List of Recent Abstracts

Slepyan, L. Dempsey, J. and Shekhtman, I., Asymptotic Solutions for Crack Closure in an Elastic Plate under Combined Extension and Bending. J. Mech. Phys. Solids, 1995, 43, No. 11, 1727-1749.

A coupled plane-bending problem is considered for an elastic Kirchhoff-Poisson plate containing a through-the-thickness or (part-through) surface crack under closure. The stress intensity factors at the ends of the crack are not zero. Asymptotic solutions are derived for cases in which the ratio of the crack length to its depth is large. As is shown, the width of the contact strip decreases as the crack length increases; the limiting contact force and moment distribution may be determined by considering an edge-cracked strip with zero stress intensity factor in the thickness direction. As is also shown, the crack surface interaction in-plane force and bending moment can be derived directly from the initial force and moment distribution acting in the intact plate on the prospective crack line. The same result is valid for a collinear system of cracks. In addition, the closure stress distribution is determined. For the latter, the width of the contact strip asymptote is derived as a function of the crack length coordinate, and the asymptotic stress distribution is found as a product of the thickness averaged closure stress and a function of a normalized plate coordinate. The latter stress distribution is unique and universal for any slow curving crack or crack system under closure.

Slepyan, L., Krylov, V. and Parnes, R., Solitary Waves in an Inextensible, Flexible, Helicoidal Fiber. Physical Review Letters, 1995, 74, No. 14, 2725-2728.

The exact solution to the nonlinear vector equations governing an inextensible, flexible, helicoidal fiber yields a one-parameter set of solitary waves. Analytical expressions are found for the displacements, internal force, axial and angular momenta and energy. This solution represents an interesting example of three-dimensional solitary waves propagating in a system devoid of potential energy.

Ryvkin, S., Slepyan, L.I. and Banks-Sills, L., On the Scale Effect in the Thin Layer Delamination Problem. Int. J. Fracture, 1995, 71, 247-271.

Influence of layer thickness on the stress distribution in the vicinity of a crack tip is examined, taking into account the fact that the conventional stress intensity factor concept becomes invalid if the thickness of the layer is not much more than the size of the fracture process zone. The eigen-problem is considered which is characterized by two asymptotes. The first is a near one; it is formed in a small vicinity of the crack tip in the layer thickness scale. The second asymptote is a far one in the same scale. The regions of validity of these asymptotes are determined and shown to depend upon material parameters and crack tip speed. The complete stress distribution in front of the crack is determined, as well. Some conclusions are made concerning the stress distribution and energy release rate for the general problem. Mode 3 crack propagation is considered in detail.

Dempsey, J., Slepyan, L. and Shekhtman, I., Radial Cracking with Closure. Int. J. Fracture, 1995, 73, 233-261.

Progressive radial cracking of a clamped plate subjected to crack-face closure is studied. The material behavior is assumed to be elastic-brittle. The cracks are assumed to be relatively long in the sense that the three dimensional contact problem can be described via a statically equivalent two-dimensional idealization. The number of cracks is supposed large enough to permit a quasi-continuum approach rather than one involving the discussion of discrete sectors. The formulation incorporates the action of both bending and stretching as well as closure effects of the radial crack face contact. Fracture mechanics is used to explore the load-carrying capacity and the importance of the role of the crack-surface-interaction. For a given crack radius, the closure contact width is assumed to be constant. Under this condition, a closed-form solution is obtained for the case of a finite clamped plate subjected to a concentrated force. Crack growth stability considerations predict that the system of radial cracks will initiate and grow unstably over a significant portion of the plate radius. The closure stress distribution is determined exactly in the case of narrow contact widths and approximately otherwise.

Krylov, V.I., Parnes, R. and Slepyan, L.I., Nonlinear Waves in an Inextensible Flexible Helix. Wave Motion, 1998, 27, 117-136.

The complete analytical solution of the governing nonlinear vector equations is found which describes periodical and solitary waves in a helicoidal fiber. The solitary wave velocity is found to be proportional to the square root of the amplitude of the internal force, and the effective wave length is shown to be independent of the amplitude. The essential influence of the rigid body rotation of the helix on the solitary wave shape is shown. Axial and angular momenta of the wave are determined as well.

Slepyan, L., Krylov, V. and Rosenau, Ph., Solitary Waves in a Flexible, Arbitrary Elastic Helix J. Engn. Mech., 1998, Vol. 124, No. 9, 966-970.

A flexible helicoidal fiber with an arbitrary, unspecified, stress-strain relations is considered, and a general, analytical, steady-state solution of the 3D nonlinear vector equation is found. This solution describes a new class of spatial motion of an elastic string which is shown to be a waveguide for subsonic solitary waves. The geometrically nonlinear problem for a linearly elastic fiber is considered in more detail.

Slepyan, L. I. and Dempsey, J. P., Radial Crack Dynamics with Closure IUTAM Symposium on Nonlinear Analysis of Fracture (Proceedings of the IUTAM Symposium held in Cambridge, U.K., 3-7 September 1995, Ed. by J.R. Willis), Kluwer Academic Publishers, 1997, 201-210.

An infinite elastic-brittle plate subjected to a transverse impact is considered. Many cracks, each of which extends through the full thickness of the plate, are assumed to grow radially from the impact point. Due to circumferential bending, the faces of these radial cracks tend to close over a portion of the plate thickness. In this paper, the dynamics of the radial crack system within the influence of a tangential unilateral constraint which prevents crack face interpenetration is examined. The analysis is simplified by assuming that the faces interact only over the lines where these faces meet a surface of the plate, rather than over some unknown area of the fracture surfaces. In other words, the local contact strain is neglected. The radial cracks are assumed to be large in number and evenly distributed so that the deformation retains axial symmetry. The governing equations are derived, and a transient, self-similar solution of these equations is constructed, relating the plate response to the impact force. The extension of the radial cracks according to the principles of fracture mechanics is explored.

A. Chiskis, R. Parnes and L. Slepyan, Nonlinear Behavior of Wavy Composites under Tension J. Mech. Phys. Solids, 1997, 45, 1357-1392.

A composite structure consisting of doubly periodic moderately wavy layers under tension is considered. The periodic cell of the composite is represented by two layers of different materials, but having the same arbitrary initial shape. Using a geometrical nonlinear Cosserat rod model to describe the layers, we take into account extension, flexure and shear deformation in the layers. The problem is reduced to a boundary value problem for a scalar nonlinear ODE of second order for the rotational degree of freedom. The determination of the deformed shape of the layers, as well as the relation between the elongation of the periodic cell and the applied tension force is then reduced to nonlinear quadratures. For the case of small initial waviness, an analytical solution is obtained for an arbitrary initial shape of the layers. For moderate initial waviness, numerical results are presented relating stress, strain and amplitude of waviness. The influence of material and structural parameters is investigated and discussed. In particular, as opposed to theories with full homogenization, the thickness of the periodic cell with respect to its length is not assumed to approach zero in the analysis. The influence of this structural parameter is shown to be especially significant %particularly if one of the materials of the layers is much weaker than the other.

V. Krylov and L. Slepyan, Binary Wave in a Helical Fiber Physical Review B, 1997, No. 21, 14067-14070.

Axial dynamic tension of a flexible helical fiber is found to lead to a specific, extraordinary nonlinear wave consisting of two different portions. The leading portion is a quasi-periodical propagating wave with rotation opposite to the initial twist of the helix, while the rear portion is a sequence of standing waves rotating in the reverse direction. The interface is the origin of two angular momentum fluxes which, being different in sign, fill up angular momenta of the leading and rear waves. The phenomenon described presents an interesting example of a zero-total-angular-momentum wave propagating in a twisted waveguide.

J. P. Dempsey, I. I. Shekhtman and L. I. Slepyan, Closure of a Through Crack in a Plate under Bending Int. J. Solids Structures, 35, Nos. 31-32, 4077-4089.

A plate with a pre-existent through crack is considered under the action of a remote in-plane force. The problem statement is reduced to the solution of two coupled integral equations with strongly singular kernels. The independent variables in the latter equations are the closure displacement and rotation angle. The corresponding closure force and moment distribution, and the contact-crack opening boundary (the closure perimeter), are found as functions of the remote bending-compression ratio. The validity of previously stated analytical asymptotics for the contact boundary is examined. The dependence of the extent of closure on the crack length-to-thickness ratio is studied. Comparisons are made with experimental results.

L. I. Slepyan, M. V. Ayzenberg and J. P. Dempsey, A Lattice Model for Viscoelastic Fracture. Mech. Time-Dependent Materials 1999, 3, 159-203.

A plane, periodic, square-cell lattice is considered, consisting of point particles connected by mass-less viscoelastic bonds. Homogeneous and inhomogeneous problems for steady-state semi-infinite crack propagation in an unbounded lattice and lattice strip are studied. Expressions for the local-to-global energy-release-rate ratios, stresses and strains of the breaking bonds as well as the crack opening displacement are derived. Comparative results are obtained for homogeneous viscoelastic materials, elastic lattices and homogeneous elastic materials. The influences of viscosity, the discrete structure, cell size, strip width and crack speed on the wave/viscous resistances to crack propagation are revealed. Some asymptotic results related to an important asymptotic case of large viscosity (on a scale relative to the lattice cell) are shown. Along with dynamic crack propagation, a theory for a slow crack in a viscoelastic lattice is derived.

L. I. Slepyan, V. I. Krylov and R. Parnes HELICAL INCLUSION IN AN ELASTIC MATRIX. J. Mech. Phys. Solids 2000, 48, 827-865.

An elastic space containing an elastic helical rod subjected to both axial and radial extension as well as torsion is considered. Due to translation-rotation helical symmetry, the resulting elastic fields in the matrix can be expressed in terms of a two-dimensional helix-associated coordinate system. In this problem, a `helical elastic foundation' as a generalization of the Winkler foundation is determined by means of which the interacting force and moment at the rod/matrix interface can be expressed in terms of the rod displacement. The matrix is assumed to be linear elastic while the geometric nonlinearity of the helical rod is taken into account. Using superposition of fundamental solutions for a homogeneous elastic space (in the absence of the rod), and the constitutive and equilibrium equations for the rod, the internal forces and moments in the rod as well as the displacement and elastic fields in the matrix are obtained. Along with the general results, two asymptotic solutions are presented. The first, corresponding to a small curvature but not too small pitch, allows an analytical integration of the rod-matrix interaction over the rod cross-section boundary. The second corresponds to an almost straight helical rod: the helix becomes a straight line, but in the limit the main normal to its axis describes a screw surface as in the case of a `genuine' helix. In this case, the helical elastic foundation has a closed-form parametric expression which is valid for a rather large range of the helix parameters. The foundation stiffness is found as a function of the helix pitch and the rod radius; the problem thus is reduced to a system of finite, nonlinear equations.

L.I. Slepyan DYNAMIC FACTOR IN IMPACT, PHASE TRANSITION AND FRACTURE. J. Mech. Phys. Solids 2000, 48, 927-960.

The related questions `how to avoid oscillations under an impact' and `why a crack or phase-transition wave can/cannot propagate slowly' are discussed. The underlying phenomenon is the dynamic overshoot which can show itself in deformation of a body under a load suddenly applied. The manifestation of this phenomenon in a unit cell of the material structure is shown to trigger a fast crack in fracture as well as a fast wave in phase transition. Two ways for the elimination of the overshoot, to obtain a static-amplitude response (SAR), are examined. The first is a proper control of the load in an initial portion of the loading time. This is illustrated by means of an example of elastic collision. In the case of fracture, such control can be envisioned as provided by a proper post-peak tensile softening of the material. Secondly, the SAR can be achieved under the influence of viscosity. In this connection, the following transient problems are considered: a viscoelastic-spring oscillator under a step excitation, a square-cell viscoelastic lattice with a crack and a two-phase viscoelastic chain as the phase-transition waveguide. For each problem, in the space of viscosity parameters, the SAR domain is separated from the dynamic-overshoot-response (DOR) domain. In the SAR domain, in contrast to the DOR domain, a slow crack or a slow phase-transition wave can exist. A structure-associated size effect in the SAR/DOR domains separation is noted.

L.I. Slepyan, Feeding and Dissipative Waves in Fracture ans Phase Transition. I. Some 1D Structures and a Square-Cell Lattice J. Mech. Phys. Solids (in press)

In the considered lattice structure, crack propagation is caused by feeding waves, carrying energy to the crack front, and accompanied by dissipative waves carrying a part of this energy away from the front (the difference is spent on the bond disintegration). The feeding waves differ by their wavenumber. A zero feeding wavenumber corresponds to a macrolevel-associated solution with the classical homogeneous-material solution as its long-wave approximation. A nonzero wavenumber corresponds to a genuine microlevel solution which has no analogue on the macrolevel. In the latter case, on the crack surfaces and their continuation, the feeding wave is located behind (ahead) the crack front if its group velocity is greater (less) than the phase velocity. Dissipative waves, which appear in both macrolevel-associated and microlevel solutions, are located in accordance with the opposite rule. (Wave dispersion is the underlying phenomenon which allows such a wave configuration to exist.) In contrast to a homogeneous material model, both these solutions permit supersonic crack propagation. Such feeding and dissipative waves and other lattice phenomena are characteristic of dynamic phase transformation as well. In the present paper, mode III of crack propagation in a square-cell elastic lattice is studied. Along with the lattice model, some simplified one-dimensional structures are considered allowing one to retrace qualitatively (with no technical difficulties) main lattice phenomena.

L.I. Slepyan, Feeding and Dissipative Waves in Fracture ans Phase Transition. II. Phase-Transition Waves J. Mech. Phys. Solids (in press)

Dscrete and homogeneous models of a structured material are considered to resolve difficulties in the analysis of dynamic phase transition. The discrete model is a chain consisting of particles connected by mass-less bonds, while the continuous model is represented by a partial differential equation with higher than the second order of coordinate-derivatives. The macrolevel constitutive law is represented by a bi-linear stress-strain relation, such that the transition from the first, stiffer phase to the second one is irreversible. Solutions of two types, macrolevel-associated and microlevel, are derived. The first type solution is characterized by a macrolevel feeding wave (the wave delivering energy to the phase-transition front is of a zero wavenumber), while the microlevel solutions correspond to a nonzero feeding wavenumber. Subsonic, intersonic and supersonic phase-transition waves are described. For the homogeneous model it is shown that the contradiction between the limiting stress and energy criteria, inherent for the macrolevel formulation of the problem, is eliminated if and only if the phase transition does not concern the highest-order modulus. Total structure- and speed-dependent dissipation, as the energy carried by microlevel waves away from the phase-transition front, as well as parameters of each dissipative wave are determined. For the fourth-order partial differential equation, the existence of the Maxwell type, dissipation-free, subsonic phase-transition wave is shown. In this case, the microstructure plays the role of a catalyst. Common and distinctive properties of the discrete and homogeneous models are discussed.

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