NUMERICAL ANALYSIS OF STOCHASTIC EFFECTIVE PROPERTIES FOR - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NUMERICAL ANALYSIS OF STOCHASTIC EFFECTIVE PROPERTIES FOR POLYMER-BASED COMPOSITES M. Ka miński 1,2 *, B. Lauke 2 1 Department of Structural Mechanics, Technical University of Łódź , Łódź , Poland, 2 Institute of Polymer Research, Dresden, Germany * Corresponding author (Marcin.Kaminski@p.lodz.pl) Keywords : effective properties, random processes, polymers, stochastic simulation 1 Introduction on the computer analysis using the Finite Element Method. Let us note also that elastomers are some The main aim of this work is a computer simulation specific composite materials, where usually more than two components are analyzed – some interface of the effective properties for the polymer-based composites reinforced with the carbon black or silica layers are inserted also between them (as Sticky nano-particles. Contrary to the general formulas Hard (SH) and Glassy Hard (GH) layers). We available in the homogenization theory addressed to analyze here the homogenization rules under the stress- softening behavior, where the cluster size ξ is the micro-macro transitions in the composites [1],      E now experimental results for the specific a deformation-dependent quantity with E compositions of the matrix and reinforcing particles being a scalar deformation related explicitly to the   are necessary to build up the new additional    E first strain tensor invariant. The function is mathematical models. Further, we study the usually determined empirically and it results in the influence of stochastic fluctuations of the reinforcing following formulas describing the coefficient f particles volumetric ratio on the overall effective varying together with the strain level changes [3]: properties using the stochastic perturbation  the exponential cluster breakdown technique implemented into the computer algebra            f E X X X exp E (2) system MAPLE. This study may have a paramount  0  importance in industrial applications because of the and  very realistic nonlinear constitutive models and final the power-law cluster breakdown      y  probabilistic moments functions of the homogenized     (3) f E X X X 1 E .  0  materials, especially in the reliability analysis. The following notation is employed here 2 Theoretical homogenization model  2  d d   w f     3 d   0 X 1 C f , As it is known from the homogenization method    b (4,5) history, one of the dimensionless techniques leading 2 to the description of the effective parameters is the     3 d X 1 C f , following relation describing the shear modulus [3]: 0 G eff      (1) f G where C and d w is the fractal dimension , 0 representing the displacement of the particle from its where G stands for the virgin, unreinforced  stands for the initial 0 original position. Because 0 material and f means the coefficient of this value of the parameter ξ , one can rewrite eqn (5) as parameter increase, related to the reinforcement 2 portion into the final mixture. A particular (6)      d d  3 d X 1 C w f f . characterization of this coefficient strongly depends 0 on the type of the reinforcement (long or short 3 Probabilistic background fibers, reinforcing particles, arrangement of this The generalized given order stochastic perturbation reinforcement etc.) [2]. It is not necessary to technique based on the Taylor series expansion with underline that the effective nonlinear behavior of random coefficients is employed. To provide this many, both traditional and nano composites, needs formulation let us denote the random variable of the much more sophisticated techniques based usually

problem by b (ω) and its probability density as   g b time being. As far as some periodic measurements . are available, one can approximate in some way The central m th probabilistic moment may be those stochastic processes moments, however a expressed as posteriori analysis is not convenient considering the          m     (7) b b E b g b db . reliability of designed structures and materials. This m   stochasticity does not need result from the cyclic The coefficient of variation, asymmetry and fatigue loading but may reflect some unpredictable flatness are introduced in the form structural accidents, aggressive environmental        b   influences etc. This problem may be also considered b b       (8) 2     3   4 b , b , b   .     in the context of the homogenization method, where 3 4 E b   b b the additional formula for effective parameters may According to the main philosophy of this method, include some stochastic processes. Considering all functions in the basic deterministic problem above one may suppose, for instance, the scalar (heat conductivity, heat capacity, temperature and strain variable E as such a process, i.e.       t its gradient as well as the material density) are    0    (13) E , t E E expressed similarly to the following finite where superscript 0 denotes here the initial random expansion of a random function G ( b ) distribution of the given parameter and dotted  n G      0    n  n (9) G b G b ... 1 b quantities stand for the random variations of those , n !  n b parameters (measured in years). From the stochastic where ε is a given small perturbation parameter and point of view it is somewhat similar to the Langevin the n th order variation is given as follows: equation approach, where Gaussian fluctuating     n  n  n   n   n  0 white noise was applied. It is further assumed that b b b b . (10) all aforementioned random variables are Gaussian Using this expansion one may derive the expected and their first two moments are given; the goal values exactly from the definition as would be to find the basic moments of the process          0 0 E G b G b f  , t to be included in some stochastic counterpart (11)    2 m 1 G b n of eqn (1). The plus sign in eqn (13) suggests that      2 m  b   the strain measure with some uncertainty should 2 m  2 m 2 m ! b  m 1 increase with time according to some initially for any natural m ;  2m is the central probabilistic unpredictable deformations; introduction of higher moment of 2 m th order and analogously one derives order polynomium is also possible here and does not the higher probabilistic moments. Usually, according lead to significant computational difficulty. A to some previous convergence studies we may limit determination of the first two moments of the aging this expansion-type approximation to the 10 th order. process is analytical, where As one may suppose, the higher order moments we (14)     t need to compute, the higher order perturbations need           0     E E , t E E E E to be included into all formulas, so that the and complexity of the computational model grows non-     2            0    (15) Var E , t Var E , t Var E , t t proportionally together with the precision and the . range of the output information needed. This method  ( eff ) Then four input parameters are needed to provide G ( ) may be applied to determe - through the the analysis for stochastic ageing of any of the differentiation of both components in the R.H.S. of models presented above; the additional eqn (1) until the given n th order computational analysis is performed with respect to       n k n ( eff ) k n n G G  f   (12)  0 the exponential and power-law cluster breakdown   models and we adopt the following data:       n k n k   0  b k b b E E 3  , k 0         2  E   1  Engineering practice in many cases leads to the E 0 . 03 year Var E 0 . 01 E E , , 0 0     conclusion that the initial values of mechanical   2  Var ( E ) 0 . 01 E E . Using this data we provide parameters decrease stochastically together with the