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TECHNICAL PAPERS

J. Appl. Mech. 1995;62(4):827-833. doi:10.1115/1.2896007.

Using an idealized planar single crystal model undergoing symmetrical double slip in tension, the effect of rate sensitivity on shear band initiation and on shear band development is analysed. The behavior of the crystal is assumed to be rigid-viscoplastic. By analysing the kinematics and statics of shear banding, the deformation modes involving shear banding pattern are formulated. By a linearized stability analysis, the critical condition for shear band initiation is obtained. To study shear band development, the formulated constitutive equations are numerically solved, and the maximum value of the localized shear is predicted. The results show three different stages of shear band development. The first corresponds to a slow progression of shear localization in the band, the second to a rapid shear localization accompanied with an unloading of surrounding material, and the third to a resumption of deformation in the surrounding material and to a progressive saturation of the shear band. All three stages depend strongly on rate sensitivity, especially the first stage which does not exist in the rigid-plastic case. Even very small rate sensitivity can delay significantly or even preclude the shear band formation. Finally a discussion of the results illustrates how a macroscopic shear band forms and propagates.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):834-840. doi:10.1115/1.2896008.

The mechanical behavior of fiber-reinforced composite thermoplastic sheets during forming processes is modelled as a viscous fluid with inextensibility and incompressibility constraints. Techniques of linear stability analysis are used to study the growth or decay of initial imperfections in plane sheets reinforced by two families of fibers subjected to biaxial strains. This theory delimits situations when buckles can be expected to form during forming operations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):841-846. doi:10.1115/1.2896009.

A new approach is presented for investigating the dispersive character of structural waves. The wavelet transform is applied to the time-frequency analysis of dispersive waves. The flexural wave induced in a beam by lateral impact is considered. It is shown that the wavelet transform using the Gabor wavelet effectively decomposes the strain response into its time-frequency components. In addition, the peaks of the time-frequency distribution indicate the arrival times of waves. By utilizing this fact, the dispersion relation of the group velocity can be accurately identified for a wide range of frequencies.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):847-852. doi:10.1115/1.2896010.

Distribution concept of physical variables in viscoelasticity theory enables to represent the stress-strain relations in the form of convolution equations, that is algebraic equations in the convolution algebra of right-sided distributions. These equations can be handled in much the same way that one handles matrix equations. Distributional correspondence principle is formulated as a transition process from the algebra of numbers (elastic solution) to the convolution algebra of distributions (viscoelastic solution). Corresponding elements and operations, respectively, in both algebras are established. Applications to a wide class of problems of the plate and shell theory are shown.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):853-859. doi:10.1115/1.2896011.

A new model is developed for predicting the internal loads in a wound roll. The principal contribution is the ability to account for large deformation. This is important for applications that require tightly packed rolls with large interlayer pressure and web compression. A nonlinear, orthotropic, plane-strain, pseudoelastic constitutive law is used. Arguments are made as to why the “material symmetry condition” does not apply to the wound roll. An efficient numerical solution scheme is developed, and a good correlation is achieved with interlayer pressure measurements available in the literature.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):860-866. doi:10.1115/1.2896012.

Eshelby’s tensor for an ellipsoidal inclusion with perfect bonding at interface has proven to have a far-reaching influence on the subsequent development of micromechanics of solids. However, the condition of perfect interface is often inadequate in describing the physical nature of the interface for many materials in various loading situations. In this paper, Airy stress functions are used to derive Eshelby’s tensor for a circular inclusion with imperfect interface. The interface is modeled as a spring layer with vanishing thickness. The normal and tangential displacement discontinuities at the interface are proportional to the normal and shear stresses at the interface. Unlike the case of the perfectly bonded inclusion, the Eshelby’s tensor is, in general, not constant for an inclusion with the spring layer interface. The normal stresses are dependent on the shear eigenstrain. A closed-form solution for a circular inclusion with imperfect interface under general two-dimensional eigenstrain and uniform tension is obtained. The possible normal displacement overlapping at the interface is discussed. The conditions for nonoverlapping are established.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):867-872. doi:10.1115/1.2896013.

Two flat isotropic elastic half-spaces, of different material properties, are pressed together and slide against each other with a constant coefficient of friction. Although a nominally steady-state solution exists, an analysis of the dynamic problem demonstrates that the steady solution can be dynamically unstable. Eigenvalues with positive real parts give rise to self-excited motion which occurs for a wide range of material pairs, coefficients of friction, and sliding velocities (including very low speeds). These self-excited oscillations are generally confined to the region near the interface and can lead either to regions of loss of contact or to areas of stick slip. The mechanism responsible for the instability is essentially one of destabilization of interfacial (slip) waves. It is expected that these vibrations might play an important role in the behavior of sliding members with dry friction.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):873-879. doi:10.1115/1.2896014.

The exact expression describing the contact force between an axially moving flexible strip and a one-sided constraint is derived using the Green’s function formulation. Discontinuity of the initial velocity at a boundary of the strip due to a disturbance, causes discontinuity in the contact force history at any constraint that is not modeled by a single spring element. The discontinuities in the contact force occur at the instants when those propagating wavefronts in the strip with nonvanishing slope interact with the constraint. A model of a magnetic tape-recording head system is analyzed. Tape-head contact loss is predicted, depending on the amplitude and frequency of a disturbance, the head location, and the preload of the tape against the head.

Topics: Strips , Force , Springs
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):880-886. doi:10.1115/1.2896015.

A procedure is presented for determining the three-dimensional elasticity solutions for free vibration analysis of simply supported thick skew plates. The exact expressions of strain and kinetic energies are derived from linear, small-strain, three-dimensional elasticity theory. To allow the treatment of soft and hard simple support conditions, sets of three-dimensional spatial displacement functions are expressed in terms of unit normals to the edges. By virtue of the three-dimensional elasticity theory, the present method does not require a special treatment for stress singularity at the obtuse corners. This method is also demonstrated to be free from shear locking phenomena. The significant difference in the vibration response of skew plates with soft and hard simple support conditions is highlighted. The influence of skew angle on the eigenvalues of thick skew plate is discussed in the context of the three-dimensional elasticity solutions.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):887-892. doi:10.1115/1.2896016.

The present paper provides an analysis of the response of a right-angled bent cantilever beam subjected to an out-of-plane impulsive load (i.e., suddenly imposed velocity) applied to concentrated mass at its tip. If T 0 and M 0 are the fully plastic torque and bending moment, respectively, of the cross section, it is shown that for the case T 0 /M 0 < 1, a double hinge mechanism is required, with a pure bending hinge in the first segment of the beam and a combined bending-torsion hinge in the second segment. The history of deformation is described following impartation of the load.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):893-898. doi:10.1115/1.2896017.

We present an analysis of the rigid-body model for frictional three-dimensional impacts, which was originally studied by Routh. Using Coulomb’s law for friction, a set of differential equations describing the progress and outcome of the impact process for general bodies can be obtained. The differential equations induce a flow in the tangent velocity space for which the trajectories cannot be solved for in a closed form, and a numerical integration scheme is required. At the point of sticking, the numerical problem becomes ill-conditioned and we have to analyze the flow at the singularity to determine the rest of the process. A local analysis at the point of sticking provides enough information about the global nature of the flow to let us enumerate all the possible dynamic scenarios for the sliding behavior during impact. The friction coefficient, and the mass parameters at the point of contact, determine the particular sliding behavior that would occur for a given problem. Once the initial conditions are specified, the possible outcome of the impact can then be easily determined.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):899-902. doi:10.1115/1.2896018.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):903-907. doi:10.1115/1.2896019.

The nonlinear equations for planar motions of a vertical cantilevered pipe conveying fluid are modified to take into account a small lumped mass added at the free end. The resultant equations contain nonlinear inertial terms; by discretizing the system first and inverting the inertia matrix, these terms are transferred into other matrices. In this paper, the dynamics of the system is examined when the added mass is negative (a mass defect), by means of numerical computations and by the software package AUTO. The system loses stability by a Hopf bifurcation, and the resultant limit cycle undergoes pitchfork and period-doubling bifurcations. Subsequently, as shown by the computation of Floquet multipliers, a type I intermittency route to chaos is followed—as illustrated further by a Lorenz return map, revealing the well-known normal form for this type of bifurcation. The period between “turbulent bursts” of nonperiodic oscillations is computed numerically, as well as Lyapunov exponents. Remarkable qualitative agreement, in both cases, is obtained with analytical results.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):908-914. doi:10.1115/1.2896021.

The effect of viscoelasticity in web handling systems is examined by introducing a viscoelastic equation of state into a model for tension control. Case studies and generalized results for a single open-span system and a double open-span system are presented to compare the results of the viscoelastic model to a model based on a purely elastic equation of state. The results show only small differences in the tension behavior for the single-span system. However, large differences in the magnitude and reversal of the sign of the tension in the second span of a two-span system are seen for even small degrees of viscoelasticity in the web material. The results clearly demonstrate that viscoelasticity must be considered in modeling multispan web handling systems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):915-923. doi:10.1115/1.2896022.

Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wave numbers, and an integrodif-ferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly varying wave packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. In all cases, the final equilibrium state consists of a single dominant mode and its harmonics of smaller amplitudes. The predicted range of wave numbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wave numbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that a linearly stable long wave can excite a short unstable wave through nonlinear wave interaction. An important implication of the existence of nonunique equilibrium states is that the torque induced by the fluid motion cannot be determined uniquely. The numerical results show that the uncertainty , associated with nonuniqueness, of using any accurately measured Taylor-vortex torque slightly above the first bifurcation point in engineering practice can be as large as ten percent. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles’ monumental contribution in 1965. A theoretical confirmation has come only 30 years later. It is worthwhile to point out that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The current study indicates that the state of flows on a stable bifurcation branch can involve any wave number within a finite band and can not be determined uniquely. The multiple solutions in Coles’ sense have also been found for mixed-convection flows (Yao and Ghosh Moulic, 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):924-929. doi:10.1115/1.2896023.

This paper discusses the transformation properties of the famous Johnsen-Hamel equations of motion of discrete mechanical systems in general nonlinear nonholonomic coordinates and constraints (i.e., the nonlinear extension of the well-known Boltzmann-Hamel equations), under general nonlinear (local) quasi-velocity transformations. It is shown that the individual kinematico-inertial terms making up the system inertia force, or system acceleration, such as the nonlinear nonholonomic Euler-Lagrange operator and nonholonomic correction (or deviation) terms, in general, do not transform as nonholonomic covariant vectors; although taken as a whole they do, as expected. This work extends and completes the work of Papastavridis (1994), and it is strongly recommended that it be read after that paper.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):930-934. doi:10.1115/1.2896024.

In this paper, a general solution of the equations in the linearized theory of magnetoe-lasticity, which was developed by Pao and Yeh (1973) on the basis of Brown’s phenomenological theory of magnetoelasticity (1966), is obtained. As in some applications, the magnetic fields caused by the mechanical singularities in a magnetized elastic half-space are considered. Using the general solution and the Mindlin state of the elastic half-space (1936), the exact three-dimensional solutions for the generated magnetic fields due to various mechanical singularities, such as a single force and a doublet, are obtained in closed form.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):935-940. doi:10.1115/1.2896025.

This paper examines the attitude motion of a cylindrical body with mass loss. It is found that mass variation can have a substantial influence on the behavior of such a system. Specifically, the initial dimensions as well as the manner in which mass loss affects system inertia are found to be key factors in the determination of the characteristics of the lateral motion of the system. In great contrast to the attitude behavior of spinning rigid bodies, oblate variable mass cylinders exhibit divergent transverse attitude motion, while the transverse motion of prolate variable mass cylinders is found to be bounded in general.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):941-946. doi:10.1115/1.2896026.

The dissipatively perturbed Hamiltonian system corresponding to primary resonance is analyzed in the case in which two competing stable periodic responses exist. The method of averaging fails as the trajectory approaches the unperturbed homoclinic orbit (separatrix). By using the small dissipation of the Hamiltonian (the Melnikov integral) near the homoclinic orbit, the boundaries of the basin of attraction are determined analytically in an asymptotically accurate way. The selection of the two competing periodic responses is influenced by small changes in the initial conditions. The analytic formula is shown to agree well with numerical computations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):947-951. doi:10.1115/1.2896027.

We consider a polycrystal constituted from orthorhombic single crystals for which one particular principal axis of the crystallites is always oriented parallel to a particular direction; in the plane perpendicular to this direction the crystallites are randomly oriented. Bounds are found for the Young’s modulus in the axial direction. The lower bound on the Young’s modulus, which is realizable, is found to be that of the individual crystallite in the aligned direction. The upper bound determined is necessarily realizable when the single crystal elastic constants satisfy a certain condition. When this condition is not satisfied a bound is found; whether or not this bound is realizable must be examined using the specific elastic constants of the crystal being considered. For all physical examples considered the upper bound was indeed found to be realizable. Thus, generally speaking, a wire constituted as above, with the stiffest direction of the individual crystallites being along the wire, will have a higher Young’s modulus than the maximum modulus of the individual crystallites of which it is composed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):952-958. doi:10.1115/1.2896028.

Under general loadings including body forces and crack-face traction, the energy release rate equation for a two-dimensional cracked body is derived by a shape design sensitivity approach. Defining the virtual crack extension (VCE) as the variation of the geometry, the virtual work principle and the material derivative concept are used to obtain the final analytical equation for the energy release rate. In contrast to the results of other researchers, the functionals which appear in the derived energy release rate equation do not involve the derivative of the displacement field on the crack surface, thereby improving the numerical accuracy in the computation of the energy release rate. Although the finite element method (FEM) is applied to crack problems in this paper, any numerical analysis method can be applied to the resulting equation. In addition, if body forces and crack-face traction are constant with respect to VCE, i.e., their material derivatives are identically zero, then the energy release rate equation is domain independent for domains which exclude the crack-tip region. Three example problems are treated which demonstrate the generality, accuracy, and domain-independent nature of the derived energy release rate equation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):959-964. doi:10.1115/1.2896029.

The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):965-969. doi:10.1115/1.2896030.

The preceding paper is extended to the analysis of size effect on strength and ductility of structures. For the case of geometrically similar structures of different sizes, the criterion of stability limit is transformed to an eigenvalue problem for a homogeneous Fredholm integral equation, with the structure size as the eigenvalue. Under the assumption of a linear softening stress-displacement relation for the cohesive crack, the eigenvalue problem is linear. The maximum load of structure under load control, as well as the maximum deflection under displacement control (which characterizes ductility of the structure), can be solved explicitly in terms of the eigenfunction of the aforementioned integral equation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):970-974. doi:10.1115/1.2896031.

This paper studies the attitude dynamics of variable mass systems that have axisymmetric mass distribution and that are subjected to continuous mass variation while in motion. The equations of rotational motion for such systems are solved analytically under the assumption of zero external torque. It is found that such systems can spin up or spin down in free motion, and that the transverse angular velocity magnitude can increase or decrease with time. The analytical conditions for growth or decay of spin rate and lateral angular speed are presented, and these conditions are related to practical design criteria for rocket-type systems.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):975-982. doi:10.1115/1.2896032.

Optimal design problems arising in mechanics of growing composite viscoelastic and elastic solids subjected to aging are considered. The growth means a continuous mass influx to the body surface. Due to this process, the size of the body increases in time. The mass influx with pretensioning causes the rise of stresses in the growing body. The purpose of the current study is to propose a new class of the optimal design problems for the growing viscoelastic composite solids subjected to aging, and to solve the mechanical design problems of this new type. In the current paper, we analyze the optimal preload distribution in the winding process for cylindrical solids. The proposed approach and the obtained new solutions are of a special interest and importance for the optimization of winding of composite pressure vessels.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):983-988. doi:10.1115/1.2896033.

Optimal design problems arising in mechanics of growing composite viscoelastic and elastic solids subjected to aging are considered. The growth means a continuous mass influx to the body surface. Due to this process, the size of the body increases in time. The mass influx with pretensioning causes the rise of stresses in the growing body. The purpose of the current study is to propose a new class of the optimal design problems for the growing viscoelastic composite solids subjected to aging, and to solve the mechanical design problems of this new type. In the current paper we analyze the optimal design of the shape of growing reinforced beams which minimizes their maximum deflection. The proposed approach and the obtained new solutions are of a special interest and importance for the design of the reinforced cantilevers and bridges.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):989-996. doi:10.1115/1.2896034.

The important phenomenon of delamination buckling is examined subjected to the condition of frictionless contact. Buckled delamination is examined in particular, because in-plane compressive loading is typical and detrimental. Two types of contact can be distinguished, local and global. The latter may occur everywhere in the plate while the local contact is limited to the crack front (negative KI stress intensity factors). Both local and global contact conditions were considered using a finite element scheme which employed nonlinear plate theory. The global contact problem is formulated as it appears in post-buckling of delamination. The case of simultaneous buckling and contact is also addressed in this paper. Two particularly interesting examples of thin film delaminations are presented. In the first, the contact at buckling is due to the material anisotropy. In this case the bucking load and the post-bucking analysis were very well supported by experiments. In the second example, contact at buckling arises because of a pin that holds down the delaminated layer at its center. The treated cases indicated that contact may significantly affect the fracture parameters along the delamination front, and is, therefore, important for delamination arrest.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):997-1004. doi:10.1115/1.2896035.

Experimental displacement and strain fields are presented for a generalized plane stress tensile specimen consisting of commercially pure titanium diffusion bonded to 6Al-4V titanium. Plastic flow initiates at the intersection of the interface with the specimen free edge. Further deformation results in concentrated shear bands emanating from the interface at both free edges. Interactions of the interface-free edge shear bands force a state of plane strain in the center of the specimen. These interface constraint effects have practical relevance on the testing of joined metals with property mismatch.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1005-1014. doi:10.1115/1.2896036.

A new analytical and numerical method is presented for modeling and analysis of cylindrical shells stiffened by circumferential rings. This method treats the shell and ring stiffeners as individual structural components, and considers the ring eccentricity with respect to the shell middle surface. Through use of the distributed transfer functions of the structural components, various static and dynamic problems of stiffened shells are systematically formulated. With this transfer function formulation, the static and dynamic response, natural frequencies and mode shapes, and buckling loads of general stiffened cylindrical shells under arbitrary external excitations and boundary conditions can be determined in exact and closed form. The proposed method is illustrated on a Donnell-Mushtari shell, and compared with finite element method and two other modeling techniques.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1015-1022. doi:10.1115/1.2896037.

A nonlinear analysis is presented for combination resonances in the symmetric responses of a clamped circular plate with the internal resonance, ω3 ≈ ω1 + 2ω2. The combination resonances occur when the frequency of the excitation are near a combination of the natural frequencies, that is, when Ω ≈ 2ω1 + ω2. By means of the internal resonance condition, the frequency of the excitation is also near another combination of the natural frequencies, that is, Ω ≈ ω1 − ω2 + ω3. The effect of two near combination resonance frequencies on the response of the plate is examined. The method of multiple scales is used to solve the nonlinear nonautonomous system of equations governing the generalized coordinates in Galerkin’s procedure. For steady-state responses, we determine the equilibrium points of the autonomous system transformed from the nonautonomous system and examine their stability. It has been found that in some cases resonance responses with nonzero-amplitude modes don’t exist, and the amplitudes of the responses decrease with the excitation amplitude. We integrate numerically the nonautonomous system to find the long-term behaviors of the plate and to check the validity of the analytical solution. It is found that there exist multiple stable responses resulting in jumps. In this case the long-term response of the plate depends on the initial condition. In order to visualize total responses depending on the initial conditions, we draw the deflection curves of the plate.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1023-1028. doi:10.1115/1.2896038.

A theoretical approach is proposed to estimate the elastic moduli of three-phase composites consisting of a matrix phase reinforced by two-phase particles. The theoretical predictions are based on a simple extension to nondilute concentrations of the mechanical concentration factors obtained from the recent analysis of the average elastic fields in a double inclusion by Hori and Nemat-Nasser (1993). The proposed micromechanics theory can account for the effects of shapes and concentrations of both the particles and the dispersed phase in the particles. Theoretical estimates of the concentration factors and the effective elastic moduli are obtained in closed form and are diagonally symmetric and fall within the Hashin-Shtrikman-Walpole bounds for all cases considered. The theoretical predictions are in excellent agreement with experimental results obtained from pulse-echo and rod-resonance measurements of the elastic moduli of a three-phase composite consisting of an aluminum matrix reinforced by mullite/alumina particles.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1029-1038. doi:10.1115/1.2896039.

In this study, a dynamic antiplane crack propagation with constant velocity in a configuration with boundary is investigated in detail. The reflected cylindrical waves which are generated from the free boundary will interact with the propagating crack and make the problem extremely difficult to analyze. A useful fundamental solution is proposed in this study and the solution is determined by superposition of the fundamental solution in the Laplace transform domain. The proposed fundamental problem is the problem of applying exponentially distributed traction (in the Laplace transform domain) on the propagating crack faces. The Cagniard’s method for Laplace inversion is used to obtain the transient solution in time domain. Numerical results of dynamic stress intensity factors for the propagation crack are evaluated in detail.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1039-1046. doi:10.1115/1.2896040.

Based on a linear comparison composite (Tandon and Weng, 1988) and an energy criterion for the effective stress of the ductile matrix (Qiu and Weng, 1992), a nonlinear theory is developed to estimate the strain potential and the overall stressstrain relations of a two-phase composite containing aligned spheroidal inclusions. The plastic state of the ductile matrix under a given external load is determined by solving two simultaneous equations, one being its constitutive equation and the other the expression of its effective stress as a function of its secant shear modulus. Then by means of the effective properties of the linear comparison composite, the overall strain and strain potential of the nonlinear system are evaluated. It is demonstrated that, for an elastically incompressible matrix containing either aligned voids or rigid inclusions, the derived strain potential is exactly equal to Ponte Castaneda’s (1991) bound or estimate, respectively, of Willis’ (1977) type. Comparison with an exact solution of a fiber-reinforced composite under the plane-strain biaxial loading also shows an excellent agreement. The theory is generally intended for the condition when the concentration is not high, and is finally applied to examine the aspect-ratio dependence of the overall response for a silicon carbide/aluminum system. It is found that, more so than the elastic behavior, the nonlinear plastic response of the twophase composite is very sensitive to the inclusion shape under most types of loading.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1047-1052. doi:10.1115/1.2896041.

A central crack which propagates symmetrically in an orthotropic composite plate is investigated using the techniques of Fourier and Laplace transforms. The crack is located along one of the principal axes of the material. Complete contour integrations are carried out in the evaluation of the Laplace inversion integrals. For the crack tips running at a constant speed, exact expressions for the dynamic crack shape and the dynamic stress distribution with singularities in the crack plane are obtained in terms of anisotropic material constants and crack speed. The dynamic expressions are evaluated numerically by using graphite/epoxy and glass/ epoxy composites and an isotropic material as sample materials. The dynamic solution reduces to the static solution at zero crack speed. During crack propagation, the deviation between dynamic and static solutions is governed by dynamic correction factors which are nondimensional functions of the ratios among anisotropic material constants and the ratio of crack speed to shear-wave speed. Values of these dynamic factors are obtained for the sample composites at a large range of crack speed. The dynamic stress intensity factor vanishes at the corresponding Rayleigh wave speed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1053-1062. doi:10.1115/1.2896042.

For elastic materials containing fluid-saturated porosity, the pore compressibility is a measure of the deformation of a unit pore volume in response to a change in fluid pressure. Rather than being measured, this quantity has been routinely set equal to an effective solid compressibility, since this equality is exact whenever a single solid component is present. However, we show that the pore compressibility and solid compressibility may be uncorrelated in general. In certain special circumstances they do not even share the same sign. Although thermodynamic and mechanical stability constraints cause solid and drained-frame bulk moduli of a porous composite to be positive and bounded by component properties, the pore compressibility is unconstrained and, therefore, can have negative values. For special realizable model materials, the value of the pore compressibility can be found using an exact expression valid for a composite made up of one fluid and two solid components, i.e., two porous components. In order to quantify how various factors affect the sign and magnitude of the pore compressibility, pore compressibilities were calculated for models that used two porous components having the microgeometry of an assemblage of concentric spheres. This model implicitly assumes the pores are on a much smaller length scale than the concentric spheres. Modeling results show that with the stiffer porous material forming the outer shells of the concentric spheres, the pore compressibility of such materials is negative when solid component bulk moduli differ by at least a factor of 5, if, in addition, the porosities and drained frame moduli of the two porous components are relatively low. Negative pore compressibilities were found for realizable models whose two porous constituents had the properties of silicon nitride and either sandstone or clay. For models using combinations of alumina and glass foam properties, pore compressibilities were non-negative but smaller than the compressibilities of the solid components.

Commentary by Dr. Valentin Fuster

BRIEF NOTES

J. Appl. Mech. 1995;62(4):1063-1065. doi:10.1115/1.2896043.

A recent article by Vigdergauz considered two-component, linearly elastic plates. It showed the existence of spatially periodic microstructures producing plates of extremal rigidity. For the microstructures which maximize or minimize the effective bulk modulus, Vigdergauz asserted that the effective behavior was isotropic. We show this is wrong: the effective tensor is in fact anisotropic.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1065-1067. doi:10.1115/1.2896044.
Abstract
Topics: Stress
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(4):1067-1069. doi:10.1115/1.2896045.

A transient thermoelasticity problem of a multilayered anisotropic medium under the state of generalized plane deformation is considered in this note. The flexibility/stiffness matrix method is adopted here to obtain the complete solution of the entire layered medium by introducing the thermal and mechanical boundary and layer interface conditions in the Fourier and Laplace transform domains. As a numerical illustration, the distributions of transient temperatures and thermal stresses in a laminated anisotropic slab subjected to a uniform surface temperature rise are presented for some stacking sequences of fiber-reinforced layers.

Commentary by Dr. Valentin Fuster

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