bluuebunny/crossref_metadata_2025 huggingface dataset:
Stochastic Response of Nonlinear Dynamic Systems Based on a Non-Gaussian Closure
Authors: Ibrahim R. A. , Soundararajan A. , Heo H.
Published: July 2009

The random response of nonlinear dynamic systems involving stochastic coefficients is examined. A non-Gaussian closure scheme is outlined and employed to resolve an observed contradiction of the results obtained by other techniques. The method is applied to a system possessing nonlinear damping. The stationary response is obtained by numerical integration of the closed differential equations of the moments (up to fourth order). An interesting feature of the numerical results reveals the existence of a jump in the response statistic functions. This new feature may be attributed to the fact that the non-Gaussian closure more adequately models the nonlinearity, and thus results in characteristics that are similar to those of deterministic nonlinear systems. The results are compared with solutions derived by the Gaussian closure and stochastic averaging method.
Bimodal Optimization of Compressed Columns on Elastic Foundations
Authors: Plaut R. H. , Johnson L. W. , Olhoff N.
Published: July 2009

We consider columns attached to elastic foundations and compressed by axial end loads. Pinned-pinned, clamped-clamped, and pinned-clamped boundary conditions are treated. The columns have rectangular sandwich cross sections with a fixed lightweight core and identical face sheets of variable thickness. For given total volume, we optimize the variation of the thickness along the column so as to maximize the buckling load. In most cases, the optimal design is bimodal (i.e., associated with two buckling modes). The optimal designs depend on the foundation stiffness, and the largest increase in buckling load relative to a column with constant thickness is 22 percent.
Stability of a Clamped-Free Rotor Partially Filled With Liquid
Authors: Hendricks S. L.
Published: July 2009

When a flexible rotor is partially filled with liquid, the motion is unstable over some operating range. The extent of this operating range depends on various system parameters such as rotor damping, fluid viscosity, the amount of fluid present, etc. If the rotor is arranged so that it must tilt as it vibrates (as in the clamped-free configuration) then the tilt of the rotor greatly complicates the stability analysis. The source of the complication is that the fluid motion becomes three-dimensional. A three-dimensional stability theory is developed here and applied to a simple clamped-free rotor. The results show that the stability boundaries are influenced by both rotor and fluid “gyroscopic stiffening” effects. Brief experimental results are also reported.
Stresses and Displacements on the Boundaries of Circular Rings Diametrically Loaded
Authors: Durelli A. J. , Lin Y. H.
Published: July 2009

The paper deals with stresses and displacements in circular rings of rectangular cross-section, loaded in the plane and perpendicular to the boundary. Values are given for all points at the inside and outside boundaries, are presented parametrically for rings for which the ratio of diameters ID/OD varies from 0 to very close to 1, and have been obtained from several sources, mainly Nelson’s equations. References to some previous contributions are included. The information presented in the paper was not available in a complete manner and will be useful in numerous structural applications. The analysis corresponding to loads applied tangentially to the boundary could be approached in a similar manner.
A Note on the Obtainment of Instantaneous Penetration Information From Final Penetration Data
Authors: Heimdahl O. E. R. , Schulz J. C.
Published: July 2009

A simple, practical technique is described for generating instantaneous penetration depth versus velocity information for a given projectile-target combination from experimentally obtained final penetration depth versus impact velocity data. The technique applies to penetrations where the resistive pressure is a function of velocity only.
Causes and Consequences of Nonuniqueness in an Elastic/Perfectly-Plastic Truss
Authors: Hodge P. G. , Bathe K.-J. , Dvorkin E. N.
Published: July 2009

A complete solution to collapse is given for a three-bar symmetric truss made of an elastic/perfectly-plastic material, using linear statics and kinematics, and the solution is found to be partially nonunique in the range of contained plastic deformation. The introduction of a first-order deviation from symmetry and/or the inclusion of first-order nonlinear terms in the equilibrium equations is found to restore uniqueness. The significance of these effects is analyzed and discussed from mathematical, physical, modelling, computational, and engineering points of view.
Boundary Integral Methods for Thermoelasticity Problems
Authors: Sharp S. , Crouch S. L.
Published: July 2009

The boundary integral method for solving transient heat flow problems is extended to calculate thermally induced stresses and displacements. These results are then corrected by means of an elastostatic solution to satisfy the boundary conditions.
Laboratory-Scale Penetration Experiments Into Geological Targets to Impact Velocities of 2.1 km/s
Authors: Forrestal M. J. , Lee L. M. , Jenrette B. D.
Published: July 2009

We obtained forces on 3.0 and 6.0 CRH, ogival-nosed projectiles penetrating a simulated geological target. A gas gun and a powder gun accelerated foundry core targets (a simulated soft sandstone) to steady velocities between 0.6–2.1 km/s, and the targets subsequently impacted 20.6 mm diameter penetrators. Rigid-body motions of the penetrators were obtained with laser interferometry and piezoelectric accelerometers. At a penetration velocity of about 1.5 km/s, the 6.0 CRH nose shape exhibited severe, permanent deformation; whereas, the 3.0 CRH nose shape remained undeformed for penetration velocities up to 2.1 km/s.
Elastic Wave Scattering From an Interface Crack in a Layered Half Space Submerged in Water: Part II: Incident Plane Waves and Bounded Beams
Authors: Gracewski S. M. , Bogy D. B.
Published: July 2009

This is Part II of a two part paper which analyzes time harmonic elastic wave scattering by an interface crack in a layered half space submerged in water. The analytic solution was derived in Part I. Also numerical results for uniform harmonic normal or shear traction applied to the liquid-solid interface were presented. These were compared with previously published results as a check on the computer program used to obtain the numerical results. Here in Part II, additional numerical results are presented. Plane waves incident from the liquid onto the solid structure are first considered to gain insight into the response characteristics of the structure. The solution for an incident beam of Gaussian profile is then presented since this profile approximates the output of an ultrasonic transducer.
Mechanics of Fatigue Damage and Degradation in Random Short-Fiber Composites, Part II—Analysis of Anisotropic Property Degradation
Authors: Wang S. S. , Chim E. S.-M. , Suemasu H.
Published: July 2009

Based on the microcrack density and cumulative distribution functions obtained in (Wang et al., 1986), cyclic fatigue degradation and associated damage-induced anisotropy of elastic properties of random short-fiber composites are studied. Constitutive equations of the fatigue-damaged composite are derived on the basis of the well-known self-consistent mechanics scheme in conjunction with a three-dimensional elliptic crack theory and the probabilistic functions of microcrack density and cumulative distribution. The anisotropic stiffness degradation is determined as a function of microcrack evolution and accumulation in the damaged composite. Theoretical predictions and experimental data of effective modulus decay during fatigue are in excellent agreement. A damage parameter is introduced to depict quantitatively the degree of homogeneous fatigue damage. The tensorial nature of anisotropic stiffness degradation and fatigue damage is examined in detail. A power-law relationship is established between the rate of damage development and the fatigue loading cycle. The rate of fatigue damage growth is found to decrease exponentially with the loading cycle — a phenomenon unique to the random short-fiber composite. The fundamental mechanics of composite fatigue damage and associated property degradation is elucidated in this paper.
Compressibility of Two-Dimensional Cavities of Various Shapes
Authors: Zimmerman R. W.
Published: July 2009

Muskhelishvili-Kolosov complex stress functions are used to find the stresses and displacements around two-dimensional cavities under plane strain or plane stress. The boundary conditions considered are either uniform pressure at the cavity surface with vanishing stresses at infinity, or a traction-free cavity surface with uniform biaxial compression at infinity. A closed-form solution is obtained for the case where the mapping function from the interior of the unit circle to the region outside of the cavity has a finite number of terms. The area change of the cavity due to hydrostatic compression at infinity is examined for a variety of shapes, and is found to correlate closely with the square of the perimeter of the hole.
Detecting Damage Surfaces in Brittle Materials Using Acoustic Emissions
Authors: Holcomb D. J. , Costin L. S.
Published: July 2009

A technique for mapping damage surfaces is developed and applied to Westerly granite. A damage surface for an elastic-brittle material is similar in concept to a yield surface for an elastic-plastic material and may be defined as the locus of stress states where micro-damage begins to occur. Using a generalization of the Kaiser effect, the onset of acoustic emissions (AE) during loading is taken to indicate the initiation of damage and, thus, contact with the damage surface. By following a number of loading paths, the locus of onset stresses can be mapped and identified as a damage surface. To demonstrate the technique, a large block of granite was loaded along one axis, establishing an initial damage state. Subsequent loading of cores taken from the large block at various angles to the initial loading axis, permitted reloading to the damage surface along different paths. The results show that the onset of AE is a strong function of the angle between the original loading axis and the axis of the core, indicating that the damage surface produced by uniaxial loading is highly anisotropic, even though the damage surface for the virgin material is isotropic.
On the Scale of the Nonlinear Effect in a Crack Problem
Authors: Knowles J. K. , Rosakis A. J.
Published: July 2009

This note provides a necessary and sufficient condition for the nonlinear effect to be “small-scale” in a Mode III crack problem for a Neo-Hookean material.
A Mixture Model for Unidirectionally Fiber-Reinforced Composites
Authors: Murakami H. , Hegemier G. A.
Published: July 2009

A binary mixture theory with microstructure is constructed for unidirectionally fiber-reinforced elastic composites. Model construction is based on an asymptotic scheme with multiple scales and the application of Reissner’s new mixed variational principle (1984). In order to assess the accuracy of the model, comparison of the mixture model predictions with available experimental data on dispersion of harmonic waves is made for boron/epoxy and tungsten/aluminum composites. Formulas for the effective moduli are also presented, and the results are compared with test data and other available predictions.
Shear Stress Intensity Factors for a Planar Crack With Slightly Curved Front
Authors: Gao Huajian , Rice James R.
Published: July 2009

Recent work (Rice, 1985a) has presented the calculations of the first order variation in an elastic displacement field associated with arbitrary incremental planar advance of the location of the front of a half-plane crack in a loaded elastic full space. That work also indicated the relation of such calculations to a three-dimensional weight function theory for crack analysis and derived an expression for the distribution of the tensile mode stress intensity factor along a slightly curved crack front, to first order accuracy in the deviation of the crack front location from a reference straight line. Here we extend the results on stress intensity factors to the shear modes, solving to similar first order accuracy for the in-plane (Mode 2) and antiplane (Mode 3) shear stress intensity factors along a slightly curved crack front. Implications of results for the configurational stability of a straight crack front are discussed. It is also shown that the concept of line tension, while qualitatively useful in characterizing the crack extension force (energy release rate) distribution exerted on a tough heterogeneity along a fracture path as the crack front begins to curve around it, does not agree with the exact first order effect that is derived here.
In-Plane Indentation and Separation of a Flat, or Rounded Rigid Stamp from an Elastic, Finite Layer
Authors: Solecki R.
Published: July 2009

This paper describes symmetric, frictionless indentation of a two-dimensional, rectangular, elastic solid, with two parallel edges constrained against displacements, by a flat, rigid, or rounded stamp pressed against one of the remaining edges. Comparison with the classical problem of indentation of the elastic half-plane by a flat rigid stamp reveals new interesting features of the present analysis: 1) For each relative stamp length a critical aspect ratio of the indented rectangle exists such that for a narrower rectangle the separation of the midsection of the stamp occurs. 2) The stress intensity factors at the ends of the stamp are found to depend strongly on the relative depth of the “beam”; they also depend, but to a lesser degree, on the relative length of the stamp. 3) Unlike in the classical problem here the penetration of the stamp is uniquely determined. Numerical results are provided. The numerical results obtained for the case of the rounded stamp are compared with the results available from earlier studies.
On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part II
Authors: Simo J. C. , Vu-Quoc L.
Published: July 2009

The numerical treatment of the methodology proposed in Part I of this paper is considered in detail. Unlike traditional approaches, a Galerkin spatial discretization of the equations of motion, now referred to the inertial frame, yields the standard form of nonlinear structural dynamics: Mq¨ + Dq˙ + P(q) = F, with M and D constant matrices. Numerical examples that involve finite vibrations coupled with large overall motions are presented. These simulations also demonstrate the capability of the present formulation in handling multibody dynamics.
A General Theory of Shells and the Complementary Potentials
Authors: Wempner G.
Published: July 2009

This theory incorporates the attributes which are essential to the approximation of shells by finite elements. It is limited only by one assumption: Displacement is a linear function of distance along the normal to a reference surface. Deformation is decomposed into rotation and strain; the rotation carries elements of the reference surface to the same surface in any subsequent state. Transverse-shear deformations accommodate simple elements. The theory is couched in the potential Pv and in the complementary potential Pc; these have the property, Pv + Pc= 0 for all admissible (equilibrated) states. The theory is also cast in the complementary functional P¯c of stress and displacement, and the functional P¯v of displacement, strain and stress; P¯c and P¯v are akin to the functionals of Hellinger-Reissner and Hu-Washizu. These alternate functionals provide the means to develop various hybrid elements.
On the Distribution of Hoop Stresses Around Circular Holes in Elastic Sheets
Authors: Kienzler R. , Zhuping Duan
Published: July 2009

By means of well-known solutions of linear plane elasticity in polar coordinates, a remarkably simple formula is derived to calculate the distribution of hoop stresses at the boundary of a circular hole in an infinite sheet subjected to arbitrary (in plane) loads. The application of the formula is demonstrated with several examples.
Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid
Authors: Irie T. , Yamada G. , Tagawa Y.
Published: July 2009

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamical energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramids in which the base is clamped and the other sides are free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically, from which the vibration characteristics arising in the pyramids are studied.
The Reverse Spaghetti Problem: Drooping Motion of an Elastica Issuing from a Horizontal Guide
Authors: Mansfield L. , Simmonds J. G.
Published: July 2009

The nonlinear equations of motion of an elastica that moves out of a horizontal guide at a constant velocity are expressed in terms a dimensionless weight-to-stiffness ratio and a dimensionless velocity. The equations are written in horizontal-vertical directions rather than tangential-normal directions to minimize algebraic complexities. The introduction of deformation potentials allows each of the linear momentum equations to be integrated once. This simplifies the remaining equations. A series solution of the equations, useful for small motions—and perhaps useful for design—is given. To facilitate numerical solution, the triangular space-time domain of the problem is transformed into a square domain in pseudo space-time. Finally, some solutions based on the finite element method are presented for typical values of the dimensionless weight-to-stiffness and velocity parameters.
A Complex-Variable Method for Two-Dimensional Internal Stress Problems and Its Applications to Crack Growth in Nonelastic Materials: Part I—Theory
Authors: Wu K. C. , Hui C. Y.
Published: July 2009

In Part I of this work, a systematic method based on a nonanalytic complex calculus is developed for two-dimensional internal stress problems. The method is employed to derive the internal stress due to a field of nonelastic strain, which can be nonzero at infinity, in an infinite uncracked or cracked body.
On the Characterization of the Transient Stress Field Near the Tip of a Crack
Authors: Ravi-Chandar K. , Knauss W. G.
Published: July 2009

The dynamic stress field near a propagating crack tip is usually characterized in terms of one parameter, namely, the dynamic stress intensity factor. While analytically this is an exact representation at the crack tip itself, under transient conditions, the domain of dominance of the stress intensity factor varies as discussed by Ma and Freund (1986). In this paper, we present experimental results which show that while the stress intensity factor may dominate the near tip stress field under transient conditions as long as the crack velocity is small, it may not be dominant over an appreciable region under other transient conditions of crack tip motion, thus making it difficult to determine this quantity experimentally.
Natural Frequencies of a Railroad Track
Authors: Patil S. P.
Published: July 2009

The natural frequency of an infinite railroad track was first determined by Timoshenko as ωR = √k/m, where k is the constant for the massless Winkler foundation and m is the mass per unit length of the rail. The natural frequencies of the track are determined here by modeling the track as a beam resting on a 3-D inertial elastic layer. It is shown that the mass of the supporting foundation has a significant effect on the natural frequencies of a railroad track. Finally, the concept of “added mass” is introduced in order to determine the natural frequency in a desired mode of vibration, by modeling the track as a beam on the massless Winkler foundation and adding the mass of the foundation to the beam.
On Using Stress Relaxation Tests to Characterize Material Behavior
Authors: Ding J. L. , Findley W. N.
Published: July 2009

Stress relaxation tests, in which the gage length of the specimen was maintained constant by servocontrol, were performed on 2618-T61 aluminum. The test results, which were independent of the stiffness of the testing machine, were converted into a relation between stress and inelastic strain rate. It was found that the contribution by the anelastic component to the total stress relaxation was significant only in the initial stage. The validity of using the obtained stress versus inelastic-strain-rate relation to characterize the material behavior is also discussed. Results do not substantiate the concept of a “hardness” flow curve, but data were well predicted from the creep data by theory based on strain hardening and viscoelasticity.
Elastic Moduli of Lungs
Authors: Budiansky B. , Kimmel E.
Published: July 2009

Macroscopic elastic moduli governing the incremental deformation of pressurized lungs are calculated on the basis of a micromechanical study of an individual lung element (alveolus). The spongy lung tissue (parenchyma) is presumed to be in an initial state of isotropic tension, and the lung alveolus is idealized as a pin-jointed truss in the shape of a regular dodecahedron. The analysis is based on a variational statement of nonlinear structural mechanics, and the results show how the moduli depend on the initial tension and the assumed constitutive behavior of the idealized truss members.
Subsurface Cracks Under Conditions of Slip, Stick, and Separation Caused by a Moving Compressive Load
Authors: Sheppard S. D. , Barber J. R. , Comninou M.
Published: July 2009

The Mode I and II stress intensity factors (KI, KII) at the two tips of a subsurface crack subjected to a moving compressive load are studied. Coulomb friction along the crack faces results in a number of history dependent slip-stick configurations and nonsymmetric variation in KI and KII. The formulation used to study this variation involves a singular integral equation in two variables which must be solved numerically, and because of the history dependence, requires an incremental solution. Crack lengths and coefficients of friction that result in as many as three zones for any load location are considered in this paper, while a previous paper (Sheppard et al., in press) was limited to configurations involving two zones only.
Energy Release Rates in a Fiber-Reinforced Strip
Authors: Mannion L. F.
Published: July 2009

This paper discusses Mode I fracture of a semi-infinite crack in a strip of fiber-reinforced material. The crack is opened by point forces applied to the crack faces. The problem is solved using two models for fiber reinforcement; these models are known as the inextensible and idealized theories and are both based on simplification of the equations of orthotropic elasticity. In the inextensible theory the problem is solved by conformal mapping while the idealized solution involves only stress resultants on free bodies. The energy release rates are calculated from the solution within each model and are then compared.
On Tensile Shock Waves in Rubber-Like Materials
Authors: Mihaˇilescu-Suliciu M. , Suliciu I.
Published: July 2009

The problem of generation of one-dimensional tensile shock waves in rubber-like materials is studied numerically and compared to the exact elastic nonlinear solution and the steady wave solution. It is shown that a rate-type semilinear visco-elastic model can describe the steepening of the wave during its propagation and a “thickness” of the wave is naturally incorporated. An energetic criterion for the numerical stability is discussed. The numerical results point out the uncertainty (difficulty) one may encounter in measuring the dynamic Young’s modulus and Maxwell-type viscosity coefficient.
Elastic Wave Scattering from an Interface Crack: Antiplane Strain
Authors: Bostro¨m A.
Published: July 2009

The two-dimensional scalar problem of scattering of elastic waves under antiplane strain from an interface crack between two elastic half-spaces is considered. The method used is a direct integral equation method with the crack-opening displacement as the unknown. Chebyshev polynomials are used as expansion functions and the matrix in the resulting equations is simplified by contour integration techniques. The scattered far field is expressed explicitly in simple functions and the expansion coefficients. The consequences of energy conservation are explored and are used as a check in the numerical implementation. For incoming plane waves numerical results are given for the total scattered energy and the far field amplitude.
A New Method for Predicting the Critical Taylor Number in Rotating Cylindrical Flows
Authors: Cruickshank J. O.
Published: July 2009

A method for determining the boundaries of dynamic stability of a fluid system, as distinct from the prediction of the subsequent motion, is presented. The method is based on well-known approaches to the problem of instability in elastic systems. The extension of these methods to fluid systems, specifically, to the stability of flow between concentric cylinders, confirms that it may be possible in some cases to determine the boundaries of stability of fluid systems without recourse to an Orr-Sommerfeld type treatment. The results also suggest that the concept of apparent (virtual) viscosity may have implications for fluid stability outside the current realm of turbulence modelling. Finally, it is also shown that flow instability may be preceded by the onset of a critical stress condition in analogy with elastic systems.
Characterization of Cracks From Ultrasonic Scattering Data
Authors: Achenbach J. D. , Sotiropoulos D. A. , Zhu H.
Published: July 2009

An inverse method for ultrasonic scattering data is proposed, to characterize a planar crack of general shape contained in an elastic solid. The method is based on an integral representation for the scattered field in the frequency domain. For a given scattered field the inverse problem has been formulated as a nonlinear optimization problem. At low frequencies its solution gives the location of the crack, the orientation of the crack-plane, and the crack-opening volumes. In addition the Mode I stress-intensity factor is obtained for a related static stress state corresponding to service loads.
A Micromechanical Theory of High Temperature Creep
Authors: Weng G. J.
Published: July 2009

Based on the mechanism of dislocation climb-plus-glide, a micromechanical theory is developed for the high-temperature creep of polycrystals. This model assumes that dislocation climb is responsible for the release of dislocations and whose subsequent glide provides the only significant contribution to the overall creep strain. Taking into consideration the forces acting on both dislocation climb and dislocation glide, a microconstitutive equation is introduced to describe the transient and steady-state creep of slip systems. Together with the self-consistent relation, the creep property of a polycrystal is determined by an averaging process over the behavior of its constituent grains. The developed micromechanical theory is then applied to model the creep behavior of lead at 0.56 Tm, under both tension and shear. Based on these micromechanical analyses, a macroscopic multiaxial theory—involving an effective normal stress to reflect the climb force on the microscale as well as the usual effective shear stress—is also developed. It is found that the effective normal stress, which is independent of the hydrostatic pressure, depends primarily on the second invariant of the deviatoric stress, and only weakly so on the third invariant. Thus despite the distinct presence of two types of microstress, the constitutive equations on the macroscale can still be reasonably described by the second invariant alone even at high temperature.
Circumferential Crack at the Fixed End of a Cylinder in Flexure
Authors: Alabi J. A.
Published: July 2009

The problem of a circumferential through-crack at the fixed end of a cylinder in flexure is considered herein. An expression for the energy-release rate, in closed form, is obtained according to linear fracture mechanics and thin shell theory. The expression consists of two terms, one of which is the energy-release due to shear and the other is the energy-release due to bending. The former is relatively negligible except for very short cylinders. A formula for the flexibility factor is also derived.
The Wrinkling of Thin Membranes: Part I—Theory
Authors: Roddeman D. G. , Drukker J. , Oomens C. W. J. , Janssen J. D.
Published: July 2009

A method to describe the stress situation in a wrinkled membrane is presented. In this paper it will be shown that a special deformation tensor can be chosen which leads to the correct stress state of a membrane after wrinkling when it is substituted in the constitutive equation. The method can be used for anisotropic membranes in geometrically and physically nonlinear analysis. The case of simple shear and stretching of a membrane is considered to illustrate the potency of the method.
General Dynamic Equations of Helical Springs With Static Solution and Experimental Verification
Authors: Lin Yuyi , Pisano Albert P.
Published: July 2009

The general dynamic equations of helical compression springs with circular wire cross section, variable pitch angle, and variable helix radius are derived. The equations are formulated by Hamilton’s principle and a variational method. In contrast to previous studies, the effects of coil flexure bending, variable pitch angle and variable helix radius are taken into account. The general equations are shown to agree with dynamic equations found in literature when the general equations are reduced to simplified forms. For a specific helical spring and static loading, the equations are solved with both the predicted radial expansion and the predicted longitudinal spring compression force in excellent agreement with experimental data.
Limit Cycle Behavior of a Flexible Truck
Authors: Whitman A. M. , Molyneux J. E.
Published: July 2009

We calculate the variation in critical speed of a flexible truck as a function of limit cycle amplitude and truck parameters (i.e., shear and bending stiffnesses, and truck geometry), by means of a perturbation procedure. We find that the creep force nonlinearity is dominant, and that it can cause the nonlinear critical speed to be either lower or higher than the linear critical speed, depending on the values of the two stiffnesses.
A Theory of Particle-Reinforced Plasticity
Authors: Tandon G. P. , Weng G. J.
Published: July 2009

A simple, albeit approximate, theory is developed to determine the elastoplastic behavior of particle-reinforced materials. The elastic, spherical particles are uniformly dispersed in the ductile, work-hardening matrix. The method proposed combines Mori-Tanaka’s concept of average stress in elasticity and Hill’s discovery of a decreasing constraint power of the matrix in polycrystal plasticity. Under a monotonic, proportional loading the latter was characterized, approximately, by the secant moduli of the matrix. The theory is established for both traction and displacement-prescribed boundary conditions, under which, the average stress and strain of the constituents and the effective secant moduli of the composite are explicitly given in terms of the secant moduli of the matrix and the volume fraction of particles. In particular, the yield stress and work-hardening modulus of the composite are shown to be inversely proportional to the deviatoric part of average stress concentration factors of the matrix, and therefore will increase (or decrease) with increasing hard (or soft) particle concentration. It is also found that, even if the matrix is plastically incompressible, the composite as a whole is not. Comparison between the theory and the experiment for a silica/epoxy system shows a reasonable agreement. The theory is also compared with a recently developed one by Arsenault and Taya; while both give the same initial yield stress for the composite, the work-hardening modulus predicted by their theory is found to be higher.
Analytical and Experimental Study of Beam Torsional Stiffness With Large Axial Elongation
Authors: Degener M. , Hodges D. H. , Petersen D.
Published: July 2009

The axial force and effective torsional stiffness versus axial elongation are investigated analytically and experimentally for a beam of circular cross section and made of an incompressible material that can sustain large elastic deformation. An approach based on a strain energy function identical to that used in linear elasticity, except with its strain components replaced by those of some finite-deformation tensor, would be expected to provide only limited predictive capability for this large-strain problem. Indeed, such an approach based on Green strain components (commonly referred to as the geometrically nonlinear theory of elasticity) incorrectly predicts a change in volume and predicts the wrong trend regarding the experimentally determined axial force and effective torsional stiffness. On the other hand, use of the same strain energy function, only with the Hencky logarithmic strain components, correctly predicts constant volume and provides excellent agreement with experimental data for lateral contraction, tensile force, and torsional stiffness—even when the axial elongation is large. For strain measures other than Hencky, the strain energy function must be modified to consistently account for large strains. For comparison, theoretical curves derived from a modified Green strain energy function are added. This approach provides results identical to those of the Neo-Hookean formulation for incompressible materials yielding fair agreement with the experimental results for coupled tension and torsion. An alternative approach, proposed in the present paper and based on a modified Almansi strain energy function, provides very good agreement with experimental data and is somewhat easier to manage than the Hencky strain energy approach.
Micromechanical Aspects of Isotropic Granular Assemblies With Linear Contact Interactions
Authors: Bathurst R. J. , Rothenburg L.
Published: July 2009

The paper presents a micromechanical analysis of plane granular assemblies of discs with a range of diameters, and interacting according to linear contact force-interparticle compliance relationships. Contacts are assumed to be fixed and indestructible. Macroscopically, the system is described in terms of a two-dimensional analogue of generalized Hooke’s law. Explicit expressions for elastic constants in terms of microstructure are derived for dense isotropic assemblies. It is shown that Poisson’s ratio for dense systems depends on the ratio of tangential to normal contact stiffnesses. The derived expression for Poisson’s ratio is verified by numerically simulating plane assemblies comprising 1000 particles. The effect of density on Poisson’s ratio is investigated using numerical simulations. The theory of dense plane systems is extended to dense three-dimensional systems comprising spheres. Finally, it is shown that Poisson’s result ν=1/4 is recovered for spherical particles with central interactions.
A Nonlinear Analysis of an Equilibrium Craze: Part II—Simulations of Craze and Crack Growth
Authors: Ungsuwarungsri T. , Knauss W. G.
Published: July 2009

In this study we investigate the effects of nonlinear fibril behavior on the mechanics of craze and crack growth. The effect of strain-softening cohesive material on crack stability is of particular interest and is examined via a craze and crack model developed in the first part of this work where the formulation and solution of the problem are discussed.1 In this second part, quasi-static growth of a craze with a central crack is analyzed for different nonlinear force-displacement (p-v) relations for the craze fibrils. A “critical crack tip opening displacement” (CTOD), or more precisely, “critical fibril extension” is employed as the criterion for fracture. The p-v relation is further assumed to be invariant with respect to the craze and crack lengths. The results are compared with the Dugdale model; the craze zone size and the energy dissipation rate approach asymptotic values in the limit of long cracks. The problem of craze growth from a precut crack under increasing far-field loading is then studied. In the case where the p-v relation is monotonically softening, the crack can start to grow in an unstable manner before the crack tip opening displacement reaches its critical value.
Plastic Analysis of Ice Contact Problems
Authors: Nordgren R. P.
Published: July 2009

The plane problem of a rigid stamp on a truncated wedge of rigid/perfectly plastic material is solved by the method of characteristics for a generalized von Mises yield function. Quadratic dependence of shear strength on mean normal stress is included in the yield function. For the corresponding three-dimensional contact problem upper bounds on contact force are obtained by limit analysis. Both crushing and splitting failure modes are treated. Numerical results are presented in dimensionless form for application to the prediction of contact stress of ice on arctic structures with sloping sides.
Some History-Dependent Problems for Dissimilar Cylinders With Finite Friction
Authors: Azarkhin A.
Published: July 2009

This paper is concerned with the problem of contact between two dissimilar elastic cylinders. The cylinders are pressed together by a compressive force, P, and loaded by a tangential force, H, which is transmitted in the form of a distribution of frictional tractions at the interface. We assume that Coulomb’s law of friction holds at the interface. We develop a numerical algorithm which is sufficiently general to deal with an arbitrary history of normal and tangential loading, P(t), H(t) and which retains the full description of the coupling between the normal and tangential tractions. The proposed algorithm gives good numerical accuracy in comparison with previous analytical and numerical solutions of a variety of problems for normal and tangential loading of two contacting cylinders with friction. It is also found to be very efficient in the use of computer time. Results for some previously unsolved problems involving the full coupling between normal and tangential contact tractions indicate that such coupling can have a significant effect on the solution, particularly in regard to the extent of slip and stick zones within the contact area.
Analysis of a Power-Law Material Containing a Single Hole Subjected to a Uniaxial Tensile Stress Using the Complex Pseudo-Stress Function
Authors: Lee Y. S. , Smith L. C.
Published: July 2009

The two-dimensional compatibility equation for time-dependent materials described by the power law is expressed in terms of the second derivative of the stress function with respect to complex conjugate variables. The equation is solved by introducing the pseudo-stress function which satisfies the biharmonic equation resulting from the compatibility equation. The relationship between the second derivative of the stress function and the pseudo-stress function is established. The mixed derivative of the stress function associated with the dilatational stress is expressed by an integral of the complicated pseudo-stress function. The velocity and strain-rate components are expressed in terms of the pseudo-stress function. Therefore, responses of power-law creep materials subjected to various boundary conditions can be obtained. Using the pseudo-stress function, the stress distribution in power-law materials, containing a single hole under plane strain and subjected to a uniaxial tensile stress, is found. The Stress Concentration Factors (SCF) on the hole surface, obtained by using the pseudo-stress function, are compared with those under plane stress obtained by another investigator. The results show that the SCF under plane stress is approximately 8 percent higher than that obtained by using the analysis techniques described herein. The maximum tangential stress for m < 0.5 is obtained away from the hole whereas for 0.5 ≤ m ≤ 1 the maximum stress is found at the hole for θ = π/2.
Nonlocal Continuum Damage, Localization Instability and Convergence
Authors: Bazˇant Zdeneˇk P. , Pijaudier-Cabot Gilles
Published: July 2009

A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.
Creep Rupture of Shell Structures Subjected to Cyclic Loading
Authors: Cocks Alan C. F. , Leckie Frederick A.
Published: July 2009

A bounding theorem is obtained for the time of creep rupture of shell structures subjected to cyclic loading. It is found that the shakedown concepts can be used in which a reference rupture stress replaces the yield stress.
Elastic-Plastic Analysis of Cracks on Bimaterial Interfaces: Part I—Small Scale Yielding
Authors: Shih C. F. , Asaro R. J.
Published: July 2009

Full-field numerical solutions for a crack which lies along the interface of an elastic-plastic medium and a rigid substrate are presented. The solutions are obtained using a small strain version of the J2-deformation theory with power-law strain hardening. In the present article, results for loading causing only small scale yielding at the crack tip are described; in subsequent articles the mathematical structure of the crack-tip fields under small scale yielding and results for contained yielding and fully plastic behavior will be presented. We find that although the near-tip fields do not appear to have a separable singular form, of the HRR-type fields as in homogeneous media, they do, however, bear interesting similarities to certain mixed-mode HRR fields. Under small scale yielding, where the remote elastic fields are specified by a complex stress-concentration vector Q = |Q|eiφ with φ being the phase angle between the two in-plane stress modes, we find that the plastic fields are members of a family parameterized by a new phase angle ξ, ≡ φ + εln(QQ/σ02L), and the fields nearly scale with the well-defined energy release rate as evaluated by the J-integral. Here σ0 is the reference yield stress and L is the total crack length (or a relevant length of the crack geometry). Numerical procedures appropriate for solving a general class of interface crack problems are also presented. A description of a numerical method for extracting the mixed mode stress intensities for cracks at interfaces and in homogeneous isotropic or anisotropic media, is included.
Inclusion and Misfit Effects in Hydrostatic Tension
Authors: Wilner Benjamin
Published: July 2009

A study of the effect of an elastic spherical particle embedded in an infinite elastic-plastic matrix subject to hydrostatic tension is presented. Two types of particles will be considered: (1) a perfectly fitting inclusion, (2) a misfitting one. A numerical solution is developed to solve the governing equations. Various combinations of elastic constants and power hardening coefficients are studied for different load levels. The analysis is concentrated on the interfacial normal stresses and its dependence on the different parameters.
Determining the Stress Distributions in Wound Reels of Magnetic Tape Using a Nonlinear Finite-Difference Approach
Authors: Willett M. S. , Poesch W. L.
Published: July 2009

A new method of calculating the internal stress distributions in wound reels of magnetic tape using a finite-difference approach is presented. A functional representation of the nonlinear radial elastic modulus is used to model reel behavior more accurately. Procedures are described to measure the radial elastic modulus and radial Poisson’s ratio. It is shown, by comparison to experimental data, that the method accurately predicts in-roll stress distributions.
Scattering From Three-Dimensional Planar Cracks by the Boundary Integral Equation Method
Authors: Budreck D. E. , Achenbach J. D.
Published: July 2009

The equations governing three-dimensional elastodynamic scattering from planar cracks are formulated and solved in the frequency domain by Boundary Integral Equation (BIE) methods. The formulation requires a regularization of all nonintegrable kernels in the representation integral for the scattered stress field. The regularization procedure is novel in that it requires an initial discretization of the crack. The resulting discretized system of integral equations can be solved explicitly for the unknown crack-opening displacements. The crack-opening displacements, in conjunction with the appropriate representation integral, have been used to calculate far-field quantities of physical interest. Numerical results are compared with those from earlier papers dealing with a penny-shaped crack under normal incidence of a longitudinal wave field. The code has also been applied to elliptic crack geometries of various aspect ratios under normal longitudinal wave incidence. Numerical results are given for crack-opening displacements, scattered far fields, and scattering cross sections. The numerical results indicate that at sufficiently high frequencies, the scattering process is significantly affected by the extra length parameter of the elliptic crack.
Construction of a Consistent Damping Matrix
Authors: Buhariwala K. J. , Hansen J. S.
Published: July 2009

A general method for constructing a material damping matrix in dynamical systems based on viscoelastic assumptions is presented. A generalization of the classical lamination theory, in particular, the consideration of viscoelasticity in the constitutive relation is considered. The discretized equations of motion for a laminated anisotropic viscoelastic plate using the finite-element method are derived. The mass, damping and stiffness matrices are completely defined and arise consistently in the formulation of motion equations. The technique is illustrated by calculating the mass, damping and stiffness matrices of a graphite-reinforced epoxy shell element. The eigenvalues are then calculated for the resulting eigenproblem.
Interpolated Cell Mapping of Dynamical Systems
Authors: Tongue B. H. , Gu K.
Published: July 2009

A method is proposed to efficiently determine the basins of attraction of a nonlinear system’s different steady-state solutions. The phase space of the dynamical system is spacially discretized and the continuous problem in time is converted to an iterative mapping. By means of interpolation procedures, an improvement in the system accuracy over the Simple Cell Mapping technique is achieved. Both basins of attraction for a representative nonlinear system and characteristic system trajectories are generated and compared to exact solutions.