A Survey of Neural Computing of Effective Properties of Random Composites

Transcription

1 A Survey of Neural Computing of Effective Properties of Random Composites ABRAHAM I. BELTZER, VICTOR A. GOTLIB Holon Academic Institute of Technology POB 305, Holon 580 ISRAEL TADANOBU SATO Disaster Prevention Research Institute yoto University Gokasho, Uji, yoto 6 JAPAN Abstract: - The effective behavior of disordered heterogeneous materials, in general, is not amenable to exact analysis because the phase geometry may not be completely specified. Besides the bounds for the effective moduli, most reported results are essentially approximate, may contradict each other or even violate the bounds. With increasing number of phases, and therefore increasing uncertainty inherent in the very statement of the problem, the investigation of effective behavior becomes less amenable to analytical treatment, in particular, by methods of boundary-value problems of mathematical physics. The present paper discusses, in light of recent publications, a methodology for investigating the effective behavior of disordered composites with the help of neural networks as well as particular results obtained for effectively isotropic and macroscopically homogeneous two-phase materials. It appears that, after incorporating the bounds and proper training, simple neural networks may describe a wide variety of the effective behavior for two-phase composites, though in general more complicated networks appear necessary. ey-words: - neural network, random composite, effective behavior, elastic properties. Introduction The present paper deals with recent investigations into the effective behavior of disordered composites with the help of a neural network, which has become one of the main paradigms of engineering research. Artificial neural networks have proven useful in treating problems characterized by incomplete a priori given information and are difficult for exact analysis, in particular, by the methods borrowed from boundary-value problems of mathematical physics. This is typical of disordered materials and therefore neural networks, which are capable of discovering a hidden regularity, may constitute a useful tool for specifying their effective behavior. For details concerning the effective static behavior of composites the reader is referred, for example, to the works [] [6]. The effective behavior of disordered heterogeneous materials is not amenable to exact analysis because the phase geometry may not be completely specified. Besides the bounds for the effective moduli, most reported results are essentially approximate, may contradict each other or even violate the bounds. Resort to a particular shape of inclusion necessary in the frameworks of such methods as the composite sphere assemblage or self-consistent schemes seems to contradict to a basic assumption of arbitrary phase geometry, which should come into play through the volume concentration only. With increasing number of phases and therefore increasing uncertainty inherent in the very statement of the problem, the investigation of effective behavior becomes less amenable to analytical treatment, in particular, by methods of boundary-value problems of

2 mathematical physics. That is why applications of neural networks to this problem is useful. The publications on neural computing of the efective behavior of composite materials include a communication [7] and papers [8] and [9]. These deal with: (i the general methodology of application of neural networks to disordered heterogeneous materials; (ii particular results derived for the effective elastic moduli and for the effective scalar property, such as conductivity, of two-phase composites. The elastic response of isotropic composites is governed by two effective moduli. One therefore may consider a problem of specifying the bulk and Young s moduli, and E, respectively, of a twophase elastic isotropic material of arbitrary geometry. Another relevant issue is the problem of effective scalar properties, such as: thermal conductivity, electrical conductivity, dielectric constant, magnetic permeability, and diffusivity in the realm of disordered composites. Below we denote all these quantities by a scalar parameter k. In this case, the phenomena are described by similar mathematical expressions, the effective property being the coefficient in the relevant equation, which relates solenoidal and irrotational vector fields. For details of the similarity among the indicated phenomena the reader is referred to the reviews [] and [3]. The lower - and upper Hashin-Shtrikman bounds for the bulk modulus for such composites are [3] c - =, ( 3c 3 4G c =, ( 3c 3 4G where and are the bulk moduli of the phases, G and G their shear moduli, c and c their volume concentrations and <, G < G, c c =. Similarly, for the shear modulus, G, the bounds are c G- = G, (3 6c ( G G G 5G (3 4G G c = G, (4 6c( G ( G G 5G (3 4G These Hashin-Shtrikman bounds are the best possible as far as arbitrary geometry is concerned, and their disparity is sensitive to the shear moduli ratio, G /G, vanishing for G /G =. The latter case, for which the exact solution does exist, is known as Hill s solid [] and is naturally excluded from the present investigation. The main body of experiments available deals with the effective moduli and E rather then and G. The bounds for E follow from the well known relation E = E(, G and expressions ( - (4: E ± 9± G± = 3 G ± ±. (5 As far as effective scalar property k is concerned, the lower, k -, and upper, k, Hashin- Shtrikman bounds, which are also,the best possible as far as arbitrary geometry is concerned, are (see, for example, Ref. [3]: k = c k, (6 c k k 3k k = c k, (7 c k k 3k where k and k are the parameters of the phases, c and c their volume concentrations and k < k, c c. = Neural Networks First, consider the case of elastic behavior. It is assumed that (i the composite microstructure is not completely known and is specified in terms of,, G, G, c, (or c = - c only; and (ii the composite may be treated as effectively isotropic and macroscopically homogeneous. In the case of a scalar property, which is denoted by the generic notation k, the difference is that the microstructure description contains three parameters only, k, k and c.

3 3 Taking into account the bias and avoiding the overfitting, one may try a single-neuron network (perceptron with six inputs in case of the elastic behavior and with four inputs in case of the effective scalar behavior, such as electrical conductivity, for example. As to the transfer function, we note that the effective moduli depend on its arguments in a nonlinear manner and are bounded from below and above. A sigmoid transfer function - F ( n = tanh( n (8 may incorporate this fundamental behavior. Here F is the network output and n = w (9 0 w wg w3 w4g w5c is the total (weighted network input in case of the elastic response. Given the value of F, the bulk modulus follows as: F F = and the inverse transformation F ( = (0 ( relates -. and the network output - F. Similar expressions hold for Young s modulus E. Evidently, the weights and biases can be different for the cases of and E. For the effective scalar property, k, the total (weighted net input is 3 n = w u = w w k w θ w c, ( i= 0 i i 0 3 where w 0 is the bias, θ is the dimensionless input parameter, θ = ( k k k, 0 < θ. k is the normalized quantity k, (k > k scaled to the interval 0. < k according to the expressions k = k 0 p, = [ log0 k ] p, (3 where the square brackets denote the floorfunction. Another parameter k is scaled using the same value of p, i.e. k = p k 0. This methodology enables one to inherently incorporate the bounds, which are shown below to also facilitate a proper specification of the training set. Besides the bounds, the relevant information consists of experimental data, which are approximate because of the experimental scatter. Indeed, some experiments fall outside the bounds and must be excluded from considerations. Also, experiments with dilute mixtures or with the phases having slight mismatch of the shear moduli are of little interest, as the Hashin-Shtrikman bounds are sufficiently stringent in this case. For the case of the elastic response use has been made of experiments with sintered alumina, ceramics, tungsten carbide-cobalt alloy and ClCu aggregate for a wide interval of the volume concentrations. For the case of a scalar property use has been made of the electrical conductivity data for various compounds, such as Cu Sb -Sb, Pb-Mg Pb, Sn-Bi, Bi Pb-Bi, Cd-Pb, and the thermal conductivity data for a sandstone and for a firebrick. The data have been divided to a training set and a test set. It is essential that training set has been specified so as to include samples displaying the effect at hand to its maximal degree. It appears that data dealing with concentrated mixtures c c 0.5 and possibly large mismatch of the shear moduli are most informative, as the uncertainty left over by the bounds is maximal under these conditions. Because, as the present methodology suggests, the bounds are fed in the network separately for each particular composite prior to the weight optimization process, it seems possible to adopt small training sets, which are mainly placed in the region of maximal disparity between the bounds. At the same time, it is essential that the training samples would represent a possibly large variety of microstructure in agreement with the assumption (i made in the above. The training sets first consisted of three samples and then of four samples These samples have been so chosen as to comply with the above comments concerning specification of the training sets. The weights arrived at by the gradient descent method, which has been chosen to perform the training, are given for the above three cases in Table.

4 4 Tab.. The values of weigths E k w w (GPa - (GPa - w (GPa - (GPa - w (GPa - (GPa - w (GPa - (GPa - w moduli: = 30 GPa, G = GPa, = GPa, G = Gpa. The values, which correspond to the differential scheme and to the self-consistent method, have been computed according to expressions discussed in detail in [] and [0]. Fig. presents the results for a scalar property k.for the values k = 5 and k = 00. It is observed from Figs. and that the assumption of a particular inclusion shape, which is inhereht in the above analytical investigations and absent in the neural computing, may strongly affect the evaluation of effective modulus. 3 Analysis of Results The neural computation shows good agreement with the experimental data. This is particularly true for the network trained with four samples. The generalization capability of the networks seems fairly impressive, particularly, in view of their extreme simplicity and the small training sets. Going over to comparison of the neural computation with analytical techniques such as the self-consistent scheme and the differential method, we note that even though the results associated with these techniques employ the phase concentration c as the only phase geometry parameter, their derivation essentially assumes a particular phase geometry (spherical inclusions embedded in a homogeneous matrix. On the other hand, the above neural computation is solely based on the Hashin- Shtrikman bounds and experiments which incorporate a variety of microstructures in agreement with the assumption made. Therefore any comparison between these approaches should not be taken literally. This comparison would be more meaningful if the training and test sets consist only of the samples of spherical inclusions in a purely homogeneous matrix. Presently, to the best knowledge of these authors, the available experimental data of this type are too limited to allow for such a procedure. Nevertheless, it would be of interest to appreciate the difference between these analytical models and the neural computation in case of composite with a particularly large gap between the Hashin-Shtrikman bounds. For example, Fig. shows bulk modulus and the effective modulus E as functions of the volume concentration, c, for a composite with the following values of elastic Fig.. Effective bulk modulus, (GPa, and effective Young s modulus, E (GPa vs. volume concentration c : B Hashin-Shtrikman bounds, N neural network, D differential scheme, S self-consistent method.

5 5 If, contrary to the assumption (i of Section, the inclusion shape is known, the training set should be reformulated so as to include proper samples, say, disk-like inclusions. The Hashin-Shtrikman bounds still hold, though, in general, they may be replaced by more stringent bounds because of the specific geometry involved. This would allow for incorporating the effect of inclusion shape. Fig.. Effective scalar property, k, vs. volume concentration c : B Hashin-Shtrikman bounds, N neural network, S self-consistent method. Further, since the Hashin-Shtrikman bounds do not distinguish between phases in the form of matrix or particles, in order to investigate yet another effect of which component is the matrix phase and which is the inclusion phase, one should design two networks trained on two different sets. One of these sets should contain only the samples with much stiffer matrix phase and the second with much stiffer particles phase. Unfortunately, the experimental data presently available do not seem sufficient for such investigations. The simplicity of the results obtained herein and a possibility of their extension to more complicated cases in the frameworks of the modern paradigm of neural networks seems attractive. 4 Conclusions The uncertainty typical of disordered heterogeneous materials increases in case of multiphase and incompletely specified microstructure. This further complicates their analysis by the conventional methods borrowed from boundary-value problems of mathematical physics and neural networks may serve as a useful alternative. The bounds play an essential role in the present methodology of applications of neural networks to the effective behavior of disordered composites. Since the experimental data concerning dilute mixtures and/or slight mismatch of the parameters of the microstucture bear little information compared to the bounds, they may be put aside while specifying a training set. Then the bounds are fed in the neural network separately for each particular composite as an integral part of the training. The generalization capability of the designed networks appears substantial, particularly in view of their extreme simplicity and the small training sets employed. Though the simple perceptrons show a wide variety of the effective behavior for two-phase composites, in general a more complicated network appears necessary. It seems that neural networks, which incorporate the relevant bounds and are trained on small but properly selected experimental sets, may provide a useful insight into the effective behavior of heterogeneous materials. References [] Christensen, R.M., A Critical Evaluation for a Class of Micromechanics Models, Journal of Mechanics and Physics of Solids, Vol. 38, No. 3, 990, pp [] Hale, D.., The Physical Properties of Composite Materials, Journal of Material Science, Vol., 976, pp [3] Hashin, Z., Analysis of Composite Materials - A Survey, Journal of Applied Mechanics, Vol. 50, 983, pp [4] Willis, J.R., Variational and Related Methods for the Overall Properties of Composites. Advances in Applied Mechanics, Vol., 98, pp [5] Beltzer, A.I., The Effective Dynamic Response of Random Composites and Polycrystals - A Survey of the Causal Approach, Wave Motion, Vol., 989, pp. -9. [6] Nemat-Nasser, S., Hori, M., Micromechanics: overall properties of heterogeneous materials, North-Holland, Amsterdam, 993.

6 [7] Beltzer, A.I., Sato, T., A Neural Network Approach to Effective Response of Heterogeneous Materials, in: Proceedings of the International Conference Mesomechanics 98, Tel Aviv, 998, p. 38. [8] Gotlib V.A., Sato T., Beltzer A.I., Neural Computations of Effective Response of Random Composites, International Journal of Solids and Structures, Vol. 37, 000, pp [9] Gotlib V.A., Sato T., Beltzer A.I., Neural Computing of Effective Properties of Random Composite Materials, Computers and Structures, Vol. 79, No., 000, pp. -6. [0] Zimmerman, R.W., Elastic Moduli of a Solid Containing Spherical Inclusions, Mechanics of Materials, Vol., 99, pp