Dual series solution for the standardized ISRM Brazilian disc test modelled as an advancing contact problem

A new analytical solution is found for the stress and displacement fields in a standardized ISRM Brazilian disc test with curved jaws, based on the solution of a set of dual trigonometric series. The disc is assumed to be linear elastic and isotropic and in frictionless advancing contact with the two jaws assumed as rigid, under plane stress loading conditions. Use is made of the Michell solution for an elastic disc in polar coordinates, whose coefficients are found by imposing the mixed boundary conditions along the disc rim, both along the free surface and the contact region. The problem is first reduced to a set of dual trigonometric series and then to a linear system of infinite equations, which is solved by truncation. The non-linear relations providing the contact angular extent and pressure distribution in terms of the applied load or jaw displacement, consequent to the progressive advance of the contact, are derived by using an inverse method. The obtained results are validated by comparison with previous theoretical and experimental results available in the literature. The study indicates that the method of dual series is simpler and more straightforward as compared to the analytical methods proposed in the literature for treating the Brazil disc test as an advancing contact problem

Mechanics of Interfacial Cracks between dissimilar Quasicrystals

We analize the steady propagation of a straight interfacial crack between two dissimilar planar quasicrystals in pure elastic setting and infinitesimal deformation regime. A closed form solution to the balance equations is furnished. Inertia is attributed only to the macroscopic motio

Non-standard contact conditions between a beam and a couple stress elastic half-plane

In the present work, the problem of a deformable Euler-Bernoulli beam of length 2a in bilateral frictionless contact with a couple stress elastic half-plane, whose constitutive parameters are the shear modulus , the Poisson coefficient and the material characteristic length l, is investigated by assuming that both contact pressure and couple stress tractions are transmitted across the contact zone. The present study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. It may be considered an extension of the works on beams in contact with an elastic half-plane performed by Shield and Kim (1992), Lanzoni and Radi (2016), and on rigid indenters in contact with an elastic couple-stress half-plane developed by Guorgiotis and Zisis (2016) and Zisis et al. (2018). The couple stress theory of elasticity requires boundary conditions on the microrotation and couple stress tractions in addition to the usual boundary conditions of the classic non-polar continuum on displacements and stress tractions. A challenging problem is thus how to extend the classic contact conditions to include the effects of the microrotation and couple stress tractions. In the proposed approach, the classical strain compatibility condition between the slope of the beam and that of the half-plane surface is imposed along the contact region. Moreover, three alternative kinds of microstructural contact conditions are considered and discussed, namely, vanishing of couple stress tractions, vanishing of microrotations and compatibility between microrotatons of the half-plane surface and slope of the beam. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, although the third boundary condition seems the most correct one. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic half-plane. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials of the first kind displaying the classical square-root singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the Chebyshev series expansion adopted for the contact pressure and couple stress tractions. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness EI of the beam and the characteristic length l of the elastic half-plane. The three alternative conditions lead to significantly different results in term of bending moment along the beam. The size effects due to the characteristic length of the half-plane and the implications of the generalized contact conditions are illustrated and discussed. The classical elastic solution is recovered as the characteristic length becomes vanishing small. Generally, the magnitude of the couple stress tractions is found to increase with the characteristic length. Although its contribution is usually smaller than that of the contact pressure and mainly restricted to the edges of the beam, it may provide a significant influence on the shear force and bending moment along the beam. Therefore, the obtained results show that the couple stress tractions exhibit a large influence on the beam internal forces and moments and display size dependent behavior when the beam length is comparable to the intrinsic characteristic length scale of the ground. Moreover, we show that accounting for the micropolar behavior of the ground, but neglecting the moment tractions in the contact region may lead to a substantial underestimation of the bending moment in the beam, in particular for the intermediate range of values of the material characterisic length (Fig. 1). The most interesting applications concern the case of beam length comparable with the microstructural characteristic length, namely for the ratio of l/a equal 0.5 and 1 considered in the plots. These results are expected to be significant and useful for engineering applications, specially in the field of micromechanics. We aspire indeed that the provided results may serve as a reference for the design of structural components in contact with heterogeneous and complex materials, not only at the macroscale, but also at the micro and nanoscale, providing a fundamental basis for the assessment of the proper microstructural contact conditions. References 1. Gourgiotis, P.A., Zisis, Th., 2016. Two-dimensional indentation of microstructure solids characterized by couple-stress elasticity. Journal of Strain Analysis and Enginering Design, 51, 1-14. 2. Lanzoni, L., Radi, E., 2016. A loaded Timoshenko beam bonded to an elastic half plane. International Journal of Solids and Structures, 92(1), 76-90. 3. Shield, T.W., Kim, K.S., 1992. Beam theory models for thin film segments cohesively bonded to an elastic half space, International Journal of Solids and Structures, 29, 1085-1103. 4. Zisis, Th., Gourgiotis, P.A., Georgiadis, H.G., 2018. Contact mechanics in the framework of couple stress elasticity. In H. Altenbach et al. (eds.), Generalized models and non-classical approaches in comple

A loaded beam in full frictionless contact with a couple stress elastic halfplane: Effects of non-standard contact conditions

The plane problem of a loaded Euler-Bernoulli beam of finite length in frictionless bilateral contact with a microstructured half-plane modelled by the couple stress theory of elasticity is considered here. The study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic half-plane. The problem is formulated by imposing compatibility of strain between the beam and the half-plane along the contact region and three alternative types of microstructural contact conditions, namely vanishing of couple stress tractions, vanishing of microrotations and compatibility between rotations of the beam cross sections and microrotations of the half-plane surface. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, without any sound motivation, although the third boundary condition seems the most correct one. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials displaying squareroot singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness of the beam and the characteristic length of the elastic half-plane. The size effects due to the characteristic length

Analytical bounds for the pull-in voltage of carbon nanotubes

Carbon nanotubes (CNTs) display a number of attractive electronic and mechanical properties that are currently exploited in a wide variety of industrial applications, such as sensors, nanoactuators, memory devices, switches, high frequncy nanoresonators and nanotweezers. Due to their tiny size they indeed display ultra-low mass and very high resonance frequency as well as the capability to carry huge electrical currents and to sustain high current densities. These properties, in conjunction with the significant progress recently made in the fabrication of carbon nanostructures, allow CNTs to become essential components in the fabrication of enhanced nano-electromechanical systems (NEMS) An accurate determination of the stable actuating range and the pull-in instability threshold is a crucial issues for designing reliable CNT based NEMS. Despite the amount of numerical or approximated investigations, analytical models and closed form expressions for pull-in instability analysis of CNT still appears to be limited. An analytical methodology for assessing accurate lower and upper bounds to the pull-in parameters of an electrostatically actuated micro- or nanocantilever has been provided in two previous works [1, 2], taking into consideration the contributions of flexible support and compressive axial load. In the present work, attention is paid to investigate the pull-in phenomenon in CNT with circular cross-section, by considering the proper expressions of the electrostatic and van der Waals forces per unit length acting on a CNT, as well as the significant reduction of the pull-in voltage induced by the charge concentration at the free end [3]. Two-side accurate analytical estimates of the pull-in parameters of a carbon nanotube switch clamped at one end under electrostatic actuation are provided by considering the effects of van der Waals interactions and charge concentration at the free end. The problem is governed by a fourth-order nonlinear boundary value problem, according to the Eulero-Bernoulli beam theory. Two-side estimates on the deflection are first derived, then very accurate lower and upper bounds to the pull-in voltage and deflection are obtained as functions of the geometrical and material parameters. The analytical predictions are then found to agree remarkably well with the numerical results provided by the shooting method

Three-Dimensional Bioprinting Materials with Potential Application in Preprosthetic Surgery

Current methods in handling maxillofacial defects are not robust and are highly dependent on the surgeon’s skills and the inherent potential in the patients’ bodies for regenerating lost tissues. Employing custom-designed 3D printed scaffolds that securely and effectively reconstruct the defects by using tissue engineering and regenerative medicine techniques can revolutionize preprosthetic surgeries. Various polymers, ceramics, natural and synthetic bioplastics, proteins, biomolecules, living cells, and growth factors as well as their hybrid structures can be used in 3D printing of scaffolds, which are still under development by scientists. These scaffolds not only are beneficial due to their patient-specific design, but also may be able to prevent micromobility, make tension free soft tissue closure, and improve vascularity. In this manuscript, a review of materials employed in 3D bioprinting including bioceramics, biopolymers, composites, and metals is conducted. A discussion of the relevance of 3D bioprinting using these materials for craniofacial interventions is included as well as their potential to create analogs to craniofacial tissues, their benefits, limitations, and their application

Lower and Upper Bound for the Pull-in Parameters of a Micro- or Nanocantilever Beam Immersed in Liquid Electrolytes

An analytical method is proposed to accurately estimate the pull-in parameters of a micro- or nanocantilever beam immersed in liquid electrolytes with a flexible support at one end. The system is actuated by electrochemical force, namely the sum of electric and osmotic forces, and is subject to Casimir or van der Waals forces according to the spacing between the two electrodes. The deflection of the beam is described by a fourth-order nonlinear boundary value problem that can be formulated by an equivalent nonlinear integral equation. At first, a priori upper and lower analytical estimates on the beam deflection are derived and then very accurate lower and upper bounds for the pull-in voltage and tip deflection are obtained. The analytical predictions are in excellent agreement with the numerical results provided by the shooting method. Finally, a simple closed-form relation is proposed for the pull-in voltage under the effect of bulk ion concentration

Resistivity contribution tensor for nonconductive sphere doublets

The distribution of the temperature and heat flux fields around a couple of unequal nonconductive tangent spherical inhomogeneities (or pores) embedded in an infinite medium under a steady-state and remotely applied heat flux is addressed in the present work. Owing to the 3D geometrical layout of the inhomogeneity, use is made of the tangent sphere coordinate system. A corrective temperature field expressed in terms of convergent integrals is superposed to the fundamental one to fulfill the BCs at the surfaces of the spheres. When the heat flux is aligned to the symmetry axis (axisymmetric problem), the solution can be found straightforwardly by introducing a stream function, which allows for transforming the Neumann BCs into a Dirichlet boundary value problem. Conversely, for the transversal heat flux (non-axisymmetric problem), the problem is formulated in terms of temperature, thus leading to a system of two ODEs which is handled numerically through a Euler shooting method, after preliminary asymptotic expansions. Once the temperature fields are known, the components of the resistivity contribution tensor are assessed varying the aspect ratio of the two spheres. It is found that the extrema of the thermal resistivity are achieved for spheres of equal size. The study allows assessing the effective thermal conductivity of a wide range of smart composites involving insulating inhomogeneities resembling sphere doublets