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TECHNICAL PAPERS

J. Appl. Mech. 1997;64(3):449-456. doi:10.1115/1.2788914.

In this study the plane elasticity problem for a nonhomogeneous layer containing a crack perpendicular to the boundaries is considered. It is assumed that the Young’s modulus of the medium varies continuously in the thickness direction. The problem is solved under three different loading conditions, namely fixed grip, membrane loading, and bending applied to the layer away from the crack region. Mode I stress intensity factors are presented for embedded as well as edge cracks for various values of dimensionless parameters representing the size and the location of the crack and the material nonhomogeneity. Some sample results are also given for the crack-opening displacement and the stress distribution.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):457-465. doi:10.1115/1.2788915.

This analysis presents the elastic field in a half-space caused by an ellipsoidal variation of normal traction on the surface. Coulomb friction is assumed and thus the shear traction on the surface is taken as a friction coefficient multiplied by the normal pressure. Hence the shear traction is also of an ellipsoidal variation. The half-space is transversely isotropic, where the planes of isotropy are parallel to the surface. A potential function method is used where the elastic field is written in three harmonic functions. The known point force potential functions are utilized to find the solution for ellipsoidal loading by quadrature. The integrals for the derivatives of the potential functions resulting from ellipsoidal loading are evaluated in terms of elementary functions and incomplete elliptic integrals of the first and second kinds. The elastic field is given in closed-form expressions for both normal and shear loading.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):466-470. doi:10.1115/1.2788916.

In this paper we present an analytical solution to calculate the stress concentrations around an elliptical void in a piezoelectric medium subjected to electrical loading. We show that the stress concentrations can be eliminated if the material properties satisfy a certain mathematical relation. While a trivial solution exists for this problem, we demonstrate that other families of solutions exist (optimal) to minimize/eliminate the stresses. The optimal families are shown to be independent of geometry and therefore are universally applicable to a specific material system. The optimal families do not limit the deformation profile and represent admissible solutions to the problem. Numerical studies demonstrate that the entire stress field in the medium vanishes and not just at the critical locations as dictated by the mathematics. Finally, we numerically demonstrate that the optimal properties are also applicable to the crack problem.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):471-479. doi:10.1115/1.2788917.

We find the elastic fields in a half-space (matrix) having a spherical inclusion and subjected to either a remote shear stress parallel to its traction-free boundary or to a uniform shear transformation strain (eigenstrain) in the inclusion. The inclusion has distinct properties from those of the matrix, and the interface between the inclusion and the surrounding matrix is either perfectly bonded or is allowed to slip without friction. We obtain an analytical solution to this problem using displacement potentials in the forms of infinite integrals and infinite series. We include numerical examples which give the local elastic fields due to the inclusion and the traction-free surface.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):480-486. doi:10.1115/1.2788918.

A tensegrity structure composed of six slender struts interconnected with 24 linearly elastic cables is used as a model of cell deformability. Struts are allowed to buckle under compression and their post-buckling behavior is determined from an energy formulation of the classical pin-ended Euler column. At the reference state, the cables carry initial tension balanced by forces exerted by struts. The structure is stretched uniaxially and the stretching force versus axial extension relationships are obtained for different initial cable tensions by considering equilibrium at the joints. Structural stiffness is calculated as the ratio of stretching force to axial extension. Predicted dependences of structural stiffness on initial cable tension and on stretching force are consistent with behaviors observed in living cells. These predictions are both qualitatively and quantitatively superior to those obtained previously from the model in which the struts are viewed as rigid.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):487-494. doi:10.1115/1.2788919.

The constitutive response and failure behavior of a W-Ni-Fe alloy over the strain rate range of 10-4 to 5 X 105 s-1 is experimentally investigated. Experiments conducted are pressure-shear plate impact, torsional Kolsky bar, and quasi-static torsion. The material has a microstructure of hard tungsten grains embedded in a soft alloy matrix. Nominal shear stress-strain relations are obtained for deformations throughout the experiments and until after the initiation of localization. Shear bands form when the plastic strain becomes sufficiently large, involving both the grains and the matrix. The critical shear strain for shear band development under the high rate, high pressure conditions of pressure-shear is approximately 1–1.5 or 6–8 times that obtained in torsional Kolsky bar experiments which involve lower strain rates and zero pressure. Shear bands observed in the impact experiments show significantly more intensely localized deformation. Eventual failure through the shear band is a combination of grain-matrix separation, ductile matrix rupture, and grain fracture. In order to understand the effect of the composite microstructure and material inhomogeneity on deformation, two other materials are also used in the study. One is a pure tungsten and the other is an alloy of W, Ni, and Fe with the same composition as that of the matrix phase in the overall composite. The results show that the overall two-phase composite is more susceptible to the formation of shear bands than either of its constituents.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):495-502. doi:10.1115/1.2788920.

In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):503-509. doi:10.1115/1.2788921.

An approximate yield criterion for porous ductile media at high strain rate is developed adopting energy principles. A new concept that the macroscopic stresses are composed of two parts, representing dynamic and quasi-static components, is proposed. It is found that the dynamic part of the macroscopic stresses controls the movement of the dynamic yield surface in stress space, while the quasi-static part determines the shape of the dynamic yield surface. The matrix material is idealized as rigid-perfectly plastic and obeying the von Mises yield. An approximate velocity field for the matrix is employed to derive the dynamic yield function. Numerical results show that the dynamic yield function is dependent not only on the rate of deformation but also on the distribution of initial micro-damage, which are different from that of the quasi-static condition. It is indicated that inertial effects play a very important role in the dynamic behavior of the yield function. However, it is also shown that when the rate of deformation is low (≤103 /sec), inertial effects become vanishingly small, and the dynamic yield function in this case reduces to the Gurson model.

Topics: Deformation , Stress , Shapes
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):510-518. doi:10.1115/1.2788922.

Bounds are investigated on the plastic deformations in a continuous solid body produced during the transient phase by cyclic loading not exceeding the shakedown limit. The constitutive model employs internal variables to describe temperature-dependent elastic-plastic material response with hardening. A deformation bounding theorem is proved. Bounds turn out to depend on some fictitious self-stresses and mechanical internal variables evaluated in the whole structure. An optimization problem, aimed to make the bound most stringent, is formulated. The Euler-Lagrange equations related to this last problem are deduced and they show that the relevant optimal bound has a local character, i.e., it depends just on some fictitious plastic deformations produced in the same region of the body where the bounded real plastic deformations are considered. The bounding technique is also generalized to the case of loads arbitrarily varying in a given domain. An application is worked out.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):519-524. doi:10.1115/1.2788923.

By combining the crystalline orientation distribution with a hardening evolution equation, a new elastic/crystalline viscoplastic material model is established. We focus our discussion on looking primarily at the texture effects on the strain localization of limit dome height (LDH) tests which are simulated using our Dynamic-Explicit finite element code. Three crystalline models in addition to the classical plastic potential and associated flow law model (J2 F) are employed. The results demonstrate that, according to our failure criterion, the random orientation model shows the earliest indication of failure. The better formability is obtained for aluminum alloy 6111-T4 and cube texture models than the random crystalline orientation model. The J2 F model shows no signs of strain localization. A comparison between numerical results also confirms that the strain localization region and crystalline rotation are different, due to the crystalline orientation distribution, which is initially set.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):525-531. doi:10.1115/1.2788924.

By utilizing the general solutions derived for the plies with arbitrary fiber orientations under uniform axial strain (Huang and Chen, 1994), the explicit solutions of the edge-delamination stress singularities for the angle-ply and cross-ply laminates are obtained. The dominant edge-delamination stress singularities for the angle-ply laminates are found to be a real constant, −1/2, and a pair of complex conjugates, −1/2 ± i/2π ln {(b + b2 − a2)/a}. For the cross-ply laminates, the significant effect of transverse shear stresses of the laminate is considered and the dominant edge-delamination stress singularities are shown as −1/2 and −1/2 ± i/2π ln {(c2 + c22 − 4c1c3)/2c1}. a, b, cl , c2 , and c3 are the corresponding combined complex constants. In addition, two elementary forms of edge-delamination stress singularity, say, r−1/2 and rδr (ln r)n (δr = n − 3/2, n = 1, 2 . . .) exist for both the angle-ply and cross-ply laminates. Excellent correlations between the present results and available solutions show the validity of the approach. The deficiencies of the solutions available in the literature are compensated. New results for other angle-ply and cross-ply laminates are also provided.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):532-537. doi:10.1115/1.2788925.

This paper describes the theory of wave domain control for the reduced control design of plate systems. A transformation, which changes the original system into an image system in which the control force is designed in wave domain control and wave control, is proposed such that the number of degrees-of-freedom of the undisturbed state in its image system is reduced. The control design in the original system is then derived by an inverse transformation. This work focuses on, first, proposing a new wave control theory and, second, applying the theory for structural control design.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):538-545. doi:10.1115/1.2788926.

A layer-wise theory of laminated plates, which accounts for piecewise constant shear strain in the thickness, is derived from the three-dimensional elasticity theory by imposing suitable constraints on the strain and stress fields. At this aim, the functional of the three-dimensional elasticity is modified according to the Lagrange multipliers theory. In fact, a nonstandard application of the Lagrange theory is presented, because of the simultaneous presence of constraints on dual spaces. The imposed constraints make reactive strain and stress fields arise. Thus, it is necessary to distinguish between elastic and total strain and stress fields. The difference between them is emphasized in a numerical application.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):546-556. doi:10.1115/1.2788927.

In this study, the transient stress fields and the dynamic stress intensity factor of a semi-infinite antiplane crack propagating along the interface between two different media are analyzed in detail. The crack is initially at rest and, at a certain instant, is subjected to an antiplane uniformly distributed loading on the stationary crack faces. After some delay time, the crack begins to move along the interface with a constant velocity, which is less than the smaller of the shear wave speed of these two materials. A new fundamental solution is proposed in this study, and the solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The exact full-field solutions and the stress intensity factor are found in the time domain by using the Cagniard-de Hoop method (de Hoop, 1958) of Laplace inversion. The near-tip fields are also obtained from the reduction of the full-field solutions. Numerical results for the dynamically extending crack are evaluated in detail. The region of the stress singular field dominated in the transient process is also discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):557-561. doi:10.1115/1.2788928.

The in-plane and out-of-plane free vibration frequencies of Archimedes-type spiral springs are computed by the transfer matrix method. Taking into account the effects of the axial and the shear deformations and the rotary inertia, the overall dynamic transfer matrix is computed up to any desired numerical accuracy by the complementary functions method. Since there are no restrictions for the number of coils and for the form of the spring (close-coiled or open-coiled), the presented method is general. After having verified the soundness of the computer program devised, the effects of the number of coils, of the axial and shear deformations, of rotary inertia and of the boundary conditions on the frequencies are also investigated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):562-567. doi:10.1115/1.2788929.

An asymptotic solution within the bounds of steady-state coupled thermoelastodynamic theory is given for the surface displacement and temperature due to a line mechanical/heat source that moves at a constant velocity over the surface of a half-space. This problem is of basic interest in the fields of contact mechanics and tribology, and an exact formulation is considered. The results may serve as a Green’s function for more general thermoelastodynamic contact problems. The problem may also be viewed as a generalization of the classical Cole-Huth problem and the associated Georgiadis-Barber correction. Asymptotic expressions are obtained by means of the two-sided Laplace transform, and by performing the inversions exactly. The range of validity of these expressions is actually quite broad, because of the small value of the thermoelastic characteristic length appearing in the governing equations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):568-575. doi:10.1115/1.2788930.

This article presents an analytical treatment of the dynamic interaction between a crack and an arbitrarily located circular inhomogeneity under antiplane incident wave. The method is based upon the use of a pseudo-incident wave technique which reduces the interaction problem into a coupled solution of a single crack and a single inhomogeneity problems. The newly proposed pseudo-incident wave technique avoids the numerical integration commonly used in the boundary element and volume integral methods and thus provides reliable and accurate analytical solutions. The resulting dynamic stress intensity factor of the crack is verified by comparison with existing results and numerical examples are provided to show the dependence of dynamic shielding and amplification upon the frequency of the incident wave, the material combination and the location of the inhomogeneity. The results show that the toughening associated with special geometric configurations under quasi-static loading may provide undesirable weakening effect upon the crack under dynamic loading in a certain frequency region.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):576-581. doi:10.1115/1.2788931.

The use of a small circular hole in elastostatic photoelasticity to determine the stress tensor for any two-dimensional general loading situation is well known. The original application required fringe-order information at four points on the boundary, on opposite sides, along the axes of symmetry or principal stress directions. Later, to obtain greater precision, it was adapted so that fringe information inside the field could be used. This led to the also limited use of fringe-order information from four points at 1.4 and two times the radius of the hole, along the principal axes of symmetry. More recent work has even allowed the use of fringe-order information, at a fixed radius, anywhere along the two principal axes of symmetry. The greatest limitation of all of these approaches is that the majority of the fringe-order information that is available, away. from the axes of symmetry, is not utilized at all. The current work presents a least-squares approach to the hole method that allows the simultaneous use of information anywhere and at any radial distance from the center of the hole inside the stress field. The objectives of this paper are: to apply the use of the least-squares approach to the hole method in photoelasticity; and, to show the consistent and practical application of this least-squares approach to the hole method. The achievement of this last objective permits the use of the values of specimen birefringence at a large number of points, taken from anywhere in the field around the hole.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):582-589. doi:10.1115/1.2788932.

Numerous dynamical systems undergo, while in motion, imposition and/or removal of constraints. Three phases of motion are involved: a phase during which the motion is defined as unconstrained, a phase during which the motion is defined as constrained, and an intermediate, transition phase, when constraints are imposed or removed. Noncontributing forces (sometimes called nonworking, reaction forces), and noncontributing impulses, namely, impulses associated with noncontributing forces in the transition phase, play a central role in the mechanics of systems undergoing such motions, and are the subject matter of the present paper. Specifically, a six-step procedure is introduced for the determination of noncontributing forces and of noncontributing impulses throughout three phase motions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):590-595. doi:10.1115/1.2788933.

Results of a transient analysis developed to model the dynamic response and establish post-buckling/post snap-thru equilibrium of discrete structures are presented. Three systems that exhibit unstable buckling characteristics are analyzed. The analysis consisted of first statically loading the structures up to there respective static limit loads. The structure is then perturbed from their critical state and a transient analysis is used to model the ensuing dynamic response. The transient formulation is first applied to two simple one-degree-of-freedom systems consisting of rigid links, springs, dampers, and lumped masses. The first of these systems was an arch with a point load applied at its vertex. This structure admits dynamic snap-thru response when loaded beyond its limit load. The second system was a model of a curved panel under an applied axial end-shortening. This system exhibited dynamic buckling behavior consisting of a large decrease in the resultant axial load when loaded beyond its limit load. The transient analysis was then applied to a finite element model of a cylindrical shell with a cutout under an applied axial compression load to model the dynamics of the global buckling response upon reaching its limit load. The results from this study illustrate the usefulness of the transient analysis in modeling the dynamics of an unstable structural response and establishing equilibrium beyond any points of instability.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):596-600. doi:10.1115/1.2788934.

The critical (resonance) velocities of a harmonically varying point load moving uniformly along an elastic layer are determined as a function of the load frequency. It is shown that resonance occurs when the velocity of the load is equal to the group velocity of the waves generated by the load. The critical depths of the layer are determined as function of the load velocity in the case the load frequency is proportional to the load velocity. This is of importance for high-speed trains where the loading frequency of the train wheel excitations is mainly determined by the ratio between the train velocity and the distance between the sleepers (ties). It is shown that the critical depths are decreasing with increasing train velocity. It is concluded that the higher the train velocity, the more important are the properties of the ballast and the border between the ballast and the substrate.

Topics: Stress , Trains , Resonance , Wheels , Waves
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):601-605. doi:10.1115/1.2788935.

This paper considers a symmetric inverse vibration problem for linear vibrating systems described by a vector differential equation with constant coefficient matrices and nonproportional damping. The inverse problem of interest here is that of determining real symmetric, coefficient matrices assumed to represent the mass normalized velocity and position coefficient matrices, given a set of specified complex eigenvalues and eigenvectors. The approach presented here gives an alternative solution to a symmetric inverse vibration problem presented by Starek and Inman (1992) and extends these results to include noncommuting (or commuting) coefficient matrices which preserve eigenvalues, eigenvectors, and definiteness. Furthermore, if the eigenvalues are all complex conjugate pairs (underdamped case) with negative real parts, the inverse procedure described here results in symmetric positive definite coefficient matrices. The new results give conditions which allow the construction of mass normalized damping and stiffness matrices based on given eigenvalues and eigenvectors for the case that each mode of the system is underdamped. The result provides an algorithm for determining a nonproportional (or proportional) damped system which will have symmetric coefficient matrices and the specified spectral and modal data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):606-612. doi:10.1115/1.2788936.

A complete solution of the well-known Mayer’s problem, which is concerned with the possibility of extending Hamilton’s principle expressed in the form valid for conservative dynamical systems to one special case of nonconservative systems (Appell, 1911), is obtained. Namely, the necessary and sufficient conditions which have to be satisfied by the coefficients of the given nonconservative generalized forces so that the Mayer’s potential (and, as a consequence, the descriptive function of the system) can be constructed, are established. This result is illustrated by an example.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):613-619. doi:10.1115/1.2788937.

A new spectral analysis for the asymptotic locations of eigenvalues of a constrained translating string is presented. The constraint modeled by a spring-mass-dashpot is located at any position along the string. Asymptotic solutions for the eigenvalues are determined from the characteristic equation of the coupled system of constraint and string for all constraint parameters. Damping in the constraint dissipates vibration energy in all modes whenever its dimensionless location along the string is an irrational number. It is shown that although all eigenvalues have strictly negative real parts, an infinite number of them approach the imaginary axis. The analytical predictions for the distribution of eigenvalues are validated by numerical analyses.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):620-628. doi:10.1115/1.2788938.

In this study, a cracked body with finite boundaries subjected to static loading and the crack propagating with a constant speed are analyzed. The interaction of the propagating crack with reflected waves generated from traction-free boundaries is investigated in detail. The methodology for constructing the scattered field by superimposing the fundamental solution in the Laplace transform domain is proposed. The fundamental solutions represent the responses of applying exponentially distributed loadings in the Laplace transform domain on the surface of a half-plane or a crack. The dynamic stress intensity factors of a propagating crack induced from the interaction with the first few reflected waves generated from the traction-free boundary are obtained in an explicit closed form. The analytical solutions of dynamic stress intensity factors are compared with available numerical and experimental results and the agreement is quite good. We find one thing very interesting: the dynamic stress intensity factor for a long time period is a universal function of the instantaneous extending rate of a crack tip times the static stress intensity factor for an equivalent stationary crack for the finite strip problem. It was also found that the reflected waves generated from free boundaries always increase the stress intensity factor, and the influence from reflected waves generated from the boundary, which is perpendicular to the crack, are weaker than those generated from the boundary, which is parallel to the crack.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):629-635. doi:10.1115/1.2788939.

The use of fractional derivatives has proved to be very successful in describing the behavior of damping materials, in particular, the frequency dependence of their parameters. In this article the three-parameter model with fractional derivatives of order 1/2 is applied to single-degree-of-freedom systems. This model leads to second-order semidifferential equations of motion for which previously there were no closed-form solutions available. A new procedure that permits to obtain simple closed-form solutions of these equations is introduced. The method is based on the transformation of the equations of motions into a set of first-order semidifferential equations. The closed-form expression of he eigenvalues and eigenvectors of an associated eigenproblem are used to uncouple the equations. Using the Laplace transform method, closed-form expressions to calculate the impulse response function, the step response function and the response to initial conditions are derived.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):636-641. doi:10.1115/1.2788940.

The Hamilton-Jacobi partial differential equation is solved for potential energy functionals of constant, linear, and quadratic form using a class of nonseparable solutions; these solutions give a geometric property to the generating solution, embedding it into the class of conics. These solutions have two basic components, that designated as a kernel component which belongs to the system regardless of the specific dynamics of the system and the primary and secondary system functions that are dependent on the specific initial conditions. Solutions are obtained for the linear oscillator, a rheonomic oscillator and a two-degree-of-freedom system, the latter suggesting an approach for general multidegree-of-freedom systems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):642-648. doi:10.1115/1.2788941.

In engineering applications when the intensity of external forces depends on the response of the system, the input is called parametric. In this paper dynamical systems subjected to a parametric deterministic impulse are dealt with. Particular attention has been devoted to the evaluation of the discontinuity of the response when the parametric impulse occurs. The usual forward difference and trapezoidal integration schemes have been shown to provide only approximated solutions of the jump of the response; hence, the exact solution has been pursued and presented under the form of a numerical series. The impulse is represented throughout the paper by means of a classical Dirac’s delta function; however, a new model of Dirac’s delta is presented and adopted in order to validate the results provided by the numerical series.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):649-657. doi:10.1115/1.2788942.

This paper addresses the response and stability of elastic-plastic steel tubes with square cross section under pure bending. An analytical model with sufficiently nonlinear kinematics to capture the development of ripples in the compression flange was developed. the results indicate that collapse of such tubes is imperfection sensitive for tubes with “high” height-to-thickness ratio (h/t), but the sensitivity decreases as h/t decreases. Experimentally, the tubes collapse due to a limit moment instability which is followed by the formation of a kink on the compression flange of the tubes. The limit moment and the development of the kink are captured well by the analytical model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):658-663. doi:10.1115/1.2788943.

The Hamilton-Jacobi partial differential equation is established for continuum systems; to do this a new concept in material distributions is introduced. The Lagrangian and Hamiltonian are developed, so that the Hamilton-Jacobi equation can be formulated and the principal function defined. Finally the principal function is constructed for the dynamics of a one-dimensional linear elastic bar; the solution for its’ vibrations is then established following the differentiation of the principal function.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):664-669. doi:10.1115/1.2788944.

This paper proposes a new version (fundamentally different from the existing ones) of finite element method for the mean and covariance functions of the displacement for bending beams with spatially random stiffness. Apart from the conventional finite element method for stochastic problems, which utilizes either perturbation or series expansion technique or the Monte Carlo simulation, the present method is based on the newly established variational principles. The finite element scheme is formulated directly with respect to the mean function and covariance function, rather than perturbed components of the displacement. It takes into account an information on joint probability distribution function of the random stiffness to obtain the covariance function of the displacement. Therefore, the accurate solution can be obtained even if the coefficient of variation of the random stiffness is large, in contrast to conventional technique. Several examples are given to illustrate the advantage of the proposed method, compared with the conventional ones.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):670-675. doi:10.1115/1.2788945.

The stochastic vibration of a flexible, articulated, and mobile manipulator is studied. The manipulator is mounted on a vehicle which is supported by a suspension system. Stochastic excitation of the manipulator is induced by the uniform horizontal motion of the vehicle on a traction surface. The power spectral density representation and the state-space representation are used to derive expressions for the covariance matrices of the manipulator tip motions. Sensitivity of the variance of the tip motion to the manipulator configuration, length, vehicle velocity, surface roughness coefficient, and structural damping and stiffness are explored. Suggestions for mobile manipulator design to minimize the influence of the stochastic base vibration on the manipulator tip motion are proposed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):676-683. doi:10.1115/1.2788946.

This paper investigates the nonlinear dynamic response of a linearly elastic string fixed at one boundary and undergoing constant speed circular motion at the other boundary. The response divides into nonlinear steady-state ballooning that is fixed relative to a rotating coordinate system and linearized vibration about the steady state. Single-loop balloons have high tension and purely imaginary eigenvalues. The single-loop vibration frequencies generally decrease with increasing balloon length. Highly extensible strings whirl in first and higher modes with forward whirling modes having lower frequencies. Axially stiff strings exhibit whirling only in higher modes. If the nondimensional string stiffness is larger than 1000, then the inextensible steady-state solutions and the lowest six vibration frequencies match the extensible results to within three percent. One-and-a-half loop balloons are divergently unstable. Long and/or sufficiently extensible strings form low-tension double-loop balloons. Inextensible double balloons are coupled mode flutter unstable. The steady-state balloons, steady-state eyelet tension, and balloon stability are experimentally verified.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):684-691. doi:10.1115/1.2788947.

A procedure for obtaining closed-form homogeneous solutions for the problem of vibration of a discrete viscoelastic system is developed for the case where the relaxation kernel characterizing the constitutive relation of the material is expressible as a sum of exponentials. The developed procedure involves the formulation of an eigenvalue problem and avoids difficulties encountered with the application of the Laplace transform approach to multi-degree-of-freedom viscoelastic systems. Analytical results computed by using the developed method are demonstrated on an example of a viscoelastic beam.

Commentary by Dr. Valentin Fuster

BRIEF NOTES

J. Appl. Mech. 1997;64(3):692-694. doi:10.1115/1.2788948.

This paper evaluates the elastic field in a transversely isotropic half-space caused by a circular flat bonded punch under torsion loading. The elastic field is found by integrating the point force potential functions. For the case of isotropy the present results agree with previous analysis.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):697-700. doi:10.1115/1.2788950.

This paper introduces an effective method for the solution of axisymmetric elastic inclusions in bonded dissimilar half-spaces. The method leads to new concise expressions for the elastic fields in terms of only two scalar Papkovich potential functions. A host of earlier solutions, which were obtained using a range of alternative techniques, involving surface stress relaxation and Hankel transforms, are shown to be subsumed in the present result.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):704-707. doi:10.1115/1.2788952.

It is shown that any two-dimensional elastic tensor can be orthogonally and uniquely decomposed into a symmetric tensor and an antisymmetric tensor. To within a scalar multiplier, the latter turns out to be equal to the right-angle rotation on the space of two-dimensional second-order symmetric tensors. On the basis of these facts, several useful results are derived for the traction boundary value problem of plane linear elasticity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):707-709. doi:10.1115/1.2788953.
Abstract
Topics: Motion
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):710-712. doi:10.1115/1.2788954.
Abstract
Topics: Eigenvalues
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):712-717. doi:10.1115/1.2788955.

The stochastic differential equations for quasi-linear systems excited by parametric non-normal Poisson white noise are derived. Then it is shown that the class of memoryless transformation of filtered non-normal delta correlated process can be reduced, by means of some transformation, to quasi-linear systems. The latter, being excited by parametric excitations, are frst converted into ltô stochastic differential equations, by adding the hierarchy of corrective terms which account for the nonnormality of the input, then by applying the Itô differential rule, the moment equations have been derived. It is shown that the moment equations constitute a linear finite set of differential equation that can be exactly solved.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1997;64(3):717-719. doi:10.1115/1.2788956.

A formulation for the determination of the order of the stress singularities at the tip of a reentrant corner for anisotropic wedges was given by Bogy (1972). Results for orthotropic wedges were obtained as a special case, and it was concluded that the order of the stress singularities at the tip of reentrant orthotropic wedges is always more severe than that of the corresponding isotropic wedge. It is shown here that the order of the stress singularities at the wedge tip can be above or below that of the corresponding isotropic wedge, depending on the material properties.

Commentary by Dr. Valentin Fuster

DISCUSSION

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

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