Johan Byström

An evaluation of different models for prediction of elastic properties of woven composites

Abstract A critical analysis of two simple and convenient analytical models for calculation of elastic properties of woven fabric composites is performed. Predictions of these models are compared with results obtained using the method of reiterated homogenization and with experimental data for plain weave glass fiber and carbon fiber polyester composites. Three different scales are identified in the analysis. The first scale predictions, which are the tow properties (obtained by applying Hashins concentric cylinder model, the Halpin–Tsai expressions or mathematical homogenization technique), are the most critical because they form the input information for woven composite modeling. It appears that the uncertainty in this information causes larger differences in predictions than the deviations between models of different degree of accuracy. This fact sets limits on the required accuracy of the models. Model comparisons reveal that the woven composite model based on isostrain assumption in the composite plane and isostress assumption for out-of-plane components is in very good agreement with both experimental data and the reiterated homogenization method, whereas the modified mosaic parallel model fails to describe composites with large interlaced regions.

Influence of the inclusions distribution on the effective properties of heterogeneous media

In this paper we investigate the effective conductivity of composite materials by means of the homogenization method. We concentrate on composites with circular or elliptic cylindrical inclusions. In particular, we are interested in the effect of the distribution of the cylinders in the continuous material on the effective properties. We compare rectangular and hexagonal distributions with random distributions for different volume fractions of the inclusions. We also study the effect of the number of inclusions in each periodic cell for the random structure as well as shape influence of the elliptical inclusions.

Some computational aspects of iterated structures

We consider some computational aspects of effective properties for some multi-scale structures. In particular, we discuss iterated square honeycombs and another type of square honeycombs containing up to 4000 small discs randomly distributed inside each square. We present some numerical methods for estimating the effective conductivity with good control of the accuracy.

On the effect of stacked fabric layers on the stiffness of a woven composite

Abstract Most micromechanical models for stiffness prediction of woven composites assume independence of the Q-matrix on the number of fabric layers in the composite. For example, the moduli of single and 10 layer composites are assumed to be equal in the case when all layers have the same in-plane orientation. Although this statement is likely to be true for isotropic materials or even for unidirectional laminated composites, it may not be valid in some cases of woven composites. This paper contains experimental and theoretical investigations of plain weave carbon fiber/polyester composites. Specimens with one single and eight layers of fabrics are tested and observable differences of mechanical properties are obtained. The theoretical part of this article consists of derivation and application of several micromechanical models on these particular composites. The use of those simplified models finally allows us to find the main mechanisms which cause the observed effects.

Correctors for Some Nonlinear Monotone Operators

Abstract In this paper we study homogenization of quasi-linear partial differential equations of the form −div (a (x, x/ε h, Du h)) = f h on Ω with Dirichlet boundary conditions. Here the sequence (ε h) tends to 0 as h → ∞ and the map a (x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions. We prove that u h → u weakly in (Ω) as h → ∞, where u is the solution of a homogenized problem of the form −div (b (x, Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (P h) such that Du h − P h (Du) → 0 in L p (Ω, R n).In this paper we study homogenization of quasi-linear partial differential equations of the form −div (a (x, x/εh, Duh)) = fh on Ω with Dirichlet boundary conditions. Here the sequence (εh) tends to 0 as h → ∞ and the map a (x, y, ξ) is periodic in y, monotone in ξ and satisfies suitable continuity conditions. We prove that uh → u weakly in W 1,p 0 (Ω) as h → ∞, where u is the solution of a homogenized problem of the form −div (b (x,Du)) = f on Ω. We also derive an explicit expression for the homogenized operator b and prove some corrector results, i.e. we find (Ph) such that Duh − Ph (Du) → 0 in L (Ω,R).

REITERATED HOMOGENIZATION OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS

The authors study homogenization of some nonlinear partial differential equations of the form -div (a(hx, h2x, Duh)) = f, where a is periodic in the first two arguments and monotone in the third. In particular the case where a satisfies degenerated structure conditions is studied. It is proved that uh converges weakly in to the unique solution of a limit problem as h → ∞. Moreover, explicit expressions for the limit problem are obtained.

Optimal Control and the Fibonacci Sequence

We bridge mathematical number theory with optimal control and show that a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady-state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock–Mirman economic growth model.

Bounds and Numerical Results for Homogenized Degenerated p-Poisson Equations

In this paper we derive upper and lower bounds on the homogenized energy density functional corresponding to degenerated p-Poisson equations. Moreover, we give some non-trivial examples where the bounds are tight and thus can be used as good approximations of the homogenized properties. We even present some cases where the bounds coincide and also compare them with some numerical results.