Showing papers in "Journal of Applied Mechanics in 1996"

Modeling of cyclic ratchetting plasticity, part i: Development of constitutive relations

Abstract: The existing plasticity models recognize that ratchetting direction strongly depends on the loading path, the stress amplitude, and the mean stresses, but their predictions deviate from experiments for a number of materials. We propose an Armstrong-Frederick type hardening rule utilizing the concept of a limiting surface for the backstresses. The model predicts long-term ratchetting rate decay as well as constant ratchetting rate for both proportional and nonproportional loadings. To represent the transient behavior, the model encompasses a memory surface in the deviatoric stress space which recalls the maximum stress level of the prior loading history. The coefficients in the hardening rule, varying as a function of the accumulated plastic strain, serve to represent the cyclic hardening or softening. The stress level effect on ratchetting and non-Masing behavior are realized with the size of the introduced memory surface. Simulations with the model checked favorably with nonproportional multiaxial experiments which are outlined in Part 2 of this paper.

Modeling of cyclic ratchetting plasticity, Part II: Comparison of model simulations with experiments

Abstract: The material constants of the new plasticity model proposed in the first part of the paper can be divided into two independent groups. The first group, c (i) and r (i) (i = 1, 2,..., M), describes balanced loading and the second group, X (i) (i = 1, 2,..., M), characterizes unbalanced loading. We define balanced loading as the case when a virgin material initially isotropic will undergo no ratchetting and/or mean stress relaxation, and unbalanced loading as the loading under which a virgin material initially isotropic will produce strain ratchetting and/or mean stress relaxation. The independence of the two groups of material constants and the interpretation of the model with a limiting surface concept facilitated the determination of material constants. We describe in detail a computational procedure to determine the material constants in the models from simple uniaxial experiments. The theoretical predictions obtained by using the new plasticity model are compared with a number of multiple step ratchetting experiments under both uniaxial and biaxial tension-torsion loading. In multiple step experiments, the mean stress and stress amplitude are varied in a stepwise fashion during the test. Very close agreements are achieved between the experimental results and the model simulations including cases of nonproportional loading. Specifically, the new model predicted long-term ratchetting rate decay more accurately than the previous models.

Electrorheological Dampers, Part I: Analysis and Design

Abstract: Electrorheological (ER) materials are suspensions of specialized, micron-sized particles in nonconducting oils. When electric fields are applied to ER materials, they exhibit dramatic changes (within milli-seconds) in material properties. Pre-yield, yielding, and post-yield mechanisms are all influenced by the electric field. Namely, an applied electric field dramatically increases the stiffness and energy dissipation properties of these materials. A previously known cubic equation which describes the flow of fluids with a yield stress through a rectangular duct can be applied to annular flow, provided that certain conditions on the material properties are satisfied. An analytic solution and a uniform approximation to the solution, for the rectangular duct Poiseuille flow case is presented. A numerical method is required to solve the flow in annular geometries. The approximation for rectangular ducts is extended to deal with the annular duct case.

Micromechanical Definition of the Strain Tensor for Granular Materials

Abstract: In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.

Theory for Multilayered Anisotropic Plates With Weakened Interfaces

Abstract: Rigorous kinematical analysis offers a general representation of displacement variation through thickness of multilayered plates, which allows discontinuous distribution of displacements across each interface of adjacent layers so as to provide the possibility of incorporating effects of interfacial imperfection. A spring-layer model, which has recently been used efficiently in the field of micromechanics of composites, is introduced to model imperfectly bonded interfaces of multilayered plates. A linear theory underlying dynamic response of multilayered anisotropic plates with nonuniformly weakened bonding is presented from Hamilton’s principle. This theory has the same advantages as conventional higher-order theories over classical and first-order theories. Moreover, the conditions of imposing traction continuity and displacement jump across each interface are used in modeling interphase properties. In the special case of vanishing interface parameters, this theory reduces to the recently well-developed zigzag theory. As an example, a closed-form solution is presented and some numerical results are plotted to illustrate effects of the interfacial weakness.

A Critical Study of the Applicability of Rigid-Body Collision Theory

Abstract: This article deals with the collision of steel bars with external surfaces. The central issue of the article is the investigation of the fundamental concepts that are used to solve collision problems by using rigid-body theory. We particularly focus on low-velocity impacts of relatively rigid steel bars to test the applicability of these concepts. An experimental analysis was conducted to study the rebound velocities of freely dropped bars on a large external surface. A high-speed video system was used to capture the kinematic data. The number of contacts and the contact time were determined by using an electrical circuit and an oscilloscope. Tests were performed by using six bar lengths and varying the pre-impact inclinations and the velocities of the bars. The experimental results were used to verify the applicability of Coulomb's law of friction and the invariance of the coefficient of restitution in the class of impacts considered in this study. Then, given the unusual variation the coefficient of restitution as a result of changing pre-impact inclinations, a theoretical model was developed to explain this variation. A discrete model of the bar was used to obtain the equations of motion during impact. Computed and experimental results were compared to establish the accuracy of numerical model. The internal vibrations of the bar and multi impacts between the bar and the surface were found to be two main factors that cause the variation of the coefficient of restitution. Furthermore, a slenderness factor was proposed to identify the subset of collision problems where the coefficient of restitution was invariant to the inclination angle.

Exact Solutions for Laminated Piezoelectric Plates in Cylindrical Bending

Abstract: Exact solutions are presented for the problem of piezoelectric laminates in cylindrical bending under an applied surface traction or potential. An arbitrary number of elastic or piezoelectric layers can be considered in this analysis. Example problems are considered for several representative cases, with resulting displacement, potential, stress, and electric displacement distributions shown to demonstrate the effects of the electroelastic coupling.

Electrorheological Dampers, Part II: Testing and Modeling

Abstract: Electrorheological (ER) materials develop yield stresses on the order of 5-10 kPa in the presence of strong electric fields. Viscoelastic and yielding material properties can be modulated within milli-seconds. An analysis of flowing ER materials in the limiting case of fully developed steady flow results in simple approximations for use in design. Small-scale experiments show that these design equations can be applied to designing devices in which the flow is unsteady. More exact models of ER device behavior can be determined using curve-fitting techniques in multiple dimensions. A previously known curve-fitting technique is extended to deal with variable electric fields. Experiments are described which illustrate the potential for ER devices in large-scale damping applications and the accuracy of the modeling technique.

Hydroelastic vibration of rectangular plates

Abstract: This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

An Exact Three-Dimensional Solution for Simply Supported Rectangular Piezoelectric Plates

Abstract: An exact three-dimensional solution for the problem of a simply supported rectangular homogeneous piezoelectric plate is obtained, in the framework of the linear theory of piezoelectricity. The plate is made of a transversely isotropic material, is earthed on the lateral boundary, and is subjected to prescribed surface charge and tractions on the end faces. The limit of this solution as the plate thickness aspect ratio approaches zero is explicitly carried out. The analytical results obtained may constitute a reference case when developing or applying two-dimensional plate theories for the analysis of more complex piezoelectric problems. A numerical investigation in the case of a square uniformly loaded plate is also performed, in order to evaluate the influence of the thickness-to-side ratio on the three-dimensional solution of the plate problem.

Vertical Vibration of a Deep Bed of Granular Material in a Container

Abstract: A deep bed of granular material (more than six layers of particles) was subjected to sinusoidal, vertical vibrations. Several phenomena were observed depending on the amplitude of excitation. These included heaping, surface waves, and arching; the transitions from one state to another involved various dynamic instabilites and bifurcations. The paper includes a description of these phenomena and the characteristic properties associated with each in addition to measurements of the transitions from one phenomena to another.

Effect of the Interphase Zone on the Bulk Modulus of a Particulate Composite

Abstract: An exact solution is found for the problem of hydrostatic compression of an infinite body containing a spherical inclusion, with the elastic moduli varying with radius outside of the inclusion. This may represent an interphase zone in a composite, or the transition zone around an aggregate particle in concrete, for example. Both the shear and the bulk moduli are assumed to be equal to a constant term plus a powerlaw term that decays away from the inclusion. The method of Frobenius series is used to generate an exact solution for the displacements and stresses. The solution is then used to estimate the effective bulk modulus ofa material containing a random dispersion of these inclusions. The results demonstrate the manner in which a localized interphase zone around an inclusion may markedly affect both the stress concentrations at the interface, and the overall bulk modulus of the material.

A Split Hopkinson Bar Technique to Evaluate the Performance of Accelerometers

Abstract: We developed a split Hopkinson bar technique to evaluate the performance of accelerometers that measure large amplitude pulses. A nondispersive stress pulse propagates in an aluminum bar and interacts with a tungsten or steel disk at the end of the bar. We measure stress at the aluminum bar-disk interface with a quartz gage and measure acceleration at the free end of the disk with an accelerometer. The rise time of the incident stress pulse in the aluminum bar is long enough and the disk length is short enough that the response of the disk can be approximated closely as rigid-body motion; an experimentally verified analytical model supports this assumption. Since the cross-sectional area and mass of the disk are known, we calculate acceleration of the rigid disk from the stress measurement and Newton's Second Law. Comparisons of accelerations calculated from the quartz gage data and measured acceleration data show excellent agreement for acceleration pulses with the peak amplitudes between 20,000 and 120,000 G (1 G = 9.81m/s 2 ), rise times as short as 20 μs, and pulse durations between 40 and 70 μs.

A Higher-Order Theory for Vibration of Doubly Curved Shallow Shells

Abstract: A higher-order shear deformation 'theory is presented for vibration analysis of thick, doubly curved shallow shells. An orthogonal curvilinear coordinate system is em- ployed to arrive at the strain components. A third-order displacement field in trans- verse coordinate is adopted. Though no transverse normal stress is assumed, the theory accounts for cubic distribution of the transverse shear strains through the shell thickness in contrast with existing parabolic shear distribution. The unsymmetric shear distribution is a physical consequence of the presence of shell curvatures where the stress and strain of a point above the mid-surface are different from its counterpart below the mid-surface. Imposing the vanishing of transverse shear strains on top and bottom surfaces, the rotation field is reduced from a six-degree to a two-degree system. The discrepancy between the existing and the present theories is highlighted.

The Folding of Triangulated Cylinders, Part III: Experiments

Abstract: This paper describes an experimental investigation ofa type of foldable cylindrical structure, first presented in two earlier papers. Three cylinders of this type were designed and manufactured, and were then tested to find the force required to fold them. The results from these tests show some discrepancies with an earlier computational simulation, which was based on a pin-jointed truss model of the cylinders. Possible explanations for these discrepancies are explored, and are then verified by new simulations using computational models that include the effect ofhinge stiffness, and the effect of geometric imperfections.

Subsonic and intersonic crack growth along a bimaterial interface

Abstract: An experimental investigation has been conducted to study the dynamic failure of bimaterial interfaces. Interfacial crack growth is observed using dynamic photoelasticity and characterized in terms of crack-tip velocity, complex stress intensity factor, and energy release rate. On the basis of crack-tip velocity two growth regimes are established, viz. the subsonic and transonic regimes. In the latter regime crack-tip velocities up to 1.3 times the shear wave velocity of the more compliant material are observed. This results in the formation of a line of discontinuity in the stress field surrounding the crack tip and also the presence of a pseudo crack tip that travels with the Rayleigh wave velocity (of the more compliant material).

Analysis of a Nonlinear System Exhibiting Chaotic, Noisy Chaotic, and Random Behaviors

Abstract: This study presents a stochastic approach for the analysis of nonchaotic, chaotic, random and nonchaotic, random and chaotic, and random dynamics of a nonlinear system. The analysis utilizes a Markov process approximation, direct numerical simulations, and a generalized stochastic Melnikov process. The Fokker-Planck equation along with a path integral solution procedure are developed and implemented to illustrate the evolution of probability density functions. Numerical integration is employed to simulate the noise effects on nonlinear responses. In regard to the presence of additive ideal white noise, the generalized stochastic Melnikov process is developed to identify the boundary for noisy chaos. A mathematical representation encompassing all possible dynamical responses is provided. Numerical results indicate that noisy chaos is a possible intermediate state between deterministic and random dynamics. A global picture of the system behavior is demonstrated via the transition of probability density function over its entire evolution. It is observed that the presence of external noise has significant effects over the transition between different response states and between co-existing attractors.

Three-Dimensional Finite Element Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loading

Abstract: Three-dimensional finite element simulations of the indentation and sliding ofa rigid sphere on a half-space with a harder and stiffer layer are presented The sphere is modeled by contact elements, thereby avoiding a priori assumptions for the pressure profile Indentations are performed to normal loads of 100 and 200 times the initial yield load of the substrate material and subsequent sliding is performed at a constant normal load to distances of approximately twice the indentation contact radius Two complete load cycles are performed in selected cases to assess the effect of repeated sliding on the surface displacements and contact stresses The effects of layer material properties, interface friction, and normal load on the sliding and residual contact stresses and forward plastic flow are examined Emphasis is given to the sliding and residual tensile stresses at the surface in order to assess the consequences for crack initiation and subsequent failure as a function of the layer material properties, the coefficient offriction, and normal load The finite element results are shown to be in good agreement with the results of analytical and experimental studies

Random Field Representation and Synthesis Using Wavelet Bases

Abstract: The paper addresses the representation and simulation of random fields using wavelet bases. The probabilistic description of the wavelet coefficients involved in the representation of the random field is discussed. It is shown that a broad class of random fields is amenable to a simplified representation. Further, it is shown that a judicious use of the local and multiscale structure of Daubechies wavelets leads to an efficient simulation algorithm. The synthesis of random field samples is based on a wavelet reconstruction algorithm which can be associated with a dynamic system in the scale domain. Implementation aspects are considered and simulation errors are estimated. Examples of simulating random fields encountered in engineering applications are discussed.

A Fractal Model for the Rigid-Perfectly Plastic Contact of Rough Surfaces

Abstract: In this study a continuous asymptotic model is developed to describe the rigid-perfectly plastic deformation of a single rough surface in contact with an ideally smooth and rigid counter-surface. The geometry of the rough surface is assumed to be fractal, and is modeled by an effective fractal surface compressed into the ideally smooth and rigid counter-surface. The rough self-affine fractal structure of the effective surface is approximated using a deterministic Cantor set representation. The proposed model admits an analytic solution incorporating volume conservation. Presented results illustrate the effects of volume conservation and initial surface roughness on the rigid-perfectly plastic deformation that occurs during contact processes. The results from this model are compared with existing experimental load displacement results for the deformation of a ground steel surface.

Feedback Control of Karman Vortex Shedding

Abstract: A computational study of the feedback control of the magnitude of the lift in flow around a cylinder is presented. The uncontrolled flow exhibits an unsymmetric Karman vortex street and a periodic lift coefficient. The size of the oscillations in the lift is reduced through an active feedback control system. The control used is the injection and suction of fluid through orifices on the cylinder; the amount of fluid injected or sucked is determined, through a simple feedback law, from pressure measurements at stations along the surface of the cylinder. The results of some computational experiments are given; these indicate that the simple feedback law used is effective in reducing the size of the oscillations in the lift.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory

Abstract: A least-squares-based pressure projection method is proposed for the nonlinear analysis of nearly incompressible hyperelastic materials. The strain energy density function is separated into distortional and dilatational parts by the use of Penn's invariants such that the hydrostatic pressure is solely determined from the dilatational strain energy density. The hydrostatic pressure and hydrostatic pressure increment calculated from displacements are projected onto appropriate pressure fields through the least-squares method. The method is applicable to lower and higher order elements and the projection procedures can be implemented into the displacement based nonlinear finite element program. By the use of certain pressure interpolation functions and reduced integration rules in the pressure projection equations, this method can be degenerated to a nonlinear version of the selective reduced integration method.

Exact Stationary Solutions of Stochastically Excited and Dissipated Integrable Hamiltonian Systems

Abstract: It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ≤ n - 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

A Comparison Between the Tiersten Model and O(H) Boundary Conditions for Elastic Surface Waves Guided by Thin Layers

Abstract: In this paper we make a comparison between the boundary conditions (BCs) derived by Tiersten and the so-called O(h) BCs for elastic surface waves guided by thin films. By a thin layer we here mean a layer for which the thickness is much smaller than the wavelengths involved. The advantage of the O(h) model is that it starts with the general three-dimensional equation of motion and derives the boundary conditions in a rational manner keeping all terms linear in the layer thickness. The Tiersten model is obtained from the approximate equations for low frequency and flexure of thin plates by neglecting the flexural stiffness. We consider straight-crested surface waves under plane-strain conditions, so-called Rayleigh-type waves (P-SV), and Love waves (SH). It is shown that for the Rayleigh type waves the O(h) BCs gives a much better approximation of the exact case than the Tiersten BCs. Even for the Tiersten model including flexural stiffness, the O(h) BCs yields more accurate results. Concerning Love waves both the Tiersten model and O(h) model reduces to the same dispersion relation which quite well approximates the exact solution.

Transient Elastic Waves in a Transversely Isotropic Plate

Abstract: The elastodynamic response of thick plate, with the axis of transverse isotropy normal to the plate surface, is calculated by double numerical inverse transforms, a method particularly well-suited for calculations of responses in the near field of layered structures. Applications of these calculations include point-source/point-receiver ultrasonics, quantitative acoustic emission measurements, and seismology. The singularities of the integrand are eliminated by the introduction of a small, but nonzero, imaginary part to the frequency. We discuss issues of numerical efficiency and accuracy in the evaluation of the resulting integrals. The method can be generalized to calculate the responses in materials of more general symmetry, in viscoelastic materials and to include the effects of finite aperture sources and receivers. The calculated responses are compared to those measured in a single crystal specimen of zinc.

Dynamic Analysis of Nonuniform Beams With Time-Dependent Elastic Boundary Conditions

Abstract: The dynamic response of a nonuniform beam with time-dependent elastic boundary conditions is studied by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of general elastically restrained nonuniform beams given by Lee and Kuo. The time-dependent elastic boundary conditions for the beam are formulated. A general form of change of dependent variable is introduced and the shifting polynomials of the third-order degree, instead of the fifth-order degree polynomials taken by Mindlin-Goodman, are selected. The physical meaning of these shifting polynomial functions are explored. Finally, the limiting cases are discussed and several examples are given to illustrate the analysis.

A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part II: Applications

Abstract: In the first part of this paper a pressure projection method was presented for the nonlinear analysis ofstructures made of nearly incompressible hyperelastic materials The main focus of the second part of the paper is to demonstrate the performance of the present method and to address some of the issues related to the analysis of engineering elastomers including the proper selection of strain energy density functions The numerical procedures and the implementation to nonlinear finite element programs are presented Mooney-Rivlin, Cubic, and Modified Cubic strain energy density functions are used in the numerical examples Several classical finite elasticity problems as well as some practical engineering elastomer problems are analyzed The need to account for the slight compressibility of rubber (finite bulk modulus) in the finite element formulation is demonstrated in the study of apparent Young's modulus of bonded thin rubber units The combined shear-bending deformation that commonly exists in rubber mounting systems is also analyzed and discussed

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

Abstract: A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications ('3:1' resonances in a two-degree-of- freedom system and '3:1' resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

The Boundary Contour Method for Three-Dimensional Linear Elasticity

Abstract: This paper presents a novel variant of the boundary element method, here called the boundary contour method, applied to three-dimensional problems of linear elasticity. In this work, the surface integrals on boundary elements of the usual boundary element method are transformed, through an application of Stokes' theorem, into line integrals on the bounding contours of these elements. Thus, in this formulation, only line integrals have to be numerically evaluated for three-dimensional elasticity problems-even for curved surface elements of arbitrary shape. Numerical results are presented for some three-dimensional problems, and these are compared against analytical solutions.

On the Eigenstrain Problem of a Spherical Inclusion With an Imperfectly Bonded Interface

Abstract: This article provides a comprehensive theoretical treatment of the eigenstrain problem of a spherical inclusion with an imperfectly bonded interface. Both tangential and normal discontinuities at the interface are considered and a linear interfacial condition, which assumes that the tangential and the normal displacement jumps are proportional to the associated tractions, is adopted. The solution to the corresponding eigenstrain problem is obtained by combining Eshelby's solution for a perfectly bonded inclusion with Volterra's solution for an equivalent Somigliana dislocation field which models the interfacial sliding and normal separation. For isotropic materials, the Burger's vector of the equivalent Somigliana dislocation is exactly determined ; the solution is explicitly presented and its uniqueness demonstrated. It is found that the stresses inside the inclusion are not uniform, except for some special cases.