Bend stiffness of laminate microstructures containing three dissimilar materials

  • Damiano Pasini


This article examines the effective flexural modulus of a multilayered micro-system evolving into alternative layered structures consisting of three dissimilar materials. A multiscale model of the bending stiffness is presented to capture the impact of changing the constituent materials, the layer architecture and the cross-section geometry. The results are plotted onto maps to show the existence of specific domains, within which fall the effective properties of all possible tri-material multilayered configurations. The potential to stiffen a bi-material system is demonstrated by integrating additional layers of a more flexible material for given constraints on the volume fraction. The proposed scheme is conducive to contrast structural alternatives in constrained and unconstrained design. A case study shows how the maps enable optimum selection among various design concepts, which may range from monolithic materials with alternative shape geometries to systems consisting of two and three materials arranged in dissimilar multiple layer architectures.


Multilayered system Flexural modulus modelling Layer geometry and material selection 



Cross sectional area


Width (m)


Internal width (m)


Dimensionless multiplicator of cross-section internal width (c = b/B)


Beam curvature


Dimensionless multiplicator of cross-section internal height (d = h/H)


Rectangular cross-section envelope


Young’s modulus (GPa)


Effective of properties of prismatic beam


Effective of properties of shaped beam


Internal height (m)


Height (m)


Second moment of area (m4)


Cross-section torsional constant (m4)


Beam length (m)


Bending moment per unit width


Mass (mg)


Exponent of Lame’ curves


Performance index


Scaling parameter of performance index


Radius of gyration (m)

u, v

Envelope multiplicators




Volume (m3)


Material density (mg/m3)


Envelope efficiency parameter


Shape transformer


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Copyright information

© Springer Science+Business Media, B.V. 2008

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringInstitute for Advanced Materials of McGill UniversityMontrealCanada

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