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IN MEMORIAM

J. Appl. Mech. 1988;55(2):259. doi:10.1115/1.3173669.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster

RESEARCH PAPERS

J. Appl. Mech. 1988;55(2):260-266. doi:10.1115/1.3173670.

The effects of Strength Differential (SD) and plastic compressibility for materials obeying the modified von Mises yield criterion were exemplified by solving two boundary-value problems. The assumptions of associated plasticity (leading to maximum plastic volume increase) and nonassociated plasticity (leading to zero plastic volume increase) were used for comparative studies on the effects of plastic compressibility. The solutions for compression processes showed that SD effects increased the pressure at initial yielding and at failure, as well as increased the capacity of the materials to withstand plastic deformations. The opposite was true for tension processes. For associated and nonassociated plasticity, upper and lower bounds for stresses and strains for load and stroke-controlled situations were indicated. The results also showed unrealistic restrictions on the Poisson’s ratio and C/T for nonassociated plasticity under certain conditions. Hence, plastic volume increase, although small, should be incorporated into a more realistic plasticity model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):267-274. doi:10.1115/1.3173671.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):275-279. doi:10.1115/1.3173672.

We developed an analytical model for the elastic-plastic response of a compressible material from the uniform expansion of a spherically symmetric cavity. Previous models consider the material as incompressible. Numerical results from both models showed the effect of compressibility.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):280-286. doi:10.1115/1.3173673.

A systematic theory to describe the anisotropic damage states of materials and a consistent definition of effective stress tensors are developed within the framework of continuum damage mechanics. By introducing a fictitious undamaged configuration, mechanically equivalent to the real damaged configuration, the classical creep damage theory is extended to the general three-dimensional states of material damage; it is shown that the damage state can be described in terms of a symmetric second rank tensor. The physical implications, mathematical restrictions, and the limitations of this damage tensor, as well as the effects of finite deformation on the damage state, are discussed in some detail. The notion of the fictitious undamaged configuration is then applied also to the definition of effective stresses. Finally, the extension of the effective stresses incorporating the effects of crack closure is discussed. The resulting effective stress tensor is employed to analyze the stress-path dependence of the elastic behavior of a cracked elastic-brittle material.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):287-293. doi:10.1115/1.3173674.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):294-298. doi:10.1115/1.3173675.

A bounding theorem is obtained for the time of creep rupture of shell structures subjected to cyclic loading. It is found that the shakedown concepts can be used in which a reference rupture stress replaces the yield stress.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):299-316. doi:10.1115/1.3173676.

Full-field numerical solutions for a crack which lies along the interface of an elastic-plastic medium and a rigid substrate are presented. The solutions are obtained using a small strain version of the J2 -deformation theory with power-law strain hardening. In the present article, results for loading causing only small scale yielding at the crack tip are described; in subsequent articles the mathematical structure of the crack-tip fields under small scale yielding and results for contained yielding and fully plastic behavior will be presented. We find that although the near-tip fields do not appear to have a separable singular form, of the HRR-type fields as in homogeneous media, they do, however, bear interesting similarities to certain mixed-mode HRR fields. Under small scale yielding, where the remote elastic fields are specified by a complex stress-concentration vector Q = |Q |eiφ with φ being the phase angle between the two in-plane stress modes, we find that the plastic fields are members of a family parameterized by a new phase angle ξ, ≡ φ + εln(QQ /σ0 2 L ) , and the fields nearly scale with the well-defined energy release rate as evaluated by the J-integral. Here σ0 is the reference yield stress and L is the total crack length (or a relevant length of the crack geometry). Numerical procedures appropriate for solving a general class of interface crack problems are also presented. A description of a numerical method for extracting the mixed mode stress intensities for cracks at interfaces and in homogeneous isotropic or anisotropic media, is included.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):317-324. doi:10.1115/1.3173677.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):325-331. doi:10.1115/1.3173678.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):332-340. doi:10.1115/1.3173679.

The finite axisymmetric deformation of a thin shell of revolution is treated in this analysis. The governing differential equations are given for a hyperelastic shell material with the Mooney-Rivlin strain-energy-density function. These equations are solved numerically using a 4th-order Runge-Kutta integration method. A generalized Newton-Raphson iteration procedure is used to systematically improve trial solutions of the differential equations. The governing differential equations are differentiated with respect to a set of generalized coordinates to derive associated rate equations. The rate equations are solved numerically to generate the tangent stiffness matrix which is used to determine the load deformation history of the shell with incremental loading. Numerical examples are presented to illustrate the major characteristics of nonlinear shell behavior.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):341-346. doi:10.1115/1.3173680.

A theoretical model for the linear elastic properties of three-dimensional open-cell foams is developed. We consider a tetrahedral unit cell, which contains four identical half-struts that join at equal angles, to represent the essential microstructural features of a foam. The effective continuum stress is obtained for an individual tetrahedral element arbitrarily oriented with respect to the principal directions of strain. The effective elastic constants for a foam are determined under the assumption that all possible orientations of the unit cell are equally probable in a representative volume element. The elastic constants are expressed as functions of compliances for bending and stretching of a strut, whose cross section is permitted to vary with distance from the joint, so the effect of strut morphology on effective elastic properties can be determined. Strut bending is the primary distortional mechanism for low-density foams with tetrahedral microstructure. For uniform strut cross section, the effective Young’s modulus is proportional to the volume fraction of solid material squared, and the coefficient of proportionality depends upon the specific strut shape. A similar analysis for cellular materials with cubic microstructure indicates that strut extension is the dominant distortional mechanism and that the effective Young’s modulus is linear in volume fraction. Our results emphasize the essential role of microstructure in determining the linear elastic properties of cellular materials and provide a theoretical framework for investigating nonlinear behavior.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):347-354. doi:10.1115/1.3173681.

This paper studies the determination of rigorous upper and lower bounds on the effective transport and elastic moduli of a transversely isotropic fiber-reinforced composite derived by Silnutzer and by Milton. The third-order Silnutzer bounds on the transverse conductivity σe , the transverse bulk modulus ke , and the axial shear modulus μe , depend upon the microstructure through a three-point correlation function of the medium. The fourth-order Milton bounds on σe and μe depend not only upon three-point information but upon the next level of information, i.e., a four-point correlation function. The aforementioned microstructure-sensitive bounds are computed, using methods and results of statistical mechanics, for the model of aligned, infinitely long, equisized, circular cylinders which are randomly distributed throughout a matrix, for fiber volume fractions up to 65 percent. For a wide range of volume fractions and phase property values, the Silnutzer bounds significantly improve upon corresponding second-order bounds due to Hill and to Hashin; the Milton bounds, moreover, are narrower than the third-order Silnutzer bounds. When the cylinders are perfectly conducting or perfectly rigid, it is shown that Milton’s lower bound on σe or μe provides an excellent estimate of these effective parameters for the wide range of volume fractions studied here. This conclusion is supported by computer-simulation results for σe and by experimental data for a graphite-plastic composite.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):355-360. doi:10.1115/1.3173682.

A study of the effect of an elastic spherical particle embedded in an infinite elastic-plastic matrix subject to hydrostatic tension is presented. Two types of particles will be considered: (1) a perfectly fitting inclusion, (2) a misfitting one. A numerical solution is developed to solve the governing equations. Various combinations of elastic constants and power hardening coefficients are studied for different load levels. The analysis is concentrated on the interfacial normal stresses and its dependence on the different parameters.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):361-364. doi:10.1115/1.3173683.

The singular behavior of the stress and strain fields at the apex of a square rigid wedge embedded in a nonlinear material under plane-strain conditions is described. Both a power law and a bilinear law for the nonlinear material are considered.

Topics: Wedges , Stress , Plane strain
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):365-371. doi:10.1115/1.3173684.

A new method of calculating the internal stress distributions in wound reels of magnetic tape using a finite-difference approach is presented. A functional representation of the nonlinear radial elastic modulus is used to model reel behavior more accurately. Procedures are described to measure the radial elastic modulus and radial Poisson’s ratio. It is shown, by comparison to experimental data, that the method accurately predicts in-roll stress distributions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):372-376. doi:10.1115/1.3173685.

A solution is given for the thermoelastic stress field due to the obstruction of a uniform heat flux by a plane crack in a generally anisotropic body. A Green’s function formulation is used to reduce the problem to a set of singular integral equations which are solved in closed form. When the crack is assumed to be traction free, the crack opening displacement is found to be negative over one half of the crack unless a sufficiently large far field tensile stress is superposed. The problem is, therefore, reformulated assuming a contact zone at one crack tip. The extent of this zone and the stress intensity factors in all three modes at each crack tip are obtained as functions of the applied stress and heat flux.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):377-382. doi:10.1115/1.3173686.

A general finite-element model is proposed to deal with dynamic theromelastic problems especially with longer transient period. The method consists of formulating and solving the problem in the Laplace transform domain by the Finite Element Method (FEM) and then numerically inverting the transformed solution to obtain the time-domain response. Therefore, the transient solutions at any time could be evaluated directly. A number of examples are presented which demonstrate the accuracy, efficiency, and versatility of the proposed method, and show the effects of relaxation times, inertia, and thermoelastic coupling terms.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):383-388. doi:10.1115/1.3173687.

A finite length 2D crack is closed by remote compression followed by remote shear, and is sufficiently rough so that it remains locked with zero initial tangential crack-face shift. Then a pair of equal and opposite concentrated line loads acting in tension across the crack plane pass over the crack, causing partial opening and shift and then reclosure with residual (locked-in) shift and associated shear Stress Intensity Factors (SIF). Analytical and numerical solutions of the governing singular integral equations are used in studies of the mixed-mode transient and residual states for various coefficients of friction.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):389-397. doi:10.1115/1.3173688.

An approximate 3-dimensional analysis to study the effects of a system of transverse and longitudinal matrix cracks in cross-ply laminates is given. The method considers a repeating unit cell containing three subcells in every one of which the displacement vector is expanded to second order. The equilibrium equations, in conjunction with continuity conditions of displacements and tractions, together with the appropriate loading and boundary conditions, provide a system of equations for the elastic field variables. The dependence of the effective axial stiffness and Poisson’s ratio of the laminate on the cross-cracks density is given. The Poisson’s ratio appears to be a good indicator for the state of damage of the laminate. The distribution of the axial displacement and stress in the cracked laminate is shown.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):398-404. doi:10.1115/1.3173689.

The wavefield radiated into an elastic half-space by an ultrasonic transducer, as well as the radiation admittance of the transducer coupled to the half-space, are studied. Two models for the transducer are used. In one an axisymmetric, Gaussian distribution of normal traction is imposed upon the surface, while in the other a uniform distribution of normal traction is imposed upon a circular region of the surface, leaving the remainder free of traction. To calculate the wavefield, each wave emitted by the transducer is expressed as a plane wave multiplied by an asymptotic power series in inverse powers of the aperture’s (scaled) radius. This reduces the wave equations satisfied by the compressional and shear potentials to their parabolic approximations. The approximations to the radiated waves are accurate at a depth where the wavefield remains well collimated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):405-412. doi:10.1115/1.3173690.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):413-418. doi:10.1115/1.3173691.

An improved version of Reissner’s theory, called Optimal version, is proposed in the case of homogeneous and isotropic plates with any edge boundary conditions. It differs from the classical theory by the value of the transverse shear deformability factor and by the boundary conditions. Three-dimensional displacement and stress distributions expressed in terms of the Optimal version are given for any point of the plate, whether within the plate itself or in the neighborhood of its edge. It is proved that these distributions are second-order approximations of the exact three-dimensional solution—relative error O(h2 /L2 ). Consequently, Optimal version is a second-order theory; therefore it is better than Kirchhoff-Love’s theory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):419-424. doi:10.1115/1.3173692.

A small deflection theory, consisting of differential equations and a total potential energy expression, is presented for determining the stresses and deformations in variable thickness elastic sandwich plates symmetric about a middle surface. The theory takes into account the contribution of the face-sheet membrane forces (by virtue of their slopes) to the transverse shear. A finite-difference formulation of the stationary total potential energy principle is presented along with an illustrative application.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):425-429. doi:10.1115/1.3173693.

This paper incorporates an analysis of the stability of orthotropic or isotropic cylindrical shells subjected to external pressure applied over all or part of their surfaces. An eighth-order governing equation for buckling of orthotropic, isotropic, and composite cylindrical shells is deduced. This governing differential equation can facilitate the analysis and enable us to resolve the buckling problem. The formulas and results, deduced for the first time in this paper, may be readily applied in determining critical loads for local loading of orthotropic, isotropic, and composite cylindrical shells.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):430-436. doi:10.1115/1.3173694.

The unbonded frictionless receding contact problem of a thin plate placed under centrally symmetric vertical loading while resting on an elastic half-space or a Winkler foundation is solved in this paper. The problem is transformed into the solution of two-coupled integral-series equations over an unknown contact region. The problem is nonlinear by virtue of unilateral contact and therefore needs to be solved iteratively. Special attention is given to the edge and corner contact pressure singularities for the plate on the elastic half-space. Comparison is made with other relevant numerical results available.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):437-442. doi:10.1115/1.3173695.

A new boundary-element formulation using particular integrals is developed for the free-vibration analysis of axisymmetric solids. The numerical results for a number of axisymmetric free-vibration problems are given and some of the results are compared with those obtained from Finite Element Method, Series Solution Method, or experimental method. Generally, agreement among all of these results is satisfactory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):443-447. doi:10.1115/1.3173696.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):448-452. doi:10.1115/1.3173697.

A new approach is proposed to predict the dynamic behavior of rotor-bearing systems in time domain using the combined methodologies of finite elements and transfer matrices. This approach makes use of the finite element method to model symmetric shafts and then transforms the system properties to transfer matrix mode. The formulation provides flexibility to include both linear and nonlinear system models, often encountered in rotor dynamic applications. Few example rotor cases had been studied and the results were compared with those obtained using finite element method. This establishes that considerable savings in computational effort can be achieved without losing any accuracy.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):453-457. doi:10.1115/1.3173698.

Due to a complicated spectrum of natural modes of vibration, flexible spinning disks are very sensitive to transverse loading, and deflections have been difficult to compute. Developed in this paper is an efficient technique for determining the forced response from space fixed loads. Using an eigenfunction expansion, we are able to identify, a priori , those modes which are most important to the overall response. By ignoring unimportant modes we are able to coverge upon a numerical solution with a speed that is orders of magnitude faster than other commonly used techniques.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):458-460. doi:10.1115/1.3173699.

A method of obtaining a sufficient almost-sure asymptotic stability condition for second-order, linear oscillatory systems with an ergodic damping coefficient is presented. In this method, the probabilistic property of the derivative process of the damping coefficient is taken into account. A sufficient condition for almost-sure asymptotic stability is derived and numerical results are presented for the case of a Gaussian noise coefficient. The results are found to be an improvement over previously available results for oscillatory systems with stochastic damping.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):461-466. doi:10.1115/1.3173700.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):467-473. doi:10.1115/1.3173701.

This paper presents an experimental investigation of the random parametric excitation of a dynamic system with nonlinear inertia. The experimental model is a rigid circular tank partially filled with an incompressible inviscid liquid. The random responses of the first antisymmetric and symmetric sloshing modes are considered for band-limited random excitations. These include the means, mean squares, and probability density functions of each sloshing mode. The response of the liquid-free surface is found to be a stationary process for test durations exceeding ten minutes. The time-history response records reveal four response characteristic regimes. Each regime takes place within a certain range of excitation spectral density level. An evidence of the jump phenomenon, which was predicted theoretically by using the non-Gaussian closure scheme, is also reported. Comparisons with analytical results, derived by three different approaches, are given for the first antisymmetric sloshing mode.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):474-481. doi:10.1115/1.3173702.

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

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):482-486. doi:10.1115/1.3173703.

The ultimate effects of the entrance region on startup flow are considered by one-dimensional steady-state analysis. By roughly accounting for the entrance loss, semiempirical formulas for the average velocity and the hydrodynamic entrance length in terms of a startup flow parameter and an entrance effect coefficient are provided. The startup flow parameter M = gHD4 /2048v2 L2 is identified as the fundamental parameter of the problem. While Szymanski’s solution is recovered in the long pipe limit, i.e., M → 0, the steady-state volume flux through short pipes (M > 0.05) is significantly reduced due to entrance region effects.

Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):488-489. doi:10.1115/1.3173705.
Abstract
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):491-492. doi:10.1115/1.3173707.
Abstract
Topics: Tension
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):492-495. doi:10.1115/1.3173708.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):495-497. doi:10.1115/1.3173709.
Abstract
Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster

BOOK REVIEWS

J. Appl. Mech. 1988;55(2):500. doi:10.1115/1.3173714.
FREE TO VIEW
Abstract
Topics: Solids
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):500-501. doi:10.1115/1.3173715.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1988;55(2):501. doi:10.1115/1.3173716.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster

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