layout representation, fine mesh and consequently costly computation should be done; moreover further post processing technique such as filtering or smoothing becomes necessary when this method is implemented. Thirdly, it must be also considered that, before economical and technological evaluations are performed, to fabricate element- wise variation of fiber content in a discretized continuum structure is still a daunting step with present available technologies. Fig.4.1schematically shows such limitations involved in FE mapping representation.
Figure 4.1: Schematic illustration of mesh dependency in element-based representation of fiber volume fraction: a) Coarse mesh b) Fine mesh
Instead of using element-based fiber volume fraction description, as has been al- ready done in other researches, the idea of utilizing quadratic NURBS basis functions in order to smoothly and continuously approximate given set of nodal points are devel- oped. Useful characteristics of NURBS basis functions such as compact support and higher order elements not only provides mesh independent distribution results but also makes it possible to use coarse meshes to decrease computational time, while maintain- ing the accuracy of the results. The presented novel computational approach combines NURBS based and gradient based optimization methodologies to get an efficient op- timization algorithm, which has been verified to be enough accurate, computationally fast and convenient for real industrial applications.
4.2 FRC homogenization methodology
Basically the aim of homogenization techniques is to determine equivalent material characteristics in a Representative Volume Element (RVE) of composite material. There are some classical approaches in order to model the material properties of composites; among which the rule of mixture, Hashin-Shtrikman type bounds [Hashin, 1962and
Hashin & Shtrikman,1963], Variational Bounding Techniques [Paul,1960], Self Con- sistency Method [Hill,1965] and Mori-Tanaka Method [Mori & Tanaka,1973] can be mentioned. The homogenization approach used in this chapter is a simplified version of recently developed mechanical model [Brighenti, 2004a], to get the FRC consti- tutive behavior based on the shear stress distribution along the fiber-matrix interface during the loading process. The adopted model for fiber homogenization can be con-
4.2 FRC homogenization methodology
sidered to be mechanically based, since the fiber contribution to the FRC mechanical properties are determined from the effective stress transfer between matrix and fibers; moreover the possibility of fiber-matrix debonding can be easily taken into account. Since the goal of this research is to focus on fiber distribution through the structure rather than developing micromechanical model, for sake of simplicity this issue is neglected in the present work. Moreover it can be considered that for not too high stressed composite elements (as followed in presented numerical examples) leading to shear fiber-matrix interface stresses well below the allowable limit shear bimaterial stress, the debonding phenomenon can reasonably assumed not to occur as well as fiber breaking. This approach is briefly summarized below; however interested reader can refer to [Brighenti,2004a], [Brighenti,2012] and [Brighenti,2004b] for more details.
The equivalent elastic properties of a fiber reinforced composite material for which the hypotheses of short, homogeneously and randomly dispersed fibers are made can be obtained by equating the virtual work rate of constituents for a RVE (it is assumed that the RVE characteristic length d is much more smaller that the structure character- istic length D) of the composite material (Fig. 4.2) with the equivalent homogenized one
w′=
composite’s work rate
z }| { Z Vκ(xxx) ˙˜εεε:σσσdV + Z Vχ(xxx) ˙˜εf·σf dV =
homogenized material’s work rate
z }| {
Z
Vεεε˙˜:σσσeqdV
(4.1)
where ˙eεf;σf are the virtual strain rate and the stress in a fiber, respectively, while the
scalar functionsκ(xxx) andχ(xxx) assume the following meaning: κ(xxx) = 1 i f (xxx) ∈ Vm 0 i f (xxx) 6∈ Vm and χ(xxx) = 1 i f (xxx) ∈ Vf 0 i f (xxx) 6∈ Vf (4.2)
allowing to identify the location of the material point xxx either in the matrix or in the reinforcing phase.
The constitutive relationships of the fibers and of the bulk material can be simply expressed through the following linear relations:
σf = Ef· (iii ⊗iii) :εεε and σσσeq(xxx) = CCCeq(xxx) :εεε (4.3)
in which Ef is the fibers’ Young’s modulus,εf is the fiber strain, CCCeqis the composite
equivalent elastic tensor whileεεε is the actual matrix strain tensor. Eq. (4.3) has been written by taking into account that the matrix strain measured in the fiber direction is given by εf = (iii ⊗iii) :εεε where iii = (sinθcosφ sinθsinφ cosθ) is the unit vector
4.2 FRC homogenization methodology
the virtual strain rate, ˜
εf = (iii ⊗iii) : ˜εεε and ε˙˜f = (iii ⊗iii) : ˙˜εεε (4.4)
By substituting the above expressions in the virtual work rate equality (Eq. (4.1)) one can finally identify the composite equivalent elastic tensor
CCCeq(xxx) = 1 V Z V{κ(xxx) CCCm+χ(xxx) Ef[QQQ ⊗QQQ]}dV =µCCCm+ηpEf Z VQQQ ⊗QQQ dV (4.5)
where the second-order tensor QQQ = (iii ⊗iii) has been introduced and the matrix and fiber volume fractionsµ =V1RVκ(xxx) dV =Vm
V andηp=V1
R
Vχ(xxx) dV = Vf
V have been used.
It can be easily deduced as the equivalent material is macroscopically homogeneous at least at the scale of the RVE with volume V - i.e. the equivalent elastic tensor CCCeq(xxx)
does not depend on the position vector, i.e. CCCeq(xxx) = CCCeq.
The calculation of the equivalent elastic tensor CCCeq through Eq. (4.5), requires to
evaluate the integral in Eq. (4.5) over a sufficiently large volume, representative of the macroscopic characteristics of the composite. The above integral can be suitably as- sessed on a hemisphere volume which allows considering all possible fiber orientations in the composite
Figure 4.2: Fiber reinforced composite material: definition of the RVE (with a char- acteristic length d, while the composite has a characteristic length D>>d) and of the fiber orientation anglesφ,θ, Ref. [Brighenti,2012]
4.3 Definition of the optimization problem