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Non-standard contact conditions between a beam and a couple stress elastic half-plane
Full text linkIn 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
Full text linkThe 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 modeling of the shape memory effect in SMA beams with rectangular cross section under reversed pure bending
Full text linkAn analytical model is developed for a prismatic SMA beam with rectangular cross section subjected to alternating bending at temperature below the austenitic transformations. The loading path consists in a loading-unloading cycle under bending and reversed bending. Two opposite martensitic variants take place, whose volume fractions evolve linearly with the axial stress. Different Young’s moduli are taken for the austenitic and martensitic phases. As the bending moment is increased, the martensitic transformation starts from the top and bottom and then it extends inwards. If the maximum applied bending moment is large enough, then the complete Martensitic transformation takes place at the upper and lower parts of the cross section. During unloading and reversed bending, reorientation of the Martensite variants into the opposite ones takes place starting from the boundary between the fully martensitic region and the intermediate transforming region. Special attention is devoted to calculate analytically the axial stress and Martensite variant distributions within the cross section at each stage of the process. A closed form moment-curvature relation is provided for loading and elastic unloading and in integral form for the rest of the process. The approach is then validated by comparison with analytical results available in the literature
Analytical estimates of the pull-in voltage in MEMS and NEMS
No full textMicro- or Nano-Electro-Mechanical Systems, MEMS-NEMS, are currently employed in a wide variety of applications, ranging from mechanical or electronic engineering to chemistry or biology. The growing interest in this technology is due to notable need for accurate ultrasmall instruments and equipment characterized by very diminutive size, low power consumption, high precision, reliability and compatibility with the integrated circuits [1]. The micro- or nanocantilever beam electrode, suspended above a conductive substrate and actuated by a voltage difference, is the fundamental component of many MEMS and NEMS devices. Moreover, due to their smart mechanical and electronic properties and the recent progress in their fabrication, carbon nanotubes are significantly exploited in industrial applications, such as sensors, nanoactuators, memory devices and nanotweezers, becoming essential components in NEMS [2]. Recent research remarks the role of micropumps in drug delivery systems able to regulate very small and accurate volumes in various industrial, chemical and biomedical applications. Electrostatic micropumps typically are composed of two parallel, thin, circular micro- or nanoplates. The membrane of the electrostatic micropump can be actuated and displaced towards the fixed electrode by applying a voltage across the electrodes. When the actuation voltage is removed, the displaced membrane releases and returns to its original position. In general, under the action of the electrostatic force and intermolecular surface forces, particularly significant at the micro- or nanoscale, the movable electrode deflects toward to the substrate, thus reducing the separation distance between the electrodes. Correspondingly, the magnitude of the attractive forces increases until at a critical voltage, named the pull-in voltage, the flexible electrode collapses onto the substrate. In this work, an analytical method is proposed for estimating the pull-in voltage and the correspondent deflection accurately, thus providing a useful tool for the effective design of innovative MEMS and NEMS devices [3]
Lower and Upper Bound for the Pull-in Parameters of a Micro- or Nanocantilever Beam Immersed in Liquid Electrolytes
Full text linkAn 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
Advancing contact of a 2D elastic curved beam indented by a rigid pin with clearance
Full text linkA two-dimensional analytical solution is presented for stresses and displacements in an elastic curved beam forming an incomplete ring in frictionless and unbonded contact with a rigid pin loaded by a point force and in the presence of clearance. The circular beam is modelled as an incomplete elastic thick ring, constrained at both ends and in a plane stress state. The stress and displacement fields within the beam are derived from a biharmonic Airy stress function, according to the Michell solution in polar coordinates. The mixed boundary value problem is reduced to a set of dual series equations and then to a non-homogeneous linear system of infinite equations, which is then solved by truncation. The non-linear relations between the applied load and the contact angle
or the pressure distribution are obtained by using an inverse method. The analytical results are compared with finite element predictions for a pin-lug connection and a reasonable agreement is
observed for several typical geometries. The peaks of contact pressure and von Mises equivalent stress and their location within the curved beam are evidenced
Resistivity contribution tensor for nonconductive sphere doublets
Full text linkThe 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
Lower and Upper Bound for the Pull-in Parameters of a Micro- or Nanocantilever Beam Immersed in Liquid Electrolytes
No full textAn 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
Elastic solution for a circular disk with a central crack under compressive diametrical load
No full textThe splitting tensile strength test (Brazilian test) is widely used because of its simplicity to assess the ultimate tensile strength and fracture behaviour of a variety of brittle materials, with specific reference to ceramics, concrete, and cementitious composites. Indeed, from the linear theory of elasticity, the distribution of the normal stress along any diameter of an uncracked disk subjected to a pair of concentrated diametrical compressive forces is known in closed form. Conversely, a Brazilian disk with a pre-existing central crack of a given length turns out to be a much more challenging problem owing to the mixed boundary conditions to be imposed inside and outside the crack along the crack direction, together with the condition about the stress field along the outer curved boundary of the disk.
Few studies deal with such a demanding layout. Among these, based on the weight function method, Dong et al. (2004) evaluated the SIFs varying the angle of the external load with respect the crack plane and the crack length as well. Critical conditions for the achievement of pure mode I and mode II loading have been found also. However, that study is restricted to the neighbouring of the crack tips.
In the present study, the full field solution of the Brazilian disk is provided analytically in terms of Airy stress function in bipolar coordinates. The study is handled by examining separately a skew-symmetric and a symmetric loading condition, representative for the mode II and mode I loadings, respectively. Both the situations lead to a Fredholm hypersingular integral equation, whose solution is found through a collocation method by expanding the unknown in series of Chebyshev polynomials. It is pointed out that for mode I loading, a closing or an opening crack may arise. Both these circumstances have been analysed in detail. Conversely to the existing studies, the proposed formulation allows assessing both the displacement and stress fields along the entire diameter of the disk in the direction of the crack for any loading angle
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