Large deformation Kirchhoff–Love shell hierarchically enriched with warping: Isogeometric formulation and modeling of alternating stiff/soft layups

This paper presents a large deformation Kirchhoff–Love shell model hierarchically enhanced with through-the-thickness warping functions, arbitrarily chosen by the user. Two unknowns are introduced for each of them, representing its amplitudes in two directions tangent to the shell surface. NURBS are used to approximate reference surface displacement and warping amplitudes in the weak form. The transverse shear strains are linear functions of the warping parameters only and naturally free from locking. A patch-wise reduced integration avoids membrane locking and improves efficiency. Particular attention is paid to the modeling of composites made up of multiple stiff layers coupled with soft interlayers. The alternating layup with high stiffness ratios induces a significant sectional warping with transverse shear strains concentrated in the soft layers. Two warping models are investigated: (WI) all stiff layers maintain the same director orthogonal to the deformed surface with independent transverse shear deformations of the soft layers; (WZ) a single zigzag function linking these deformations. The numerical tests confirm the great accuracy of the hierarchic shell model in reproducing the solid solution with a small number of discrete parameters, provided that the correct warping model is chosen. WI is reliable for all alternating layups. WZ reduces the unknowns to five per surface point, regardless of the number of layers, and is accurate for uniform soft layers.