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RESEARCH PAPERS

J. Appl. Mech. 1989;56(1):1-9. doi:10.1115/1.3176046.

Plane strain compression of a rectangular block is used as a model problem to investigate the dynamics of shear band development from an internal inhomogeneity. The material is characterized as a von Mises elastic-viscoplastic solid, with a hardness function that exhibits a local maximum. Regardless of whether the material is hardening or softening, plastic strain development involves the evolution of fingerlike contours emanating from the inhomogeneity at 45 deg to the compression axis. Once a given strain contour crosses the specimen, it fans out about its initial direction of propagation. For a softening solid, this fanning out ceases for some strain level greater than the strain at the hardness maximum and further straining takes place in an ever narrowing band. Many of the qualitative features of shear band development under dynamic loading conditions are the same as under quasistatic loading conditions, but a significant retardation of shear band development due to inertial effects is found.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):10-14. doi:10.1115/1.3176029.

An algorithm based on a combination of the upper bound method and finite element repesentation has been developed. The algorithm is applied to the problem of a rigid indenter ploughing through a rigid/perfectly-plastic material. Numerical examples are given and the results are compared with previous approximate solutions. Limitations of the upper bound method are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):15-24. doi:10.1115/1.3176038.

A two-phase continuum mixture model is used to analyze steady compaction waves in porous materials. It is shown that such a model admits both subsonic and supersonic steady compaction waves in response to a piston-driven boundary condition when a Tait equation is used to describe a solid matrix material and a generic static compaction relation is used to describe collapse of the matrix. Parameters for the Tait equation are chosen to match shock and compaction wave data. The model is able to predict compaction wave speed, final pressure, and final volume fraction in porous HMX. The structure of the compaction wave is also studied. A shock preceding the compaction wave structure is predicted for compaction waves travelling faster than the ambient sound speed of the solid. For subsonic compaction waves no leading shock is predicted. The compaction zone length is studied as a function of initial volume fraction, piston velocity, and compaction viscosity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):25-32. doi:10.1115/1.3176061.

The development and numerical implementation of a constitutive model for jointed rock media is the subject of investigation in this paper. The constitutive model is based on the continuum assumption of strain-partitioning among the elastic rock matrix and joint sets with nonlinear normal and shear responses. Rate equations for the stress-strain response of the jointed media have been formulated. A numerical incremental solution scheme to these equations has been developed. It has been implemented into the finite element code JAC as an additional material model. Several sample problems have been solved for demonstration purposes. Interpretation and discussion of these results are presented.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):33-39. doi:10.1115/1.3176062.

We examine the plane strain cold rolling problem of thin strip implementing a perturbation scheme upon the governing equations. The analysis accounts for transverse variations in the flow and inhomogeneous work hardening. We establish deformation regimes based upon the parameters μ, the friction coefficient, τ, the ratio of the initial yield stress and the maximum roll pressure, and δ, a measure of the small gap reduction. We see that the inclusion of these inhomogeneous effects gives mildly different results for values of μ/τ > 0(1) or τ δ/μ = 0(1). We gauge these effects by examining the roll pressure, shear stress, longitudinal stress, yield stress, front and back tensions and torque to see how they are affected by the inclusion of inhomogeneous flow effects.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):40-46. doi:10.1115/1.3176063.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):47-50. doi:10.1115/1.3176064.

Melting of a disk is facilitated by rotation. The problem is governed by a nondimensional parameter α which represents the relative importance of injection (melt) rate and rotation times viscosity. The nonlinear governing equations are solved by perturbations for small α and numerical integration for arbitrary α. Torque and heat transfer rates are found. The solution is one of the rare exact similarity solutions of the Navier-Stokes equations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):51-56. doi:10.1115/1.3176065.

The discussion centers on the formulation of a constitutive model for brittle solids satisfying two conflicting requirements: simplicity and accuracy. Owing to the inherent complexity oif the problem, stemming from the random distribution of interacting defects, analytical expressions relating macrofields are possible only under some restrictive conditions emphasizing modest levels of defect concentrations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):57-62. doi:10.1115/1.3176066.

The approximate, closed-form solutions for the inelastic strain and compliances are derived for some simple plane stress and plane strain cases. The computations are performed for a model-ignoring crack interaction as well as for the case of the self-consistent model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):63-69. doi:10.1115/1.3176067.

A general mathematical formulation to analyze cracks in a multilayered medium is constructed. First, a matrix for a single layer is formed in the Hankel transformed domain. Then a global matrix is formed by assembling together each layer matrix. The displacement and stress fields are obtained by inverting the Hankel transform. Finally, the planar crack problem is solved by the boundary integral method. The results given are the crack opening displacement and the stress intensity factors.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):70-76. doi:10.1115/1.3176068.

A series solution is presented for a hemispheroidal elastic inhomogeneity at the free surface of an elastic half space. The loading is either all around tension at infinity, perpendicular to the axis of symmetry of the inhomogeneity, or uniform, nonshear type eigenstrains sustained by the inhomogeneity. The displacement potentials of Boussinesq are used to represent the solution and several numerical calculations are performed to illustrate the results.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):77-82. doi:10.1115/1.3176069.

A test specimen capable of measuring the fracture resistance of bimaterial interfaces has been devised. A finite element approach has been used to characterize trends in the stress intensities and center point displacement with specimen dimensions, elastic properties, and crack length. The utility of the specimen has been demonstrated by conducting experiments on the model system, Al/PMMA.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):83-88. doi:10.1115/1.3176070.

The Mori-Tanaka method is considered in the context of both scalar thermal conductivity and anisotropic elasticity of multiphase composites, and some general properties are deduced. Particular attention is given to its relation to known general bounds, and to the differential scheme. It is shown that the moduli predicted by the method always satisfy the Hashin-Shtrikman and Hill-Hashin bounds for two-phase composites. This property does not generalize to multiphase composites. A specific example illustrates that the method can predict moduli in violation of the Hashin-Shtrikman bounds for a three-phase medium. However, if the particle shapes are all spheres, then the prediction for the multiphase composite is coincident with the Hashin-Shtrikman bounds if the matrix material is either the stiffest or the most compliant phase. It is also shown that the generalized differential effective medium method yields the same moduli as the Mori-Tanaka approximation if certain conditions are satisfied in the differential scheme. Thus, it is required that at each stage in the differential process, and for each phase j (j = 1, 2, [[ellipsis]], n) of new material, the average field in the incrementally added phase j material must be the same as the average field in the bulk phase j . For two phase media, n = 1, this condition reduces to the less stringent requirement that the ratio of the field in the incrementally added material to the average field in the matrix material is the same as the dilute concentration ratio. The cumulative findings of this paper, particularly those concerning bounds, suggest that the Mori-Tanaka approximation be used with caution in multiphase applications, but is on firmer ground for two-phase composites.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):89-95. doi:10.1115/1.3176071.

The induced magnetic fields generated by a line mechanical singularity in a magnetized elastic half plane are investigated in this paper. One version of linear theory for soft ferromagnetic elastic solids which has been developed by Pao and Yeh (1973) is adopted to analyze the plane strain problem undertaken. By applying the Fourier transform technique, the exact solutions for the generated magnetic inductions due to various mechanical singularities such as a single force, a dipole, and single couple are obtained in a closed form. The distributions of the generated inductions on the surface are shown with figures.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):96-104. doi:10.1115/1.3176072.

The quasi-static loading of a curved strip compressed by a flat, rigid plate is considered, with particular reference to large deformations and the ensuing buckling behavior. Experiments were performed on curved strips of constant width but of different thickness. The strips were initially deformed to a fixed radius of curvature and stress relieved before pinning the ends. The span was held constant at about 305 mm. The deformation characteristics have been analyzed using an incremental finite element technique. Particular attention has been paid to modeling the situation when a node contacts the plate and the condition for separation of the strip from the plate. The predicted loads and deformation modes agreed well with experimental results from tests on steel and aluminum specimens. The experimental and theoretical procedures are pertinent to the study of dent resistance of sheet metal stampings, particularly automotive panels.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):105-112. doi:10.1115/1.3176028.

The elastic response of a complete spherical shell under the influence of concentrated loads (normal point loads, concentrated tangential loads, and concentrated surface moments) which apply in a self-equilibrating fashion is obtained. The mathematical analysis incorporates the classical uncoupled system of equations for the transverse displacement W and a stress function F . The solution formulae for all three types of singular loading are in closed form and they are expressed in terms of complex Legendre and other elementary functions. The two latter portions of the analysis are associated with a multivalued stress function F which leads to a single-valued stress and displacement formulae. The intricacies of the solutions and their singular character are also discussed. Lastly, some representative shell problems are evaluated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):113-120. doi:10.1115/1.3176030.

A method is developed for predicting crush behavior of multicorner prismatic columns subjected to an axial compressive load. The corner element of an arbitrary angle is analyzed first using rigorous methods of structural plasticity with finite deformations and rotations. On that basis, crush predictions are made for multicorner columns with an even number of corners. Static crush tests on square, hexagonal, and rhomboidal thin-walled columns are also reported here. Good correlation between the theory and experiments was obtained for the magnitude of a mean crushing force and kinematic parameters describing the process of progressive folding.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):121-126. doi:10.1115/1.3176031.

The stability of thin composite layered anisotropic cylindrical shells under axial compression is considered for the case of nonuniform boundary conditions. Such conditions are employed to model the situation where there is edge damage to the shell. The influence of weakening or a crack at an edge on the critical buckling load of a variety of single and multilayered shells is investigated. Results indicate that isotropic shells exhibit a rather sudden steep reduction in the critical buckling load for relatively small edge damage. However, some anisotropic composite shells may not be so sensitive and, in contrast, only a gradual reduction may be brought about by the edge damage. The degree of sensitivity to edge damage appears to be dependent, in some complex fashion, on the various geometric and physical shell parameters.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):127-132. doi:10.1115/1.3176032.

The axisymmetric behavior of both shallow, and deep, ring-loaded spherical caps, which are simply supported but otherwise unconstrained at their edges, is investigated using a large-strain shell theory based on a variational principle. A numerical technique is used to extract the solution. Particular attention is paid to highly nonlinear phenomena such as snapthrough, single and multiple snapback, and load-free everted states.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):133-138. doi:10.1115/1.3176033.

A general approach is presented for solving the problem of the collision of two rigid bodies at a point. The approach overcomes the difficulties encountered by others on the treatment of contact velocity reversals and negative energy losses. A classical problem is solved; the initial velocities are presumed known and the final velocities unknown. The interaction process between the two bodies is modeled using two coefficients. These are the classical coefficient of restitution, e , and the ratio, μ, of tangential to normal impulses. The latter quantity can be a coefficient of friction as a special case. The paper reveals that these coefficients have a much broader intepretation than previously recognized in the solution of collision problems. The appropriate choice of values for μ is related to the energy loss of the collision. It is shown that μ is bounded by values which correspond to no sliding at separation and conservation of energy. Another bound on μ combined with limits on the coefficient e , provides an overall bound on the energy loss of a collision. Examples from existing mechanics literature are solved to illustrate the significance of the coefficients and their relationship to the energy loss of collisions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):139-145. doi:10.1115/1.3176034.

It is shown in this paper that Euler was first to derive the finite rotation formula which is often erroneously attributed to Rodrigues, while Rodrigues was responsible for the derivation of the composition formulae for successive finite rotations and the so-called Euler parameters of finite rotation. Therefore, based upon historical facts, the following nomenclature is suggested: Euler’s finite rotation formula, Rodrigues’ composition formulae of finite rotations, and Euler-Rodrigues parameters. The text of the paper contains modern symbols and formula forms, while the Appendices contain brief summaries from relevant historical sources with minor alterations in symbols at the most.

Topics: Rotation
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):146-148. doi:10.1115/1.3176035.

Forms of the variable-heat-conductivity coefficient function in the one-dimensional heat equation are determined which yield a standard harmonic eigenvalue sequence as in the case of homogeneity. The continuous case is found to correspond to a four-thirds power law dependence on coordinate. For the stepped case, the condition on the ratio of segmental heat conductivities in terms of the junction location is presented.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):149-154. doi:10.1115/1.3176036.

A method is proposed for analyzing the steady-state response of nonlinear dynamic systems. The method iterates to obtain the discrete Fourier transform of the system response, returning to the time domain at each iteration to take advantage of the ease in evaluating nonlinearities there—rather than analytically describing the nonlinear terms in the frequency domain. The updated estimates of the nonlinear terms are transformed back into the frequency domain in order to continue iterating on the frequency spectrum of the steady-state response. The method is demonstrated by solving a problem with friction damping in which the excitation has multiple discrete frequencies.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):155-161. doi:10.1115/1.3176037.

The stability of bifurcated normal modes in coupled nonlinear oscillators is investigated, based on Synge’s stability in the kinematico-statical sense, utilizing the calculus of variations and Floquet’s theory. It is found, in general, that in a generic bifurcation, the stabilities of two bifurcated modes are opposite, and in a nongeneric bifurcation, the stability of continuing modes is opposite to that of the existing mode, and the stabilities of the two bifurcated modes are equal but opposite to that of the continuing mode. Some examples are illustrated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):162-167. doi:10.1115/1.3176039.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):168-174. doi:10.1115/1.3176040.

The dynamic response of a two-degree-of-freedom impacting system is considered. The system consists of an inverted pendulum with motion limiting stops attached to a sinusoidally excited mass-spring system. Two types of periodic response for this system are analyzed in detail; existence, stability, and bifurcations of these motions can be explicitly computed using a piecewise linear model. The appearance and loss of stability of very long period subharmonics is shown to coincide with a global bifurcation in which chaotic motions, in the form of Smale horseshoes, arise. Application of this device as an impact damper is also briefly discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):175-178. doi:10.1115/1.3176041.

The almost-sure stability of linear second-order systems which are parametrically excited by ergodic, “nonwhite,” random processes is studied by an extension of the method of Infante. In this approach, a positive-definite quadratic function of the form V = x′ Px is assumed and a family of stability boundaries depending on the elements of the matrix P is obtained. An envelope of these boundaries is then solved for by optimizing the stability boundary with respect to the elements of P . It is found that the optimum matrix P in general depends not only on the system constants but also on the excitation intensities. This approach is, in principle, applicable to study systems involving two or more random processes. The results reported in previous investigations are obtained as special cases of the present study.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):179-185. doi:10.1115/1.3176042.

The methods of Gaussian linearization along with a new Gaussian Criterion used in the prediction of the stationary output variances of stable nonlinear oscillators subjected to both stochastic parametric and external excitations are presented. The techniques of Gaussian linearization are first derived and the accuracy in the prediction of the stationary output variances is illustrated. The justification of using Gaussian linearization a priori is further investigated by establishing a Gaussian Criterion. The non-Gaussian effects due to system nonlinearities and/or large noise intensities in a Duffing oscillator are also illustrated. The validity of employing the Gaussian Criterion test for assuring accuracy of Gaussian linearization is supported by performing the Chi-square Gaussian goodness-of-fit test.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):186-191. doi:10.1115/1.3176043.

A statistical method for the identification of nonlinear random vibration systems is presented. The first step in the identification process is to obtain a discrete time version of a random vibration model using a local linearization approach. It is shown that the discrete time version thus obtained may be utilized in the identification of original random vibration model. The method is applied to some real ship engineering data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):192-195. doi:10.1115/1.3176044.

A linear stochastic differential equation of order N excited by an external random force and whose coefficients are white noise random processes is studied. The external force may be either white or colored noise random process. Given the statistical properties of the coefficients and of the force, equivalent statistics are obtained for the response. The present solution method is based on the derivation of the equation governing the response autocorrelation function. The simplifying assumption that the response is stationary when the coefficients and input force are stationary is introduced. Another simplification occurs with the assumption that the response is uncorrelated from the random coefficients. Closed-form solutions for the response autocorrelation function and spectral density are derived in conjunction with a stability bound.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):196-201. doi:10.1115/1.3176045.

A linear stochastic differential equation of order N with colored noise random coefficients and random input is studied. An approximate expression for the autocorrelation of the response is derived in terms of the statistical properties of the random coefficients and input. This is achieved by using an expansion method known as the Born expansion (Feynman, 1962). Feynman diagrams are used as a short hand notation. In the particular case where the coefficients are white noise processes, the expansion method yields identical results to those obtained using an alternate method in a companion paper (Benaroya and Rehak, 1989). The expansion method is also used to demonstrate that white noise coefficients are statistically uncorrelated from the response.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):202-207. doi:10.1115/1.3176047.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):208-210. doi:10.1115/1.3176048.

An integral variational equation can adequately describe heat, mass, and momentum transfer in a moving chemically reactive fluid. The Euler-Lagrange equations corresponding to the suggested variational equation are identical to the equations of entropy, momentum, angular momentum, and mass balance. The constructed Lagrangian density relates energy change in the system to the work and energy dissipation of the system. For steady-state processes, the Lagrangian density includes convective energy flow through the system boundary, energy dissipation in the system, and work of the system. The proposed variational equation is equivalent to the expansion of the principle of minimum energy dissipation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):211-217. doi:10.1115/1.3176049.

A boundary integral analysis of the creeping motion of a long inviscid bubble in a liquid filled tube is presented. The effects of interfacial surface tension are included in the stress balance across the liquid-bubble interface. Velocities, stresses, and the bubble profile are obtained as a function of the capillary number. Computed values of the thickness of the liquid film between the bubble and tube wall are in excellent agreement with published experimental measurements. The results are of interest in exposing the role of surface tension in multiphase flow in capillary tubes and porous materials.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):218-221. doi:10.1115/1.3176050.

A similarity solution may be obtained for a flow between two parallel disks which, at time t* , are spaced a distance H (1 − αt* )1/2 apart and a magnetic field proportional to B 0 (1 − αt* )−1/2 is applied perpendicular to the disks. Approximate analytic solutions are given and a numerical solution to the resulting nonlinear ordinary differential equations is presented. The effects of the magnetic forces on the velocity profiles and on the normal forces which the fluid exerts on the disks are studied. It has been found that by increasing the magnetic force a considerable increase in the load can be achieved.

Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Appl. Mech. 1989;56(1):222-224. doi:10.1115/1.3176051.
Abstract
Topics: Chain
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):224-226. doi:10.1115/1.3176052.
Abstract
Topics: Eigenvalues
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):226-228. doi:10.1115/1.3176053.

A thermodynamical inequality is obtained for mixtures of interacting continua; it represents the analogue of a corresponding inequality for single phase media contained in a paper of Green and Naghdi (1984) and is less restrictive than one proposed earlier (1978b). Also, the use of such inequalities in conjunction with the so-called reciprocal relations is briefly discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):228-231. doi:10.1115/1.3176054.

Studies making use of higher vibration modes and frequencies have indicated a need for a more accurate beam theory. Equations of motion are developed here that give a more accurate representation of the dynamic behaivor of a beam than conventional beam theory. Results are obtained using these equations for the natural vibrations of simply-supported aluminum beams of rectangular cross-sections. These results are compared to results from conventional beam theory, and they are examined to identify where various effects are important.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):231-233. doi:10.1115/1.3176055.
Abstract
Commentary by Dr. Valentin Fuster

DISCUSSIONS

BOOK REVIEWS

J. Appl. Mech. 1989;56(1):235. doi:10.1115/1.3176058.
FREE TO VIEW
Abstract
Topics: Elastic waves
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):235-236. doi:10.1115/1.3176059.
FREE TO VIEW
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1989;56(1):236-237. doi:10.1115/1.3176060.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster

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