Bifurcation of Elastic Multilayers

Abstract

The occurrence of a bifurcation during loading of a multilayer sets a limit on structural deformability, and therefore represents an important factor in the design of composites. Since bifurcation is strongly influenced by the contact conditions at the interfaces between the layers, mechanical modelling of these is crucial. The theory of incremental bifurcation is reviewed for elastic multilayers, when these are subject to a finite strain before bifurcation, corresponding to uniform tension/compression and finite bending. The interlaminar contact is described by introducing linear imperfect interfaces. Results are critically discussed in view of applications and available experiments.

Finite bending of a multilayered cylindrical nanosector with residual deformations

“…These films, for simplicity, are assumed to be made of the same homogeneous, isotropic hyperelastic material with elastic constants different from those of the phases 1 and 2 . From [24], the boundary value problem for this domain looks as follows: (2) − σ (1)…”

“…Here σ (1) and σ (2) are the stresses in solids 1 and 2 , respectively, while σ s represents the surface/interface Cauchy stresses; ∇ and ∇ s are the usual (three-dimensional) and tangential differential (two-dimensional) nabla operators in the current configuration. Normals to the corresponding surfaces and interface are given by n j , n s j , n i ; and t j , t s j , t i are the prescribed tractions ( j = 1, 2).…”

“…where µ ( j) = {µ (1) , µ (2) , µ (3) } = {µ out , µ mid , µ out }μ are the shear moduli of the corresponding layers with nondimensional factors µ out and µ mid , and the dimensional shear modulusμ is used together with radius R 1 and height H to non-dimensionalize all quantities of the problem. Unless otherwise mentioned, we will be dealing with non-dimensional quantities.…”

“…Consider how the residual deformations in the layers λ ( j) = {λ (1) , λ (2) , λ (3) } = {λ out , λ mid , λ out } affect the resultant bending moment and bending stiffness of the multilayered sectors with layers made of the same materials. For the sake of simplicity, we will primarily focus on cases where only the middle layer is residually deformed, while the outer layers remain unchanged, that is, λ out = 1.…”

“…Different size-scale layered structures have high proportions of interfaces, due to which they display unique mechanical, thermal, electrical optical and magnetic properties that are not found in their components alone [1]. In this study we will concentrate on changes in the mechanical response of a sector of a cylindrical tube undergoing finite bending due not only to difference in layers’ materials (as it is done, for example, in [2] for a rectangular block), but also due to the effect of residual stresses characterized for all size-scales and surface/interface effects appearing at smaller scales only.…”