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RESEARCH PAPERS

J. Appl. Mech. 1991;58(3):612-616. doi:10.1115/1.2897239.

The constitutive equations of linear poroelasticity presented by Biot (1955) and Biot and Willis (1957) extended the description of rock behavior into the realm of saturated porous rocks. For isotropic material behavior, Rice and Cleary (1976) gave a formulation which involved material constants whose physical interpretation was particularly simple and direct; this is an aid both to their measurement and to the interpretation of predictions from the theory. This paper treats anisotropic poroelasticity in terms of material tensors with interpretations similar to those of the constants employed by Rice and Cleary. An effective stress principle is derived for such anisotropic material. The material tensors are defined, rigorously, from the stress field and pore fluid content changes produced by boundary displacements compatible with a uniform mean strain and uniform pore pressure increments. Such displacements and pore pressure increments lead to homogeneous deformation on all scales significantly larger than the length scale of microstructural inhomogeneities. This macroscopic behavior is related to the microscopic behavior of the solid skeleton. The tensors which describe the microscopic behavior of the solid skeleton would be difficult, even impossible, to measure, but their introduction allows relationships between measurable quantities to be identified. The end product of the analysis is a set of constitutive equations in which the parameters are all measurable directly from well-accepted testing procedures. Relationships exist between measurable quantities that can be used to verify that the constitutive equations described here are valid for the rock under consideration. The case of transverse isotropy is discussed explicitly for illustration.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):617-622. doi:10.1115/1.2897240.

A phenomenological corner theory was proposed for elastic-plastic materials by the authors in the previous paper (Goya and Ito, 1980). The theory was developed by introducing two transition parameters, μ (α) and β (α), which, respectively, denote the normalized magnitude and direction angle of plastic strain increments, and both monotonously vary with the direction angle of stress increments. The purpose of this report is to incorporate the Bauschinger effect into the above theory. This is achieved by the introduction of Ziegler’s kinematic hardening rule. To demonstrate the validity and applicability of a newly developed theory, we analyze the bilinear strain-path problem using the developed equation, in which, after some linear loading, the path is abruptly changed to various directions. In the calculation, specific functions, such as μ (α) = Cos (.5πα/αmax ) and β (α) = (αmax - .5π) α/αmax , are chosen for the transition parameters. As has been demonstrated by numerous experimental research on this problem, the results in this report also show a distinctive decrease of the effective stress just after the change of path direction. Discussions are also made on the uniqueness of the inversion of the constitutive equation, and sufficient conditions for such uniqueness are revealed in terms of μ(α), β(α) and some work-hardening coefficients.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):623-630. doi:10.1115/1.2897241.

A simple elastic-plastic constitutive model based on the two-surface theory is developed to describe deformation behavior of austenitic stainless steels under multiaxial cyclic loading. Dependency of saturated stress range both on strain range and the proportionality of loading is considered. To establish a precise procedure for determination of material constants for nonproportional loading, the intervariable relation in the axial-torsional circular strain-path condition is studied in detail. A full procedure is then developed for determination of all material parameters. Finally, the effectiveness of the present model is demonstrated by application to axial-torsional cyclic tests for type 304 stainless steel at 550°C.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):631-638. doi:10.1115/1.2897242.

The void growth occurring during tensile testing of uniaxial and notched specimens of free-cutting brass has been determined experimentally. This material contains a globular lead phase which tears or bursts to nucleate voids during deformation. Using quantitative metallographic data from specimens whose deformation was interrupted prior to failure, histories of void volume fraction and void aspect ratio were determined. The measured stress-strain response from the tensile tests was shown to be close to predictions from a finite element model incorporating Gurson’s constitutive model for a porous plastic solid. Predicted void growth rates agreed well with experiment for uniaxial specimens but were less than the measured growth rates in notched, high triaxiality specimens.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):639-643. doi:10.1115/1.2897243.

In the present work, the propagation of elasto-damage longitudinal stress waves in thin rods is investigated. The material behavior is characteristic to that of certain monolithic ceramics. The damage constitutive relation that characterizes this type of materials gives rise to certain dynamic behavior which is somewhat different from dynamic plastic behavior. Plastic and damage dynamic response are compared through an example.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):644-650. doi:10.1115/1.2897244.

A viscoplastic model for extrusion is discussed which simultaneously predicts the deformation field, optimal die geometry, and plastic boundaries. The die geometry and plastic boundaries are expressed in terms of chosen trial functions that satisfy certain geometrical and physical constraints. The variational power integral is minimized in the trial plastic domain using FEM technique to determine the deformation field and shape coefficients for the die contour and plastic boundaries. The proposed method is implemented for the optimal design of an axisymmetric streamlined die. The predicted values are in reasonable agreement with the experimental observations and are in conformity with the results published earlier.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):651-657. doi:10.1115/1.2897245.

Analytical solutions are presented for the diffuse and localized bifurcations of compressible solids subjected to plane-strain loadings. The solutions generalize the works for incompressible solids of Biot (1965) and Hill and Hutchinson (1975). They are verified by comparing them to results previously established for incompressible solids and elastoplastic Mohr-Coulomb materials.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):658-665. doi:10.1115/1.2897246.

The stability and structure of shear bands and how they relate to initial imperfections is studied within the framework of a one-dimensional boundary value problem. It is shown that in strain-softening viscoplasticity the structure of the band depends on the structure of the imperfection. A Fourier analysis shows that the width of the shear band depends directly on the width of the imperfection, suggesting that the imperfection scales the response of the viscoplastic material. For continuously differentiable imperfections, the shear band is continuously differentiable, whereas when the imperfection is C ° at the maximum, the shear band is C °, and cusp-shaped. For step function imperfections, the shear band is shown to be a step function, but it is shown that this solution is unstable.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):666-679. doi:10.1115/1.2897247.

Two circular cylinders consisting of a rigid core which is covered by an arbitrary number of homogeneous, isotropic, viscoelastic coats of arbitrary, but uniform thickness are pressed together so that a contact area in the form of a strip forms between them, and subsequently rolled in the presence of dry friction. A Maxwell model of viscoelasticity is employed; the friction is finite and modeled by Coulomb’s law; partial slip in the contact area is allowed. It is required to find the viscoelastic field in the cylinder, notably in the contact strip, when the compressive force and the creepage in rolling direction are specified. The proposed method works almost equally fast in the case of pure elasticity and of viscoelasticity. It is akin to the method of Bentall et al. (1968), but automated, modernized, and extended.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):680-687. doi:10.1115/1.2897248.

The paper establishes the range of in-plane fracture mode mixtures and contact zone sizes that can be obtained from an edge-cracked bimaterial strip under biaxial applied displacements. The development of a suitable loading device for and the application of crack opening interferometry to interfacial crack initiation experiments is described. The crack initiation process under bond-normal loading is examined in detail for a glass/epoxy interface in order to establish a hybrid optical interference/finite element analysis technique for extracting mixed-mode fracture parameters.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):688-694. doi:10.1115/1.2897249.

Reflection and transmission of Rayleigh surface waves by a juncture normal to the free surface, between identical or different materials, has been investigated. The juncture, which may be an interface containing defects or a thin layer, is represented by a layer of extensional and shear springs. The mathematical statement of the problem is reduced to a system of singular integral equations for the displacements on the free surface and the tractions and the displacements across the juncture. Numerical solutions of this system have been computed by the use of the boundary element method. Expressions for the reflection and transmission coefficients have subsequently been obtained by the use of half-plane Green’s functions in conjunction with an elastodynamic representation integral. Results are presented for selected values of the elastic constants of the joined bodies and the stiffness parameters of the juncture.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):695-702. doi:10.1115/1.2897250.

The scattering of elastic waves by a circular crack situated in a transversely isotropic solid is studied here. The axis of material symmetry and the axis of the crack coincides. The incident wave is taken as a plane longitudinal wave propagating perpendicular to the crack surface. A Hankel transform representation of the scattered field is used, and after some manipulations using the boundary conditions this leads to an integral equation over the crack for the displacement jump across the crack. This jump is expanded in a series of Legendre polynomials which fulfill the correct edge condition and the integral equation is projected on the same set of Legendre polynomials. The far field is computed by the stationary phase method. A few numerical computations are carried out for both isotropic and anisotropic solids. Results for the isotropic solid compare favorably with those available in the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):703-709. doi:10.1115/1.2897251.

The problem considered here is the antiplane response of an elastic solid containing a half-plane crack subjected to suddenly applied concentrated point forces acting at a finite distance from the crack tip. A fundamental solution for the dynamic dislocation is obtained to construct the dynamic fracture problem containing a characteristic length. Attention is focused on the time-dependent full-field solutions of stresses and stress intensity factor. It is found that at the instant that the first shear wave reaches the crack tip, the stress intensity factor jumps from zero to the appropriate static value. The stresses will take on the appropriate static value instantaneously upon arrival of the shear wave diffracted from the crack tip, and this static value is thereafter maintained. The dynamic stress intensity factor of a kinked crack from this stationary semi-infinite crack after the arrival of shear wave is obtained in an explicit form as a function of the kinked crack velocity, the kink angle, and time. A perturbation method, using the kink angle as the perturbation parameter, is used. If the maximum energy release rate is accepted as the crack propagation criterion, then the crack will propagate straight ahead of the original crack when applying point load at the crack face.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):710-715. doi:10.1115/1.2897252.

The paper describes a theoretical and experimental study pertaining to torsional stress wave motion in an axisymmetric waveguide whose cross-sectional area varies periodically as a function of the axial coordinate. Dispersion relations for the phase speed are obtained for both nonresonant and resonant conditions, using perturbation techniques for small amplitude, sinusoidal modulation. Resonant conditions exist when the modulation wave number is proportional to the sum or difference of wave numbers corresponding to various modes of the torsional stress wave. The experiments consist of measuring the stress wave speed in waveguides with threaded surfaces. The experimental observations verify the general trends predicted by the theory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):716-723. doi:10.1115/1.2897253.

A partial differential equation model of a cantilevered beam with a tip mass at its free end is used to study damping in a composite. Four separate damping mechanisms consisting of air damping, strain rate damping, spatial hysteresis, and time hysteresis are considered experimentally. Dynamic tests were performed to produce time histories. The time history data is then used along with an approximate model to form a sequence of least squares problems. The solution of the least squares problem yields the estimated damping coefficients. The resulting experimentally determined analytical model is compared with the time histories via numerical simulation of the dynamic response. The procedure suggested here is compared with a standard modal damping ratio model commonly used in experimental modal analysis.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):724-728. doi:10.1115/1.2897254.

Frequencies of vibration of an elliptic plate, clamped along the edge, are determined by means of a perturbation scheme based on a boundary perturbation method (B.P.M.). Eigenvalues are obtained corresponding to higher modes of vibration containing elliptic nodes, in addition to the fundamental mode. Comparison with previously derived values in the fundamental mode reveals that the present scheme leads to accurate results for moderately elliptic plates.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):731-737. doi:10.1115/1.2897256.

We model the lung parenchyma as an elastic half-space and the pleura as a taut elastic membrane in smooth or in welded contact with the half-space. In each instance we deduce that the presence of a sufficiently high surface tension T in the pleural membrane will lead to the existence of a cutoff frequency f0 for the Rayleigh-type surface waves, and we derive an equation which gives T in terms of f0 and parameters that characterize the layered medium. We performed experiments on four inflated horse lungs at transpulmonary pressures of 5, 10, and 15 cmH2 O. A comparison of the experimental results and the theoretical predictions provides an empirical test to the validity of the modeling.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):738-742. doi:10.1115/1.2897257.

An understanding of the response of pipework systems to high levels of seismic excitation is required to enable aseismic design methods to be securely based. Theoretical and experimental modeling of simple systems and components demonstrate that plasticity in the pipe wall controls the vibration response level and that, because of an unexpected level of material strain hardening, in pressurized pipes, simple elastic modal and frequency analysis are satisfactory. Given the correct material properties an energy balance approach correctly predicts the steady.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):743-748. doi:10.1115/1.2897258.

The theoretical basis of two related but distinctly different dynamic buckling criteria are summarized with the objective of demonstrating the range of applicability of each, so that together they cover the entire range of dynamic pulse loads from nearly impulsive loads to step loads of infinite duration. The example chosen is a cylindrical shell under elastic axial loads but the approach is applicable more generally. A critical amplification-of-imperfections criterion with a linear shell theory is shown to be applicable for short duration loads, for which a threshold nonlinear divergence criterion gives loads an order of magnitude too conservative. Conversely, the linear theory is inapplicable for long duration loads, for which critical loads are lower than the linear static buckling load because of imperfection sensitivity. In this range the threshold nonlinear divergence criterion is used. For loads of intermediate duration, an extended critical amplification criterion is used with equations that conservatively assume zero static buckling load but give an unchanged formula for critical load amplitude-duration combinations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):749-753. doi:10.1115/1.2897259.

The aim of the paper is to set up a scheme for efficient computation of the small-displacement response of a plane assembly of rigid links, frictionless joints, and elastic springs to static external forces applied at the joints. The particular assembly of Fig. 1 is used as an example. The conventional “stiffness method”-which becomes singular when, as here, the links are rigid-is abandoned in favor of a method which describes the current state of the assembly in terms of the amplitudes of m (here = 3) independent infinitesimal modes of inextensional deformation of the assembly; and the calculation boils down to the solving of an m x m (here 3 x 3) set of algebraic equations. The method is particularly straightforward if the inextensional modes (as here) may be obtained by inspection; but a general algorithm is presented for obtaining the inextensional modes of an arbitrary assembly of the same general kind. A major advantage over the conventional stiffness method-which requires, of course, the replacement of rigid links by (stiff) elastic members is that the number of variables may be reduced substantially. This can be very important for large assemblies.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):754-758. doi:10.1115/1.2897260.

The observation by Thomas Kane a few years ago, that long-used relationships for predicting post-collision motion of a system of rigid bodies can imply a significant increase in kinetic energy during collision, has revived interest in this type of problem. This paper is intended to clarify understanding of the sources of this difficulty, and to suggest an alternative to some of the previously used assumptions for making such predictions. An organization of the pertinent equations of kinetics is presented, which provides a more direct means of examining the aforementioned question and of obtaining rebound predictions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):759-765. doi:10.1115/1.2897261.

A mode synthesis approach is presented to calculate the eigenproperties of a structure from the eigenproperties of its substructures. The approach consists of synthesizing the substructures sequentially, one degree-of-freedom at a time. At each coupling stage, the eigenvalue is obtained as the solution of a characteristic equation, defined in closed form in terms of the eigenproperties obtained in the preceding coupling stage. The roots of the characteristic equation can be obtained by a simple Newton-Raphson root finding scheme. For each calculated eigenvalue, the eigenvector is defined by a simple closed-form expression. The eigenproperties obtained in the final coupling stage provide the desired eigenproperties of the coupled system. Thus, the approach avoids a conventional solution of the second eigenvalue problem. The approach can be implemented with the complete set or a truncated number of substructure modes; if the complete set of modes is used, the calculated eigenproperties would be exact. The approach can be used with any finite element discretization of structures. It requires only the free interface eigenproperties of the substructures. Successful application of the approach to a moderate size problem (255 degrees-of-freedom) on a microcomputer is also demonstrated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):766-775. doi:10.1115/1.2897262.

This paper presents a multi-flexible-body dynamics formulation incorporating a recently developed theory for capturing motion-induced stiffness for an arbitrary structure undergoing large rotation and translation accompanied by small vibrations. In essence, the method consists of correcting dynamical equations for an arbitrary flexible body, unavoidably linearized prematurely in modal coordinates, with generalized active forces due to geometric stiffness corresponding to a system of 12 inertia forces and 9 inertia couples distributed over the body. Computation of geometric stiffness in this way does not require any iterative update. Equations of motion are derived by means of Kane’s method. A treatment is given for handling prescribed motions and calculating interaction forces. Results of simulations of motions of three flexible spacecraft, involving stiffening during spinup motion, dynamic buckling, and a slewing maneuver, demonstrate the validity and generality of the theory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):776-783. doi:10.1115/1.2897263.

The steady-state responses of linear flexible rotor-bearing systems are analyzed by the modified transfer matrix method. The transfer matrix has the advantage of solving the problems in frequency domain with fixed matrix size. This makes the method more economical in analyzing a large degree-of-freedom rotor system than many time-marching integrating methods. In this paper, the modifications of transfer matrix method include that the transfer matrix of shaft is derived from the “continuous system” concept instead of conventional “lumped system” concept, and the paper tries to extend the transfer matrix method to fit synchronous elliptical orbit and nonsynchronous multi-lobed whirling orbit. To demonstrate the applications of the method, three examples are presented; two synchronous and one nonsynchronous.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):784-791. doi:10.1115/1.2897264.

Although the galloping of an iced electrical conductor has been considered by many researchers, no special attention has been given to the galloping’s sensitivity to alternations in the system’s parameters. A geometrical method is presented in this paper to describe these instability trends and to provide compromises for controlling an instability. The conventional but uncontrollable parameter of the wind speed is chosen as the basis for obtaining the critical conditions under which bifurcations occur for a representative two degrees-of-freedom model. Variations in these critical conditions are found in a two-dimensional parameter space in order to determine the trends for the initiation of galloping as well as to evaluate the stability of the ensuring periodic vibrations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):792-803. doi:10.1115/1.2897265.

Experiments on continuous, steady flows of granular materials down an inclined channel or chute have been conducted with the objectives of understanding the characteristics of chute flows and of acquiring information on the rheological behavior of granular material flow. Two neighboring fiber-optic displacement probes provide a means to measure (1) the mean velocity by cross-correlating two signals from the probes, (2) the unsteady or random component of the particle velocity in the longitudinal direction by a procedure of identifying particles, and (3) the mean particle spacing at the boundaries by counting the frequency of passage of the particles. In addition, a strain-gauged plate built into the chute base has been employed to make direct measurement of shear stress at the base. With the help of these instruments, the vertical profiles of mean velocity, velocity fluctuation, and linear concentration were obtained at the sidewalls. Measurements of some basic flow properties such as solid fraction, velocity, shear rate, and velocity fluctuation were analyzed to understand the characteristics of the chute flow. Finally, the rheological behavior of granular materials was studied with the experimental data. In particular, the rheological models of Lun et al. (1984) for general flow and fully developed flow were compared with the present data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):804-811. doi:10.1115/1.2897266.

The stability conditions of a hollow rotor partially filled with a Newtonian liquid are investigated. The rotor is considered here to be a rigid body, supported by springs and dampers, and exposed to an external dynamic force in the shape of actions of the encountered liquid. The system has two degrees-of-freedom, defined by deflection in two mutually orthogonal fixed directions perpendicular to the rotor axis. The fluid motions are described by Navier-Stokes equations and comparison is made between the inviscid and viscous case in connection with their predictions of the stability conditions. Experiments are performed with two different rigidity ratios and results are found to be in agreement with theoretical data.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):812-819. doi:10.1115/1.2897267.

The entrainment of lubricant at the entrance of a lubrication zone, such as that of a partially starved slider bearing, is analyzed in a closed system using the method of matched asymptotic expansions. A sphere falling together with a small lens of lubricant in a closely fitting tube is shown to fall under gravity at a speed

V = (Mg − Fc)[(RC − RS)/RC]/(16π2μRC)
, where M denotes the total mass of the system, sphere plus lubricant, g the acceleration of gravity, F c the differential contact force, μ the viscosity of the lubricant, and R C and R S the radii of the tube and the sphere, respectively. Potential biological applications and experimental verification are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):820-824. doi:10.1115/1.2897268.

A semi-analytical solution for plane velocity fields describing steady-state incompressible flow of nonlinearly viscous fluid into an elliptical opening is presented. The flow is driven by hydrostatic pressure applied at infinity. The solution is obtained by minimizing the rate of energy dissipation on a sufficiently flexible incompressible velocity field in elliptical coordinates. The medium is described by a power creep law and solutions are obtained for a range of exponents and ellipse eccentricites. The obtained solutions compare favorably with results of finite element analysis.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):825-833. doi:10.1115/1.2897269.

A new theory of viscous fluid wakes behind rod-like bodies is presented and is used to study the onset and downstream development of vortex street flows. Analytical solutions are obtained for the evolution of wave number, mean centerline velocity, vortex velocity, and vortex “spacing ratio” as a function of downstream distance in a laminar vortex street. A simple criterion for the onset of oscillations in the far wake, which slightly precede vortex street initiation, is also obtained. All of these solutions account for the action of viscous diffusion in spreading the street, and they are found to compare quite well with available experimental results.

Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Appl. Mech. 1991;58(3):834-836. doi:10.1115/1.2897270.

A method based on the eigenfunction (Williams stress functions) expansion is developed to examine the stress distribution around the tip region of a macrocrack which has some specific micro configuration. In particular, a macrocrack running into a micro hole under mode I condition is analyzed in detail. The coefficients associated with each eigenfunction are determined by a collocation procedure and the convergence of the numerical results is shown to be quite satisfactory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):836-840. doi:10.1115/1.2897271.

Altmann’s equations for describing the residual stresses in center-wound rolled webs are solved to determine the winding stress necessary to produce prescribed residual stress distributions in the finished roll. A solution for constant circumferential stress is expanded to control the peak winding stress. Two example winding problems are discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):842-846. doi:10.1115/1.2897273.

The title problem was first considered by Knowles and Wang (1960) and was shown to be related to the solution given by the classical plate theory. This solution is actually the outer solution of a singular perturbation problem, and therefore is valid only away from the crack-tip region. Within a boundary layer of order h/a, where h is the plate thickness and a is the half-crack length, the two theories differ considerably. In this study the leading order solution is obtained for h/a - 0 and it is shown that the limiting stress intensity factor given by the Reissner plate theory is more than 50 percent higher than the asymptotic result (1 + v)/(3 + v) which is obtained from the displacement field as given by the classical plate theory.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):846-848. doi:10.1115/1.2897274.

Matrix cracking is a major pattern of the failure of composite materials. A crack can form in the matrix during manufacturing, or be produced during loading. Erdogan, Gupta, and Ratwani (1974) first considered the interaction between an isolated circular inclusion and a line crack embedded in infinite matrix. As commented by Erdogan et al., their model is applicable to the composite materials which contain sparsely distributed inclusions. For composites filled with finite concentration of inclusions, it is commonly understood that the stress and strain fields near the crack depend considerably on the microstructure around it. One notable simplified model is the so-called three-phase model which was introduced by Christensen and Lo (1979). The three-phase model considers that in the immediate neighborhood of the inclusion there is a layer of matrix material, but at certain distance the heterogeneous medium can be substituted by a homogeneous medium with the equivalent properties of the composite. Thus, for the problems of which the interest is in the field near the inclusion, it can reasonably be accepted as a good model. The two-dimensional version of the three-phase model consists of three concentric cylindrical layers with the outer one, labeled by 3, extended to infinity. The external radii a and b of the inner and intermediate phases, labeled by 1 and 2, respectively, are related by (a/b) 2 =c , where c is the volume fraction of the fiber in composite.

Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):851-853. doi:10.1115/1.2897276.
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster
Commentary by Dr. Valentin Fuster

BOOK REVIEWS

J. Appl. Mech. 1991;58(3):860. doi:10.1115/1.2897280.
FREE TO VIEW
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1991;58(3):860. doi:10.1115/1.2897281.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster

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