Time-harmonic analysis of antiplane crack in couple stress elastic materials

The time harmonic response of a rectilinear and semi-infinite crack in a couple stress (CS) elastic solid under Mode III loading conditions is investigated in the present work. The full-field solution of the dynamic crack problem obtained in [1] through Fourier integral transforms and the Wiener\u2013Hopf technique is generalized here by considering more general loading conditions, consisting in arbitrary reduced stress and couple stress tractions applied at the crack faces. The solution for quasistatic Mode III crack in indeterminate CS elastic materials was given in [2]. Later, the problem of steady-state Mode III crack propagation was investigated in [3]. In the present work, a travelling wave loading, applied in the form of generalized reduced tractions at the crack faces, is considered as the forcing term. As a result, a complex wave pattern appears, which differs significantly from the Mode III classical elastic solution. The results of the present analysis may be used as a building block to address, by means of superposition, the problem of arbitrary antiplane wave propagation in a cracked CS solid. Resonance is triggered when the applied loading is fed into the crack-tip at Rayleigh speed. Elastodynamic stress intensity factors are given, which generalize the corresponding results presented in [2] for the qusistatic framework. They incorporate the effect of the applied loading frequency and thereby account for the interplay of the diffracted waves. A remarkable wave pattern appears which consists of entrained waves extending away from the crack, reflected Rayleigh waves moving along the crack surfaces, localized waves irradiating from the crack-tip and body waves scattered around the crack-tip. Interestingly, the localized wave solution may be greatly advantageous for defect detection through acoustic emission

Indentation of a free beam resting on an elastic substrate with an internal lengthscale

The plane strain problem of a slender and weightless beam-plate loaded by a transversal point force in unilateral contact with a couple stress elastic foundation is investigated. The study aims to explore the consequences of the material internal lengthscale on the contact mechanics. In particular, compatibility between the beam and the foundation surface demands that both displacement and rotation match along the contact line. To this aim, couple tractions are exchanged besides the traditional contact pressure until separation between the beam and the foundation occurs. The problem is formulated making use of the Green's functions for a point force and a point couple acting atop of a couple stress elastic half-plane. A pair of coupled integral equations is thus derived, that governs the distribution of contact pressure and couple tractions, with one of them being immediately solved to provide an explicit relation between the two unknowns. In this sense, we retrieve the concept of a mechanically equivalent action, as it is the case of the Kirchhoff shear for plates. The remaining integral equation sets a cubic eigenvalue problem, whose linear term accounts for the microstructure. Its numerical solution is sought by expanding the equivalent contact pressure in series of Chebyshev polynomials vanishing at the contact region ends points, namely the lift-off points, and then applying a collocation strategy. The contact length, the distributions of contact pressure and couple tractions under the beam and the shearing force and bending moment along the beam are then obtained as a function of the material characteristic length. Results clearly indicate that accounting for the material internal lengthscale is mainly realized through exchange of the couple tractions, in the lack of which results much resemble those of the classical solution. Specifically, greater contact lengths and a stronger focusing effect about the loading point are encountered, which become very significant when the contact length approaches the internal lengthscale

A new Rayleigh-like wave in guided propagation of antiplane waves in couple stress materials

Motivated by the unexpected appearance of shear horizontal Rayleigh surface waves, we investigate the mechanics of antiplane wave reflection and propagation in couple stress (CS) elastic materials. Surface waves arise by mode conversion at a free surface, whereby bulk travelling waves trigger inhomogeneous modes. Indeed, Rayleigh waves are perturbations of the travelling mode and stem from its reflection at grazing incidence. As well known, they correspond to the real zeros of the Rayleigh function. Interestingly, we show that the same generating mechanism sustains a new inhomogeneous wave, corresponding to a purely imaginary zero of the Rayleigh function. This wave emerges from "reflection" of a bulk standing mode: This produces a new type of Rayleigh-like wave that travels away from, as opposed to along, the free surface, with a speed lower than that of bulk shear waves. Besides, a third zero of the Rayleigh function may exist, which represents waves attenuating/exploding both along and away from the surface. Since none of these zeros correspond to leaky waves, a new classification of the Rayleigh zeros is proposed. Furthermore, we extend to CS elasticity Mindlin’s boundary conditions, by which partial waves are identified, whose interference lends Rayleigh-Lamb guided waves. Finally, asymptotic analysis in the thin-plate limit provides equivalent 1-D models

Effective thermal properties of fibre reinforced materials

The thermal behaviour of an elastic matrix reinforced with synthetic micro or macro fibres subjected to a constant heat flow is investigated in the present work. Steady-state condition for the heat flux is considered and isotropic thermal conductivity for both the matrix and fibres is assumed. Owing to the geometry of the system, reference is made to bipolar cylindrical coordinates. Various boundary conditions can be considered on the contours of the fibres. In particular, for a matrix reinforced with two fibres taken as insulated inclusions, a vanishing heat flow across the contour of the fibres must be imposed. After the temperature field has benn determined analytically, a homogeneization procedure is performed in order to find the equivalent thermal properties of the fibre reinforced composite material

Couple stress effects in a thin film bonded to a half-space

This study investigates the contact mechanics of a thin film laying on an elastic substrate within the context of couple-stress elasticity. It aims to introduce the effects of material internal length scale, which has proven an effective way of modeling structures at micro to nano-scales, allowing to capture their size dependent behavior. Specifically, stress analysis for a thin film bonded to a couple stress elastic half-space is considered under plane strain loading conditions by assuming that both shear stress and couple tractions are exchanged between the thin film and the substrate. The problem is converted to a singular integral equation, which is solved by expanding the shear stress tractions as a Chebyshev series. The results show that the introduction of couple tractions decreases the shear stress tractions and the axial load in the thin film. When the characteristic length is sufficiently small, but still finite, the results for classical elastic behavior are approached

On the edge-wave of a thin elastic plate supported by an elastic half-space

In this contribution, we consider edge-wave propagating in a thin elastic semiinfinite plate which is bilaterally supported by a homogenenous isotropic elastic half-space. The problem is formulated in terms of a eigenproblem constituted by a system of five linear PDEs in the plate transverse displacement and in the scalar and vector elastic potentials subject to mixed boundary conditions accounting for plate-fundation displacement continuity under the plate and zero normal stress outside. Zero tangential stress is envisaged throughout. The problem could be reduced to an inhomogenenous Wiener-Hopf functional equation in terms of the half-space surface displacement and of the plate-to-fundation contact pressure only. The kernel function is analyzed and the Rayleigh wave speed is obtained together with a novel dispersion equation. Finally, kernel factorization is performed

Experimental characterization of pull-in parameters for an electrostatically actuated cantilever

MEMS-NEMS applications extensively use micro-nano cantilever structures as actuation system, thanks to their intrinsically simple end efficient configuration. Under the action of an electrostatic actuation voltage the can- tilever deflects, until it reaches the maximum value of the electrostatic actuation voltage, namely the pull-in voltage. This limits its operating point and is a critical issue for the switching of the actuator. The present work aims to experimentally measure the variation of the pull-in voltage and the tip deflection for different geometri- cal parameters of an electrostatically actuated cantilever. First, by relying on a nonlinear differential model from the literature, we designed and built a macro-scale cantilever switch, which can be simply adapted to different configurations. Second, we experimentally investigated the effect of the free length of the suspended electrode, and of the gap from the ground, on the pull-in response. The experimental results always showed a close agree- ment with the analytical predictions, with a maximum relative error lower that 10% for the pull-in voltage, and a relative difference lower than 18% for the pull-in deflection

On fracture criteria for dynamic crack propagation in elastic materials with couple stresses

The focus of the article is on fracture criteria for dynamic crack propagation in elastic materials with microstructures. Steady-state propagation of a Mode III semi-infinite crack subject to loading applied on the crack surfaces is considered. The micropolar behavior of the material is described by the theory of couple-stress elasticity developed by Koiter. This constitutive model includes the characteristic lengths in bending and torsion, and thus it is able to account for the underlying microstructures of the material. Both translational and micro-rotational inertial terms are included in the balance equations, and the behavior of the solution near to the crack tip is investigated by means of an asymptotic analysis. The asymptotic fields are used to evaluate the dynamic J-integral for a couple-stress material, and the energy release rate is derived by the corresponding conservation law. The propagation stability is studied according to the energy-based Griffith criterion and the obtained results are compared to those derived by the application of the maximum total shear stress criterion.Comment: 31 pages, 6 figure

Integral identities for a semi-infinite interfacial crack in anisotropic elastic bimaterials

The focus of the article is on the analysis of a semi-infinite crack at the interface between two dissimilar anisotropic elastic materials, loaded by a general asymmetrical system of forces acting on the crack faces. Recently derived symmetric and skew-symmetric weight function matrices are introduced for both plane strain and antiplane shear cracks, and used together with the fundamental reciprocal identity (Betti formula) in order to formulate the elastic fracture problem in terms of singular integral equations relating the applied loading and the resulting crack opening. The proposed compact formulation can be used to solve many problems in linear elastic fracture mechanics (for example various classic crack problems in homogeneous and heterogeneous anisotropic media, as piezoceramics or composite materials). This formulation is also fundamental in many multifield theories, where the elastic problem is coupled with other concurrent physical phenomena.Comment: 29 pages, 4 figure