J. Cent. South Univ. Technol. (2008) 15(s1): 047-050
DOI: 10.1007/s11771-008-312-4
An elasto-plastic constitutive model of moderate sandy clay based on BC-RBFNN
PENG Xiang-hua(彭相华)1, WANG Zhi-chao(王智超)2, LUO Tao(罗 涛)3, YU Min(余 敏)3, LUO Ying-she(罗迎社)3
(1. Swan College of Central South University of Forestry and Technology, Changsha 410004, China;
2. College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China;
3. Institute of Rheological Mechanics and Material Engineering, Central South University of Forestry and Technology, Changsha 410004, China)
Abstract: Application research of neural networks to geotechnical engineering has become a hotspot nowadays. General model may not reach the predicting precision in practical application due to different characteristics in different fields. In allusion to this, an elasto-plastic constitutive model based on clustering radial basis function neural network(BC-RBFNN) was proposed for moderate sandy clay according to its properties. Firstly, knowledge base was established on triaxial compression testing data; then the model was trained, learned and emulated using knowledge base; finally, predicting results of the BC-RBFNN model were compared and analyzed with those of other intelligent model. The results show that the BC-RBFNN model can alter the training and learning velocity and improve the predicting precision, which provides possibility for engineering practice on demanding high precision.
Key words: elasto-plastic constitutive model; artificial neural network; BC-RBFNN (based on clustering radial basis function neural network); moderate sandy clay
1 Introduction
With the development of economic, engineering items related to stress/strain analysis of geotechnical materials under high stress and complex stress path have increased. Here, conventional constitutive model theory cannot satisfy the increasing engineering requirements, which demands us to investigate new methods and models for researching stress/strain status of geotechnical materials.
Neural computing models have favorable ability of self-organizing, including self-adaptation and self- learning, even self-growth and self-duplication[1]. They bear input/output relationship, which can approach the given nonlinear mapping[2]. Geotechnical material is an important engineering geologic material and presents multiphase, anisotropy, non-homogenization and non-continuity in its composition and construction[3]. Its constitutive relationship is always high nonlinear. The high complex and nonlinear geotechnical materials’ constitutive relationship can be modeled directly using the strong nonlinear mapping ability of ANN (artificial neural network) and combining the measured stress/ strain information, which will own strong pertinence and applicability[4].
Application research of neural computing models to geotechnical materials has evolved in recent decade, which provides a good flat for the research on geotechnical materials’ constitutive models[5]. However, general neural computing models may not reach perfect predicting precision in practical application due to different characteristics in different fields. Thus, application research of neural computing models to special field has become keystone and hotspots for researchers.
Based on a set of routine triaxial compression test data of saturation sandy sample in Ref.[6], an elasto-plastic constitutive model based on BC-RBFNN (clustering radial basis function neuron network) was established for moderate sandy clay considering its mechanical properties and taking use of the ability of approaching arbitrary nonlinear function of RBFNN (radial basis function neural network).
2 Construction principle of BC-RBFNN
BC-RBFNN, a feedforward neural network established on clustering analysis of samples, can improve the properties of network through programming data sets of samples in reason[7]. BC-RBFNN has the same system structure as RBFNN, being tri-layer structure of “Sensory-Associative-Response”; but they have different inner structures in each layer. The system structure of BC-RBFNN is shown in Fig.1. In Fig.1, C represents the center vector; W represents the weight matrix; CA represents clustering analysis neuron.
Fig.1 System structure of BC-RBFNN
Compared with typical RBFNN, there is a CA neuron used for clustering analysis in the sensory layer of BC-RBFNN; the sensory layer of BC-RBFNN is composed of many subnetworks and each subnetwork is formed by many associative neurons.
The function mechanism of BC-RBFNN and RBFNN resembles with that of BP networks. Working course of neurons in the same layer is parallel while that in different layer is serial, being the sequence of S→A→R[8]. Owing to the variation of the system structure of BC-RBFNN, working flow in each layer has a certain change. The function mechanism can be reduced to the following steps.
(i) Sensory layer (S)
S(xi) = xi (i=1, 2, …, N);
(1)
(ii) Associative layer (A)
It works on the basis of the corresponding networks selected by 
; the output values 
of A are to be calculated:
(j=1, 2, …,
; k=) (2)
(iii) Response layer (R)
The output values 
of R are to be calculated:
(j=1, 2,…, m; k=
) (3)
3 Elasto-plastic constitutive model based on BC-RBFNN
3.1 Numerical model
Research on elasto-plastic theory for geotechnical materials has been developed greatly. ZHENG et al[9] proposed generalized plasticity theory. Thereinto, the double yield surface theory can roundly reflect the strain hardening process of clay using the shear yield surface and the volumetric yield surface relatively, where the isolines of the octahedron shear strain
were selected as the shear yield surface while those of the plastic volumetric strain
as the volumetric yield surface. And the stress-strain relationship is denoted by the following two functions[9]:
(4)
(5)
The stress/stain relationship can not only reflect the intercross influence coupling relationship of the volumetric strain
with the generalize shear strain
and the mean stress p with the generalize shear stress q, but also reflect the influence of stress path on constitutive relationship. Because parameters are obtained by testing simulation according to practical stress path of clay, Eqn.(4) and Eqn.(5) could be simplified as:
(6)
(7)
3.2 Elasto-plastic constitutive model based on BC- RBFNN
It can be known from Fig.1 that BC-RBFNN is a data mapping from problem space to solution space. The design of models depends on accurate data. Therefore, a set of routine triaxial compression testing data of saturation sandy samples in Ref. [6] was analyzed and processed and the correlation data were extracted from them. Then, the testing data were normalized and clustered[10]; moreover, they were to be divided into training samples (
= 50, 100, 200, 250 kPa) and testing samples (=150 kPa). Finally, the knowledge base was formed.
Combined the design principle of BC-RBFNN, an elasto-plastic constitutive model for moderate sandy clay based on BC-RBFNN has been set up depended on the two functions of clay’s stress—strain relationship (Eqns.(6) and (7)). The corresponding system structure is shown in Fig.2 and its work flow chart is shown in Fig.3.
Fig.2 System structure of elasto-plastic constitutive model
4 Emulation analysis of elasto-plastic constitutive model based on BC-RBFNN
The elasto-plastic constitutive model based on BC-RBFNN was trained/learned and emulated/ tested. Firstly, the training samples composed of 97 groups of data were trained/learned by the model; then the testing samples composed of 25 groups of data were emulated/tested by the model. The emulating results are listed in Table 1.
Fig.3 Work flow chart of elasto-plastic constitutive model
Table 1 Emulating results of BC-RBFNN constitutive model
Fig.4 Fitting results of shear strain
Fig.5 Fitting results of volumetric strain
It can be seen from the fitting results that the fitting curves of BC-RBFNN model are smoother and much more approach the testing curves compared to other two models, which account for that BC-RBFNN model possesses higher predicting precision and strong convergence ability.
It can be seen from Table 1 that the errors of emulating results are small and the predicting errors are basically controlled in the range of 0-0.014, which shows that the BC-RBFNN model has strong function approximating ability.
To validate the superiority of the model, the fitting results obtained by BC-RBFNN model are compared and analyzed with those by RBFNN model and BPNN (back propagation) model, which are shown in Figs.4-5.
5 Conclusions
1) The elasto-plastic constitutive model based on BC-RBFNN for moderate clay sandy can make the best of the sample information, and simplify the determination of parameters in constitutive model.
2) The elasto-plastic constitutive model owns high predicting precision and can factually reflect the characteristics of moderate sandy clay’s constitutive model, which provides possibility for engineering practice demanding high precision.
References
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(Edited by YANG Hua)
Foundation item: Project(07031B) supported by the Scientific Research Fund of Central South University of Forestry and Technology; Project(06C843) supported by the Scientific Research Fund of Hunan Provincial Education Department
Received date: 2008-06-25; Accepted date: 2008-08-05
Corresponding author: YU Min, Lecture; Tel: +86-731-5623376; E-mail: yumin1999@163.com
