Showing papers in "Journal of Applied Mechanics in 2005"

Electrostatic Forces and Stored Energy for Deformable Dielectric Materials

Abstract: An isothermal energy balance is formulated for a system consisting of deformable dielectric bodies, electrodes, and the surrounding space. The formulation in this paper is obtained in the electrostatic limit but with the possibility of arbitrarily large deformations of polarizable material. The energy balance recognizes that charges may be driven onto or off of the electrodes, a process accompanied by external electrical work; mechanical loads may be applied to the bodies, thereby doing work through displacements; energy is stored in the material by such features as elasticity of the lattice, piezoelectricity, and dielectric and electrostatic interactions; and nonlinear reversible material behavior such as electrostriction may occur. Thus the external work is balanced by (I) internal energy consisting of stress doing work on strain increments, (2) the energy associated with permeating free space with an electric field, and (3) by the electric field doing work on increments of electric displacement or, equivalently, polarization. For a conservative system, the internal work is stored reversibly in the body and in the underlying and surrounding space. The resulting work statement for a conservative system is considered in the special cases of isotropic deformable dielectrics and piezoelectric materials. We identify the electrostatic stress, which provides measurable information quantifying the electrostatic effects within the system, and find that it is intimately tied to the constitutive formulation for the material and the associated stored energy and cannot be independent of them. The Maxwell stress, which is related to the force exerted by the electric field on charges in the system, cannot be automatically identified with the electrostatic stress and is difficult to measure. Two well-known and one novel formula for the electrostatic stress are identified and related to specific but differing constitutive assumptions for isotropic materials. The electrostatic stress is then obtained for a specific set of assumptions in regard to a piezoelectric material. An exploration of the behavior of an actuator composed of a deformable, electroactive polymer is presented based on the formulation of the paper.

The Mechanical Response of Freestanding Circular Elastic Films Under Point and Pressure Loads

Abstract: This paper provides a comprehensive description of the mechanical response of freestanding circular elastic films subjected to point and pressure loads. Regimes of behavior, such as plate, linear membrane, and nonlinear membrane, are identified in terms of two dimensionless variables that allow the creation of a single map that indicates appropriate closed-form solutions. This map provides a theoretical framework to design experiments and interpret film behavior for all orders of magnitude of: film thickness-to-span ratio, deflection, loads, prestretch, and elastic properties. The normalization approach provides the means to quickly identify appropriate simplifications to the nonlinear governing equations, and identify applicable analytical solutions. Numerical results are used to illustrate behavior in transition regions, e.g., the transition for a given plate thickness from small to large deflections under increasing load. Critical loads, thickness and prestretch are identified which indicate when asymptotic plate or membrane solutions are accurate. Asymptotic and numerical results are presented which illustrate finite-sized regions of bending-influenced deformation near point loads and clamped edges. Theoretical predictions for the width of these regions enable us to estimate the validity of analytical strain distributions, and in turn the maximum strains in the film. These results help avoiding brittle fracture or ductile yielding of the film by identifying physical parameters that limit strains to an acceptable level.

Frequency Analysis of the Tuned Mass Damper

Abstract: The damping properties of the viscous tuned mass damper are characterized by dynamic amplification analysis as well as identification of the locus of the complex natural frequencies. Optimal damping is identified by a combined analysis of the dynamic amplification of the motion of the structural mass as well as the relative motion of the damper mass. The resulting optimal damper parameter is about 15% higher than the classic value, and results in improved properties for the motion of the damper mass. The free vibration properties are characterized by analyzing the locus of the natural frequencies in the complex plane. It is demonstrated that for optimal frequency tuning the damping ratio of both vibration modes are equal and approximately half the damping ratio of the applied damper, when the damping is below a critical value corresponding to a bifurcation point. This limiting value corresponds to maximum modal damping and serves as an upper limit for damping to be applied in practice.

A Nonlinear Model for Dielectric Elastomer Membranes

Abstract: The material and geometrical nonlinearities of novel dielectric elastomer actuators make them more difficult to model than linear materials used in traditional actuators. To accurately model dielectric elastomers, a comprehensive mathematical formulation that incorporates large deformations, material nonlinearity, and electrical effects is derived using Maxwell-Faraday electrostatics and nonlinear elasticity. The analytical model is used to numerically solve for the resultant behavior of an inflatable dielectric elastomer membrane, subject to changes in various system parameters such as prestrain, external pressure, applied voltage, and the percentage electroded membrane area. The model can be used to predict acceptable ranges of motion for prescribed system specifications. The predicted trends are qualitatively supported by experimental work on fluid pumps [A. Tews, K. Pope, and A. Snyder, Proceedings SPIE, 2003)]. For a potential cardiac pump application, it is envisioned that the active dielectric elastomer membrane will function as the motive element of the device.

A Fast Boundary Element Method for the Analysis of Fiber-Reinforced Composites Based on a Rigid-Inclusion Model

Abstract: A new boundary element method (BEM) is developed for three-dimensional analysis of fiber-reinforced composites based on a rigid-inclusion model. Elasticity equations are solved in an elastic domain containing inclusions which can be assumed much stiffer than the host elastic medium. Therefore the inclusions can be treated as rigid ones with only six rigid-body displacements. It is shown that the boundary integral equation (BIE) in this case can be simplified and only the integral with the weakly-singular displacement kernel is present. The BEM accelerated with the fast multipole method is used to solve the established BIE. The developed BEM code is validated with the analytical solution for a rigid sphere in an infinite elastic domain and excellent agreement is achieved. Numerical examples of fiber-reinforced composites, with the number of fibers considered reaching above 5800 and total degrees of freedom above 10 millions, are solved successfully by the developed BEM. Effective Young’s moduli of fiber-reinforced composites are evaluated for uniformly and ‘‘randomly’’ distributed fibers with two different aspect ratios and volume fractions. The developed fast multipole BEM is demonstrated to be very promising for large-scale analysis of fiber-reinforced composites, when the fibers can be assumed rigid relative to the matrix materials. @DOI: 10.1115/1.1825436#

Plane-Strain Propagation of a Fluid-Driven Fracture: Small Toughness Solution

Abstract: The paper considers the problem of a plane-strain fluid-driven fracture propagating in an impermeable elastic solid, under condition of small (relative) solid toughness or high (relative) fracturing fluid viscosity. This condition typically applies in hydraulic fracturing treatments used to stimulate hydrocarbons-bearing rock layers, and in the transport of magma in the lithosphere. We show that for small values of a dimensionless toughness K, the solution outside of the immediate vicinity of the fracture tips is given to O1 by the zero-toughness solution, which, if extended to the tips, is characterized by an opening varying as the 2/3 power of the distance from the tip. This near tip behavior of the zero-toughness solution is incompatible with the Linear Elastic Fracture Mechanics (LEFM) tip asymptote characterized by an opening varying as the 1/2 power of the distance from the tip, for any nonzero toughness. This gives rise to a LEFM boundary layer at the fracture tips where the influence of material toughness is localized. We establish the boundary layer solution and the condition of matching of the latter with the outer zero-toughness solution over a lengthscale intermediate to the boundary layer thickness and the fracture length. This matching condition, expressed as a smallness condition on K, and the corresponding structure of the overall solution ensures that the fracture propagates in the viscosity-dominated regime, i.e., that the solution away from the tip is approximately independent of toughness. The solution involving the next order correction in K to the outer zero-toughness solution yields the range of problem parameters corresponding to the viscosity-dominated regime. DOI: 10.1115/1.2047596

Consistent Formulations of the Interaction Integral Method for Fracture of Functionally Graded Materials

Abstract: The interaction integral method provides a unified framework for evaluating fracture parameters (e.g., stress intensity factors and T stress) in functionally graded materials. The method is based on a conservation integral involving auxiliary fields. In fracture of nonhomogeneous materials, the use of auxiliary fields developed for homogeneous materials results in violation of one of the basic relations of mechanics, i.e., equilibrium, compatibility or constitutive, which naturally leads to three independent formulations: nonequilibrium, incompatibility, and constant-constitutive-tensor. Each formulation leads to a consistent form of the interaction integral in the sense that extra terms are added to compensate for the difference in response between homogeneous and nonhomogeneous materials. The extra terms play a key role in ensuring path independence of the interaction integral. This paper presents a critical comparison of the three consistent formulations and addresses their advantages and drawbacks. Such comparison is made both from a theoretical point of view and also by means of numerical examples. The numerical implementation is based on finite elements which account for the spatial gradation of material properties at the element level (graded elements).

Solutions for the Inclined Borehole in a Porothermoelastic Transversely Isotropic Medium

Abstract: A porothermoelastic solution of the general problem of the inclined borehole in a transversely isotropic porous material is presented herein and compared with the isotropic porothermoelastic solution. The governing equations are outlined for the case of general anisotropy and specialized for a transversely isotropic poroelastic material under nonhvdrostatic and nonisothermal in situ conditions. A superposition scheme is employed to obtain the analytical solutions within the isotropic and transversely isotropic poromechanics theory. The borehole generator is assumed to coincide with the material axis of symmetry, in the case of transverse isotropy, yet subjected to a three-dimensional state of stress. A systematic analysis has been carried out to evaluate the effect of the anisotropy of the poromechanical material parameters as well as the thermal material properties on stress and pore pressure distributions and the potential impact on the overall stability of deep wellbore drilling.

Terahertz vibration of short carbon nanotubes modeled as timoshenko-beams

Abstract: Short carbon nanotubes of smaller aspect ratio (say, between 10 and 50) are finding significant application in nanotechnology. This paper studies vibration of such short carbon nanotubes whose higher-order resonant frequencies fall within terahertz range. Because rotary inertia and shear deformation are significant for higher-order modes of shorter elastic beams, the carbon nanotubes studied here are modeled as Timoshenko beams instead of classical Euler beams. Detailed results are demonstrated for double-wall carbon nanotubes of aspect ratio 10, 20, or 50 based on the Timoshenko-beam model and the Euler-beam model, respectively. Comparisons between different single-beam or double-beam models indicate that rotary inertia and shear deformation, accounted for by the Timoshenko-beam model, have a substantial effect on higher-order resonant frequencies and modes of double-wall carbon nanotubes of small aspect ratio (between 10 and 20). In particular, Timoshenoko-beam effects are significant for both large-diameter and small-diameter double-wall carbon nanotubes, while double-beam effects characterized by noncoaxial deflections of the inner and outer tubes are more significant for small-diameter than large-diameter double-wall carbon nanotubes. This suggests that the Timoshenko-beam model, rather than the Euler-beam model, is relevant for terahertz vibration of short carbon nanotubes.

Negative Poisson’s Ratios in Anisotropic Linear Elastic Media

Abstract: Poisson 's ratio for an anisotropic linear elastic material depends on two orthogonal directions n and m. Materials with negative Poisson's ratios for all (n,m) pairs are called completely auxetic while those with positive Poisson's ratios for all (n,m) pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some (n,m) pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.

Flaw Tolerance in a Thin Strip Under Tension

Abstract: Recent studies on hard and tough biological materials have led to a concept called flaw tolerance which is defined as a state of material in which pre-existing cracks do not propagate even as the material is stretched to failure near its limiting strength. In this process, the material around the crack fails not by crack propagation, but by uniform rupture at the limiting strength. At the failure point, the classical singular stress field is replaced by a uniform stress distribution with no stress concentration near the crack tip. This concept provide an important analogy between the known phenomena and concepts in fracture mechanics, such as notch insensitivity, fracture size effects and large scale yielding or bridging, and new studies on failure mechanisms in nanostructures and biological systems. In this paper, we discuss the essential concept for the model problem of an interior center crack and two symmetric edge cracks in a thin strip under tension. A simple analysis based on the Griffith model and the Dugdale-Barenblatt model is used to show that flaw tolerance is achieved when the dimensionless number A n =ΓE/(S 2 H) is on the order of 1, where r is the fracture energy, E is the Young s modulus, S is the strength, and H is the characteristic size of the material. The concept of flaw tolerance emphasizes the capability of a material to tolerate cracklike flaws of all sizes.

Numerical Analysis of Nanotube Based NEMS Devices — Part II: Role of Finite Kinematics, Stretching and Charge Concentrations

Abstract: In this paper ci nonlinear analysis of nanotuhe based nano-electromechanical systems is reported. Assuming continuum mechanics, the complete nonlinear equation of the elastic line of the nanotube is derived and then numerically solved. In particular, we study singly and doubly clamped nanotubes under electrostatic actuation. The analysis emphasizes the importance of nonlinear kinematics effects in the prediction of the pull-in voltage of the device, a key design parameter. Moreover, the nonlinear behavior associated with finite kinematics (i.e., large deformations), neglected in previous studies, as well as charge concentrations at the tip of singly clamped nanotuhes, are investigated in detail. We show that nonlinear kinematics results in an important increase in the pull-in voltage of doubly clamped nanotube devices, but that it is negligible in the case of singly damped devices. Likewise, we demonstrate that charge concentration at the tip of singly clamped devices results in a significant reduction in pull-in voltage. By comparing numerical results to analytical predictions, closed form formulas are verified. These formulas provide a guide on the effect of the various geometrical variables and insight into the design of novel devices.

Analytical Solution of a Dynamic System Containing Fractional Derivative of Order One-Half by Adomian Decomposition Method

Abstract: The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI λ D μ controller for the control of dynamical systems, etc Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and materials science are also described by differential equations of fractional order The solution of the differential equation containing a fractional derivative is much involved Instead of an application of the existing methods, an attempt has been made in the present analysis to obtain the solution of an equation in a dynamic system whose damping behavior is described by a fractional derivative of order 1/2 by the relatively new Adomian decomposition method The results obtained by this method are then graphically represented and compared with those available in the work of Suarez and Shokooh [Suarez, L E, and Shokooh, A, 1997, An Eigenvector Expansion Method for the Solution of Motion Containing Fraction Derivatives, ASME J Appl Mech, 64, pp 629-635] A good agreement of the results is observed

Elastic Fields due to Eigenstrains in a Half-Space

Abstract: Engineering components inevitably encounter various eigenstrains, such as thermal expansion strains, residual strains, and plastic strains. In this paper, a set of formulas for the analytical solutions to cases of uniform eigenstrains in a cuboidal region-influence coefficients, is presented in terms of derivatives of four key integrals. The linear elastic field caused by arbitrarily distributed eigenstrains in a half-space is thus evaluated by the discrete correlation and fast Fourier transform algorithm, along with the discrete convolution and fast Fourier transform algorithm. By taking advantage of both the convolution and correlation characteristics of the problem, the formulas of influence coefficients and the numerical algorithms are expected to enable efficient and accurate numerical analyses for problems having nonuniform distribution of eigenstrains and for contact problems.

Nonlinear Vibration of Parametrically Excited, Viscoelastic, Axially Moving Strings

Abstract: The dynamic response of parametrically excited, axially moving viscoelastic belts is investigated in this paper. Results are compared to previous work in which the partial, not material, time derivative was used in the viscoelastic constitutive relation. It is found that this added steady state dissipation greatly affects both the existence and amplitudes of nontrivial limit cycles. The discrepancy increases with increasing translation speed. To limit the comparison to the additional physics included in the model, the solution procedure of Zhang and Zu [1,2], who applied the method of multiple scales to the governing equations prior to discretization, is retained. The excitation here is provided by physically stretching the belt. In this case, viscoelastic behavior and excitation frequency also affects the amplitude of the tension fluctuations.

A Second Look at the Higher-Order Theory for Periodic Multiphase Materials

Abstract: In this communication, we present a reformulation, based on the local/global stiffness matrix approach, of the recently developed higher-order theory for periodic multiphase materials, Aboudi et al. [Linear Thermoelastic Higher-Order Theory for Periodic Multiphase Materials, J. Appl. Mech., 68(5), pp. 697-707]. This reformulation reveals that the higher-order theory employs an approximate, and standard, elasticity approach to the solution of the unit cell problem of periodic multiphase materials based on direct volumeaveraging of the local field equations and satisfaction of the local continuity conditions in a surface-averge sense. This contrasts with the original formulation in which different moments of the local equilibrium equations were employed, suggesting that the theory is a variant of a micropolar, continuum-based model. The reformulation simplifies the derivation of the global system of equations governing the unit cell response, whose size is substantially reduced through elimination of redundant continuity equations employed in the original formulation, allowing one to test the theory's predictive capability in most demanding situations. Herein, we do so by estimating the elastic moduli of periodic composites characterized by repeating unit cells obtained by rotation of an infinite square fiber array through an angle about the fiber axis. Such unit cells possess no planes of material symmetry in the rotated coordinate system, and may contain a few or many fibers, depending on the rotation angle, which the reformulated theory can easily accommodate. The excellent agreement with the corresponding results obtained from the standard transformation equations confirms the new model's previously untested predictive capability for a class of periodic composites characterized by nonstandard, multi-inclusion repeating unit cells lacking planes of material symmetry. Comparison of the effective moduli and local stress fields with the corresponding results obtained from the original Generalized Method of Cells, which the higher-order theory supersedes, confirms the need for this new model, and dramatically highlights the original model's shortcomings for a certain class of unidirectional composites.

Numerical Analysis of Nanotube-Based NEMS Devices—Part I: Electrostatic Charge Distribution on Multiwalled Nanotubes

Abstract: The charge distribution on the surface of a biased conductive, finite-length, cylindrical nanotube, free standing above an infinite grounded plane, is investigated. The diameter range of the cylinder tube under study is 20-60 nm, which is much larger than the screening length, meaning the quantum and statistical effects on the charge distribution are negligible. The relationship between the charge distribution and the geometry of the nanotube is examined in detail by classical electrostatics using full three-dimensional numerical simulations based on the boundary element method. A model of the concentrated charge at the end of nanotubes is proposed. The charge distribution for a clamped cantilever nanotube is also computed and discussed. The findings here reported are of particular usefulness in the design and modeling of electrostatic actuated nanotube/ nanowire based nano-electromechanical systems.

Steady Mechanics of Belt-Pulley Systems

Abstract: Steady state analysis of a two-pulley belt drive is conducted where the belt is modeled as a moving Euler-Bernoulli beam with bending stiffness. Other factors in the classical creep theory, such as elastic extension and Coulomb friction with the pulley, are retained, and belt inertia is included. Inclusion of the bending stiffness leads to nonuniform distribution of the tension and speed in the belt spans and alters the belt departure points from the pulley. Solutions for these quantities are obtained by a numerical iteration method that generalizes to n-pulley systems. The governing boundary value problem (BVP), which has undetermined boundaries due to the unknown belt-pulley contact points, is first converted to a standard fixed boundary form. This form is readily solvable by general purpose BVP solvers. Bending stiffness reduces the wrap angles, improves the power efficiency, increases the span tensions, and reduces the maximum transmissible moment.

Evolution of Wrinkles in Elastic-Viscoelastic Bilayer Thin Films

Abstract: This paper develops a model for evolving wrinkles in a bilayer thin film consisting of an elastic layer and a viscoelastic layer. The elastic layer is subjected to a compressive residual stress and is modeled by the nonlinear von Karman plate theory. A thin-layer approximation is developed for the viscoelastic layer. The stability of the bilayer and the evolution of wrinkles are studied first by a linear perturbation analysis and then by numerical simulations. Three stages of the wrinkle evolution are identified: initial growth of the fastest growing mode, intermediate growth with mode transition, and, finally, an equilibrium wrinkle state.

A Comparison of the Structural Response of Clamped and Simply Supported Sandwich Beams With Aluminium Faces and a Metal Foam Core

Abstract: Plastic collapse modes for clamped sandwich beams have been investigated experimentally and theoretically for the case of aluminium face sheets and a metal foam core. Three initial collapse mechanisms have been identified and explored with the aid of a collapse mechanism map. It is shown that the effect of clamped boundary conditions is to drive the deformation mechanism towards plastic stretching of the face sheets. Consequently the ultimate strength and level of energy absorption of the sandwich beam are set by the face sheet ductility. Limit load analyses have been performed and simple analytical models have been developed in order to predict the postyield response of the sandwich beams; these predictions are validated by both experiments and finite elements simulations. It is shown experimentally that the ductility of aluminium face sheets is enhanced when the faces are bonded to a metal foam core. Finally, minimum weight configurations for clamped aluminium sandwich beams are obtained using the analytical formulas for sandwich strength, and the optimal designs are compared with those for sandwich beams with composite faces and a polymer foam core.

Dynamic Spherical Cavity Expansion in an Elastoplastic Compressible Mises Solid

Abstract: The elastoplastic field induced by a self-similar dynamic expansion of a pressurised spherical cavity is investigated for the compressible Mises solid. The governing system consists of two ordinary differential equations for two stress components where radial velocity and density are known functions of these stresses. Numerical illustrations of radial profiles of field variables are presented for several metals. We introduce a new solution based on expansion in powers of the nondimensionalized cavity expansion velocity, for both elastic/perfectly plastic response and strain-hardening behavior. A Bernoulli-type solution for the dynamic cavitation pressure is obtained from the second-order expansion along with a more accurate third-order solution. These solutions are mathematically closed and do not need any best fit procedure to numerical data, like previous solutions widely used in the literature. The simple solution for elastic/perfectly plastic materials reveals the effects of elastic-compressibility and yield stress on dynamic response. Also, an elegant procedure is suggested to include strain-hardening in the simple elastic/perfectly plastic solution. Numerical examples are presented to demonstrate the validity of the approximate solutions. Applying the present cavitation model to penetration problems reveals good agreement between analytical predictions and penetration depth tests.

A Stabilized Mixed Finite Element Method for Nearly Incompressible Elasticity

Abstract: We present a new multiscale/stabilized finite element method for compressible and incompressible elasticity. The multiscale method arises from a decomposition of the displacement field into coarse (resolved) and fine (unresolved) scales. The resulting stabilized-mixed form consistently represents the fine computational scales in the solution and thus possesses higher coarse mesh accuracy. The ensuing finite element formulation allows arbitrary combinations of interpolation functions for the displacement and stress fields. Specifically, equal order interpolations that are easy to implement but violate the celebrated Babushka-Brezzi inf-sup condition, become stable and convergent. Since the proposed framework is based on sound variational foundations, it provides a basis for a priori error analysis of the system. Numerical simulations pass various element patch tests and confirm optimal convergence in the norms considered.

Using Series-Series Iwan-Type Models for Understanding Joint Dynamics

Abstract: In mechanical assemblies, the energy loss induced by joints and interfaces can account for a significant portion of the overall structural dissipation. This work considers the dynamical behavior of an elastic rod on a frictional foundation as a model for the dissipation introduced by micro-slip in mechanical joints. In a quasi-static loading limit, the deformation of the rod and hence the frictional dissipation can be solved in closed form. The resulting model is a continuum model of series arrangements of parallel Jenkins elements. For a general class of normal load distributions, the resulting energy loss per forcing cycle follows a power-law and is qualitatively similar to observed experimental findings. Finally, these results are compared with those obtained from a discrete formulation of the rod including inertial effects. For loading conditions that are consistent with mechanical joints, the numerical results from the discrete model are consistent with the closed form predictions obtained in the quasistatic limit.

Strength Analyses of Sandwich Pipes for Ultra Deepwaters

Abstract: Design requirements for pipelines regarding both ultimate strength and flow assurance in ultra deepwater scenarios motivated the development of a new sandwich pipe which is able to combine high structural and thermal insulation properties. In this concept, the annulus is filled with low cost materials with adequate thermal insulation properties and good mechanical resistance. The aim of this research work is to perform small-scale laboratorial tests and to develop a finite element model to evaluate the structural performance of such sandwich pipes with two different options of core material. After calibrated in view of the experimental results, a three-dimensional finite element model incorporating nonlinear geometric and material behavior is employed to perform strength analyses of sandwich pipes under combined external pressure and longitudinal bending. Ultimate strength envelopes for sandwich pipes are compared with those generated for single-wall steel pipes with equivalent collapse pressures. The study shows that sandwich pipe systems with either cement or polypropylene cores are feasible options for ultra deepwater applications.

Stochastic dynamics of impact oscillators

Abstract: The purpose of this work is to develop an averaging approach to study the dynamics of a vibro-impact system excited by random perturbations. As a prototype, we consider a noisy single-degree-of-freedom equation with both positive and negative stiffness and achieve a model reduction, i.e., the development of rigorous methods to replace, in some asymptotic regime, a complicated system by a simpler one. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative Hamiltonian system with either center type or saddle type fixed points. We achieve the model-reduction through stochastic averaging. Examination of the reduced Markov process on a graph yields mean exit times, probability density functions, and stochastic bifurcations.

Coupled Plastic Wave Propagation and Column Buckling

Abstract: The plastic buckling of columns is explored in a regime where plastic wave propagation and lateral buckling are nonlinearly coupled. Underlying the work is the motivation to understand and quantify the dynamic crushing resistance of truss cores of all-metal sandwich plates where each truss member is a clamped column. Members are typically fairly stocky such that they buckle plastically and their load carrying capacity decreases gradually as they buckle, even at slow loading rates. In the range of elevated loading rates of interest here, the columns are significantly stabilized by lateral inertia, resisting lateral motion and delaying buckling and loss of load carrying capacity to relatively large overall plastic strains. The time scale associated with dynamic axial behavior, wherein deformation spreads along the column as a plastic wave, is comparable to the time scale associated with lateral buckling such that the two phenomena are coupled. Several relevant problems are analyzed using a combination of analytical and numerical procedures. Material strain-rate dependence is also taken into account. Detailed finite element analyses are performed for axially loaded columns with initial imperfections, as well as for inclined columns in a truss core of a sandwich plate, with the aim of determining the resistance of the column to deformation as dependent on the loading rate and the relevant material and geometric parameters. In the range of loading rates of interest, dynamic effects result in substantial increases in the reaction forces exerted by core members on the faces of the sandwich plate with significant elevation in energy absorption. DOI: 10.1115/1.1825437

Buckling of long sandwich cylindrical shells under external pressure

Abstract: The paper deals with the theoretical prediction of buckling loads for sandwich long cylindrical shells with laminated facings and foam core. The loading is a uniform hydrostatic pressure, which means that the loading remains normal to the deflected surface during the buckling process. Several fiber materials are used in the laminated facings. The materials are: Boron/Epoxy, Graphite/Epoxy and Kevlar/Epoxy laminates with 0° orientation with respect to the hoop direction. These various materials are employed to provide comparative data that can be used in design. Shell theory results are generated with and without accounting for the transverse shear effect. Moreover, results based on three-dimensional elasticity are also generated for comparison purposes. The effect of the ratio of radius to thickness is assessed.

An Efficient and Accurate Numerical Method of Stress Intensity Factors Calculation of a Branched Crack

Abstract: Based on the analytical solution of Crouch to the problem of a constant discontinuity in displacement over a finite line segment in an infinite elastic solid, in the present paper, the crack-tip displacement discontinuity elements, which can be classified as the left and the right crack-tip elements, are presented to model the singularity of stress near a crack tip. Furthermore, the crack-tip elements together with the constant displacement discontinuity elements presented by Crouch and Starfied are used to develop a numerical approach for calculating the stress intensity factors (SIFs) of general plane cracks. In the boundary element implementation, the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called the hybrid displacement discontinuity method (HDDM). Numerical examples are given and compared with the available solutions. It can be found that the numerical approach is simple, yet very accurate for calculating the SIFs of branched cracks. As a new example, cracks emanating from a rhombus hole in an infinite plate under biaxial loads are taken into consideration. The numerical results indicate the efficiency of the present numerical approach and can reveal the effect of the biaxial load on the SIFs. In addition, the hybrid displacement discontinuity method together with the maximum circumferential stress criterion (Erdogan and Sih) becomes a very effective numerical approach for simulating the fatigue crack propagation process in plane elastic bodies under mixed-mode conditions. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the HDDM. Crack propagation is simulated by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented.

Veering Phenomena in Systems With Gyroscopic Coupling

Abstract: The sharp divergence of two root-loci for a critical value of the parameters is called veering. Veering phenomena are interesting since they involve relevant energetic exchanges between the eigenmodes and strongly affect the undamped forced response of the system. A straightforward perturbation approach has already been used in the literature to analyze the dependence of the eigensprectrum on a system parameter and formulate a veering criterion. This perturbation approach and other ideas are generalized to the study of veering in discrete and continuous systems with gyroscopic operators of internal coupling and the results applied to a real electromechanical interaction.

Connections Between Stability, Convexity of Internal Energy, and the Second Law for Compressible Newtonian Fluids

Abstract: In this note we provide proofs of the following statements for a compressible Newtonian fluid: (i) internal energy being a convex function of entropy and specific volume is equivalent to nonnegativity of both specific heat at constant volume and isothermal bulk modulus; (ii) convexity of internal energy together with the second law of thermodynamics imply linear stability of the rest state; and (iii) linear stability of the rest state together with the second law imply convexity of internal energy.