Thin Elastic Strip Research Articles

On the refined boundary condition at the edge of a thin elastic strip supported by a Winkler-type foundation under antiplane shear deformation

The derivation of the boundary conditions is the most challenging part of the asymptotic techniques underlying low-dimensional models for thin elastic structures. At the moment, these techniques do not take into consideration the effect of the environment, e.g., a Winkler foundation, when tackling boundary conditions, and have to be amended. In this paper as an example we consider an antiplane problem for a thin elastic strip contacting with a relatively compliant Winkler foundation. Refined boundary conditions at an edge loaded by prescribed stresses are established using a properly adjusted Saint-Venant’s principle. They appear to be useful for advanced structure modelling including analysis of the static equilibrium under self-equilibrated loading.

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

Experimental and finite element vibration analysis of CNT reinforced MR elastomer sandwich beam

In this study, the free vibration analysis of carbon nanotubes reinforced magnetorheological elastomer (CNT-MRE) sandwich beam has been presented. The sandwich specimens are made of thin elastic strips integrated with CNT-MRE core. The viscoelastic properties of MRE and CNT-MRE are identified using ASTM E756-05 standard test. The natural frequencies and damping factors of with and without CNT reinforced MRE sandwich beams have been measured experimentally at various magnetic fields under cantilever condition. The measured natural frequencies are validated through the 3D finite element model. Further, the variations of natural frequencies with the influence of different levels of magnetic fields, boundary conditions, and core-skin thickness ratios of the CNT-MRE sandwich beam are studied. The analysis results demonstrate that the presence of carbon nanotubes improves the vibration behavior of MR elastomer.

Buckling of a clamped strip‐like beam with a linear pre‐stress distribution

AbstractA thin linear elastic strip is clamped at both ends and subjected to a linear stress distribution across its width. We use Kirchhoff beam theory to study this problem. If displacements out of the strips own plane are prohibited, the straight configuration remains stable as long as the compression is not too high. With the three‐dimensional spatial description of the rod theory, we find possible buckling modes even in the case of average tensile stresses in the beam. Comparison with shell and beam finite elements shows excellent agreement with the analytical investigation, also with respect to the supercritical behavior.

A numerical solution framework for simultaneous peeling of thin elastic strips from a rigid substrate

Simultaneous peeling of multiple strips is commonly observed particularly at small-scale detachment processes. Although theoretical treatment of this problem is addressed, numerical solution procedures for geometrically arbitrary multiple-peeling problems are still missing. In this paper, a finite element-based numerical solution procedure for 3-D large displacement multiple-peeling problems is presented. Loading/unloading of peeling strips are expressed in the form of optimality conditions, and the current positions of the peeling fronts are determined locally adapting the multiplicative decomposition and the return mapping algorithm of finite strain plasticity theories. Within an incremental-iterative solution framework, peeling fronts and the current position of other nodes are determined in a staggered way instead of using an active set-based solution algorithm. The effectiveness of the approach is demonstrated by a series of example problems including multiple peeling of an assembly of randomly oriented strips.

Characteristic features of the process of wear of a punch and a thin elastic strip with non-uniform friction and wear parameters

Abstract An analytical solution of the problem on the wear of a punch sliding on a thin elastic strip under the condition that they are in complete contact is given for variable friction and wear coefficients. The particular case of a wavy punch (periodic problem) is considered. Some distinguishing features of the process of wear in the transient regime are revealed. It is established, for example, that the contact pressure distribution curve can have a narrow local maximum (peak), significantly exceeding the average level of the contact pressure in magnitude.

Peano modes at the D = 2 delocalisation transition

.We study the elasticity of the plane-filling Peano chain in terms of a spring network, modelling a flat and thin elastic strip, smoothly curved so as to follow the chain pattern.This network sustains normal modes, named Peano modes; in the same way as acoustic waves in crystals are sustained by the matching of translations with rotations, these modes arise when discrete dilatations match with rotations. The pattern shares this eightfold symmetry with the octagonal quasicrystal, but is much looser and prone to disruption; its relaxation marks the saddle point separating the solid from the liquid. In terms of the equivalent quantum mechanical problem the above corresponds to the D = 2 quantum delocalisation transition.Some experimental observations, regarding a priori uncorrelated systems such as colloids, bilayer water and chromosomal DNA come together, at least qualitatively, within the viewpoint proposed here. With reference to the D = 2 colloids the model accounts for the bosonic peak found in their phononic spectrum and for the relaxation of the excitations. The latter typically evolve towards patterns of increasingly large size, as in the inverse energy cascade of D = 2 turbulence. As for bilayer water, it undergoes a liquid to quasicrystal transition surprisingly similar to the phase transition found here. Finally, considering the chromosomal DNA, the model unveils the mixed nature of its compact conformations, lying at the boundary between a liquid and a solid, similar to colloids and quasicrystals.

Equilibrium Shapes with Stress Localisation for Inextensible Elastic Möbius and Other Strips

We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Mobius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order one-dimensional problem posed on the centreline of the strip. We derive Euler–Lagrange equations for this problem in Euler–Poincare form and formulate boundary-value problems for closed symmetric one- and two-sided strips. Numerical solutions for the Mobius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Mobius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami.

Inverted bi-prism phononic crystals for one-sided elastic wave transmission applications

This work presents the realization of one-sided wave transmission by using a specially engineered phononic crystal structure. It is an inverted bi-prism phononic crystal engineered for a horizontally incident elastic wave at a specific frequency. The incident wave along one direction is shown to be totally reflected by the bi-prism while the incident wave along the opposite direction transmitted through it with refraction, also evident from experiments. An application of the proposed bi-prism may be found in thin elastic strips.

Hyperbolic non-Euclidean elastic strips and almost minimal surfaces

We study equilibrium configurations of thin and elongated non-Euclidean elastic strips with hyperbolic two-dimensional reference metrics ā which are invariant along the strip. In the vanishing thickness limit energy minima are obtained by minimizing the integral of the mean curvature squared among all isometric embeddings of ā. For narrow strips these minima are very close to minimal surfaces regardless of the specific form of the metric. We study the properties of these "almost minimal" surfaces and find a rich range of three-dimensional stable configurations. We provide some explicit solutions as well as a framework for the incorporation of additional forces and constraints.

Peeling Pressure-Sensitive Adhesive Tape from Thin Elastic Strip

In some applications of peeling of a pressure-sensitive adhesive tape, the substrate is thin and flexible, such as fabric, paper, leather, rubber, or skin. A thin strip, fixed at its ends, is considered as the substrate in this study. The strip and tape are assumed to be linearly elastic with negligible bending stiffness. The peel force depends on the adhesive fracture energy, the axial stiffnesses of the tape and strip, the initial slack of the strip, the initial location and length of the tape attached to the strip, and the peel angle of the force with respect to the line connecting the supports (or the local peel angle with respect to the side of the strip to which the tape is attached). Additional analyses involving an inextensible tape or an inextensible strip are included, along with an experimental result. The effects of various nondimensional parameters on the peel force are investigated.

Triangular buckling patterns of twisted inextensible strips

When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been found in solutions to a new set of geometrically exact equations describing the equilibrium shape of thin inextensible elastic strips. Here, we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the force–extension and moment–twist behaviour of these strips by varying the mode number n of triangular facets and find critical loads with jumps to higher modes.

Identifying a single biological nano-sized particle using a quartz crystal microbalance. A mathematical model

The effect of a point load on the oscillations of the quartz crystal microbalance (QCM) is represented with a simple one-dimensional model constructed specially for the determination of the basic physical characteristics of solitary nanoparticles. Due to the analytical form of solution, the model is highly selective and sensitive to small mass load and is quite suitable for the identification of solitary viruses. Thickness shear mode in a quartz crystal microbalance is modelled with longitudinal oscillations of a pair of thin elastic strips, with the free parameters calibrated on the basis of experiment. This approach reduces a complicated three-dimensional problem to a one-dimensional one and assesses the effect of distributed and localized mass load on the frequency response of a real QCM. A change of the frequency shift depending on the weight and position of a point body is demonstrated; the ratios of signal intensities at different harmonics accompanying the rupture of the body from the surface due to inertia in the air and in a viscous liquid are considered. The model is developed for the determination of parameters of solitary biological nano-sized bodies: mass, size, and bond rupture force.

Cohesive zone laws for void growth — II. Numerical field projection of elasto-plastic fracture processes with vapor pressure

Modeling ductile fracture processes using Gurson-type cell elements has achieved considerable success in recent years. However, incorporating the full mechanisms of void growth and coalescence in cohesive zone laws for ductile fracture still remains an open challenge. In this work, a planar field projection method, combined with equilibrium field regularization, is used to extract crack-tip cohesive zone laws of void growth in an elastic–plastic solid. To this end, a single row of void-containing cell elements is deployed directly ahead of a crack in an elastic–plastic medium subjected to a remote K-field loading; the macroscopic behavior of each cell element is governed by the Gurson porous material relation, extended to incorporate vapor pressure effects. A thin elastic strip surrounding this fracture process zone is introduced, from which the cohesive zone variables can be extracted via the planar field projection method. We show that the material's initial porosity induces a highly convex traction–separation relationship — the cohesive traction reaches the peak almost instantaneously and decreases gradually with void growth, before succumbing to rapid softening during coalescence. The profile of this numerically extracted cohesive zone law is consistent with experimentally determined cohesive zone law in Part I for multiple micro-crazing in HIPS. In the presence of vapor pressure, both the cohesive traction and energy are dramatically lowered; the shape of the cohesive zone law, however, remains highly convex, which suggests that diffusive damage is still the governing failure mechanism.

Bending of a thin strip by a circular tool

We present a model of transverse bending of a wide thin elastic, elastoplastic, or rigid-plastic strip by a circular tool in the case of large displacements. The bending of the initial rectilinear strip is modeled by small increments in the load on the tool with analysis of the forces, moments, curvature, and the midline equation. The final form of the strip and the residual stresses are determined after elastic unloading. The model describes technological operations of sheet slab bending in dies and on roller-bending machines.

Qualitative analysis of initial-value problems for a thin elastic strip

The 2D initial-value problems for a thin elastic strip are subjected to an asymptotic analysis in the case of long-wave initial data with arbitrary distribution along the strip thickness. The effect of an antisymmetric initial displacement field is investigated. Composite asymptotic expansions are utilized, taking into account various asymptotic dynamic models. Along with low-frequency models that correspond to the classical theory of strip bending and its refinements, high-frequency approximations are included; the latter govern long-wave motions being near the thickness stretch and shear resonance frequencies. The coincidence of the stretch and shear thickness resonance frequencies is also taken into consideration. Asymptotically consistent initial conditions are derived and related 1D initial-value problems are formulated. In addition, the two-scale nature of the high-frequency long-wave motions in the vicinities of thickness resonance frequencies is discussed.

Propagation of a longitudinal acoustic wave in a thin elastic strip with a low-modulus coating

Propagation of a longitudinal acoustic wave in a thin elastic strip with a low-modulus coating

Propagation of an acoustic wave in a thin elastic strip embedded in a low-modulus matrix

Propagation of an acoustic wave in a thin elastic strip embedded in a low-modulus matrix

Propagation of longitudinal acoustic waves in a thin elastic strip coated by a layer of a viscous liquid

In [1, 2], we observed an effect of decreasing prop-agation velocity of longitudinal sound waves in a fineelastic rod after its immersion in a liquid. The decreasein the sound velocity is governed by the so-calledboundary layer of the liquid, which vibrates togetherwith the rod as an associated mass. The thickness ofthis layer depends on both the vibration frequency andboth the density and viscosity of the liquid. At a soundsignal frequency of about 100 kHz, the typical value ofthe boundary layer thickness for technical oils is 10 to50 µ m [1]. Therefore, the decrease in the sound veloc-ity is considerable only for thin samples, i.e., sampleswhose thickness does not exceed a few hundreds ofmicrons.Since the thickness of the boundary layer dependson the viscosity of a liquid, it is appropriate to employthe above effect for determining the viscosity. Anadvantage of this method is the possibility of perform-ing high-rate viscosity measurements that attain a rateof up to several thousands per second. Such high ratesmake it possible to measure the viscosity even in thecase when it varies with time.The problem of the effect of a liquid on the propa-gation velocity and the damping of acoustic waves in athin strip immersed into an infinite volume of a liquidwas solved analytically [1]. However, there exists theproblem of controlling the behavior of polymeric resinsdeposited in liquid form on a metal surface and solidi-fied on it.The goal of this study is to theoretically solve theproblem related to the propagation of a longitudinalharmonic acoustic wave in a thin elastic strip coated bya layer of an incompressible Newtonian liquid.We introduce a system of coordinates such that thefree surface of a liquid coincides with the plane X = 0.The longitudinal vibrations of the elastic strip along the Y -axis are described by the equation [3–5] (1) Here, u is the displacement of a small strip elementfrom the equilibrium position; ρ

Galerkin approximations to static and dynamic localization problems

A new Galerkin-type procedure is established which, unlike the classical approach, does not rely on the final shape being composed of linearly independent modes. The procedure is applied to the evolution of a localized buckle of a thin elastic strip within a visco-elastic medium. Unlike the related elastic problem, no clear-cut linear eigenvalues exist to model wavelength and exponential growth/decay in the tails of the buckle pattern. The new procedure introduces variables to measure these effects, and allows them to change in time. This results in a more natural evolutionary process than with fixed mode shapes. Analysis is run within an algebraic manipulator (M aple) and checked against that of a numerical boundary-value solver (COLPAR).