A fuzzy stochastic multiscale model for fiber composites

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110 I Babus ka M Motamed Comput Methods Appl Mech Engrg 302 2016 109 130. a Plies with fibers in different directions b Plies are stacked to form a laminate. Fig 1 A cross plied fiber composite laminate made by stacking five plies Different plies contain many unidirectional fibers in different directions. aligned in a matrix, unidirectional fibers are aligned in a thin ply To achieve high stiffness a few plies are stacked together each having. fibers oriented in a certain direction Such a stack is termed a cross plied laminate see Fig 1. Fiber composites are fabricated based on their response to external forces Two principal responses are deformation. and fracture In a non testing environment we need accurate and viable mathematical and computational models. for computing stress distributions and predicting damage and failure This requires an understanding of both micro. structural effects and uncertainty variability in the manufacturing process High fidelity models must therefore account. for two components multiple scales and uncertainty On the one hand the problem involves multiple length scales. ranging from the diameter of fibers 10 m to the laminate thickness and length 1 m On the other hand the. model is subject to uncertainty due to the random character of the size and distribution of fibers and the intrinsic. variability in material properties and fracture parameters High levels of confidence in the predictions require a. process called multiscale uncertainty quantification UQ taking into account the interaction between the two involved. components i e multiple scales and uncertainty, A chronological development of mathematical and computational tools for solving multiscale problems such as. fiber composites may in general be classified into four groups. 1 Mathematical theory of homogenization The term homogenization was first coined by Babus ka 1 2 Ho. mogenization is an analytical approach to replace the multiscale problem with heterogeneous coefficients by an. equivalent problem with homogeneous coefficients known as the homogenized problem Homogenization the. ory is well studied in the case of periodic or locally periodic microscale coefficients 3 For the more general. case of non periodic coefficients homogenization theory is carried out by studying the G convergence 4 and. H convergence 5 of solution operates, 2 Multiscale numerical methods Numerical methods that approximate the highly oscillatory solution of a multiscale. problem by solving an effective problem and including local microscale oscillations instead of directly solving the. original problem are called multiscale numerical methods Numerical homogenization 6 9 is the earliest multi. scale method where the finite element method was employed to solve the homogenized problem and some mi. croscale features were added as correctors to the homogenized solution Since then many multiscale methods have. been proposed including the generalized finite element method 10 11 the variational multiscale method 12. the multiscale finite element method 13 projection based numerical homogenization 14 15 the heterogeneous. multiscale method 16 17 and the equation free method 18 We also refer to 19 21 for detailed discussions on. a wide range of multiscale methods All these techniques usually seek the approximate solution everywhere inside. the computational domain Another class of multiscale methods is based on a global local approach 22 24. where the solution or other quantities of interest QoIs are required inside a relatively small subdomain In. these methods a global homogenized solution is employed to recover the microscale solution inside the local. 3 Multiscale methods in engineering There is a vast literature on multiscale methods in engineering especially in the. field of materials science These methods have been developed based on the same ideas and principles employed in. the applied mathematics community They have been proposed to account for microstructural heterogeneity for in. stance in complex materials such as composites and porous structures They include the unit cell or representative. I Babus ka M Motamed Comput Methods Appl Mech Engrg 302 2016 109 130 111. volume element RVE method multi level approaches using the finite element method and the Voronoi cell fi. nite element method and continuous discontinuous homogenization see for instance 25 34 These multiscale. methods treat both linear and non linear mechanical and thermomechanical responses of complex materials. 4 Probabilistic treatment of multiscale problems The literature on UQ for multiscale problems is rather sparse and. focuses more on stochastic models Stochastic homogenization 35 38 can be considered as a generalization of. classical homogenization Theoretical aspects of stochastic homogenization is well studied in the case of stationary. and ergodic random fields However numerical approaches based on stochastic homogenization are cumbersome. and not well studied particularly because the homogenized problem is set on the whole space not on a finite cell. Analogous to the case of deterministic problems a variety of stochastic multiscale methods have been proposed. within the framework of variational multiscale methods multiscale finite element methods the heterogeneous mul. tiscale methods and the global local approach see e g 39 44 Also in engineering community an increasing. number of papers are attempting to address UQ in multiscale simulations see e g 45 49. In the particular case of composite materials a majority of multiscale models are deterministic see references in. item 3 above Such models are not capable of fully describing the mechanical behavior of composites partially due to. the ignorance of uncertainty which is an important component that must be included in the model More recently. there have been efforts to include and describe uncertainty by precise probabilities 50 where the model input. parameters i e material properties such as the modulus of elasticity are described by often stationary Gaussian or. log normal random fields see references in item 4 above Despite recent advances with stochastic multiscale models. and UQ methodologies five decades after the pioneering work of Kachanov 51 52 on continuum damage mechanics. one question still remains open are there viable models for the sound investigation of composite responses As. Rohwer 53 has recently remarked A fully satisfying model for describing damage and failure of fiber composites is. not yet available Consequently for the time being a real test remains the authentic way to secure structural strength. Beside the deficiency of accurate models for damage mechanisms one reason for the inapplicability of current. stochastic multiscale models is the presence of both random statistical or aleatoric and non random systematic. or epistemic uncertainties in the problem and the scarcity of noisy measurements that are used to characterize. uncertainties In order to describe both types of uncertainties we need to develop more sophisticated uncertainty. models beyond precise probabilities, In the present paper we are concerned with mathematical and computational models for computing the deformation. of fiber composites due to external forces Since realistic models must be designed based on and backed by real. experimental data we consider a small piece of a real fiber composite plate taken from 54 consisting of four. plies and containing 13 688 unidirectional fibers with a volume fraction of 63 The measured raw data include the. material constants of fibers and the matrix and a map of the size and distribution of fibers obtained by an optical. microscope The map is considered as a prototype of fiber distributions in fiber composites We process the raw data. and convert them into a form suitable for statistical analysis and show that the current stochastic models are not capable. of correctly characterizing uncertainty in fiber composites Instead we study and show the applicability of imprecise. uncertainty models for fiber composites We propose a new hybrid fuzzy stochastic model by integrating probability. theory 50 and fuzzy set theory 55 through a calibration validation approach We finally present a numerical method. in a fuzzy stochastic framework for propagating uncertainty through the proposed model and predicting output QoIs. The numerical method utilizes the concept of RVEs and homogenization and is based on a global local approach. where a global solution is used to construct a local solution that captures the microscale features of the problem For. simplification and to motivate and establish the main concepts of the proposed model we consider a one dimensional. problem The present work is a preparation for studying fiber composites in two and three dimensions which will be. presented elsewhere, The main contributions of this paper include 1 showing the deficiency of stochastic models to the reliable predic.
tion of fiber composite responses 2 motivating the applicability of imprecise uncertainty models and constructing. a novel hybrid fuzzy stochastic model for the uncertainty characterization of fiber composites and 3 developing a. global local numerical method in a fuzzy stochastic framework for efficiently computing the output QoIs. The rest of the paper is organized as follows In Section 2 we present the real data formulate the problem and. briefly address different models for characterizing uncertainty In Section 3 we perform statistical analysis and moti. vate the deficiency of current stochastic models We outline the basic concepts of fuzzy set theory and its combination. with probability theory in Section 4 In Section 5 we discuss the construction of the new hybrid fuzzy stochastic. 112 I Babus ka M Motamed Comput Methods Appl Mech Engrg 302 2016 109 130. Fig 2 Left A 1 7 0 5 mm2 rectangular orthogonal cross section of a small piece of a fiber composite laminate consisting of four uni directional. plies containing 13 688 fibers with a volume fraction of 63 Right A binary image of a small part of the whole micrograph. Material constants for the composite under,consideration. Composite phases a,Fiber 24 GPa 0 24,Matrix 3 6 GPa 0 3. model In Section 6 we present the numerical multiscale method Finally we summarize our conclusions and outline. future works in Section 7,2 Problem statement, Reliable mathematical and computational models for predicting the response of fiber composites due to external. forces must be designed based on and backed by real experimental data In this section we first present the real data. that is used throughout this work We then formulate a simple one dimensional problem describing the deformation. of fiber composites Finally we briefly address different models for characterizing uncertainty in the problem. 2 1 Real data, The real data that we use are obtained from a small piece of a HTA 6376 carbon fiber reinforced epoxy composite. plate 54 44 with a rectangular cross section of size 1 7 0 5 mm2 and consisting of four plies containing 13 688. unidirectional fibers with a volume fraction of 63 Fiber diameters vary between 4 m and 10 m Fig 2 shows a map. of the size and position of fibers in an orthogonal cross section of the composite obtained by an optical microscope. In the present work this particular map serves as a prototype of fiber distributions in fiber composites. The Young s modulus of elasticity and Poisson s ratio of the fiber composite under consideration are given in. 2 2 Mathematical formulation a one dimensional problem. The deformation of elastic materials is given by the elastic partial differential equations PDEs in three dimensions. In the particular case of plane strain where the length of structures in one direction is very large compared to the size. of structures in the other two directions the problem may approximately be reduced to a two dimensional problem In. the present work however for simplification and to motivate and establish the main concepts of the proposed model. we consider a one dimensional problem Problems in higher dimensions will be presented elsewhere. We consider the elastic equation with homogeneous Dirichlet and non homogeneous Neumann boundary. conditions in one dimension,a x x f x x 0 1 1a,u 0 0 a 1 1 1 1b.
I Babus ka M Motamed Comput Methods Appl Mech Engrg 302 2016 109 130 113. where x is location u x is the displacement a x is the modulus of elasticity of the composite and f x is a force. term given for instance by, Here the unit of length is assumed to be meter m and the unit of force both external force f and the boundary. force and modulus of elasticity a is assumed to be giga Pascal GPa. The main goal of computations is to obtain composite deformations due to external forces We therefore need to. solve problem 1 and compute displacements u x stresses a x u x and or other QoIs for example functionals. of the displacement u x We note that in the one dimensional model problem considered here stresses are smooth. functions and do not oscillate with the small scale of fiber sizes We therefore set our main goal as the prediction of. the solution at a given point say at x0 0 75 We then introduce the QoI. Q u x0 x0 0 75 2, Here the solution to the problem 1 is analytically given by. 0 0 1 0 5 1, To obtain the solution and the QoI in 2 we need to know the parameter a x which describes the mechanical. property of the composite Therefore we first need to characterize the modulus of elasticity a x which is directly. given by the size and position of fibers in the matrix and by the modulus of elasticity of fibers and matrix listed in. A fuzzy stochastic multiscale model for fiber composites A one dimensional study Ivo Babu ska a Mohammad Motamedb a Institute for Computational Engineering and Sciences The University of Texas at Austin USA b Department of Mathematics and Statistics The University of New Mexico Albuquerque USA Received 10 September 2015 received in revised form 16 December 2015 accepted 17

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