New blending constraints and a stack-recovery strategy for the multi-scale design of composite laminates
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RESEARCH PAPER
New blending constraints and a stack-recovery strategy for the multi-scale design of composite laminates Marco Picchi Scardaoni1,2 · Marco Montemurro2
· Enrico Panettieri2 · Anita Catapano3
Received: 4 May 2020 / Revised: 17 July 2020 / Accepted: 14 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This work presents a new strategy for dealing with blending requirements in the composite structures design. Firstly, new analytical expressions of blending constraints, in the polar parameters space, are derived. Secondly, a dedicated numerical strategy for the recovery of blended stacking sequences is presented. The proposed approach is implemented in the framework of the multi-scale two-level optimisation strategy (MS2LOS) for composite laminates design. The theoretical aspects of this work are supported by the application of the proposed methodology to a numerical benchmark taken from the literature. The results obtained by means of the MS2LOS based on the polar formalism outperform those reported in the literature. Keywords Blending · Ply-drop · Composite materials · Optimisation · Anisotropy · Polar method Nomenclature Acronyms BCs CLT CNLPP
Boundary conditions Classic laminate theory Constrained non-linear Programming problem
DOFs FE FSDT FLP GA LPs MICNLPP
Responsible Editor: Ming Zhou Marco Montemurro
MS2LOS
[email protected] Marco Picchi Scardaoni [email protected] Enrico Panettieri [email protected] Anita Catapano [email protected] 1
Department of Civil and Industrial Engineering, University of Pisa, Aerospace division, via G. Caruso 8, 56122, Pisa, Italy
2
I2M UMR 5295, Arts et M´etiers Institute of Technology, Universit´e de Bordeaux, CNRS, INRA, Bordeaux INP, HESAM Universit´e, F-33405 Talence, France
3
I2M UMR 5295, Bordeaux INP, Universit´e de Bordeaux, Arts et M´etiers Institute of Technology, CNRS, INRA, HESAM Universit´e, F-33405 Talence, France
PPs RSS SR SS SLP SST UNLPP
Degrees of freedom Finite element First-order shear deformation theory First-level problem Genetic algorithm Lamination parameters Mixed-integer constrained non-linear programming problem Multi-scale two-level optimisation strategy Polar parameters Recovery stacking sequence Stiffness recovery Stacking sequence Second-level problem Stacking sequence table Unconstrained non-linear programming problem
Main symbols ξ θ Φ {ρ0 , ρ0K , ρ1 , φ1 } A(A∗ )
Design variables vector Orientation Objective function Dimensionless PPs (normalised) Membrane stiffness tensor
M. Picchi Scardaoni et al.
B(B∗ ) B C(C∗ )
D(D∗ ) H(H∗ ) h Klam N M N
n0 R R {T , R, Φ}
{T0 , T1 , R0 , R1 , Φ0 , Φ1 }
(normalised) Membrane/bending coupling stiffness tensor Blending operator (normalised) Homogeneity stiffness tensor (normalised) Bending stiffness tensor (normalised) Out-of-plane shear stiffness tensor Laminate thickness Stiffness tensor of a laminate number of plies Modulus to normalise coupling and homogeneity tensors Set of ply indices not in
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