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IN THIS ISSUE

### TECHNICAL PAPERS

J. Appl. Mech. 1995;62(2):273-281. doi:10.1115/1.2895928.

The three-dimensional problem of a multilayered composite containing an arbitrarily oriented crack is considered in this paper. The crack problem is analyzed by the equivalent body force method, which reduces the problem to a set of singular integral equations. To compute the kernels of the integral equations, the stiffness matrix for the layered medium is formulated in the Hankel transformed domain. The transformed components of the Green’s functions and derivatives are determined by solving the stiffness matrix equations, and the kernels are evaluated by performing the inverse Hankel transform. The crack-opening displacements and the three modes of the stress intensity factor at the crack front are obtained by numerically solving the integral equations. Examples are given for a penny-shaped crack in a bimaterial and a three-material system, and for a semicircular crack in a single layer adhered to an elastic half-space.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):282-288. doi:10.1115/1.2895929.

This paper is to determine vibrational eigensolutions m2, vm(r)]m = 1 of a three-dimensional, finite, linear, elastic solid C containing cracks in terms of crack configuration σc and eigensolutions n2, un(r)n = 1 of a perfect elastic solid P without the cracks. Use of Betti reciprocal theorem and the Green’s function of P expands vm(r) in terms of an infinite series of un(r). Substitution of the vm(r) series representation into the Kamke quotient of C and stationarity of the quotient result in a Fredholm integral equation whose nontrivial solutions predict λm2, and vm(r) of C . Finally, natural frequencies and mode shapes of a circular shaft of finite length containing a circumferential crack under torsional vibration are predicted through a two-term Ritz approximation of the Fredholm integral equation. The results differ significantly from those predicted by the method of flexibility matrices, when the ratio of the shaft length to the shaft radius is small.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):289-293. doi:10.1115/1.2895930.

The problem of a crack terminating at an interface between two materials which is governed by Coulomb’s law of friction is studied using Mellin integral transforms. Depending on the relative slip directions of the two wedges that are created by the crack, both the case of the two wedges moving in opposite directions and that of the two wedges moving in the same direction are treated. The characteristic equations which yield the order of the crack-tip singularity are obtained in terms of the Dundurs constants, the inclination of the crack and the coefficient of friction. For the special case when the crack is perpendicular to the interface and the two wedges slip in opposite directions, it is shown that the problem decouples into mode I (symmetric) and mode II (antisymmetric) cracks, and numerical results are presented for this case.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):294-305. doi:10.1115/1.2895931.

A previously developed linear elastic crack-tip element analysis is reviewed briefly, and then extended and refined for practical applications. The element provides analytical expressions for total energy release rate and mode mix in terms of plate theory force and moment resultants near the crack tip. The element may be used for cracks within or between homogeneous isotropic or orthotropic layers, as well as for delamination of laminated composites. Classical plate theory is used to derive the equations for total energy release rate and mode mix; a “mode mix parameter,” Ω, as obtained from a separate continuum analysis is necessary to complete the mode mix decomposition. This parameter depends upon the elastic and geometrical properties of the materials above and below the crack plane, but not on the loading. A relatively simple finite element technique for determining the mode-mix parameter is presented and convergence in terms of mesh refinement is studied. Specific values of Ω are also presented for a large number of cases. For those interfaces where a linear elastic solution predicts an oscillatory singularity, an approach is described which allows a unique, physically meaningful value of fracture mode ratio to be defined. This approach is shown to provide predictions of crack growth between dissimilar homogeneous materials that are equivalent to those obtained from the oscillatory field solution. Application of the approach to delamination in fiber-reinforced laminated composites is also discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):306-311. doi:10.1115/1.2895932.

The plane elasticity problem of dislocation inside, outside, or on the interface of an anisotropic elliptical inclusion in an unbounded anisotropic matrix is considered. A general solution for the stresses and deformations in the entire domain is obtained by applying the Stroh’s formalism and the method of analytical continuation. Since the general solutions have not been found in the literature, in order to verify the present solution, comparison is made with some special cases of which the analytical solutions exist, which shows that our results are exact and universal.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):312-319. doi:10.1115/1.2895933.

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):320-328. doi:10.1115/1.2895934.

An analytical method for the assessment of failure probability of brittle materials exhibiting progressive cracking prior to cleavage fracture is presented. The underlying fracture mechanism is based on the assumption that instability of a critical flaw no longer leads to failure and causes redistribution of the local stresses. The fracture process progresses by consecutive unstable propagation of the surviving flaws up to total failure. A limiting distribution for the fracture stress, which is identical with the first asymptotic distribution of smallest values, is derived on the basis of a chain-of-bundles probability model. Numerical procedures for calculating the parameters of the limiting distribution are also described. Due to the nature of the resulting distribution, the method employs the maximum likelihood estimation of parameters combined with a finite element solution to the crack-tip fields. An application of the present model to analyze the effect of notch depth on fracture toughness values obtained from single-edge notch bend (SENB) specimens is also included.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):329-337. doi:10.1115/1.2895935.

Couple stress theory is used in the prediction of the size of the kink band width that occurs in the compressive failure of a fiber composite by microbuckling. The composite is assumed to be inextensible in the fiber direction, and to deform as a Ramberg-Osgood solid in shear and in transverse tension. Predictions are given for the kink width as a function of the fiber diameter, modulus and strength; the material nonlinearity of the composite; and the amplitude and wavelength of fiber waviness. The kink width scales with fiber diameter but is fairly insensitive to variations in other material properties and in the amplitude and wavelength of initial fiber waviness. For typical polymer matrix composites, the predicted kink width is of the order of 10–15 fiber diameters, in agreement with observed values. The couple stress theory is also used to assess the role of fiber bending resistance in the compressive strength of fiber composites that fail by microbuckling. It is found that although the compressive strength is sensitive to the amplitude of the initial waviness, it is not very sensitive to its wavelength.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):338-345. doi:10.1115/1.2895936.

The initial post-buckling behavior of thick rings under external uniform hydrostatic pressure is investigated. In the analysis, no assumptions are placed upon the relative magnitudes of the elongations and rotations, and the ring is assumed to be elastic and extensional. The importance of including certain nonlinear terms in the initial post-buckling stability analysis and the effects of nonzero shearing strains on the buckling load and the initial post-buckling stability are examined. It is shown that the classical Kirchhoff assumptions, when employed for a ring lead to nonvanishing through thickness strains, εzz and εz θ , with the latter being proportional to the through thickness coordinate z . An approximate first order shear deformation analysis and a two-dimensional elasticity analysis (without beam-type kinematical assumptions) of the initial post-buckling behavior of thick rings are presented and the thickness effects on the buckling load and the initial post-buckling behavior are examined. The formulation for the composite ring was reduced to that of an isotropic ring and the results thus obtained were compared with published one-dimensional results in the literature. It is found from both the shear deformation and the two-dimensional analysis that the initial post-buckling behavior of the isotropic ring and the composite rings studied are stable. The influence of thickness on the degree of stability in the immediate post-buckling response is characterized.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):346-355. doi:10.1115/1.2895937.

The bifurcation of equilibrium of a compressed transversely isotropic bar is investigated by using a three-dimensional elasticity formulation. In this manner, an assessment of the thickness effects can be accurately performed. For isotropic rods of circular cross-section, the bifurcation value of the compressive force turns out to coincide with the Euler critical load for values of the length-over-radius ratio approximately greater than 15. The elasticity approach predicts always a lower (than the Euler value) critical load for isotropic bodies; the two examples of transversely isotropic bodies considered show also a lower critical load in comparison with the Euler value based on the axial modulus, and the reduction is larger than the one corresponding to isotropic rods with the same length over radius ratio. However, for the isotropic material, both Timoshenko’s formulas for transverse shear correction are conservative; i.e., they predict a lower critical load than the elasticity solution. For a generally transversely isotropic material only the first Timoshenko shear correction formula proved to be a conservative estimate in all cases considered. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one and therefore, a more precise estimate of the transverse shear effects. Furthermore, by performing a series expansion of the terms of the resulting characteristic equation from the elasticity formulation for the isotropic case, the Euler load is proven to be the solution in the first approximation; consideration of the second approximation gives a direct expression for the correction to the Euler load, therefore defining a new, revised, yet simple formula for column buckling. Finally, the examination of a rod with different end conditions, namely a pinned-pinned rod, shows that the thickness effects depend also on the end fixity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):356-361. doi:10.1115/1.2895938.

A theoretical analysis for the instability of a fluid film around a cylinder in an immiscible fluid is presented. A general dispersion equation, relating wave number to growth rate, is derived with consideration of the effects of the cylindrical interface and a finite film thickness. Application of the dispersion equation to the breakup of a liquid film around a cylindrical body in still air leads to a prediction of the dominant wavelength by λ = 2π2 / (1/ R2 + ρ1g/q), where R is the radius of the cylinder, ρ1 is the density of the liquid, g is the acceleration due to gravity, σ is the surface tension. Experiments showed good agreement with the present analysis. The dominant wavelength decreases with a decrease in the radius of the cylindrical body. A previous report on the breakup of a liquid film around a horizontal cylindrical body is shown to be in error.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):362-367. doi:10.1115/1.2895939.

A problem of frictionless contact between the running Rayleigh wave and a rigid strip is investigated. The corresponding mixed boundary value problem of elastodynamics is reduced to a system of dual series equations involving trigonometric functions. On the base of the closed-form solution obtained, explicit analytic expressions for distributions of normal displacements and stresses and of tangential velocities on the surface have been derived.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):368-372. doi:10.1115/1.2895940.

A problem of adhesive contact between the running surface wave and a rigid strip is investigated. The mixed boundary-value problem of elastodynamics is reduced to a singular integral equation for a complex combination of stresses and an exact closed-form solution of it has been derived. Analysis of variation of contact area dimensions, stress distribution and rotor velocity on the frequency of excitation displayed significant differences between the results corresponding to conditions of adhesion and slipping in contact area. The origin of these differences is discussed.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):373-379. doi:10.1115/1.2895941.

The Saint-Venant semi-inverse method generalization for the problem of torsion under large deformations is presented. The case where a prism cross-section possesses central symmetry is regarded. The torsion problem is reduced to a two-dimensional nonlinear boundary value problem. Differential balance equations and lateral conditions are satisfied by solving the boundary value problem. End conditions are implemented so that the stress system is equivalent to the torsion moment, and to the axial force passing through the cross-section center of inertia. The energy method, used to solve the torsion problem under small twist angles, is extended to the case of finite deformations. Approximate solutions of the torsion problem for elliptical, rectangular, and quadrantal cylinders made of Treloar and Blatz-Ko materials are obtained.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):380-389. doi:10.1115/1.2895942.

The problem of several indentors moving on a viscoelastic half-plane is considered in the noninertial approximation. The solution of this mixed boundary value problem is formulated in terms of a coupled system of integral equations in space and time. These are solved numerically in the steady-state limit for the case of two indentors. The phenomena of hysteretic friction and interaction between the two indentors are explored.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):390-397. doi:10.1115/1.2895943.

Based on an energy minimization principle, a mathematical/numerical model has been developed to study the impact of design and process variations associated with flip-chip solder joint on its ability to align in lateral and axial direction. The minimum-energy shape needed for joint evaluation is computed by a novel numerical method based on motion by mean curvature. The analysis shows that (1) the magnitude of the reaction force in lateral and axial direction reduces with increase in solder volume, (2) the normal reaction is an order of magnitude higher compared to the lateral reaction (restoring force) thus making the joint more susceptible to lateral misalignment compared to the axial misalignment, and (3) the axial misalignment is primarily dictated by the accuracy of the solder deposition height.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):398-406. doi:10.1115/1.2895944.

This paper is concerned with the constitutive behavior of a particular orthogonal fiber network under biaxial dead loading. We also describe a new kind of biaxial, dead-loading machine which is applicable to anisotropic materials. The machine does not require that the loads are exerted along symmetry axes of the material. A specimen of the cloth was loaded by different loading paths to the same equibiaxial dead-load, and two different final deformations were observed. A related observation was reported by Treloar for rubber in 1948. In order to understand this instability, we experimentally determined the energy function for the cloth. The energy function is then used in a variational calculation to explain this instability.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):407-413. doi:10.1115/1.2895945.

The method of substructure synthesis, originally conceived for undamped and viscously damped systems, has been extended to systems with viscoelastic damping in the hereditary integral form. Based on a new variational principle, the substructure synthesis method is formulated in the time domain. The displacement in each substructure is represented by a set of real admissible trial vectors. The traditional state space formulation is avoided by the proposed method so that the approach is independent of the form of viscoelastic models. Effectiveness of the method is illustrated through numerical examples.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):414-418. doi:10.1115/1.2895946.

A solenoidal part of displacement field appearing in a plate made of material with constraints has been determined. The semi-inverse method has been applied to description. The results being obtained together with the already known biharmonic representation (Jemielita, 1992) might be useful in a new micropolar plate theory formulation.

Topics: Displacement
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):419-422. doi:10.1115/1.2895947.

The classical inverse problem of the plane theory of elasticity, in which the stress along the boundary contour is rendered uniform, is considered. The exact solution to the problem delivering symmetrical optimum shapes with four infinite branches along the axes of symmetry is constructed. The method enables one to find many other optimum shapes with two, three, four, five, etc., infinite branches.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):423-428. doi:10.1115/1.2895948.

Two-dimensional problems of anisotropic piezoelectric composite wedges and spaces are studied. The Stroh formalism is employed to obtain the basic real-form solution in terms of two arbitrary constant vectors for a particular wedge. Explicit real-form solutions are then obtained for (i) a composite wedge subjected to a line force and a line charge at the apex of the wedge and (ii) a composite space subjected to a line force, line charge, line dislocation, and an electric dipole at the center of the composite space. For the composite wedge the surface traction on any radial plane θ = constant and the electric displacement D θ normal to the radial plane θ = constant vanish everywhere. For the composite space these quantities may not vanish but they are invariant with the choice of the radial plane.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):429-440. doi:10.1115/1.2895949.

An anisotropic rotationally inhomogeneous wedge bent by either a concentrated couple applied at the tip (Carothers problem) or uniform surface loadings (Levy problem) is considered. The existence criteria for homogeneous solutions describing stresses and strains in both problems are established. In the Levy problem there are two types of critical wedge angles, at which homogeneous solutions break down and become infinite. The first type critical wedge angles of Levy’s problem are shown to be critical also for Carothers’problem whatever the rotational inhomogeneity. Particular solutions to both problems are obtained at the critical wedge angle. The form of these solutions is established to depend on two factors: the multiplicity degree of roots of some eigenvalue equation and the number of independent eigenvectors of some real matrix. It is shown also that the eigenvalue equation does not provide an alternative way to calculate the critical angles and in the first-order perturbation theory results in just the same equations for the critical angles.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):441-449. doi:10.1115/1.2895950.

Recently, taking the second law of thermodynamics as a starting point, a theoretical framework for an exact calculation of the elastothermodynamic damping in metal-matrix composites has been presented by the authors (Kinra and Milligan, 1994; Milligan and Kinra, 1993). Using this work as a foundation, we solve two canonical boundary value problems concerning elastothermodynamic damping in continuous-fiber-reinforced metal-matrix composites: (1) a fiber in an infinite matrix, and (2) using the general methodology given by Bishop and Kinra (1993), a fiber in a finite matrix. In both cases the solutions were obtained for the following loading conditions: (1) uniform radial stress and (2) uniform axial strain.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):450-458. doi:10.1115/1.2895951.

A phenomenological algorithm, motivated by the “mode I” microcrack opening and closing mechanism, is developed for the deactivation and reactivation of the damage effects as modeled by certain continuum damage mechanics theories. One-dimensional formulations with and without coupled plasticity are used to elucidate concepts associated with damage deactivation and to suggest multidimensional deactivation formulations applicable to continuum damage theories that employ a second-order tensor as the damage measure. Strain-based projection operators are used to deactivate the damage effects in the damage tensor. Motivated by observations from one-dimensional coupled formulations, both the total and elastic strains must be compressive for the damage to be rendered inactive. By introducing smooth functions to represent the transition from the active to the inactive state, discontinuities in the response are avoided. To illustrate the aspects associated with deactivation, a consistent set of examples using a strain-controlled one-cycle uniaxial stress loading is given for each formulation.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):459-465. doi:10.1115/1.2895952.

A pure-bending apparatus is used to measure the constitutive relationship between applied pure bending moments and the resulting curvatures of a few superelastic alloy wires. The sample nickel-titanium alloy (NiTi) wires change phase when ample bending moments are imposed. Like the material’s uniaxial tension stress-strain relationship, the measured moment-curvature relationship shows plateaus of constant moment and hysteresis. The bent shape is circular, except in the mixed phase region where it is composed of a phase mixture of circles. An example of the applications of the measured moment-curvature relations is presented in Part II of this paper where the three-point bending problem is considered.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):466-470. doi:10.1115/1.2895953.

The constitutive relationship between applied pure bending moment and the resulting curvature of a few superelastic alloy wires is applied to the three-point bending problem. Three-point bending experiments on hard and soft loading machines are described. The relationship between the applied deflection and the resulting force in three-point bending is calculated from a nonlinear Euler-Bernoulli rod theory. A numerical procedure used to solve the three-point bending problem for both loading and unloading is briefly described and numerical results are compared with experiment.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):471-478. doi:10.1115/1.2895954.

Ordinary, generalized, and pseudo-variational equations of motion in three-dimensional theories of nonlinear elasticity and piezoelectricity are presented systematically. These are applied to the derivations of plate equations of the classical type. In contrast to the derivations of plate equations that include thickness and higher-order effects, it is shown that the volume and surface integrals in a three-dimensional ordinary variational equation of motion must now be used jointly in a coupled manner. Details are demonstrated by first treating a classical linear plate. Equations of the classical type for large deflections of laminated composite and piezoelectric plates are then derived, with the famous von Kármán equations of an isotropic homogeneous plate deducible as a special case. Interrelationship among various plate equations is emphasized.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):479-488. doi:10.1115/1.2895955.

A Galerkin projection of the equations of equilibrium for a recent theory of geometrically exact sandwich beams that allow finite rotations and shear deformation in each layer is presented. The continuity of the displacement across the layers is exactly satisfied. The resulting finite element formulation can accommodate large deformation. The number of layers is variable, with layer lengths and thicknesses not required to be the same, thus allowing the modeling of sandwich structures with ply drop-off. Numerical examples are presented which underline the salient features of the formulation. Saint-Venant principle is demonstrated for very short sandwich beams.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):489-495. doi:10.1115/1.2895956.

A method for global analysis of nonlinear dynamical oscillating systems was developed. The method is based on the idea of introducing a Poincare section into a multidimensional state space of the dynamical system and combine it with an interpolation procedure within the cells which constitute the discretized problem domain of interest. The proposed method was applied to study the global behavior of two nonlinear coupled van der Pol oscillators. Significant saving in calculation time, in comparison with both direct numerical integration and Poincare-like simple cell mapping, is demonstrated.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):496-504. doi:10.1115/1.2895957.

We investigate the noise-induced transitions between the oscillatory steady states of a class of weakly nonlinear oscillators excited by resonant harmonic forcing. We begin by deriving a set of averaged equations governing slow variables of the system when the latter is perturbed by both additive white Gaussian noise and by random phase fluctuations of the resonant excitation. We then examine in detail the behavior of the reduced system in the case of cubic stiffness and viscous damping forces. Three regimes are examined: the case of weak damping, the case of near-bifurcation and the more general case when neither of the first two situations apply. In each case we predict the quasi-stationary probability density of the response and the mean time taken by the trajectories to pass from one basin of attraction to the other. These theoretical predictions are based on averaging of a near-Hamiltonian system in the weak damping limit, on center-manifold theory in the near-bifurcation case, or on Wentzell-Kramers-Brillouin (WKB) singular perturbation expansions in the more general case. These predictions are compared with digital simulations which show excellent agreement. We can then determine the probability of a transition for each state and for all parameter values. For this, we compute contour curves of the activation energy of each attractor in the parameter plane to yield a complete picture of the survivability of the system subject to random perturbations.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):505-510. doi:10.1115/1.2895958.

A new viewpoint is suggested for expressing the governing equations of analytical mechanics. This viewpoint establishes a convenient framework for examining the relationships among Lagrange’s equations, Hamilton’s equations, and Kane’s equations. The conditions which must be satisfied for the existence of an energy integral in the context of Kane’s equations are clarified, and a generalized form of Hamilton’s Principle is presented. Generalized speeds replace generalized velocities as the velocity variables in the formulation. The development considers holonomic systems in which the generalized forces are derivable from a potential function.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):511-516. doi:10.1115/1.2895959.

Draw resonance instability in a bicomponent fiber spinning flow is investigated using a model in which a Newtonian and an upper-convected Maxwell fluid are the core and skin layers, respectively. This model is to consider a situation where the two fluids have very different extensional rheology. The results indicate a significant influence of the viscoelastic skin layer on the overall flow mechanics even when its relative flow rate is small. It is also predicted that under certain conditions the stability of the bicomponent fiber can be maintained to a higher draw ratio than obtainable with either fluids individually.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):517-522. doi:10.1115/1.2895960.

Traditionally V(z) curves are generated by acoustic microscopes. However, because of the high costs of the commercially available acoustic microscopes, their use is rather limited. In this paper it is shown how V(z) curves, which contain quantitative information about the material under inspection, can be generated using two ultrasonic transducers instead of an acoustic microscope. A theoretical analysis is given to synthesize V(z) curves of orthotropic plates by this technique. A basic mechanics problem of the reflection of plane waves by an orthotropic plate immersed in a fluid is solved for this purpose. Theoretically synthesized V(z) values are compared with experimental results.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):523-526. doi:10.1115/1.2895961.

Calculations of two types of fractal dimension are reported, regarding the elastic-plastic response of a two-degree-of-freedom beam model to short pulse loading. The first is Mandelbrot’s (1982) self-similarity dimension, expressing independence of scale of a figure showing the final displacement as function of the force in the pulse loading; these calculations were made with light damping. These results are equivalent to a microscopic examination in which the magnification is increased by factors of 102 ; 104 ; and 106 . It is found that the proportion and distribution of negative final displacements remain nearly constant, independent of magnification. This illustrates the essentially unlimited sensitivity to the load parameter, and implies that the final displacement in this range of parameters is unpredictable . The second fractal number is the correlation dimension of Grassberger and Procaccia (1983), derived from plots of Poincare intersection points of solution trajectories computed for the undamped model. This fractional number for strongly chaotic cases underlies the random and discontinuous selection by the solution trajectory of the potential well leading to the final rest state, in the case of the lightly damped model.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):527-532. doi:10.1115/1.2895962.
Abstract
Commentary by Dr. Valentin Fuster

### BRIEF NOTES

J. Appl. Mech. 1995;62(2):533-535. doi:10.1115/1.2895963.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):535-537. doi:10.1115/1.2895964.

The thermoelastic problem of an infinite elastic plane containing a partly bonded circular inhomogeneity of different thermomechanical properties is considered. Based upon the solution of a perfectly bonded inhomogeneity established in the current work, the complex stress intensity factor of the interfacial crack problem is obtained for full heat-conductive conditions of an “open” crack and for a linear temperature change at infinity.

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):537-539. doi:10.1115/1.2895965.
Abstract
Topics: Dislocations
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):539-540. doi:10.1115/1.2895966.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):540-542. doi:10.1115/1.2895967.
Abstract
Topics: Solids
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):542-544. doi:10.1115/1.2895968.
Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):544-547. doi:10.1115/1.2895969.

The titled problem is studied numerically by finite element calculation and analytically by three-mode eigenfunction expansion. It is found that divergence instability of the coupled system is induced only when two times the number of nodal diameters 2n is equal to a multiple of the number of stationary springs N , but n itself is not a multiple of N .

Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):547-549. doi:10.1115/1.2895970.
Abstract
Commentary by Dr. Valentin Fuster

### DISCUSSIONS

J. Appl. Mech. 1995;62(2):550. doi:10.1115/1.2895971.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):550. doi:10.1115/1.2895972.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):551. doi:10.1115/1.2895973.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):551. doi:10.1115/1.2895974.
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Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):551. doi:10.1115/1.2895975.
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Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):552. doi:10.1115/1.2895976.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):552. doi:10.1115/1.2895977.
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Abstract
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):552-553. doi:10.1115/1.2895978.
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Abstract
Topics: Geometry
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):553. doi:10.1115/1.2895979.
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Abstract
Topics: Equations
Commentary by Dr. Valentin Fuster
J. Appl. Mech. 1995;62(2):553. doi:10.1115/1.2895980.
FREE TO VIEW
Abstract
Commentary by Dr. Valentin Fuster