Homogenization and Design of Functionally Graded Composites for Stiffness and Strength

Optimization of composite structures subject to local stress constraints

“…This geometry typifies the junctions between composite substructures and possesses reentrant corners seen in lap joints and junctions of struts. We conclude by noting that the theoretical basis for the approach given here has been established for three dimensional structural design using multiphase locally periodic composites in the presence of point wise stress constraints see ( [21], Theorems 5.1 and 5.2). For locally layered microstructures the corresponding theory is presented in [29].…”

“…This Proposition is established in [21]. The homogenized design formulation together with Proposition 2.1 provide an inverse homogenization method for identifying microstructures that satisfy point wise stress constraints while delivering a torsional rigidity close to that given by the optimal designθ f for the homogenized design problem.…”

“…The homogenized design problem HP is well posed and there is an optimal designθ f provided T is chosen large enough so that D Θ contains at least one design, see [21].…”

“…Because of this we will refer to it as an inverse homogenization design method. The rigorous theoretical basis for this approach is given in [21] and [29].…”

Representation formulas forL∞norms of weakly convergent sequences of gradient fields in homogenization

“…We pick an open subset S ⊂ Ω of interest and the gradient constraint for the multi-scale problem is written in terms of the modulation function. We set When the constraint M is chosen such that there exists a control β ∈ Ad γ for which C i (β) ≤ M then an optimal design β * exists for the design problem (6.8), this is established in [28], [22]. The optimal design β * specifies characteristic functions χ i * (x, y) = χ i (β * (x), y) from which we recover continuously graded microgeometries χ i * kj (x, n j x) and coefficient matrices A * ,kj (x, n j x) of the form (5.5).…”