Bifurcation of elastic solids with sliding interfaces

Abstract

Lubricated sliding contact between soft solids is an interesting topic in biomechanics and for the design of small-scale engineering devices. As a model of this mechanical set-up, two elastic nonlinear solids are considered jointed through a frictionless and bilateral surface, so that continuity of the normal component of the Cauchy traction holds across the surface, but the tangential component is null. Moreover, the displacement can develop only in a way that the bodies in contact do neither detach, nor overlap. Surprisingly, this finite strain problem has not been correctly formulated until now, so this formulation is the objective of the present paper. The incremental equations are shown to be non-trivial and different from previously (and erroneously) employed conditions. In particular, an exclusion condition for bifurcation is derived to show that previous formulations based on frictionless contact or ‘spring-type’ interfacial conditions are not able to predict bifurcations in tension, while experiments—one of which, ad hoc designed, is reported—show that these bifurcations are a reality and become possible when the correct sliding interface model is used. The presented results introduce a methodology for the determination of bifurcations and instabilities occurring during lubricated sliding between soft bodies in contact.

Tensile material instabilities in elastic beam lattices lead to a bounded stability domain

“…The results shown in the present article are directly connected to the discovery of tensile buckling [28,29] and are obtained through a rigorous application of homogenization theory providing an energetic match between a preloaded lattice (§2) and an effective elastic continuum (§3). Examples of materials characterized by a bounded stability domain and their characteristics are provided (§4), followed by the analysis of the ‘re-stabilization’ occurring in the effective continuum, while the elastic grid is subject to local instabilities (§5).…”

Tensile material instabilities in elastic beam lattices lead to a bounded stability domain

“…The results shown in the present article are directly connected to the discovery of tensile buckling [28,29] and are obtained through a rigorous application of homogenization theory providing an energetic match between a preloaded lattice (Section 2) and an effective elastic continuum (Section 3). Examples of materials characterized by a bounded stability domain and their characteristics are provided (Section 4), followed by the analysis of the 're-stabilization' occurring in the effective continuum, while the elastic grid is subject to local instabilities (Section 5).…”

The dynamics of a shear band

“…The above equations show that the shear band is modelled as a (null-thickness) discontinuity surface, which is more general than a crack (because a shear band can carry a finite compressive tractions across his faces), but may represent a dislocation [2,55,56]; in metals the null-thickness assumption is strongly motivated by the experimental observation [41,47,57] that a shear band thickness-to-length ratio is of the order 10 −3 since lengths of shear bands can reach millimeters, while their thickness is confined to only a few micrometres. In the absence of prestress, the shear band model reduces to a weak surface whose faces can freely slide and at the same time are constrained to remain in contact, but when a prestress is present, the shear band model differs from that of a sliding planar surface [5]. The prescriptions ( 29)- (31) have been directly borrowed from those defining the onset of a shear band in a material [4].…”