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IN THIS ISSUE

### TECHNICAL PAPERS

J. Appl. Mech. 2005;74(2):177-180. doi:10.1115/1.2062828.

An exact periodic solution for the time dependent flow of a viscoelastic fluid in the presence of transverse magnetic field is derived. It is assumed that on one plate the fluid is injected with certain velocity and that it is sucked off at the other plate with the same velocity. Both plates are oscillating with a known velocity in their own plane. A perturbation method has been employed by treating the viscoelastic parameter to be small. Effects of viscoelastic parameter, cross-flow Reynolds number, frequency parameter, and Hartmann number on the velocity as well as wall shear stress of the flow are discussed here with graphs.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):181-190. doi:10.1115/1.2188016.

This paper is concerned with the T-stress change before and after crack kinking in two-dimensional elastic solids. By using asymptotic analysis and the Westergaard stress function method, approximate analytical formulas for calculating the T-stress as well as stress intensify factors of an infinitesimal kink are given. Contributions from the T-stress before crack kinking, to the T-stress and the stress intensity factors of the kinked crack, are clearly described. It is noted that since the sign of the T-stress of a kinked open crack might be different from that of a main crack, simply using the sign of the T-stress before crack kinking is not sufficient to determine crack growth stability as observed in recent experiments.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):191-202. doi:10.1115/1.2188535.

Inaccuracies in the modeling assumptions about the distributional characteristics of the monitored signatures have been shown to cause frequent false positives in vehicle monitoring systems for high-risk aerospace applications. To enable the development of robust fault detection methods, this work explores the deterministic as well as variational characteristics of failure signatures. Specifically, we explore the combined impact of crack damage and manufacturing variation on the vibrational characteristics of beams. The transverse vibration and associated eigenfrequencies of the beams are considered. Two different approaches are used to model beam vibrations with and without crack damage. The first approach uses a finite difference approach to enable the inclusion of both cracks and manufacturing variation. The crack model used with both approaches is based on a localized decrease in the Young’s modulus. The second approach uses Myklestad’s method to evaluate the effects of cracks and manufacturing variation. Using both beam models, Monte Carlo simulations are used to explore the impacts of manufacturing variation on damaged and undamaged beams. Derivations are presented for both models. Conclusions are presented on the choice of modeling techniques to define crack damage, and its impact on the monitored signal, followed by conclusions about the distributional characteristics of the monitored signatures when exposed to random manufacturing variations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):203-209. doi:10.1115/1.2188536.

The objective of this paper is to extend the framework of the continuum theory so that it can capture the properties that are embedded in the microstructure or nanostructure and still keep its simplicity and efficiency. The model thus developed is capable of accounting for local deformation of microstructures in solids especially their micro- (local) inertia effect. The essence underlying this approach is the introduction of a set of bridging functions that relate the local deformation of microstructures to the macrokinematic variables. Once the solution of the macroscopically homogeneous continuum is obtained, the solutions in the microstructures are obtained through the use of these bridging functions. Propagation of waves of different wavelengths is considered and the dispersion curve is used to evaluate the accuracy of the model. The model is also employed to study wave reflection and transmission at the boundary of two media with microstructures of very different scales.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2005;74(2):210-222. doi:10.1115/1.2188537.

This paper examines the effects of relaxing the assumption of classical linear elasticity that the loads act in their entirety on the undeformed shape. Instead, loads here are applied incrementally as deformation proceeds, and resulting fields are integrated. A formal statement of the attendant integrated elasticity theory is provided. A class of problems is identified for which this formulation is amenable to solution in closed form. Some results from these configurations are compared with linear elasticity and experimentally measured data. The comparisons indicate that, as deformation increases, integrated elasticity is capable of tracking the physical response better than linear elasticity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):223-230. doi:10.1115/1.2188538.

This paper addresses the identification of linear time-varying multi-degrees-of-freedom systems. The identification approach is based on the Hilbert transform and the empirical mode decomposition method with free vibration response signals. Three-different types of time-varying systems, i.e., smoothly varying, periodically varying, and abruptly varying stiffness and damping of a linear time-varying system, are studied. Numerical simulations demonstrate the effectiveness and accuracy of the proposed method with single- and multi-degrees-of-freedom dynamical systems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):231-239. doi:10.1115/1.2189873.

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):240-248. doi:10.1115/1.2190228.

Delamination is a common type of damage in laminated fiber-reinforced polymer (FRP) composites. As FRP composites are becoming popular in upgrading and strengthening of civil concrete structures, the specific delamination damage, i.e., the FRP-concrete debonding, is considered more critical than inter-laminar delamination occurring in the FRP composites. A finite element formulation on the FRP-bonded concrete plate with this type of delamination fault is developed in the context of non-destructive evaluation from vibration measurements and compared with a two-layer solid element model. An adhesive interface where possible debonding could occur is introduced between the FRP and the concrete plates. A scalar damage parameter characterizing the delamination is incorporated into the formulation of a finite element model that is compatible with the vibration-based damage identification procedure. The formulated model is then applied to the prediction of FRP-concrete delaminations from modal test results based on the sensitivity analysis of uniform load surface curvature, which has been previously proposed by the authors. The validity of the methodology is demonstrated in two numerical examples. The first one is used to check the model accuracy, while the second one assesses the efficiency of the model-based identification method.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):249-255. doi:10.1115/1.2190230.

In this study, the dynamic behavior of an elastic sphere-plane contact interface is studied analytically and experimentally. The analytical model includes both a continuous nonlinearity associated with the Hertzian contact and a clearance-type nonlinearity due to contact loss. The dimensionless governing equation is solved analytically by using multi-term harmonic balance method in conjunction with discrete Fourier transforms. The accuracy of the dynamic model and solution methods is demonstrated through comparisons with experimental data and numerical solutions for both harmonic amplitudes of the acceleration response and the phase difference between the response and the force excitation. A single-term harmonic balance approximation is used to derive a criterion for contact loss to occur. The influence of harmonic external excitation $f(τ)$ and damping ratio $ζ$ on the steady state response is also demonstrated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):256-258. doi:10.1115/1.2190231.

Given a specific set of Euler angles, it is common to ask what representations conservative moments and constraint moments possess. In this paper, we discuss the role that a non-orthogonal basis, which we call the dual Euler basis, plays in the representations. The use of the basis is illustrated with applications to potential energies, constraints, and Lagrange’s equations of motion.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2005;74(2):259-268. doi:10.1115/1.2198243.

The transverse compression and shear collapse mechanisms of a second order, hierarchical corrugated truss structure have been analyzed. The two competing collapse modes of a first order corrugated truss are elastic buckling or plastic yielding of the truss members. In second order trusses, elastic buckling and yielding of the larger and smaller struts, shear buckling of the larger struts, and wrinkling of the face sheets of the larger struts have been identified as the six competing modes of failure. Analytical expressions for the compressive and shear collapse strengths in each of these modes are derived and used to construct collapse mechanism maps for second order trusses. The maps are useful for selecting the geometries of second order trusses that maximize the collapse strength for a given mass. The optimization reveals that second order trusses made from structural alloys have significantly higher compressive and shear collapse strengths than their equivalent mass first order counterparts for relative densities less than about 5%. A simple sheet metal folding and dip brazing method of fabrication has been used to manufacture a prototype second order truss with a relative density of about 2%. The experimental investigation confirmed the analytical strength predictions of the second order truss, and demonstrate that its strength is about ten times greater than that of a first order truss of the same relative density.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):269-278. doi:10.1115/1.2198248.

Cyclic undulation of the gear tooth surface is one of the important sources of gear noise and vibration. It has been known that vibration caused by this source can appear at the nonmesh harmonic frequency components (ghost components). As there are no relationships between the frequency of this vibration and any gear specifications, the gear noise source is hard to detect. This paper proposes the utilization of the synchronous averaging technique for diagnosis of the source of nonmesh harmonic vibration components on a gear pair, and shows the possibility of using this technique for inspection of tooth surface undulation. The method for practically applying this technique is discussed in detail. Results demonstrated in the form of spectrum showed good agreement with the undulation assessed from precise tooth surface measurement over the whole surface of every tooth. The effect of the direction of the arrangement of cyclic undulation on tooth surface and gear vibration is also discussed in this paper. Finally the limitation to the synchronous averaging technique was discussed with respect to gear ratio.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):279-290. doi:10.1115/1.2198249.

Accurate steady and unsteady numerical solutions of the full two-dimensional (2D) governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 $(Reδ⩽60)$, establishes various features of the well-known steady solution and reveals the interesting phenomena of stability, instability, and nonlinear wave effects. It is shown that intrinsic flow instabilities cause the wave effects to grow over the well-known experiments-based range of $Reδ⩾30$. The wave effects due to film flow’s sensitivity to ever-present minuscule transverse vibrations of the condensing surface are also described. The results suggest some ways of choosing wall noise—through suitable actuators—that can enhance or dampen wave fluctuations and thus increase or decrease heat transfer rates over the laminar-to-turbulent transition zone.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):291-297. doi:10.1115/1.2198250.

Two models for predicting the stress-strain curve of porous NiTi under compressive loading are presented in this paper. Porous NiTi shape memory alloy is considered as a composite composed of solid NiTi as matrix and pores as inclusions. Eshelby’s equivalent inclusion method and Mori-Tanaka’s mean-field theory are employed in both models. Two types of pore connectivity are investigated. One is closed cells (model 1); the other is where the pores are interconnected to each other forming an open-cell microstructure (model 2). We also consider two different shapes of pores, spherical and ellipsoidal. The stress-strain curves of porous shape memory alloy with spherical pores and ellipsoidal pores are compared. It is found that the ellipsoidal shape assumption is more reasonable than the assumption of spherical pores. Comparison of the stress-strain curves of the two models shows that use of open-cell microstructure (model-2) makes the predictions more agreeable to the experimental results of porous NiTi whose microstructure exhibits open-cell microstructure.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):298-306. doi:10.1115/1.2198251.

The principle of an interferometric strain/slope rosette (ISSR) is based on interference of laser beams reflected from three microindentations on a specimen surface. The ISSR can simultaneously measure the in-plane strains and the out-of-plane slopes. Ring-core cutting is a mechanical stress relief method. When used with the ISSR technique for residual stress measurement, the ring core can be made much smaller than used with the resistance strain rosette. Thus, more localized residual stresses can be measured. The theories of the ISSR/ring-core cutting method are described in this paper. Both mechanical and finite element models are developed for the incremental ring-core cutting process with the application of the ISSR technique. The stress-strain coefficients of the ISSR/ring-core method are calculated and nondimensionalized for general applications. A test example is given to demonstrate how residual stress distribution is determined by using the stress-strain coefficients and the ISSR data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):307-314. doi:10.1115/1.2198252.

The theoretical development of the interferometric strain/slope rosette (ISSR) and ring-core cutting method is described in Part I of the paper [K. Li and W. Ren, ASME J. Appl. Mech.74(2), 298–306 (2007)]. In Part II, experiments are presented to demonstrate the applicability of the method. The procedures of experimentation are developed. An ISSR/ring-core cutting system was established and its measurement stability and accuracy were examined in a two-step measurement program. By repeating the two-step measurement procedures, several incremental ring-core cutting experiments were conducted. Residual stress distribution is calculated from the measured ISSR data by using the relaxation coefficients calibrated in Part I of the paper. Measurement resolution, accuracy, and sensitivity of the ISSR/ring-core method are evaluated. Tests on a titanium block show the reliability of the method in comparison with the results obtained by using other measurement methods. The new method is also applied on a laser weld which demonstrates its uniqueness to measure residual stresses in small areas with high stress gradients. The experiments show advantages of the ISSR/ring-core method, such as miniature size, noncontacting nature, and high sensitivity. The method can be effectively used to measure residual stress distributions with depth on various manufactured components.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):315-324. doi:10.1115/1.2198253.

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):325-331. doi:10.1115/1.2198546.

Quasi-periodic response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external sinusoidal forcing in a vicinity of main (1:1) resonance is studied analytically and numerically. It is shown that the quasi-periodic response is exhibited in well-defined amplitude-frequency range of the external force. Two qualitatively different regimes of the quasi-periodic response are revealed. The first appears as a result of linear instability of the steady-state regime of the oscillations. The second one occurs due to interaction of the dynamical flow with invariant manifold of damped-forced nonlinear normal mode of the system, resulting in hysteretic motion of the flow in the vicinity of this mode. Parameters of external forcing giving rise to the quasi-periodic response are predicted by means of simplified analytic model. The model also allows predicting that the stable quasi-periodic regimes appear for certain range of damping coefficient. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):332-346. doi:10.1115/1.2198547.

This paper introduces the idea of using mechanical steering compensators to improve the dynamic behavior of high-performance motorcycles. These compensators are seen as possible replacements for a conventional steering damper and comprise networks of springs, dampers, and a less familiar component called the inerter. The inerter was recently introduced to allow the synthesis of arbitrary passive mechanical impedances, and finds a potential application in the present work. The design and synthesis of these compensation systems make use of the analogy between passive electrical and mechanical networks. This analogy is reviewed alongside the links between passivity, positive reality, and network synthesis. Compensator design methods that are based on classical Bode-Nyquist frequency-response ideas are presented. Initial designs are subsequently optimized using a sequential quadratic programing algorithm. This optimization process ensures improved performance over the machine’s entire operating regime. The investigation is developed from an analysis of specific mechanical networks to the class of all biquadratic positive real functions. This aspect of the research is directed to answering the question: “What is the best possible system performance achievable using any simple passive mechanical network compensator?” The study makes use of computer simulations, which exploit a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The results show that, compared to a conventional steering damper, it is possible to obtain significant improvements in the dynamic properties of the primary oscillatory modes, known as “wobble” and “weave.”

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):347-351. doi:10.1115/1.2198548.

This paper employs the atomic-scale finite element method (AFEM) to study critical strain of axial buckling for carbon nanotubes (CNTs). Brenner “second-generation” empirical potential is used to model covalent bonds among atoms. The computed energy curve and critical strain for (8, 0) single-walled CNT (SWNT) agree well with molecular dynamics simulations. Both local and global buckling are achieved, two corresponding buckling zones are obtained, and the global buckling behavior of SWNT with a larger aspect ratio approaches gradually to that of a column described by Euler’s formula. For double-walled CNTs with smaller ratio of length to outer diameter, the local buckling behavior can be explained by conventional shell theory very well. AFEM is an efficient way to study buckling of CNTs.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):352-364. doi:10.1115/1.2198549.

The finite element method is used to evaluate the underwater blast resistance of monolithic beams and sandwich beams containing prismatic lattice cores (Y-frame and corrugated core) and an ideal foam core. Calculations are performed on both free-standing and end-clamped beams, and fluid-structure interaction effects are accounted for. It is found that the degree of core compression in the free-standing sandwich beam is sensitive to core strength, yet the transmitted impulse is only mildly sensitive to the type of sandwich core. Clamped sandwich beams significantly outperform clamped monolithic beams of equal mass, particularly for stubby beams. The Fleck and Deshpande analytical model for the blast response of sandwich beams is critically assessed by determining the significance of cross-coupling between the three stages of response: in stage I the front face is accelerated by the fluid up to the point of first cavitation, stage II involves compression of the core until the front and back faces have an equal velocity, and in stage III the sandwich beam arrests by a combination of beam bending and stretching. The sensitivity of the response to the relative magnitude of these time scales is assessed by appropriately chosen numerical simulations. Coupling between stages I and II increases the level of transmitted impulse by the fluid by 20–30% for a wide range of core strengths, for both the free-standing and clamped beams. Consequently, the back face deflection of the clamped sandwich beam exceeds that of the fully decoupled model. For stubby beams with a Y-frame and corrugated core, strong coupling exists between the core compression phase (stage II) and the beam bending/stretching phase (stage III); this coupling is beneficial as it results in a reduced deflection of the back (distal) face. In contrast, the phases of core compression (stage II) and beam bending/stretching (stage III) are decoupled for slender beams. The significance of the relative time scales for the three stages of response of the clamped beams are summarized on a performance map that takes as axes the ratios of the time scales.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):365-372. doi:10.1115/1.2200653.

This paper presents a novel asymptotic framework to obtain detailed solutions describing the propagation of hydraulic fractures in an elastic material. The problem consists of a system of nonlinear integro-differential equations and a free boundary problem. This combination of local and nonlocal effects leads to transitions on a small scale near the crack tip, which control the behavior across the whole fracture profile. These transitions depend upon the dominant physical process(es) and are identified by simultaneously scaling the associated parameters with the distance from the tip. A smooth analytic solution incorporating several physical processes in the crucial tip region can be constructed using this new framework. In order to clarify the exposition of the new methodology, this paper is confined to considering the impermeable case in which only the two physical processes of viscous dissipation and structure energy release compete.

Commentary by Dr. Valentin Fuster

### TECHNICAL BRIEFS

J. Appl. Mech. 2006;74(2):373-374. doi:10.1115/1.2188017.

Hertz’s theory, developed in 1881, remains the foundation for the analysis of most contact problems. In this paper, we consider the axisymmetric normal contact of two elastic bodies, and the body profiles are described by polynomial functions of integer and noninteger positive powers. It is an extension of Hertz’s solution, which concerns the contact of two elastic spheres. A general procedure on how to solve this kind of problem is presented. As an example, we consider the contact between a cone and a sphere. The relations among the radius of the contact area, the depth of the indentation, the total load, and the contact pressure distribution are derived.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):375-377. doi:10.1115/1.2189875.

This paper discusses the extraction of the wavelet-based impulse response function from the acceleration response using the discrete wavelet transform (DWT). The analytical formulation of the sensitivity of the DWT coefficient of the impulse response function with respect to a system parameter is then presented for structural damage detection. A numerical example with a $31-bar$ plane truss structure is used to verify the proposed method with different damage scenarios with or without model error and noise effect.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):378-381. doi:10.1115/1.2190232.

An integral equation method is presented to determine dynamic elastic $T$-stress. Special attention is paid to a single crack in an infinite elastic plane subjected to impact loading. By using the Laplace and Fourier transforms, the associated initial-boundary value problem is transformed to a Fredholm integral equation. The dynamic $T$-stress in the Laplace transform domain can be expressed in terms of its solution. Moreover, an explicit expression for initial $T$-stress is derived in closed form. Numerically solving the resulting equation and performing the inverse Laplace transform, the transient response of $T$-stress is determined in the time space, and the response history of the $T$-stress is shown graphically. Results indicate that $T$-stress exhibits apparent transient characteristic.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 2006;74(2):382-387. doi:10.1115/1.2198545.

In this paper, based on the theory of elastic thin plates, applying the image method and the wave function expansion method, multiple scattering of elastic waves and dynamic stress concentration in semi-infinite plates with a circular cutout are investigated, and the general solutions of this problem are obtained. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of scattered waves and dynamic stress in semi-infinite plates are different from those in infinite plates when the distance ratio $b∕a$ is comparatively small. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of scattered body with $θ=π$, but not on the side of cutout with $θ=π∕2$. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of cutout becomes little, and the dynamic stress on the shadow side becomes great.

Commentary by Dr. Valentin Fuster