bluuebunny/crossref_metadata_2025 huggingface dataset:
Discharge of a Compressible Fluid Through a Control Valve
Authors: Weng Z. M. , Ting Alice A. L. , Chow W. L.
Published: July 2009

The original analysis on incompressible flow discharge from a vessel through an axisymmetric control valve has been extended to the discharge of a compressible fluid. The inviscid anlaysis is based on the method of hodograph transformation. While it is simple to account for the effect of compressibility for conditions of subcritical pressure ratios, special treatment must be applied to establish the sonic line and the free jet boundaries under conditions of supercritical pressure ratios. Discharge characteristics have been established for different pressure ratios and positions of the control valve. This series of investigations provides ample evidence that the hodograph transformation coupled with numerical computations is effective in dealing with problems of this nature.
Utilizing Dynamic Stability to Orient Parts
Authors: Singer N. C. , Seering W. P.
Published: July 2009

The intent of this research is to study the dynamic behavior of a solid body resting on a moving surface. Results of the study are then used to propose methods for controlling the orientation of parts in preparation for automatic assembly. Two dynamic models are discussed in detail. The first examines the impacts required to cause reorientation of a part. The second investigates the use of oscillatory motion to selectively reorient parts. This study demonstrates that the dynamic behaviors of solid bodies, under the conditions mentioned above, vary considerably with small changes in geometry or orientation.
A Beam Theory for Large Global Rotation, Moderate Local Rotation, and Small Strain
Authors: Danielson D. A. , Hodges D. H.
Published: July 2009

Kinematical relations are derived to account for the finite cross-sectional warping occurring in a beam undergoing large deflections and rotations due to deformation. The total rotation at any point in the beam is represented as a large global rotation of the reference triad (a frame which moves nominally with the reference cross section material points), a small rotation that is constant over the cross section and is due to shear, and a local rotation whose magnitude may be small to moderate and which varies over a given cross section. Appropriate variational principles, equilibrium equations, boundary conditions, and constitutive laws are obtained. Two versions are offered: an intrinsic theory without reference to displacements, and an explicit theory with global rotation characterized by a Rodrigues vector. Most of the formulas herein have been published, but we reproduce them here in a new concise notation and a more general context. As an example, the theory is shown to predict behavior that agrees with published theoretical and experimental results for extension and torsion of a pretwisted strip. The example also helps to clarify the role of local rotation in the kinematics.
On the Threshold Force for Chaotic Motions for a Forced Buckled Beam
Authors: Tang D. M. , Dowell E. H.
Published: July 2009

The effects of higher modes on the chaotic oscillations of a buckled beam under forced external excitation are studied. Of principal interest are the threshold force required for chaotic motions and the influence of damping on the system response. A comparison is also presented of results from numerical simulations with experimental data.
Dynamics of Symmetrizable Nonconservative Systems
Authors: Inman D. J. , Olsen C. L.
Published: July 2009

This work examines a subclass of nonconservative and nonself-adjoint linear distributed parameter systems modeled by partial differential equations subject to various boundary conditions. It has been previously illustrated that certain nonself-adjoint operators arising in mechanics can be shown to be self-adjoint with respect to a particular self-adjoint operator. The work here extends and formalizes this approach to include systems with viscous damping. The extension presented here follows the lumped parameter case presented earlier and places emphasis on the existence of eigenfunctions for use in performing modal analysis.
Some Aspects of the Behavior of High Order Mode Internal Gravity Waves on Interfaces With Finite Thickness
Authors: Muller M. R. , Shang P. C.
Published: July 2009

The role of interface thickness on the propagation of high order internal wave modes is examined using simple analytical models for an unmixed, thick interface and one that has been vigorously mixed. It is shown that for the unmixed case, a local maximum in wave speed occurs as the interface grows from zero thickness, but that only for mode-one waves does the maximum point occur within areas of practical significance. When these models are applied to three wave resonant interactions, it is found that vigorous mixing will augment energy transfer rates when the interface thickness surpasses a certain value.
Thermoelastic Solutions for Thermal Distributions Moving Over Half Space Surfaces and Application to the Moving Heat Source
Authors: Bryant M. D.
Published: July 2009

A method is developed for obtaining fundamental thermal and thermoelastic solutions for thermal distributions moving over the surface of an elastic half space. This method uses the concept of a moving temperature wave along with a novel form of an exponential Fourier transform. The technique is developed and then demonstrated on the example of a moving heat source. Exact results are matched with results from Carslaw and Jaeger (1959) and Barber (1984).
On the Mechanical Modeling of the Interfacial Region in Bonded Half-Planes
Authors: Delale F. , Erdogan F.
Published: July 2009

In this paper the crack problem for two bonded dissimilar homogeneous elastic half-planes is considered. It is assumed that the interfacial region can be modeled by a very thin layer of nonhomogeneous material. Even though the formulation given is rather general, in the particular model used the elastic properties of the interfacial layer are assumed to vary continuously between that of the two semi-infinite planes. The layer is assumed to have a series of collinear cracks parallel to the nominal interface. The related mixed boundary problem is formulated for arbitrary crack surface tractions which can be used to accommodate any general external loading condition through a proper superposition. A single crack problem for two different material combinations is solved as examples, and Modes I and II stress-intensity factors, the energy release rate and the direction of a probable crack growth are calculated.
A Crystallographic Model for Nickel Base Single Crystal Alloys
Authors: Dame L. T. , Stouffer D. C.
Published: July 2009

The purpose of this research is to develop a tool for the mechanical analysis of nickel-base single-crystal superalloys, specifically Rene N4, used in gas turbine engine components. This objective is achieved by developing a rate-dependent anisotropic constitutive model and implementing it in a nonlinear three-dimensional finite-element code. The constitutive model is developed from metallurgical concepts utilizing a crystallographic approach. An extension of Schmid’s law is combined with the Bodner-Partom equations to model the inelastic tension/compression asymmetry and orientation-dependence in octahedral slip. Schmid’s law is used to approximate the inelastic response of the material in cube slip. The constitutive equations model the tensile behavior, creep response and strain-rate sensitivity of the single-crystal superalloys. Methods for deriving the material constants from standard tests are also discussed. The model is implemented in a finite-element code, and the computed and experimental results are compared for several orientations and loading conditions.
The Singularity at the Apex of a Rigid Wedge Embedded in a Nonlinear Material
Authors: Duva J. M.
Published: July 2009

The singular behavior of the stress and strain fields at the apex of a square rigid wedge embedded in a nonlinear material under plane-strain conditions is described. Both a power law and a bilinear law for the nonlinear material are considered.
Thermal Stresses Due to a Plane Crack in General Anisotropic Material
Authors: Sturla F. A. , Barber J. R.
Published: July 2009

A solution is given for the thermoelastic stress field due to the obstruction of a uniform heat flux by a plane crack in a generally anisotropic body. A Green’s function formulation is used to reduce the problem to a set of singular integral equations which are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is, therefore, reformulated assuming a contact zone at one crack tip. The extent of this zone and the stress intensity factors in all three modes at each crack tip are obtained as functions of the applied stress and heat flux.
Three-Dimensional Analysis of Laminates With Cross Cracks
Authors: Aboudi J. , Lee S. W. , Herakovich C. T.
Published: July 2009

An approximate 3-dimensional analysis to study the effects of a system of transverse and longitudinal matrix cracks in cross-ply laminates is given. The method considers a repeating unit cell containing three subcells in every one of which the displacement vector is expanded to second order. The equilibrium equations, in conjunction with continuity conditions of displacements and tractions, together with the appropriate loading and boundary conditions, provide a system of equations for the elastic field variables. The dependence of the effective axial stiffness and Poisson’s ratio of the laminate on the cross-cracks density is given. The Poisson’s ratio appears to be a good indicator for the state of damage of the laminate. The distribution of the axial displacement and stress in the cracked laminate is shown.
Unbonded Contact of a Square Plate on an Elastic Half-Space or a Winkler Foundation
Authors: Li Hui , Dempsey J. P.
Published: July 2009

The unbonded frictionless receding contact problem of a thin plate placed under centrally symmetric vertical loading while resting on an elastic half-space or a Winkler foundation is solved in this paper. The problem is transformed into the solution of two-coupled integral-series equations over an unknown contact region. The problem is nonlinear by virtue of unilateral contact and therefore needs to be solved iteratively. Special attention is given to the edge and corner contact pressure singularities for the plate on the elastic half-space. Comparison is made with other relevant numerical results available.
A Practical One-Parameter Integral Method for Laminar Incompressible Boundary Layer Flow With Transpiration
Authors: Thomas L. C. , Amminger W. L.
Published: July 2009

A practical one-parameter polynomial type integral method is developed in this paper for laminar incompressible plane and thin axisymmetric boundary layer flow with transpiration and pressure gradient. The method features the use of approximations for the velocity distribution that are based on second and third order polynomial approximations for the distribution in shear stress. These approximations are used to develop solutions to the integral momentum equation for similar and nonsimilar flows. The accuracy of the method is generally within about 3 percent, except near separation where the error can reach 10 to 15 percent. The range of conditions for which the method applies covers a fairly wide range of blowing and suction rates and pressure gradients which encompasses plane and axisymmetric stagnation flows and extends to separation. Because of its fundamental nature, the approach provides a basis for generalization to heat and mass transfer and turbulent flow, and provides a framework for the development of more accurate multiple parameter integral methods for transpired boundary layer flow.
Effect of Entrance Region on Laminar Startup Flow in Pipes
Authors: Andersson H. I. , Kristoffersen R.
Published: July 2009

The ultimate effects of the entrance region on startup flow are considered by one-dimensional steady-state analysis. By roughly accounting for the entrance loss, semiempirical formulas for the average velocity and the hydrodynamic entrance length in terms of a startup flow parameter and an entrance effect coefficient are provided. The startup flow parameter M = gHD4/2048v2L2 is identified as the fundamental parameter of the problem. While Szymanski’s solution is recovered in the long pipe limit, i.e., M → 0, the steady-state volume flux through short pipes (M > 0.05) is significantly reduced due to entrance region effects.
Penetration of 6061-T651 Aluminum Targets With Rigid Long Rods
Authors: Forrestal M. J. , Okajima K. , Luk V. K.
Published: July 2009

We developed engineering models for forces on rigid, long rods with spherical, ogival, and conical noses that penetrated rate independent, elastic-perfectly plastic targets. The spherical and cylindrical, cavity-expansion approximations simplified the target analyses, so we obtained closed-form penetration equations. To verify our models, we performed terminal-ballistic experiments with 7.1-mm dia., 0.024 kg, marging steel rods and 152-mm dia., 6061-T651 aluminum targets. The models predicted penetration depths that were in reasonable agreement with the data for impact velocities between 0.4-1.4 km/s.
On a Paradoxical Phenomenon Related to Beams on Elastic Foundation: Could External Compliant Leads Reduce the Strength of a Surface-Mounted Device?
Authors: Suhir E.
Published: July 2009

Compliant external electrical leads are often utilized as strain buffers in surface-mounted device technology to provide the necessary stress relief for the device, when the substrate is subjected to bending. A paradoxical situation was observed, however, during testing of compliant leaded hybrid integrated circuits (HIC): In some tests the bending moment, applied to the printed wire board (PWB) and causing HIC fracture, turned out smaller (not greater), when leads of greater compliance were installed. We show that such a paradoxical situation is due to the redistribution of lead reactions at certain combinations of HIC length, HIC and PWB flexural rigidity, and spring constant of the elastic attachment. Our analysis has indicated, that only sufficiently compliant leads can essentially reduce the stresses, while leads of moderate compliance can result in even greater stresses in the HIC than stiff leads. We suggest an easy-to-calculate governing parameter, which characterizes the mechanical behavior of HIC/PWB and similar assemblies with compliant attachments.
Theory of Elasticity of Filament Wound Anisotropic Ellipsoids With Specialization to Torsion of Orthotropic Bars
Authors: Kardomateas G. A.
Published: July 2009

The problem of determining the state of stress in a homogeneous ellipsoidal body (as can be produced by filament winding on a mandrel of elliptical cross-section) is formulated. The material is assumed to possess curvilinear anisotropy, referred to the coordinate system that is inherent to the geometry of the filament wound body. The differential equations that govern the stress functions are derived for the general anisotropy and loading cases. Then the results for the special case of torsion of an orthotropic body are derived.
Low Velocity Impact of an Elastic Plate Resting on Sand
Authors: Chen H. L. , Lin W. , Keer L. M. , Shah S. P.
Published: July 2009

This article describes the measurement and analysis of plate and soil response under low velocity impact. A free-drop impact system was developed to generate the dynamic loading on the plate free surface. The radial strain of the target plate, the longitudinal wave speed and the acceleration of the sand were measured. The measured wave speed data were then used to evaluate the elastic constants of the sand. An analysis based on linear elastodynamics was developed for transient waves on a thin plate resting on an elastic half space. The contact stresses and the normal displacements of the plate were taken as unknown functions. The contact between the plate and the half space were assumed frictionless. The experimental results of the radial strain at the bottom of the target plate and the acceleration of the sand beneath the center of the target plate were compared with the analytical solution. The arrival time, the duration, and the magnitude have good correlation between the analysis and experiment. The overall results appear good and provide an understanding of the transmission of impact load through the plate, the interaction between the plate and the sand, and the propagation of the load into the sand.
Chaotic Motions of a Torsional Vibration Absorber
Authors: Shaw S. W. , Wiggins S.
Published: July 2009

We consider large amplitude motions of a pendulum-type centrifugal vibration absorber which is used for the reduction of torsional oscillations in rotating machinery. The basic two degree-of-freedom model is shown to possess chaotic dynamics for certain ranges of parameter values. The method used is a variation of Melnikov’s method (cf., Guckenheimer and Holmes, (1983), Chapter 4) developed for slowly varying oscillators (Wiggins and Holmes (1987), Wiggins and Shaw (1988)).
The Thickness of Steady Plane Shear Flows of Circular Disks Driven by Identical Boundaries
Authors: Hanes D. M. , Jenkins J. T. , Richman M. W.
Published: July 2009

We employ balance laws and constitutive relations for rapid, dense, plane flows of identical circular disks together with boundary conditions at a bumpy wall to analyse steady shearing flows maintained by the relative motion of two identical, parallel walls. The disks and the walls are assumed to be frictionless and nearly elastic. Given the properties of the flowing disks, those of the boundary, and the ratio of the tangential and normal tractions applied to the boundary, we determine what the distance between the walls must be for a steady solution to be possible. For these steady solutions we relate the velocity of the walls to the normal and tangential tractions applied to them. We find that in certain circumstances steady motions may be maintained even when the ratio of tangential to normal traction is much less than its value in a homogeneous simple shear. In the Appendix, the corresponding results for spheres are outlined.
Deflection and Snapping of Ring-Loaded Spherical Caps
Authors: Brodland G. Wayne , Cohen H.
Published: July 2009

The axisymmetric behavior of both shallow, and deep, ring-loaded spherical caps, which are simply supported but otherwise unconstrained at their edges, is investigated using a large-strain shell theory based on a variational principle. A numerical technique is used to extract the solution. Particular attention is paid to highly nonlinear phenomena such as snapthrough, single and multiple snapback, and load-free everted states.
Order and Chaos in a Discrete Duffing Oscillator: Implications on Numerical Integration
Authors: Reinhall P. G. , Caughey T. K. , Storti D. W.
Published: July 2009

In this paper, we study the dynamics of some two-dimensional mappings which arise when standard numerical integration schemes are applied to an unforced oscillator with a cubic stiffness nonlinearity, i.e., the Duffing equation. While the continuous time problem is integrable and is solved analytically in terms of Jacobi elliptic functions, the discrete versions of this simple system arising from standard integration schemes exhibit very complicated dynamics due to the presence of homoclinic tangles. We present an alternative scheme for discretizing the nonlinear term which preserves the integrable dynamics of the continuous system and derive analytic expressions for the orbits and invariant curves of the resulting mapping.
Probability Distribution of the Eigenvalues of the Random String Equation
Authors: Iyengar R. N. , Manohar C. S.
Published: July 2009

The probability distribution of the eigenvalues of a second-order stochastic boundary value problem is considered. The solution is characterized in terms of the zeros of an associated initial value problem. It is further shown that the probability distribution is related to the solution of a first-order nonlinear stochastic differential equation. Solutions of this equation based on the theory of Markov processes and also on the closure approximation are presented. A string with stochastic mass distribution is considered as an example for numerical work. The theoretical probability distribution functions are compared with digital simulation results. The comparison is found to be reasonably good.
An Integral Variational Equation for Transport Processes in a Moving Fluid
Authors: Geskin E. S.
Published: July 2009

An integral variational equation can adequately describe heat, mass, and momentum transfer in a moving chemically reactive fluid. The Euler-Lagrange equations corresponding to the suggested variational equation are identical to the equations of entropy, momentum, angular momentum, and mass balance. The constructed Lagrangian density relates energy change in the system to the work and energy dissipation of the system. For steady-state processes, the Lagrangian density includes convective energy flow through the system boundary, energy dissipation in the system, and work of the system. The proposed variational equation is equivalent to the expansion of the principle of minimum energy dissipation.
A Similar Flow Between Two Disks in the Presence of a Magnetic Field
Authors: Hamza E. A.
Published: July 2009

A similarity solution may be obtained for a flow between two parallel disks which, at time t*, are spaced a distance H(1 − αt*)1/2 apart and a magnetic field proportional to B0(1 − αt*)−1/2 is applied perpendicular to the disks. Approximate analytic solutions are given and a numerical solution to the resulting nonlinear ordinary differential equations is presented. The effects of the magnetic forces on the velocity profiles and on the normal forces which the fluid exerts on the disks are studied. It has been found that by increasing the magnetic force a considerable increase in the load can be achieved.
Growth of an Infinitesimal Cavity in a Rate-Dependent Solid
Authors: Abeyaratne Rohan , Hou Hang-sheng
Published: July 2009

This study examines the effect of rate dependence on growth of an infinitesimal cavity in a homogeneous, isotropic, incompressible material. Specifically, a sphere containing a traction-free void of infinitesimal initial radius is considered, its outer surface being subjected to a prescribed uniform radial nominal stress p, which is suddenly applied and then held constant. The sphere is composed of a particular class of rate-dependent materials. The large strains which occur in the vicinity of the void are accounted for in the analysis, and the problem is reduced to a nonlinear initial value problem, which is then studied qualitatively through a phase plane analysis. The principal results of this paper consist of two equations that are derived between the applied stress p and the cavity radius b: p = pˆ(b) and p = p(b). The first of these describes a curve which separates the (p, b)-plane into regions where cavitation does and does not occur. The second describes a curve which further subdivides the former subregion—the post-cavitation region—into domains where void expansion occurs slowly and rapidly.
Analysis of a Vertical Crack in a Multilayered Medium
Authors: Lin W. , Keer L. M.
Published: July 2009

A general mathematical formulation to analyze cracks in a multilayered medium is constructed. First, a matrix for a single layer is formed in the Hankel transformed domain. Then a global matrix is formed by assembling together each layer matrix. The displacement and stress fields are obtained by inverting the Hankel transform. Finally, the planar crack problem is solved by the boundary integral method. The results given are the crack opening displacement and the stress intensity factors.
The Sternberg-Koiter Conclusion and Other Anomalies of the Concentrated Couple
Authors: Dundurs John , Markenscoff Xanthippi
Published: July 2009

A notable conclusion reached by Sternberg and Koiter 30 years ago is that an elastic wedge loaded by a concentrated couple at its vertex remains a well-posed elasticity problem only as long as the wedge does not degenerate into a reentrant corner. A closely related problem is that of a concentrated couple approaching the tip of a crack. It is shown in this article that the corresponding limits in the stress fields do not exist, but that the Bueckner weight function is obtained by appropriately diminishing the magnitude of the couple. It is also shown that the elastic fields, when a concentrated couple approaches an interface between two materials, depend on the direction from which the interface is approached. Moreover, the elastic wedge embedded in another material is discussed.
Maximal Crack Tip Shielding by Microcracking
Authors: Ortiz M. , Giannakopoulos A. E.
Published: July 2009

It is shown here that the shielding effect of a distribution of microcracks, i.e., the extent to which they alleviate the severity of a near-tip singular field, is maximized when the microcracks develop normal to the direction of maximum tension. In passing, we derive mode I asymptotic solutions for a class of anisotropic damage models.
Obtaining Residual Stresses in Metal Forming After Neglecting Elasticity on Loading
Authors: Lee Yong-Shin , Dawson Paul R.
Published: July 2009

A methodology for computing residual stresses in forming operations is examined in which the elasticity is neglected during the loading phase of the operation. The elastic response is recovered on unloading through the analysis of an initially-stressed body. Two examples are presented which provide a quantitative assessment of the accuracy of the approach. The first is the axisymmetric expansion of a thick-walled tube. In this case the residual stresses are compared to those computed with an elastic-plastic analysis for both the loading and unloading phases. The second example is a ring upsetting application that has been analyzed using a finite element formulation and for which there are experimental data available for comparison.
Axisymmetric Elastic Deformations of a Finite Circular Cylinder With Application to Low Temperature Strains and Stresses in Solder Joints
Authors: Suhir E.
Published: July 2009

We consider an axially symmetric elastic problem for a finite circular cylinder whose end planes are subjected to identical radial deformations. The obtained solution is used to evaluate low temperature strains and stresses for solder joints in assemblies having matched thermal expansion coefficients of the components. The thermomechanical behavior of solder joints is determined in this case by the mismatch between the solder and the soldered materials. It is shown that the arising strains and stresses, especially in the axial direction, can be rather great and could possibly result in the failure of the joint. They can be brought down by increasing its height-to-diameter ratio. The stresses can also be reduced if low modulus solder materials, such as 95 percent Pb/5 percent Sn solders or indium-based alloys, are used.
Uncoupled Wave Systems and Dispersion in an Infinite Solid Cylinder
Authors: Kim Yoon Young
Published: July 2009

In this study, it is shown that there exist uncoupled wave systems for general non-axisymmetric wave propagation in an infinite isotropic cylinder. Two cylindrical surface conditions corresponding to the uncoupled wave systems are discussed. The solutions of the uncoupled wave systems are shown to provide proper bounds of Pochhammer’s equation for a free cylindrical surface. The bounds, which are easy to construct for any Fourier number in the circumferential direction, can be used to trace the branches of Pochhammer’s equation. They also give insight into the modal composition of the branches of Pochhammer’s equation at and between the intersections of the bounds. More refined dispersion relations of Pochhammer’s equation are possible through an asymptotic analysis of the itersections of the branches of Pochhammer’s equation with one family of the bounds. The asymptotic nature of wave motion corresponding to large wave numbers, imaginary or complex, for Pochhammer’s equation is studied. The wave motion is asymptotically equivoluminal for large imaginary wave numbers, and is characterized by coupled dilatation and shear for large complex wave numbers.
Bending of Simply-Supported Elliptic Plates: B.P.M. Solutions With Second-Order Derivative Boundary Conditions
Authors: Parnes R.
Published: July 2009

A higher-order boundary perturbation method (B.P.M.) is formulated to treat a class of problems defined in an elliptic domain with associated boundary conditions expressed in terms of second-order derivatives. The method is applied to study a simply-supported elliptic plate subjected to a central lateral point load. The accuracy is investigated and the B.P.M. solution is found to yield highly accurate results for moderately elliptic domains.
Bending of Plates on Thin Elastomeric Foundations
Authors: Dillard D. A.
Published: July 2009

Closed-form and series solutions are presented for the bending of plates bonded to a thin elastomeric foundation which is in turn bonded to a rigid substrate. The standard fourth-order governing differential equation of a classical Winkler elastic foundation becomes a sixth-order equation for the case of an incompressible foundation. Oscillation decay rates are shown to be significantly different from those of the Winkler solution due to the incompressibility of the elastomer.
An Energy Approach to Anomalous Damped Elastic-Plastic Response to Short Pulse Loading
Authors: Borino G. , Perego U. , Symonds P. S.
Published: July 2009

In beams with full-end constraints, loaded transversely by short pressure pulses, the effect of extensional plastic deformation is to make possible instabilities related to snap buckling in the elastic-plastic recovery after the first peak displacement (Symonds and Yu, 1985). In the present paper we make use of a damped, Shanley-type model to study the calculation of the final displacement, reached asymptotically. We show that plots of the elastic strain energy and of the total energy as functions of the displacement help to guide thinking. They provide clarification of previously observed phenomena (Genna and Symonds, 1988) that appear complex at small damping, and lead to lower and upper bounds on the load parameter such that anomalous responses are observed. The response is calculable with the usual accuracy in problems where bifurcations are concerned.
Unloading Waves in a Plucked Hyperelastic String
Authors: Wegner J. L. , Haddow J. B. , Tait R. J.
Published: July 2009

The governing equations for the finite deformation plane motion of a hyperelastic string are obtained in conservation form. These equations and the corresponding jump relations are used to investigate the response of a symmetrically-plucked string when it is suddenly released. Similarity solutions, which are valid until the first reflection occurs at a fixed end, are obtained for two strain energy functions. Justification is given for the use of isothermal strain energy functions.
An Exact Transient Study of Dislocation Emission and its Effects on Dynamic Fracture Initiation
Authors: Brock L. M. , Wu J.-S.
Published: July 2009

Closed-form transient solutions for the micromechanical process of screw dislocation emission from a stationary crack which is subjected to SH-wave diffraction are presented. The dislocations are allowed to leave the crack edge in arbitrary directions, either singly or in pairs. Imposition of an emission criterion that is both based on the dislocation force concept and is similar to criteria applied to quasi-static emission studies allows expressions for the instants of dislocation emission and arrest and the distance traveled by the dislocation to be obtained. These expressions are studied for their dependence on parameters such as emission direction and speed, and several distinctive dynamic effects are observed. A standard fracture criterion is then imposed, and conditions for determining whether fracture will precede or follow emission are established in terms of real time. Finally, some estimates for the orders of magnitudes of the parameters involved in this micromechanical process are given.
Elastic Yield Zone Around an Interfacial Crack Tip
Authors: Zywicz Edward , Parks David M.
Published: July 2009

A closed-form approximate solution for a small-scale yielding (SSY) plastic zone around a planar interfacial crack tip, occurring between two dissimilar ideally-bonded elastic half spaces, is obtained by equating the elastically-calculated Mises equivalent stress with the material yield strength, σys. The dimensionless parameter ζ(θ), which is defined as ζ(θ) = ∠K + εlnrp (θ), where ∠K is the phase angle of the complex stress intensity factor K, ε is the bimaterial constant, and rp (θ), is the polar representation of the plastic zone radius, naturally arises. The SSY interfacial load angle (ILPA), defined as ζ0 = ∠K + εln(KK/σ2ysπcosh2(πε)), leads to periodic zone growth. The ILPA characterizes the overall applied load phase by combining the oscillatory radial phase shift, attributable to the increase in zone size due to increased loading, with ∠K. At a particular angle θ0 from the uncracked interface, the plastic zone radius thus calculated is independent of ∠K, proportional to KK, and has no oscillatory radial phase dependence. The derived plastic zone expression reproduces the shape characteristics, and it modestly reproduces the zone size when compared with solutions for an elastic/perfectly-plastic solid adjoint to an elastic solid. As the strain-hardening exponent in the plastically deforming medium decreases, agreement between the approximation and various accurate numerical solutions improves. In the limiting case when ε = 0, the well-known homogeneous elastic solutions for pure Mode I and Mode II are recovered, as well as all possible mixed-mode combinations. Approximate validity conditions for the existence of Williams-type asymptotic fields (traction-free crack faces) are presented.
How to Model a Bonded Joint
Authors: Villaggio Piero
Published: July 2009

The problem is considered of a semi-infinite plane region bonded to a rigid region, with the boundary of contact being in the shape of a cosine curve. It is shown that, when a rigid displacement is applied to the boundary of the elastic region, there is a particular value of the amplitude of the contact curve that minimizes the sum of the strain energy and adhesion energy.
The Extended Free Formulation of Finite Elements in Linear Elasticity
Authors: Felippa Carlos A.
Published: July 2009

The free formulation of Bergan and Nyga˚rd (1984) has been successfully used in the construction of high-performance finite elements for linear and nonlinear structural analysis. In its original form the formulation combines nonconforming internal displacement assumptions with a specialized version of the patch test. The original formulation is limited, however, by strict invertibility conditions linking the assumed displacement field to the nodal displacements. The present paper lifts those restrictions by recasting the free formulation within the framework of a mixed-hybrid functional that allows internal stresses, internal displacements, and boundary displacements to vary independently. This functional contains a free parameter and includes the potential energy and the Hellinger-Reissner principles as special cases. The parameter appears in the higher-order stiffness of the element equations.
An Alternative Perturbation Procedure of Multiple Scales for Nonlinear Dynamics Systems
Authors: Lau S. L. , Cheung Y. K. , Chen Shuhui
Published: July 2009

An alternative perturbation procedure of multiple scales is presented in this paper which is capable of treating various periodic and almost periodic steady-state vibrations including combination resonance of nonlinear systems with multiple degrees-of-freedom. This procedure is a generalization of the Lindstedt-Poincare´ method. To show its essential features a typical example of cubic nonlinear systems, the clamped-hinged beam, is analyzed. The numerical results for the almost periodic-free vibration are surprisingly close to that obtained by the incremental harmonic balance (IHB) method, and the analytical formulae for steady-state solution are, in fact, identical with that of conventional method of multiple time scales. Moreover, detail calculations of this example revealed some interesting behavior of nonlinear responses, which is of significance for general cubic systems.
On Realization of Program Constraints: Part I—Theory
Authors: Parczewski Jan , Blajer Wojciech
Published: July 2009

The problem of realization of program constraints is considered. The classical theory approach based on replacing the constriant reactions by adequate control forces has been generalized to the case when the control forces are not collinear with program constraint vectors or, in the extreme, when the control forces do not project in these directions at all (control forces are tangent to constraint manifolds). A classification of possible ways of program constraint realization is proposed and a general solution of the problem is presented.
On Realization of Program Constraints: Part II—Practical Implications
Authors: Blajer Wojciech , Parczewski Jan
Published: July 2009

Practical implications of a general mathematical model of realization of program constraints are investigated. Illustrative examples of different types of this realization are demonstrated and discussed. Some general conclusions concerning the problem of program constraint realization are drawn.
A Study of the Effects of Baffles on Rotating Compressible Flows
Authors: Gunzburger Max D. , Wood Houston G. , Wayland Rosser L.
Published: July 2009

Onsager’s pancake equation for the fluid dynamics of a gas centrifuge is modified for the case of centrifuges with baffles which render the flow domain doubly connected. A finite element algorithm is used for solving the mathematical model and to compute numerical examples for flow fields induced by thermal boundary conditions and by mass injection and extraction.
Nonlinear Analysis of Elastomeric Keyboard Domes
Authors: Lau John H. , Jeans Albert H.
Published: July 2009

The large deflection of an elastomeric dome is studied using the finite element method. The material properties of the elastomer are described by a hyperelastic model in order to capture the strain energy stored in the dome during deformation. The nonlinear responses are determined by the modified Riks procedure, and the calculated load-deflection curve agrees well with experimental results. In addition, a pressurized thick-walled spherical hyperelastic shell is analyzed and the stress results obtained by the finite element method are in excellent agreement with the closed-form solutions. The results provide a better understanding of the mechanical behavior of elastomeric keyboard domes and demonstrate the feasibility of using the finite element method to design such structures.
Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part II—Structure of Small-Scale Yielding Fields
Authors: Shih C. F. , Asaro R. J.
Published: July 2009

In Part I we found that although the near tip fields of cracks on bimaterial interfaces do not have a separable form of the HRR type, they appear to be nearly separable in an annular zone within the plastic zone. Furthermore, the fields bear strong similarities to mixed mode HRR fields for homogeneous medium. Based on our numerical results, we have been able to identify a clear mathematical structure. We found that the small-scale yielding crack tip fields are members of a family parameterized by a near tip phase angle ξ, and that the fields nearly scale with the value of the J-integral. In Part II, the original derivation of the mathematical structure of the small-scale yielding fields is elaborated upon. The issue of crack face contact is addressed and the phenomenology is described in terms of the phase parameter ξ. Crack tip plastic deformation results in an open crack for a range of ξ which is nearly symmetric about the state corresponding to pure remote tension. Plane-strain plastic zones and crack tip fields for the complete range of ξ are presented. Over distances comparable to the size of the dominant plastic zone, the stress levels that can be achieved are limited by the yield stress of the weaker (lower yield strength) material. On the other hand, the stresses well within the plastic zone are governed by the strain-hardening behavior of the more plastically compliant (lower strain-hardening) material. We observe that the extent of the annular zone where the fields are nearly separable (i.e., of the HRR form) is dependent on the remote load combinations and the material combination. When the tractions on the interface are predominantly tensile, there are no indications of crack face contact over any length scale of physical relevance. Instead, the crack tip opens smoothly and crack tip fields as well as the crack opening displacement are scaled by the J-integral. The paper concludes with a discussion on the range of load combinations which could be applied to two fracture test specimen geometries to obtain valid fracture toughness data.
Representations of Stress Intensity Factors by Path-Independent Integrals
Authors: Wu Kuang-Chong
Published: July 2009

Stress intensity factors are represented by path independent integrals for linear elastic materials. Two types of representation are given. The first type of integrals are expressed by integration over contours surrounding a crack tip. Those of the second type are integrated over contours enclosing a finite crack. The path independent integrals are applied to determine the stress intensity factors due to a body force and a dislocation for a finite crack in an infinite anisotropic body.
An Inverse Transient Thermoelastic Problem for a Transversely-Isotropic Body
Authors: Noda Naotake , Ashida Fumihiro , Tsuji Tomoaki
Published: July 2009

The present paper discusses an analytical method for an inverse problem of three-dimensional transient thermoelasticity in a transversely-isotropic solid. The inverse thermoelastic problem consists of the determination of the condition of heating when the conditions of displacements and stresses are given at some points of the solid considered. Applying the Laplace and Fourier transforms as well as the new potential function method, the temperature, displacements, and stresses are represented by the potential functions alone, and they are determined from the prescribed conditions. The heating condition is obtained from the boundary condition for the temperature field. As a practical example of an inverse problem, the heating temperature of a transversely-isotropic infinite circular cylinder is determined in the case where the radial displacement is given at an arbitrary cylindrical section and the radial and shear stresses are free on the lateral surface of the cylinder. Numerical calculations are carried out to illustrate the heating temperature of the cylinder as well as the temperature and stresses on the lateral surface of the cylinder.
Crack Fronts Trapped by Arrays of Obstacles: Numerical Solutions Based on Surface Integral Representation
Authors: Fares Nabil
Published: July 2009

This paper addresses the trapping of the front of a planar crack as it impinges upon a row of periodically-emplaced tough obstacles. The initial penetration of the crack between obstacles, under increasing load, as well as the ultimate unstable joining of penetrating segments so as to surround and by-pass the obstacles, are analyzed. The formulation used for the associated three-dimensional elasticity problems of half-plane cracks with nonuniform, curved fronts is a Boundary Element Method (BEM). This incorporates a specialized fundamental solution for an opening (prismatic) dislocation source ahead of a half-plane crack with a straight front (Rice, 1985a). The implementation of this BEM and associated mesh moving with the front is first discussed after which a series of case studies are carried out. The first two case studies evaluate the accuracy of previously obtained linear perturbation results (Rice (1985b), Gao and Rice (1988)). The last study is a crack growth simulation around a periodic array of circular obstacles with a particle size to spacing ratio of 0.5. The simulation shows in that case that crack trapping achieves an effective toughening ratio of 2.35 when the particle-to-matrix-toughness ratio (Kcp/Kc) is greater than 3.52. The simulation also gives lower bounds on the net toughening when Kcp/Kc < 3.52.
Effects of Lateral Surface Conditions in Time-Harmonic Nonsymmetric Wave Propagation in a Cylinder
Authors: Kim Yoon Young , Steele Charles R.
Published: July 2009

The present work is a part of the effort toward the development of an efficient method of solution to handle general nonsymmetric time-harmonic end conditions in a cylinder with a traction-free lateral surface. Previously, Kim and Steele (1989a) develop an approach for the general axisymmetric case, which utilizes the well-known uncoupled wave solutions for a mixed lateral wall condition. For the case of a traction-free lateral wall, the uncoupled wave solutions provide: (1) a convenient set of basis functions and (2) approximations for the relation between end stress and displacement which are asymptotically valid for high mode index numbers. The decay rate with the distance from the end is, however, highly dependent on the lateral wall conditions. The present objective was to demonstrate that the uncoupled solutions of the nonsymmetric waves discussed by Kim (1989), which satisfy certain mixed lateral wall conditions, can be utilized in an analogous manner for the asymptotic analysis of the traction-free case. Results for the end displacement/stress due to various end conditions, computed by the present method and by a more standard collocation method, were compared. The present method was found to reduce the computational effort by orders of magnitude.
Optimal, Coupled-Modal Control of Distributed Parameter Systems With Discrete Actuators
Authors: Tadjbakhsh Iradj G. , Su Yuan-an
Published: July 2009

Optimal control of flexible continuous structures subjected to arbitrary time-varying distributed loads is considered. The control is to be implemented by discrete sets of sensors and actuators that monitor the response and apply the necessary forces. The dynamics of the uncontrolled structure is assumed to be governed by a linear, self-adjoint partial differential equation. The control forces at any time are determined on the basis of minimization of the total energy of the system at that time. This leads to a causal optimal algorithm whereby control forces are determined solely on the basis of information available up to the time at which control is being implemented. The effectiveness of the algorithm is demonstrated by applying it to a beam subjected to an impulse.
Period Doubling and Chaos in Unsymmetric Structures Under Parametric Excitation
Authors: Szemplin´ska-Stupnicka W. , Plaut R. H. , Hsieh J.-C.
Published: July 2009

Nonlinear oscillations of a single-degree-of-freedom, parametrically-excited system are considered. The stiffness involves quadratic and cubic nonlinearities and models a shallow arch or buckled mechanism. The excitation frequency is assumed to be close to twice the natural frequency of the system. Numerical integration is used to obtain phase plane portraits, power spectra, and Poincare´ maps for large-time motions. Period-doubling bifurcations and several types of limit cycles and chaotic behavior are observed. Approximate analytical techniques are applied to analyze some of the limit cycles and transitions of behavior. The results are used to estimate the parameter region in which chaos may occur.