, l. Next, let us assume that this composite is periodic which means that there exist the smallest repeatable part of this structure denoted by X which covers by translation the whole Y. The element so characterized is called Representative Volume Element (RVE) or simply a periodicity cell. The purpose of further investigations is to ®nd a homogeneous material which will be equivalent to the real composite in the sense of energy norm.
253
For this purpose the coecients of the following linear second-order elliptic problem are considered. ÿ div C e e ue f ; eij ue
1 e ui;j uej;i ; 2
x 2 X;
1
x 2 X;
C e ve p xi C e p ; where v
e p
xi
2 3
1; xi 2 Xp ; 0; elsewhere;
4
with boundary conditions ue 0;
x 2 oX:
5
The eective (homogenized) tensor is such a tensor that replacing Ce with C0 in the above system gives `u' as a solution, which is a weak limit of ue when e ! 0. It should be mentioned that without any other assumptions on X microgeometry the whole so eff called G set of Cijkl is generated. Moreover, it can be proved that there exist such inf Cijkl and sup Cijkl tensors that eff
inf Cijkl 6 Cijkl 6 sup Cijkl :
6
It is well known [1] that the theorem of minimum potential energy gives the upper bounds of the effective tensor while the minimum complementary energy gives the lower ones. Thanks to the Eshelby formula the explicit equations for the eective elasticity tensor bounds can be obtained as follows: " #ÿ1 N X ÿ1 Cr ju jr ÿ ju ; sup j r1
sup l
" N X r1
7
#ÿ1 Cr lu lr ÿ1
ÿ lu ;
where ju , lu have the following form: ju 43 lmax ; ÿ1 3 1 10 : lu 2 lmax 9jmax 8lmax
8
Further, lower bounds for the elasticity tensor are obtained as
254
M. Kami nski / Computational Materials Science 11 (1998) 252±260
"
N X inf j Cr jl jr ÿ1
"
r1
N X Cr ll lr ÿ1 inf l r1
#ÿ1 ÿ jl ; 9
#ÿ1 ÿ ll ;
where it holds jl 43 lmin ; ÿ1 3 1 10 ; ll 2 lmin 9jmin 8lmin
10
and N is a total number of composite constituents, where cr ; 1 6 r 6 N denote their volume fractions. It should be noted that if the input elastic characteristics of the composite constituents are given in the form of Young moduli and Poisson coecients we have e ; 11 j 3 1 ÿ 2t l
e ; 2 1 t
12
k j ÿ 23 l:
13
Finally, by the use of the well-known elasticity tensor de®nition we can obtain the components of the upper and lower bounds of this tensor as follows: inf Cijkl x dij dkl inf k x ÿ dik djl dil djk inf l x; 14 sup Cijkl x dij dkl supk x ÿ 15 dik djl dil djk supl x: The upper and lower bounds presented above are compared in the numerical analysis with the classical formulation of the Voigt±Reuss bounds given explicitly as follows: 1 PN
6j6
1 PN
6l6
cr r1 jr
cr r1 lr
N X
cr jr ;
16
cr lr :
17
r1 N X r1
Having the upper and lower bounds for the elastic properties of composite constituents we can ®nd the bounds assuming randomness of these properties. It can be done by the use of pure mathematical approach in the context of respective integration over probability space or numerically, by the use of one of the methods cited above. For the needs of both of these methods we should assume the type of probability density function (PDF) of our input variables. In further analyses we will consider that the Young moduli and Poisson coecients of the superconductor component materials are uncorrelated Gaussian random variables with the ®rst two probabilistic moments speci®ed. Moreover, it should be noted that from the engineering point of view the most interesting is the eff eectiveness of such a characterization of Cijkl in the sense of dierence between the upper and lower bounds. On the other hand, the sensitivity of the eective tensor probabilistic chatacterization with respect to the total number of numerical random trials should be veri®ed to optimize the time of computations with respect to given input accuracy. To verify the correctness and eectiveness of the homogenized characteristics computed in the way described above, the probabilistic moments of these characteristics are compared with the respective moments of the classical Voigt±Reuss bounds. The results of computational simulation have been described in detail in the next section. 3. Numerical analysis 3.1. Computational implementation The Monte-Carlo simulation technique has been used to compute the probabilistic moments of the eective elasticity tensor components. Generally, as it was described in [4,5] this technique consists of three steps: 1. generation of the random distributions according to given expected values and other probabilistic characteristics and probability density function of the input random ®elds; 2. the set of problem equations computed for all random quantities generated (in our case (Eqs. (7)±(17)));
M. Kami nski / Computational Materials Science 11 (1998) 252±260
3. statistical estimation of the set of solutions obtained. All these numerical procedures have been implemented into the MCCEFF system, which has been used previously and generally designed to the deterministic as well as probabilistic homogenization procedure [3±5]. The equations listed below are implemented in statistical estimation procedure to compute the probabilistic moments with respect to M which denotes the total number of Monte-Carlo random trials performed. The expected values of the eective elasticity tensor bounds components are given as M 1X m C C ijkl E Cijkl M m1 ijkl
and the variances are obtained as M 2 ÿ 1 X m Cijkl ÿ C ijkl : Var Cijkl M ÿ 1 m1
18
The standard deviation of the variable are obtained from Eqs. (18) and (19) as follows: ÿ q ÿ r Cijkl Var Cijkl : 20 Generally, we can estimate the rth order moment of the variable Cijkl in the following way: M r ÿ 1X m mr Cijkl Cijkl M m1
· coecient of variation a, a h xk ; b
21
and the central moment of the same variable as ÿ ÿ lr Cijkl m1 Cijkl ÿ mr Cijkl : 22 In case of the Gaussian probability density function N(m, r) any odd-order probabilistic moments are equal to 0 while the ®rst three even moments are ÿ r2 l2 Cijkl ; m
23
ÿ 3r4 l4 Cijkl 2 ; m
24
ÿ 15r6 l6 Cijkl 3 : m
25
On the basis of the central moments estimators the following coecients can be de®ned and computed:
r h x k ; b ; Eh xk ; b
26
· coecient of asymmetry b, b h xk ; b
l3 h xk ; b ; r3 h xk ; b
27
· coecient of concentration c, c h xk ; b
l4 h xk ; b : r4 h x k ; b
28
Considering the Central Limit Theorem (CLT) for the total number of Monte-Carlo iterations performed M tending to 1 we obtain for the Gaussian probability distribution function that lim b n 0;
29
lim c n 3:
30
n!1
19
255
n!1
Eqs. (29) and (30) are very useful (together with the respective PDF estimator) in recognizing the character of probabilistic distribution functions of the output variables. Considering the CLT described above we can recognize for example Gaussian variables. This is very important considering the fact that theoretical considerations in this matter are rather complicated and not always possible. 3.2. Eective moduli estimators for the superconductor The superconducting cable consists of ®bers made of a superconductor placed around a thinwalled pipe (tube) covered with a jacket and insulating material. The RVE of the superconducting composite structure analyzed is shown in Fig. 1. The experimental data describing elastic characteristics of the composite constituents are collected in Table 1. The expected values and standard deviations of Young moduli and Poisson coecients are de®ned in respective columns, according to data presented in [16]. Because of negligible dierences in elastic properties of Incoloy (between `annealed' and `cold worked' state) the `annealed' state of superconductor is further considered. All the results obtained
256
M. Kami nski / Computational Materials Science 11 (1998) 252±260
As it was mentioned above, the eective properties described in the previous sections (sup, inf in Table 2) have been compared with the Voigt±Reuss ones (sup-VR, inf-VR in Table 2). Considering the results obtained, it should be noticed that these ®rst bounds are generally more restrictive than the Voigt±Reuss ones. Further, it can be observed that respective deterministic values are with acceptable accuracy equal to respective expected values. Thus, for relatively small standard deviations of the input elastic characteristics, random character of the effective characteristics can be neglected. Finally, it can be noticed that more restrictive bounds are a good approximation of the eective elasticity tensor. Taking as a basis the arithmetic average of the upper and lower bounds, the dierence between these bounds is in the range of 13% for CJJJJ bounds component, 19% for CJKJK bounds component and 8% for CJKKJ bounds component. Figs. 2±10 contain the results of the convergence analysis of the coecient of variation, asymmetry and concentration with respect to increasing total number of the Monte-Carlo random trials. All these coecients are presented for CJJJJ bounds in Figs. 2±4, for CJKJK bounds in Figs. 5±7 and for CJKKJ in Figs. 8±10. On the horizontal axes of these ®gures the total number of
Fig. 1. The RVE of the superconducting cable tested.
in the computational experiments have been collected in Table 2 and Figs. 2±13. The eective elasticity tensor components and their respective expected values (in GPa) have been collected in Table 2. Because of the fact that the expected values appeared to be rather nonsensitive to the total number of random trials in the Monte-Carlo simulations, the results of respective convergence tests have been omitted. The values of the expectations considered have been collected in Table 2 for M 10 000 random trials. Table 1 Elastic characteristics of the superconductor components Material
E[e] GPa
316LN Incoloy 908 Annealed Cold worked Titanium Insulation G10-CR
205 182 184 126 36
r(e) GPa
E[m]
r(m)
8
0.265
0.010
) ) 12 )
0.303 0.299 0.311 0.21
) ) 0.012 )
Table 2 Eective elasticity tensor components and their expected values (in GPa) Eective property type
Analysis type Deterministic
sup-VR sup inf inf-VR
Probabilistic
CJJJJ
CJKKJ
CJKJK
CJJJJ
CJKKJ
CJKJK
189.56 178.44 156.99 137.93
81.83 76.07 62.70 51.86
53.86 51.18 47.14 43.03
189.94 178.57 156.68 137.54
82.30 76.37 62.61 51.71
53.82 51.10 47.03 42.92
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 2. The coecients of variation of CJJJJ bounds.
Fig. 6. The coecients of asymmetry of CJKJK bounds.
Fig. 3. The coecients of variation of CJKJK bounds.
Fig. 7. The coecients of asymmetry of CJKKJ bounds.
Fig. 4. The coecients of variation of CJKKJ bounds.
Fig. 5. The coecients of asymmetry of CJJJJ bounds.
257
Fig. 8. The coecients of concentration of CJJJJ bounds.
Fig. 9. The coecients of concentration of CJKJK bounds.
258
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 10. The coecients of concentration of CJKKJ bounds.
Fig. 11. The probability densities of CJJJJ bounds.
Monte-Carlo random trials M is marked, while on the vertical one there are respective values of the coecients computed. The general observation is that the CJKJK bounds are the most sensitive with respect to the randomness of input elastic characteristics. This conclusion can be stated comparing results in Figs. 2±4. These coecients for CJKJK bounds appeared to be the greatest and next we obtain the coecients for CJJJJ and CJKKJ , respectively. Next, it can be mentioned that the estimators of coecients of variation show good convergence to their limits. The respective values eff of the coecients for dierent Cijkl bounds are obtained for M equal to 2500 random trials. Generally, we can observe that the coecients of variation of the eective elasticity tensor ful®ll the hierarchy of the respective expected values. The greatest coecients are obtained for Reuss bounds, followed by the upper and lower bounds proposed in the paper, and the smallest for the Voigt lower bounds. Observing the results presented in Figs. 5±10 it can be observed that all coecients of asymmetry eff of Cijkl veri®ed tend to 0 on increasing the total number of random trials. Comparing CJJJJ and CJKJK against CJKKJ bounds it can be stated that the ®rst two variables have minimum positive asymmetry, while the last have negative one. It should be noticed that for such probabilistic distributions with non-zero coecients of asymmetry the expected value is not equal to the most probable one. Moreover, taking into account the convergence of coecients of asymmetry it can be noticed that they generally converge more slowly than coecients of variation estimators. M larger than 5000 is required to compute these estimators with satisfactory accuracy. Analogous to the coe-
cients of variation, the hierarchy of the expected eff values of Cijkl , which has been discussed above, is ful®lled. Figs. 8±10 present the coecients of concentration for dierent components of the eective elasticity tensor. The estimator convergence analysis proves that M equal to almost 10 000 is needed to compute these coecients properly. The convergence of these estimators is more complex than the previous ones, but generally their values are greater than 3, which is characteristic for the Gaussian eff variables. Thus it can be stated that the Cijkl probabilistic distributions obtained are more concentrated around their expected values than the Gaussian variables, but this dierence is not greater than a maximum of 15% for the CJKJK bounds. Figs. 11±13 illustrate the probability density functions of the upper and lower bounds for CJJJJ , CJKJK and CJKKJ components of the eective elasticity tensor. On the horizontal axes of these ®gures the values computed of these components are marked, while on the vertical ones the respective density of probability is given. The probability density functions of the eff Cijkl computed together with the respective coecients of asymmetry and concentration b, c show that these PDFs have distributions quite similar
Fig. 12. The probability densities of CJKJK bounds.
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 13. The probability densities of CJKKJ bounds.
to the bell-shaped Gaussian distribution curve. Thus, in further analyses proposed in the conclusions, we will assume that for the input random variables, in the form of elastic characteristics Young moduli and Poisson coecient, being Gaussian uncorrelated random variables, the upper and lower bounds computed also have Gaussian distribution. 4. Concluding remarks (1) The results of numerical tests performed lead us to the conclusion that the probabilistic upper and lower bounds of the eective elasticity tensor may be very ecient in characterization of superconducting composites with randomly de®ned elastic characteristics because of negligible relative dierences between the upper and lower bounds. Considering the computational time cost they appear to be much more useful in engineering practice than other FEM-based direct methods. (2) The numerical experiments carried out prove that the coecients of variation of the bounds computed are in the range of the input random variables of the problem. Considering further analyses of superconducting coils homogenized, this fact con®rms the need of the application of the SFEM in such computations, considering the time savings. (3) The probabilistic sensitivity of the eective elastic characteristics with respect to probabilistic material parameters should be veri®ed computationally in future analysis as the eect of regression analysis. Such an analysis enables to ®nd out these parameters of composite constituents elastic characteristics which are crucial for superconductor behaviour.
259
(4) The procedure of eective elastic properties approximation seems to be the only method which can be succesfully applied to the homogenization of stochastic interface defects. Such an approach will make the elastic properties of interphases much more sensitive to the presence of structural defects than it was in case of the Probabilistic Averaging Method. Considering this the bounds presented should be implemented into numerical analysis of stochastic structural defects.
Acknowledgements The author wishes to thank Prof. B.A. Schre¯er from Instituto di Scienza e Tecnica delle Construzioni de Padova in Italy and M. Le®k, his colleague, for all the information connected with the superconducting coil cable which has been manufactured in Max-Planck-Institut f ur Plasmaphysik in M unchen.
References [1] R.M. Christensen, Mechanics of Composite Materials, Wiley, New York, 1979. [2] K. Sab, On the homogenization and the simulation of random materials, Eur. J. Mech. Sol. 11 (1992) 585±607. [3] M. Kami nski, Homogenization of random elastic media, Comput. Ass. Mech. Engrg. Sci. 3 (1996) 9±21. [4] M. Kami nski, M. Kleiber, Numerical homogenization of n-phase composites including stochastic interface defects, Int. J. Numer. Meth. Engrg., submitted. [5] M. Kami nski, Stochastic interphase defects in periodic ®ber composites (in Polish), Ph.D. Thesis, t odz, 1996. [6] M. Kami nski, M. Kleiber, Stochastic ®nite element method in random non-homogeneous media, in: J.A. Desideri et al. (Eds.), Numerical Methods in Engineering '96, Wiley, New York, 1996, pp. 35±41. [7] M. Kami nski, M. Kleiber, Stochastic structural interface defects in ®ber composites, Int. J. Sol. Struct. 33 (1996) 3035±3056. [8] M. Ostoja-Starzewski, C. Wang, Linear elasticity of planar Delaunay networks: Random ®eld characterization of eective moduli, Acta Mech. 80 (1989) 61±80. [9] S. Ghosh et al, Multiple scale analysis of heterogeneous elastic structures using homogenization theory an Voronoi cell ®nite element method, Int. J. Sol. Struct. 32 (1995) 27±62. [10] M. Le®k, B.A. Schre¯er, Application of the homogenisation method to the analysis of superconducting coils, Fusion Engrg. Design 24 (1994) 231±255.
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M. Kami nski / Computational Materials Science 11 (1998) 252±260
[11] M. Le®k, B.A. Schre¯er, Homogenized material coecients for 3D elastic analysis of superconducting coils, in: P. Ladeveze, O.C. Zienkiewicz (Eds.), New Advances in ComputationalStructuralMechanics,Elsevier,Amsterdam,1992. [12] P. Suquet, A dual method in homogenization: Application to elastic media (in French), J. Mech. Theor. Appl. (1982) 79±98. [13] M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47 (1987) 1216±1228.
[14] G.W. Milton, R.V. Kohn, Variational bounds on the eective moduli of anisotropic composites, Int. J. Mech. Phys. Sol. 27 (1988) 598±630. [15] J.S. Bendat, A.G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley, New York, 1971. [16] N.J. Simon, R.P. Reed, Structural Materials for Superconducting Magnets I, AISI 316 Stainless Steel, NIST, Boulder, Colorado, 1982.
Probabilistic bounds on eective elastic moduli for the superconducting coils M. Kami nski Faculty of Civil Engineering, Arch. and Env. Engineering, Division of Mechanics of Materials, Technical University of è odz, Al.Politechniki 6, 93-590 è odz, Poland Received 11 December 1997; accepted 6 February 1998
Abstract The main purpose of the paper presented is probabilistic characterization of the eective elastic characteristics of a superconducting four-component composite. The probabilistic moments of these characteristics up to the fourth-order have been estimated on the basis of experimentally measured expected values and standard deviations of component materials elastic properties (Young moduli and Poisson coecients). The Monte-Carlo simulation technique has been used to carry out all computational experiments. The results computed have been discussed in details in order to verify the sensitivity of respective probabilistic moments to input random data and to total number of random trials used to estimate these values. Ó 1998 Elsevier Science B.V. All rights reserved. Keywords: Superconductors; Eective properties; Homogenization; Monte-Carlo simulation
1. Introduction The main idea of the paper is to present the method and the results of probabilistic simulation of the eective mechanical properties of periodic composite superconducting microstructure. It is well known, considering manufacturing processes and experimentation, that physical and mechanical properties of both composites and their constituents have random, in fact stochastic, character [1,2]. There are many probabilistic computational methods enabling simulation of random elastic materials and composite structures, such as Monte-Carlo Simulation (MCS) method [2±4], the Stochastic Finite Element Method (SFEM) [5±7], Delaunay networks [8], Voronoi Cell Finite Element Method (VCFEM) [9] or wide range of different types of stochastic polynomial expansions [5]. The ®rst one seems to be the most eective, es0927-0256/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 0 0 6 - 8
pecially in very complicated models of microstructures or materials. Another problem of designing and numerical simulation of ®eld problems in microsystems and microstructures is the scale eect caused by the dierence between micro and macro scale. The homogenization method consisting in approximation of the eective (homogenized) behaviour appeared to be one of the most ecient method during the 80's and the 90's [3±5]. The fundamental idea is to obtain a new material equivalent to the real composite in the sense of energy principles (i.e. the total elastic strain and complementary energy for the eective elasticity tensor). This method found its signi®cant applications in the computational analyses of superconducting coils [10,11]. There are many dierent homogenization approaches which have been developed; the asymptotic approach [10] and the eective modules
M. Kami nski / Computational Materials Science 11 (1998) 252±260
method (direct approximation of the homogenized material) [3,4,12] with their further innovations as well as dierent types of the variational upper and lower bounds of the eective physical and mechanical composite properties [1,13,14]. Considering all these facts, the stochastic formulation of the homogenization problem seems to be very useful and applicable in the engineering of composite microstructures [3,4]. To obtain a mathematical model quite consistent with given experimental results, the Young moduli and Poisson coecients of the composite constituents have been assumed as uncorrelated Gaussian random variables with expected values and standard deviations speci®ed. The probabilistic version of respective variational upper and lower bounds [1] of the homogenized constitutive tensor are presented. The computational implementation into the MCCEFF [4±6] MCS and FEM-based system is used to test the example of the rectangular Representative Volume Element (RVE) of a superconducting coil cable. The probabilistic moments and coecients [15] of the eective elasticity tensor approximations are compared and discussed in detail. The probabilistic sensitivity problem of these tensor components to the input random variables is analyzed in the numerical analysis. The application of the probabilistic simulation approach to the computational analysis and designing of the superconducting coils cables [10,11] is proposed as well as to the stochastic structural interface defects in general composite materials [5,7]. 2. Eective characteristics of periodic composites Let us consider a composite structure Y built up with n linear elastic, isotropic and homogeneous components. The elastic properties of these components are de®ned deterministically by the Lame constants k, l. Next, let us assume that this composite is periodic which means that there exist the smallest repeatable part of this structure denoted by X which covers by translation the whole Y. The element so characterized is called Representative Volume Element (RVE) or simply a periodicity cell. The purpose of further investigations is to ®nd a homogeneous material which will be equivalent to the real composite in the sense of energy norm.
253
For this purpose the coecients of the following linear second-order elliptic problem are considered. ÿ div C e e ue f ; eij ue
1 e ui;j uej;i ; 2
x 2 X;
1
x 2 X;
C e ve p xi C e p ; where v
e p
xi
2 3
1; xi 2 Xp ; 0; elsewhere;
4
with boundary conditions ue 0;
x 2 oX:
5
The eective (homogenized) tensor is such a tensor that replacing Ce with C0 in the above system gives `u' as a solution, which is a weak limit of ue when e ! 0. It should be mentioned that without any other assumptions on X microgeometry the whole so eff called G set of Cijkl is generated. Moreover, it can be proved that there exist such inf Cijkl and sup Cijkl tensors that eff
inf Cijkl 6 Cijkl 6 sup Cijkl :
6
It is well known [1] that the theorem of minimum potential energy gives the upper bounds of the effective tensor while the minimum complementary energy gives the lower ones. Thanks to the Eshelby formula the explicit equations for the eective elasticity tensor bounds can be obtained as follows: " #ÿ1 N X ÿ1 Cr ju jr ÿ ju ; sup j r1
sup l
" N X r1
7
#ÿ1 Cr lu lr ÿ1
ÿ lu ;
where ju , lu have the following form: ju 43 lmax ; ÿ1 3 1 10 : lu 2 lmax 9jmax 8lmax
8
Further, lower bounds for the elasticity tensor are obtained as
254
M. Kami nski / Computational Materials Science 11 (1998) 252±260
"
N X inf j Cr jl jr ÿ1
"
r1
N X Cr ll lr ÿ1 inf l r1
#ÿ1 ÿ jl ; 9
#ÿ1 ÿ ll ;
where it holds jl 43 lmin ; ÿ1 3 1 10 ; ll 2 lmin 9jmin 8lmin
10
and N is a total number of composite constituents, where cr ; 1 6 r 6 N denote their volume fractions. It should be noted that if the input elastic characteristics of the composite constituents are given in the form of Young moduli and Poisson coecients we have e ; 11 j 3 1 ÿ 2t l
e ; 2 1 t
12
k j ÿ 23 l:
13
Finally, by the use of the well-known elasticity tensor de®nition we can obtain the components of the upper and lower bounds of this tensor as follows: inf Cijkl x dij dkl inf k x ÿ dik djl dil djk inf l x; 14 sup Cijkl x dij dkl supk x ÿ 15 dik djl dil djk supl x: The upper and lower bounds presented above are compared in the numerical analysis with the classical formulation of the Voigt±Reuss bounds given explicitly as follows: 1 PN
6j6
1 PN
6l6
cr r1 jr
cr r1 lr
N X
cr jr ;
16
cr lr :
17
r1 N X r1
Having the upper and lower bounds for the elastic properties of composite constituents we can ®nd the bounds assuming randomness of these properties. It can be done by the use of pure mathematical approach in the context of respective integration over probability space or numerically, by the use of one of the methods cited above. For the needs of both of these methods we should assume the type of probability density function (PDF) of our input variables. In further analyses we will consider that the Young moduli and Poisson coecients of the superconductor component materials are uncorrelated Gaussian random variables with the ®rst two probabilistic moments speci®ed. Moreover, it should be noted that from the engineering point of view the most interesting is the eff eectiveness of such a characterization of Cijkl in the sense of dierence between the upper and lower bounds. On the other hand, the sensitivity of the eective tensor probabilistic chatacterization with respect to the total number of numerical random trials should be veri®ed to optimize the time of computations with respect to given input accuracy. To verify the correctness and eectiveness of the homogenized characteristics computed in the way described above, the probabilistic moments of these characteristics are compared with the respective moments of the classical Voigt±Reuss bounds. The results of computational simulation have been described in detail in the next section. 3. Numerical analysis 3.1. Computational implementation The Monte-Carlo simulation technique has been used to compute the probabilistic moments of the eective elasticity tensor components. Generally, as it was described in [4,5] this technique consists of three steps: 1. generation of the random distributions according to given expected values and other probabilistic characteristics and probability density function of the input random ®elds; 2. the set of problem equations computed for all random quantities generated (in our case (Eqs. (7)±(17)));
M. Kami nski / Computational Materials Science 11 (1998) 252±260
3. statistical estimation of the set of solutions obtained. All these numerical procedures have been implemented into the MCCEFF system, which has been used previously and generally designed to the deterministic as well as probabilistic homogenization procedure [3±5]. The equations listed below are implemented in statistical estimation procedure to compute the probabilistic moments with respect to M which denotes the total number of Monte-Carlo random trials performed. The expected values of the eective elasticity tensor bounds components are given as M 1X m C C ijkl E Cijkl M m1 ijkl
and the variances are obtained as M 2 ÿ 1 X m Cijkl ÿ C ijkl : Var Cijkl M ÿ 1 m1
18
The standard deviation of the variable are obtained from Eqs. (18) and (19) as follows: ÿ q ÿ r Cijkl Var Cijkl : 20 Generally, we can estimate the rth order moment of the variable Cijkl in the following way: M r ÿ 1X m mr Cijkl Cijkl M m1
· coecient of variation a, a h xk ; b
21
and the central moment of the same variable as ÿ ÿ lr Cijkl m1 Cijkl ÿ mr Cijkl : 22 In case of the Gaussian probability density function N(m, r) any odd-order probabilistic moments are equal to 0 while the ®rst three even moments are ÿ r2 l2 Cijkl ; m
23
ÿ 3r4 l4 Cijkl 2 ; m
24
ÿ 15r6 l6 Cijkl 3 : m
25
On the basis of the central moments estimators the following coecients can be de®ned and computed:
r h x k ; b ; Eh xk ; b
26
· coecient of asymmetry b, b h xk ; b
l3 h xk ; b ; r3 h xk ; b
27
· coecient of concentration c, c h xk ; b
l4 h xk ; b : r4 h x k ; b
28
Considering the Central Limit Theorem (CLT) for the total number of Monte-Carlo iterations performed M tending to 1 we obtain for the Gaussian probability distribution function that lim b n 0;
29
lim c n 3:
30
n!1
19
255
n!1
Eqs. (29) and (30) are very useful (together with the respective PDF estimator) in recognizing the character of probabilistic distribution functions of the output variables. Considering the CLT described above we can recognize for example Gaussian variables. This is very important considering the fact that theoretical considerations in this matter are rather complicated and not always possible. 3.2. Eective moduli estimators for the superconductor The superconducting cable consists of ®bers made of a superconductor placed around a thinwalled pipe (tube) covered with a jacket and insulating material. The RVE of the superconducting composite structure analyzed is shown in Fig. 1. The experimental data describing elastic characteristics of the composite constituents are collected in Table 1. The expected values and standard deviations of Young moduli and Poisson coecients are de®ned in respective columns, according to data presented in [16]. Because of negligible dierences in elastic properties of Incoloy (between `annealed' and `cold worked' state) the `annealed' state of superconductor is further considered. All the results obtained
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M. Kami nski / Computational Materials Science 11 (1998) 252±260
As it was mentioned above, the eective properties described in the previous sections (sup, inf in Table 2) have been compared with the Voigt±Reuss ones (sup-VR, inf-VR in Table 2). Considering the results obtained, it should be noticed that these ®rst bounds are generally more restrictive than the Voigt±Reuss ones. Further, it can be observed that respective deterministic values are with acceptable accuracy equal to respective expected values. Thus, for relatively small standard deviations of the input elastic characteristics, random character of the effective characteristics can be neglected. Finally, it can be noticed that more restrictive bounds are a good approximation of the eective elasticity tensor. Taking as a basis the arithmetic average of the upper and lower bounds, the dierence between these bounds is in the range of 13% for CJJJJ bounds component, 19% for CJKJK bounds component and 8% for CJKKJ bounds component. Figs. 2±10 contain the results of the convergence analysis of the coecient of variation, asymmetry and concentration with respect to increasing total number of the Monte-Carlo random trials. All these coecients are presented for CJJJJ bounds in Figs. 2±4, for CJKJK bounds in Figs. 5±7 and for CJKKJ in Figs. 8±10. On the horizontal axes of these ®gures the total number of
Fig. 1. The RVE of the superconducting cable tested.
in the computational experiments have been collected in Table 2 and Figs. 2±13. The eective elasticity tensor components and their respective expected values (in GPa) have been collected in Table 2. Because of the fact that the expected values appeared to be rather nonsensitive to the total number of random trials in the Monte-Carlo simulations, the results of respective convergence tests have been omitted. The values of the expectations considered have been collected in Table 2 for M 10 000 random trials. Table 1 Elastic characteristics of the superconductor components Material
E[e] GPa
316LN Incoloy 908 Annealed Cold worked Titanium Insulation G10-CR
205 182 184 126 36
r(e) GPa
E[m]
r(m)
8
0.265
0.010
) ) 12 )
0.303 0.299 0.311 0.21
) ) 0.012 )
Table 2 Eective elasticity tensor components and their expected values (in GPa) Eective property type
Analysis type Deterministic
sup-VR sup inf inf-VR
Probabilistic
CJJJJ
CJKKJ
CJKJK
CJJJJ
CJKKJ
CJKJK
189.56 178.44 156.99 137.93
81.83 76.07 62.70 51.86
53.86 51.18 47.14 43.03
189.94 178.57 156.68 137.54
82.30 76.37 62.61 51.71
53.82 51.10 47.03 42.92
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 2. The coecients of variation of CJJJJ bounds.
Fig. 6. The coecients of asymmetry of CJKJK bounds.
Fig. 3. The coecients of variation of CJKJK bounds.
Fig. 7. The coecients of asymmetry of CJKKJ bounds.
Fig. 4. The coecients of variation of CJKKJ bounds.
Fig. 5. The coecients of asymmetry of CJJJJ bounds.
257
Fig. 8. The coecients of concentration of CJJJJ bounds.
Fig. 9. The coecients of concentration of CJKJK bounds.
258
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 10. The coecients of concentration of CJKKJ bounds.
Fig. 11. The probability densities of CJJJJ bounds.
Monte-Carlo random trials M is marked, while on the vertical one there are respective values of the coecients computed. The general observation is that the CJKJK bounds are the most sensitive with respect to the randomness of input elastic characteristics. This conclusion can be stated comparing results in Figs. 2±4. These coecients for CJKJK bounds appeared to be the greatest and next we obtain the coecients for CJJJJ and CJKKJ , respectively. Next, it can be mentioned that the estimators of coecients of variation show good convergence to their limits. The respective values eff of the coecients for dierent Cijkl bounds are obtained for M equal to 2500 random trials. Generally, we can observe that the coecients of variation of the eective elasticity tensor ful®ll the hierarchy of the respective expected values. The greatest coecients are obtained for Reuss bounds, followed by the upper and lower bounds proposed in the paper, and the smallest for the Voigt lower bounds. Observing the results presented in Figs. 5±10 it can be observed that all coecients of asymmetry eff of Cijkl veri®ed tend to 0 on increasing the total number of random trials. Comparing CJJJJ and CJKJK against CJKKJ bounds it can be stated that the ®rst two variables have minimum positive asymmetry, while the last have negative one. It should be noticed that for such probabilistic distributions with non-zero coecients of asymmetry the expected value is not equal to the most probable one. Moreover, taking into account the convergence of coecients of asymmetry it can be noticed that they generally converge more slowly than coecients of variation estimators. M larger than 5000 is required to compute these estimators with satisfactory accuracy. Analogous to the coe-
cients of variation, the hierarchy of the expected eff values of Cijkl , which has been discussed above, is ful®lled. Figs. 8±10 present the coecients of concentration for dierent components of the eective elasticity tensor. The estimator convergence analysis proves that M equal to almost 10 000 is needed to compute these coecients properly. The convergence of these estimators is more complex than the previous ones, but generally their values are greater than 3, which is characteristic for the Gaussian eff variables. Thus it can be stated that the Cijkl probabilistic distributions obtained are more concentrated around their expected values than the Gaussian variables, but this dierence is not greater than a maximum of 15% for the CJKJK bounds. Figs. 11±13 illustrate the probability density functions of the upper and lower bounds for CJJJJ , CJKJK and CJKKJ components of the eective elasticity tensor. On the horizontal axes of these ®gures the values computed of these components are marked, while on the vertical ones the respective density of probability is given. The probability density functions of the eff Cijkl computed together with the respective coecients of asymmetry and concentration b, c show that these PDFs have distributions quite similar
Fig. 12. The probability densities of CJKJK bounds.
M. Kami nski / Computational Materials Science 11 (1998) 252±260
Fig. 13. The probability densities of CJKKJ bounds.
to the bell-shaped Gaussian distribution curve. Thus, in further analyses proposed in the conclusions, we will assume that for the input random variables, in the form of elastic characteristics Young moduli and Poisson coecient, being Gaussian uncorrelated random variables, the upper and lower bounds computed also have Gaussian distribution. 4. Concluding remarks (1) The results of numerical tests performed lead us to the conclusion that the probabilistic upper and lower bounds of the eective elasticity tensor may be very ecient in characterization of superconducting composites with randomly de®ned elastic characteristics because of negligible relative dierences between the upper and lower bounds. Considering the computational time cost they appear to be much more useful in engineering practice than other FEM-based direct methods. (2) The numerical experiments carried out prove that the coecients of variation of the bounds computed are in the range of the input random variables of the problem. Considering further analyses of superconducting coils homogenized, this fact con®rms the need of the application of the SFEM in such computations, considering the time savings. (3) The probabilistic sensitivity of the eective elastic characteristics with respect to probabilistic material parameters should be veri®ed computationally in future analysis as the eect of regression analysis. Such an analysis enables to ®nd out these parameters of composite constituents elastic characteristics which are crucial for superconductor behaviour.
259
(4) The procedure of eective elastic properties approximation seems to be the only method which can be succesfully applied to the homogenization of stochastic interface defects. Such an approach will make the elastic properties of interphases much more sensitive to the presence of structural defects than it was in case of the Probabilistic Averaging Method. Considering this the bounds presented should be implemented into numerical analysis of stochastic structural defects.
Acknowledgements The author wishes to thank Prof. B.A. Schre¯er from Instituto di Scienza e Tecnica delle Construzioni de Padova in Italy and M. Le®k, his colleague, for all the information connected with the superconducting coil cable which has been manufactured in Max-Planck-Institut f ur Plasmaphysik in M unchen.
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