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TECHNICAL PAPERS

J. Appl. Mech. 1997;64(4):729-737. doi:10.1115/1.2788976.

The self-similar crack expansion method is developed to calculate stress intensity factors for three-dimensional cracks in an infinite medium or semi-infinite medium by the boundary integral element technique. With this method, the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface, and the crack expansion rate, which is related to the crack energy release rate, is estimated by using a technique based on self-similar (virtual) crack extension. For elements on the crack surface, regular integrals and singular integrals are evaluated based on closed-form expressions, which improves the accuracy. Examples show that this method yields very accurate results for stress intensity factors of penny-shaped cracks and elliptical cracks in the full space, with errors of less than one percent as compared with exact solutions. The stress intensity factors of subsurface cracks are in good agreement with other numerical solutions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):738-742. doi:10.1115/1.2788977.

This paper studies the stress redistribution in a tensile hybrid composite sheet due to the breakage of a high modulus fiber. Employing a continuous distribution of dislocations, a set of singular integral equations is established to analyze the fiber crack impinging upon weakly bonded fiber-matrix interfaces. After solving the integral equations numerically, the stress concentration factors of both high modulus and low modulus fibers are evaluated as a function of loading stress and interfacial parameters. The results are compared with those obtained from shear-lag model solution.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):743-750. doi:10.1115/1.2788978.

A finite strain analysis is presented for the pressurized spherical cavity embedded in a Drucker-Prager medium. Material behavior is modeled by a nonassociated deformation theory which accounts for arbitrary strain-hardening. The governing equations of spherically symmetric response are reduced to a single differential equation with the effective stress as the independent variable. Some related topics are discussed including the elastic-perfectly plastic solid, the thin-walled shell, and the Mohr-Coulomb material. Spontaneous growth (cavitation limit) of an internally pressurized cavity is treated as a self-similar process and a few numerical examples are presented. These illustrate, for different hardening characteristics, the pressure sensitivity of material response and that deviations from normality always reduce the caviation pressure.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):751-762. doi:10.1115/1.2788979.

Traditional averaging and homogenization techniques, developed to predict the macroscopic properties of heterogeneous media, typically ignore microstructure related scale effects—that is, the influence of the size of the representative volume, relative to the size of the unit cell. This issue is presently investigated by exploring the behavior of a nonlinearly elastic, planar, lattice model, which is subjected to general macroscopic deformations. For these materials, scale effects may be due to nonuniformities in the macroscopic strain field throughout the specimen, or alternatively, to the presence of microstructural imperfections that may be either geometric or constitutive in nature. For the case of macroscopic strain nonuniformities, it is shown that the microstructure related scale effects can be accounted for by the presence of higher order gradient terms in the macroscopic strain energy density of the model. For the case of microstructural imperfections, the difference between the respective macroscopic properties of the perfect and imperfect models are shown to depend on the relative size of the specimen, and on the imperfection amplitude and wavelength, while being nearly insensitive to the imposed macroscopic strain. For all of the cases considered, several analytical approximations are proposed to predict the influence of scale on the macroscopic properties, and the accuracy of each method is examined.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):763-771. doi:10.1115/1.2788980.

Using the nonlinearly elastic planar lattice model presented in Part I, the influence of scale (i.e., the size of the representative volume, relative to the size of the unit cell) on the onset of failure in periodic and nearly periodic media is investigated. For this study, the concept of a microfailure surface is introduced—this surface being defined as the locus of first instability points found along radial load paths through macroscopic strain space. The influence of specimen size and microstructural imperfections (both geometric and constitutive) on these failure surfaces is investigated. The microfailure surface determined for the infinite model with perfectly periodic microstructure, is found to be a lower bound for the failure surfaces of perfectly periodic, finite models, and an upper bound for the failure surfaces of finite models with microstructural imperfections. The concept of a macrofailure surface is also introduced—this surface being defined as the locus of points corresponding to the loss of ellipticity in the macroscopic (homogenized) moduli of the model. The macrofailure surface is easier to construct than the microfailure surface, because it only requires calculation of the macroscopic properties for the unit cell, at each loading state along the principal equilibrium path. The relation between these two failure surfaces is explored in detail, with attention focused on their regions of coincidence, which are of particular interest due to the possible development of macroscopically localized failure modes.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):772-780. doi:10.1115/1.2788981.

Thermal residual stresses introduced during the manufacturing process and their effect on the buckling load of stringer reinforced composite plates is investigated. The principal idea is to include stiffeners on the perimeter of the plate and thereby, during manufacture, induce a favorable thermal residual-stress state in the structure; these stresses arise by considering the difference in thermal expansion coefficients and elastic properties of the plate and the stiffeners. In this manner, it is shown that thermal residual stresses can be tailored to significantly enhance the performance of the structure. The analysis is taken within the context of an enhanced Reissner-Mindlin plate theory and the finite element technique is used to analyze the problem. A 16 node bi-cubic Lagrange element is implemented in a FORTRAN code to determine the buckling load of the composite plate in the presence of thermal residual stresses. Three different plate-stiffener geometries are used as illustrations. The analyses indicate that buckling loads can be significantly increased by properly tailoring the thermal residual stresses. Therefore it may be concluded that an evaluation of these stresses and a judicious analysis of their effects must be included in the design procedure for this class of composite structure.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):781-786. doi:10.1115/1.2788982.

A micromechanically based composite model is proposed to study the viscoelastic behavior of solid-filled rubber composites. A nonlinear So-Chen’s (1991) mechanical model which describes the viscoelastic behavior of the rubber matrix is proposed to relate volume-average deformation and stress within the two-phase composite inclusion to the remote (macroscopic) fields. The influence of the volume fractions of inclusions on the overall creep strain of a rubber-matrix composite is investigated at the level of dilute concentration. The creep rate of the rubber matrix, which depends nonlinearly on the creep strain and the primary creep and secondary creep resulting from the viscous flow of creep deformation, is also considered in addition to the usual steady-state, or secondary, creep. The method developed for the calculation of the incremental process is based upon Eshelby’s (1957) equivalence principle of an inhomogeneity-transformation problem and Mori-Tanaka’s (1973) idea of mean-field stress. In order to examine the applicability of the model as well as the nonlinear stretch parameter, a series of experiments on solid-filled silicone rubbers has been carried out, which included constant rate of tensile tests and creep tests. It is demonstrated that this simple, albeit approximate micromechanical modeling is capable of predicting the volume fraction dependence of the time dependent creep, with characteristic consistency with the known elastic behavior.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):787-794. doi:10.1115/1.2788983.

A micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented. The foam structure is based on the geometry of a Kelvin soap froth with flat faces: 14-sided polyhedral cells contain six squares and eight hexagons. Four struts meet at every joint in the perfectly ordered, spatially periodic, open-cell structure. All of the struts and joints have identical shape. Strut-level force-displacement relations are expressed by compliances for stretching, bending, and twisting. We consider arbitrary homogeneous deformations of the foam and present analytic results for the force, moment, and displacement at each strut midpoint and the rotation at each joint. The effective stress-strain relations for the foam, which has cubic symmetry, are represented by three elastic constants, a bulk modulus, and two shear moduli, that depend on the strut compliances. When these compliances are evaluated for specific strut geometries, the shear moduli are nearly equal and therefore the elastic response is nearly isotropic. The variational results of Hashin and Shtrikman are used to calculate the effective isotropic shear modulus of a polycrystal that contain grains of Kelvin foam.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):795-803. doi:10.1115/1.2788984.

A “multicontinuum” approach to structural analyses of composites is described. A continuum field is defined to represent each constituent material along with the traditional continuum field associated with the composite. Finite element micromechanics is used to establish relationships between composite and constituent field variables. These relationships uncouple the micromechanics from structural solutions and render an efficient means of extracting constituent information during the course of a finite element structural analysis. Equations are developed for the case of a linear elastic reinforcing material embedded in a linear viscoelastic matrix and verified by comparison with results of finite element micromechanics.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):804-810. doi:10.1115/1.2788985.

The problem of calculating the energy release rate for crack growth in an arbitrary composite in the presence of residual stresses is considered. First, a general expression is given for arbitrary, mixed traction, and displacement boundary conditions. This general result is then applied to a series of specific problems including statistically homogeneous composites under traction or displacement boundary conditions, delamination of double cantilever beam specimens, and microcracking in the transverse plies of laminates. In many examples, the energy release rate in the presence of residual stresses can be reduced to finding the effect of damage on the effective mechanical properties of the composite. Because these effective properties can be evaluated by isothermal stress analysis, the effect of residual stresses on the energy release rate can be evaluated without recourse to any thermal elasticity stress analyses.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):811-818. doi:10.1115/1.2788986.

Fracture initiated at a corner between two different isotropic materials is considered. A “small” crack, well within the region dominated by the asymptotic stress fields of the noncracked corner, is modeled and the stress intensities associated with the tip of the small crack are determined. Different criteria for the direction of crack propagation are studied.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):819-827. doi:10.1115/1.2788987.

In this study, the transient response of a propagating in-plane crack interacting with half-plane boundaries is investigated in detail. The reflected waves which are generated from traction-free boundaries will interact with the propagating crack and make the problem extremely difficult to analyze. The complete transient solutions are constructed by superimposing fundamental solutions in the Laplace transform domain. The fundamental solutions represent the responses of applying exponentially distributed loadings in the Laplace transform domain on the surface of a half-plane or the propagating crack faces. We focus our attention on the determination of the dynamic stress intensity factor. The dynamic stress intensity factors of a propagating crack in a configuration with boundaries and subjected to dynamic loadings are obtained in an explicit closed form. The transient solutions obtained in this study are in agreement with the experimental results from the literature. Some interesting phenomena observed in the published experimental works are also identified and discussed. It is concluded that the reflected waves generated from the boundary parallel to the crack have much stronger influence on the propagating crack than those generated from the boundary perpendicular to the crack. When the reflected waves generated from the boundary parallel to the crack return to the moving crack tip, the stress intensity factor will increase rapidly.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):828-834. doi:10.1115/1.2788988.

Following a classical plate bending theory of magneto-elasticity, we consider the scattering of time-harmonic flexural waves by a through crack in a conducting plate under a uniform magnetic field normal to the crack surface. An incident wave giving rise to moments symmetric about the crack plane is applied. It is assumed that the plate has the electric and magnetic permeabilities of the free space. By the use of Fourier transforms we reduce the problem to solving a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. The dynamic moment intensity factor versus frequency is computed and the influence of the magnetic field on the normalized values is displayed graphically. It is found that the existence of the magnetic field produces higher singular moments near the crack tip.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):835-841. doi:10.1115/1.2788989.

A viscoelastic model of finitely deforming rubber is proposed and its nonlinear finite element approximation and numerical simulation are carried out. This viscoelastic model based on continuum mechanics is an extended model of Johnson and Quigley’s one-dimensional model. In the extended model, the kinematic configurations and measures based on continuum mechanics are rigorously defined and by using these kinematic measures, constitutive relations are introduced. The obtained highly nonlinear equations are approximated by the nonlinear finite element method, where a mixture of the total and updated Lagrangian descriptions is used. To verify the theory and the computer code, uniaxial stretch tests are simulated for various stretch rates and compared with actual experiments. As a practical example, an axisymmetric rubber plate under various time-dependent pressure loading conditions is analyzed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):842-846. doi:10.1115/1.2788990.

This paper presents an elastic post-buckling analysis of an axially loaded beam-plate with two central across-the-width delaminations located at arbitrary depths. The analysis is based on the nonlinear beam equations, combined with the appropriate kinematic continuity and equilibrium conditions. A perturbation technique is employed, which transforms the nonlinear equations into a sequence of linear equations. An asymptotic solution of the post-buckling behavior of the plate is thus obtained. It is shown that with two delaminations, both the maximum deflection and the internal load of the first buckled (top) subplate increase as the external load increases. Of particular interest is the redistribution of load among subplates, which keeps the increase rate of internal load of the top buckled subplate much less than that of the external load. In other words, the load of the buckled subplate is close to the critical value even though the externally applied load is much larger than the critical load. In addition to the two-delamination configuration, a single delamination case is studied based on the present approach in order to verify the accuracy of the method. Also, a comparison with available finite element results is performed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):847-852. doi:10.1115/1.2788991.

A study on the problem of linear buckling of piezoelectric circular cylindrical shells subjected to external pressure as well as on an electric field is presented. In this paper, the structure is treated as a three-dimensional one. The results reveal that the piezoelectric effect has significant effect on the critical load, while the stress due to the uniformly applied electric field alone is not likely to cause elastic buckling. In addition, they can also be used to assess the limitation of shell theories in predicting buckling of piezoelectric smart shell structures.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):853-860. doi:10.1115/1.2788992.

Plane-strain contact analysis is presented for compositionally graded materials with power-law strain hardening. The half-space, y ≤ 0, is modeled as an incompressible, nonlinear elastic material. The effective stress, σe , and the effective total strain, εe , are related through a power-law model, σe = K0εeμ; 0 < μ ≤ min (1, (1 + k)). The material property K0 changes with depth, |y|, as K0 = A|y|k; A > 0, 0 ≤ |k| < 1. This material description attempts to capture some features of the plane-strain indentation of elastoplastic or steady-state creeping materials that show monotonically increasing or decreasing hardness with depth. The analysis starts with the solution for the normal line load (Flamant’s problem) and continues with the rigid, frictionless, flat-strip problem. Finally, the general solution of normal indentation of graded material by a convex, symmetric, rigid, and frictionless two-dimensional punch is given. Applications of the present results range from surface treatments of engineering structures, protective coatings for corrosion and fretting fatigue, settling of beam type foundations in the context of soil and rock mechanics, to bioengineering as well as structural applications such as contact of railroad tracks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):861-870. doi:10.1115/1.2788993.

The reproducing kernel particle method (RKPM) has attractive properties in handling high gradients, concentrated forces, and large deformations where other widely implemented methodologies fail. In the present work, a multiple field computational procedure is devised to enrich the finite element method with RKPM, and RKPM with analytical functions. The formulation includes an interaction term that accounts for any overlap between the fields, and increases the accuracy of the computational solutions in a coarse mesh or particle grid. By replacing finite element method shape Junctions at selected nodes with higher-order RKPM window functions, RKPM p-enrichment is obtained. Similarly, by adding RKPM window functions into a finite element method mesh, RKPM hp-enrichment is achieved analogous to adaptive refinement. The fundamental concepts of the multiresolution analysis are used to devise an adaptivity procedure.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):871-876. doi:10.1115/1.2788994.

A method for calculating the internal stress distribution in a wound roll subjected to a homogeneous temperature change is presented. The nonlinear constitutive behavior apparent in the stack-wise direction of the roll is represented by a function dependent upon pressure that is incrementally updated during the analysis. In addition to the mechanics of the model, procedures for measuring the circumferential and radial coefficients of thermal expansion are described. Particular examples are presented that show, by comparison with experimental data, the model described herein accurately predicts interlayer pressures in a wound pack. In addition, the marked effect of the core expansion coefficient on the stress distribution is shown and this example illustrates the potential for optimizing the selection of core materials such that the effects of environmental changes on roll structure are minimized.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):877-884. doi:10.1115/1.2788995.

The edge-stress problem in multilayered composite laminates under uniform axial extension is analyzed through an alternative method based on a boundary integral formulation. The basic equations of the formulation are discussed and solved by the multiregion boundary element method. Generalized orthotropic elasticity analytic fundamental solutions are employed to establish the integral equations governing the problem. The formulation is absolutely general with regard to the laminate stacking sequence and the section geometry and it does not require any aprioristic assumption on the elastic response nature. This makes the formulation suitable for an investigation of the singular behavior of the stress field at the free edge in composite laminates. The interlaminar normal and shear stress distributions are examined in detail with the aim of calculating the stress singularity at the interlaminar free edge. The singularity parameters, i.e., power and strength, are determined for two family of laminates in order to ascertain the effectiveness of the method for the free edge-stress problem.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):885-896. doi:10.1115/1.2788996.

The initial post-buckling behavior of moderately thick orthotropic shear deformable cylindrical shells under external pressure is studied by means of Koiter’s general post-buckling theory. To this extent, the objective is the calculation of imperfection sensitivity by relating to the initial post-buckling behavior of the perfect structure, since it is generally recognized that the presence of small geometrical imperfections in some structures can lead to significant reductions in their buckling strengths. A shear deformation theory, which accounts for transverse shear strains and rotations about the normal to the shell midsurface, is employed to formulate the shell equations. The initial post-buckling analysis indicates that for several combinations and geometric dimensions, the shell under external pressure will be sensitive to small geometrical imperfections and may buckle at loads well below the bifurcation predictions for the perfect shell. On the other hand, there are extensive ranges of geometrical dimensions for which the shell is insensitive to imperfections, and, therefore it would exhibit stable post-critical behavior and have a load-carrying capacity beyond the bifurcation point. The range of imperfection sensitivity depends strongly on the material anisotropy, and also on the shell thickness and whether the end pressure loading is included or not. For example, for the circumferentially reinforced graphite/epoxy example case studied, it was found that the structure is not sensitive to imperfections for values of the Batdorf length parameter z̃ above ≃270, whereas for the axially reinforced case the structure is imperfection-sensitive even at the high range of length values; for the isotropic case, the structure is not sensitive to imperfections above z̃ ≃ 1000.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):897-904. doi:10.1115/1.2788997.

It is well known that the in-plane stress and displacement distributions in a stationary annular disk under stationary edge tractions can be obtained through the use of Airy stress function in the classical theory of linear elasticity. By using Lame’s potentials, this paper extends these solutions to the case of a spinning disk under stationary edge tractions. It is also demonstrated that the problem of stationary disk-spinning load differs from the problem of spinning disk-stationary load not only by the centrifugal effect, but also by additional terms arising from the Coriolis effect. Numerical simulations show that the amplitudes of the stress and displacement fields grow unboundedly as the rotational speed of the disk approaches the critical speeds. As the rotational speed approaches zero, on the other hand, the in-plane stresses and displacements are shown, both numerically and analytically, to recover the classical solutions derived through the Airy stress function.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):905-915. doi:10.1115/1.2788998.

The focus of this paper is the development of linear, asymptotically correct theories for inhomogeneous orthotropic plates, for example, laminated plates with orthotropic laminae. It is noted that the method used can be easily extended to develop nonlinear theories for plates with generally anisotropic inhomogeneity. The development, based on variational-asymptotic method, begins with three-dimensional elasticity and mathematically splits the analysis into two separate problems: a one-dimensional through-the-thickness analysis and a two-dimensional “plate” analysis. The through-the-thickness analysis provides elastic constants for use in the plate theory and approximate closed-form recovering relations for all truly three-dimensional displacements, stresses, and strains expressed in terms of plate variables. In general, the specific type of plate theory that results from variational-asymptotic method is determined by the method itself. However, the procedure does not determine the plate theory uniquely, and one may use the freedom appeared to simplify the plate theory as much as possible. The simplest and the most suitable for engineering purposes plate theory would be a “Reissner-like” plate theory, also called first-order shear deformation theory. However, it is shown that construction of an asymptotically correct Reissner-like theory for laminated plates is not possible in general. A new point of view on the variational-asymptotic method is presented, leading to an optimization procedure that permits a derived theory to be as close to asymptotical correctness as possible while it is a Reissner-like. This uniquely determines the plate theory. Numerical results from such an optimum Reissner-like theory are presented. These results include comparisons of plate displacement as well as of three-dimensional field variables and are the best of all extant Reissner-like theories. Indeed, they even surpass results from theories that carry many more generalized displacement variables. Although the derivation presented herein is inspired by, and completely equivalent to, the well-known variational-asymptotic method, the new procedure looks different. In fact, one does not have to be familiar with the variational-asymptotic method in order to follow the present derivation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):916-922. doi:10.1115/1.2788999.

As a web is wound at speed onto a roll, a thin layer of air becomes entrapped between it and the incoming web stream. The resulting spiral-shaped air bearing separates adjacent web layers and can extend many wraps into the roll. The air entrained during the winding process increases the propensity for lateral interlayer slippage and damage to the edges of the web. In the present paper, an in situ technique is developed for measuring the thickness of the entrained air film during winding, and parameter studies quantify the effects of such winding variables as tension, width, transport speed, and surface roughness. With a view towards evaluating different transport designs and operating conditions, three measures of air entrainment are discussed: (i) the cumulative thickness of all air layers, (ii) the thickness of the outermost air layer at the nip, and (iii) the rate at which air bleeds from the roll once it comes to rest. Measured values of the first two metrics are compared with those predicted by a derived two-dimensional reduced-order model for steady-state winding. The analysis treats the two bounding configurations of symmetric and asymmetric stacking of web layers by specifying appropriate cross-web pressure profiles.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):923-928. doi:10.1115/1.2789000.

Stability problem of bearing pin-jointed assemblies, in which the number of equilibrium equations is greater than the equilibrium matrix rank (underconstrained structures), is investigated. Local and overall stability of initial and loaded states are discussed. Theoretical considerations are accompanied by numerical examples.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):929-934. doi:10.1115/1.2789001.

The deformation, using linear poroelasticity, of a two-dimensional box of porous material due to fluid flow from a line source is considered as a model of certain filtration processes. Analytical solutions for the steady-state displacement, pressure, and fluid velocity are derived when the side walls of the filter have zero solid stress. A numerical solution for the case where the porous material adheres to the side walls is also found. It will be shown, however, that simpler approximate solutions can be derived which predict the majority of the deformation behavior of the filter.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):935-939. doi:10.1115/1.2789002.

The fundamental theory of wire ropes developed by Costello and Phillips is utilized to obtain closed-form expressions for maximum contact stresses in single strand cables with fibrous cores. These should be useful for gaining an insight into the influence of various parameters of the cable as well as the material properties on its strength and hence design.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):940-945. doi:10.1115/1.2789003.

The localized modes of periodic systems with infinite degrees-of-freedom and having one or two nonlinear disorders are examined by using the Lindstedt-Poincare (L-P) method. The set of nonlinear algebraic equations with infinite number of variables is derived and solved exactly by the U-transformation technique. It is shown that the localized modes exist for any amount of the ratio between the linear coupling stiffness kc and the coefficient γ of the nonlinear disordered term, and the nonsymmetric localized mode in the periodic system with two nonlinear disorders occurs as the ratio kc /γ, decreasing to a critical value depending on the maximum amplitude.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):946-950. doi:10.1115/1.2789004.

By adopting the orthogonal transformations provided by the generalized real Schur decomposition, it is shown that every nonclassical linear system in state space can be transformed into block upper triangular form, to which the quasi-decoupling solution can be progressively carried out by solving the either first or second-order component equations with the “back substitution.” The distinct characteristics of generalized eigenvalue problems from those of standard ones are discussed. Favorable properties of the proposed method include: no inverting of any system matrix, indiscriminate applicability to both defective and nondefective systems, the simultaneous decoupling of the adjoint problem, and numerical stability. Illustrative examples are also presented.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):951-956. doi:10.1115/1.2789005.

Three-dimensional vibrations of a Euler-Bernoulli beam on an elastic half-space are investigated. In the model the beam has a finite width and the half-space and beam deflections are equal along the centre line of the beam. It is shown that the vertical and longitudinal beam vibrations are uncoupled from the lateral ones. The dispersion relations for the lateral and vertical-longitudinal waves in the beam are derived and the respective dispersion curves are plotted. These curves can cross each other due to the different equivalent stiffnesses of the half-space in vertical and lateral directions and different vertical and lateral bending stiffnesses of the beam. The existence of a crossing point implies that if the vertical-longitudinal and lateral beam vibrations are coupled for some reason (half-space inhomogeneity, beam asymmetry, etc.), the energy of the vertical vibrations of the beam can be resonantly transferred into the energy of lateral vibrations. This transfer will take place if the frequency of vibrations is close to the frequency determined by the crossing point. The dependency of the frequency of the crossing point on axial compressional stresses in the beam is studied. It is shown that this frequency decreases as the stresses increase.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):957-964. doi:10.1115/1.2789006.

This paper is concerned with the dynamical analysis of a sagged cable having small equilibrium curvature and horizontal supports under both distributed and concentrated loads. The loads are applied in vertical as well as horizontal directions. Based on a free vibration analysis, a transfer matrix method is generalized for solving coupled, nonhomogeneous differential equations to obtain closed-form solutions for the natural frequencies and the associated vibration mode shapes in vertical, horizontal, and longitudinal directions. It is shown that two sets of independent mode shapes associated with two sets of independent frequencies always exist and can be obtained via an equation of one variable only. This method demonstrates its advantages in dealing with interactions of modes in different directions, complex arrangement of concentrated loads, and high-order modes oscillations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):965-968. doi:10.1115/1.2789007.

We show that one may construct a Lyapunov function for any classically damped linear system. The explicit nature of the construction permits us to show that it remains a Lyapunov function under both perturbation of the linear part and introduction of a nonlinear term. We apply our findings to a stability analysis of the discrete, as well as continuous, damped mechanical transmission line.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):969-974. doi:10.1115/1.2789008.

In this paper, expressions are established for certain relative rotations which arise in motions of rigid bodies. A comparison of these results with existing relations for geometric phases in the motions of rigid bodies provides alternative expressions of, and computational methods for, the relative rotation. The computational aspects are illustrated using several examples from rigid-body dynamics: namely, the moment-free motion of a rigid body, rolling disks, and sliding disks.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):975-984. doi:10.1115/1.2789009.

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):985-991. doi:10.1115/1.2789010.

This is a comprehensive treatment of the time-integral variational “principles” of mechanics for systems subject to general nonlinear and possibly nonholonomic velocity constraints (i.e., equations of the form f(t, q, q̇) = 0, where t = time and q/q̇ = Lagrangean coordinates/velocities), in general nonlinear nonholonomic coordinates. The discussion is based on the Maurer-Appell-Chetaev-Hamel definition of virtual displacements and subsequent formulation of the corresponding nonlinear transitivity (or transpositional) equations. Also, a detailed analysis of the latter supplies a hitherto missing clear geometrical interpretation of the well-known discrepancies between the equations of motion obtained by formal application of the calculus of variations (mathematics) and those obtained from the principle of d’Alembert-Lagrange (mechanics); i.e., admissible adjacent paths (mathematics) are locally nonvirtual; and adjacent paths built from locally virtual displacements (mechanics) are not admissible. (These discrepancies, although revealed about a century ago, for systems under Pfaffian constraints (Hertz (1894), Hölder (1896), Hamel (1904), Maurer (1905), and others) seem to be relatively unknown and/or misunderstood among today’s engineers.) The discussion includes all relevant nonlinear nonholonomic variational principles, in both unconstrained and constrained forms of their integrands, and the corresponding nonlinear nonholonomic equations of motion. Such time-integral formulations are useful both conceptually and computationally (e.g., multibody dynamics).

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):992-999. doi:10.1115/1.2789011.

The problem of an interface crack between a circular fiber and the surrounding matrix is considered. The problem is formulated and solved with the help of complex variable methods. It is essential to take into account the existence of contact zones at the crack tips. The solution procedure relies on the use of crack opening displacements as the primary variables. Ultimately the governing equations are shown to consist of two coupled singular integral equations together with contact and single valuedness conditions. In general these equations must be solved by numerical methods. Attention is focused on the lengths of the contact zones. It is shown that the lengths of these contact zones are essentially independent of one of the Dundurs parameters.

Commentary by Dr. Valentin Fuster

BRIEF NOTES

J. Appl. Mech. 1997;64(4):1000-1004. doi:10.1115/1.2788962.

A general solution to the thermoelastic problem of a circular inhomogeneity in an infinite matrix is provided. The thermal loadings considered in this note include a point heat source located either in the matrix or in the inclusion and a uniform heat flow applied at infinity. The proposed analysis is based upon the use of Laurent series expansion of the corresponding complex potentials and the method of analytical continuation. The general expressions of the temperature and stress functions are derived explicitly in both the inclusion and the surrounding matrix. Comparison is made with some special cases such as a circular hole under remote uniform heat flow and a circular disk under a point heat source, which shows that the results presented here are exact and general.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1006-1008. doi:10.1115/1.2788964.
Abstract
Topics: Deflection
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1008-1010. doi:10.1115/1.2788965.
Abstract
Topics: Motion
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1010-1014. doi:10.1115/1.2788966.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1014-1016. doi:10.1115/1.2788967.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1017-1019. doi:10.1115/1.2788968.

The titled problem is studied analytically by an eigenfunction expansion method. In the early stage following the excitation, symmetric deformation is observed when the rotation speed is around the first critical speed. At a rotation speed ten times higher than the first critical speed, at which the convective velocity of the media is comparable to the wave propagation speed, asymmetric deformation becomes apparent.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(4):1019-1021. doi:10.1115/1.2788969.
Abstract
Topics: Friction , Waves
Commentary by Dr. Valentin Fuster

DISCUSSION

Commentary by Dr. Valentin Fuster

BOOK REVIEW

J. Appl. Mech. 1997;64(4):1029. doi:10.1115/1.2788975.
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Abstract
Commentary by Dr. Valentin Fuster

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