Leonid I. Slepyan

Dynamics of chains with non-monotone stress–strain relations. I. Model and numerical experiments

Abstract We discuss dynamic processes in materials with non-monotonic constitutive relations. We introduce a model of a chain of masses joined by springs with a non-monotone strain–stress relation. Numerical experiments are conducted to find the dynamics of that chain under slow external excitation. We find that the dynamics leads either to a vibrating steady state (twinkling phase) with radiation of energy, or (if dissipation is introduced) to a hysteresis, rather than to an unique stress–strain dependence that would correspond to the energy minimization.

Feeding and dissipative waves in fracture and phase transition: I. Some 1D structures and a square-cell lattice

Abstract In the lattice structure considered here, 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 non-zero 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 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) the main lattice phenomena.

Dynamics of chains with non-monotone stress-strain relations. II. Nonlinear waves and waves of phase transition

We investigate the dynamics of a one dimensional mass-spring chain with non-monotone dependence of the spring force vs. spring elongation. For this strongly nonlinear system we find a family of exact solutions that represent nonlinear waves. We have found numerically that this system displays a dynamical phase transition from the stationary phase (when all masses are at rest) to the twinkling phase (when the masses oscillate in a wave motion). This transition has two fronts which propagate with different speeds. We study this phase transition analytically and derive relations between its quantitative characteristics.

Band gap Green's functions and localized oscillations

We consider some typical continuous and discrete models of structures possessing band gaps, and analyse the localized oscillation modes. General considerations show that such modes can exist at any frequency within the band gap provided an admissible local mass variation is made. In particular, we show that the upper bound of the sinusoidal wave frequency exists in a non-local interaction homogeneous waveguide, and we construct a localized mode existing there at high frequencies. The localized modes are introduced via the Greens functions for the corresponding uniform systems. We construct such functions and, in particular, present asymptotic expressions of the band gap anisotropic Greens function for the two-dimensional square lattice. The emphasis is made on the notion of the depth of band gap and evaluation of the rate of localization of the vibration modes. Detailed analysis of the extremal localization is conducted. In particular, this concerns an algorithm of a ‘neutral’ perturbation where the total mass of a complex central cell is not changed

Dynamic factor in impact, phase transition and fracture

Abstract 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.

Principle of maximum energy dissipation rate in crack dynamics

Abstract The principle of maximum energy dissipation rate is introduced as an energy criterion for crack dynamics. That allows us to explain observed limiting crack speeds in brittle materials, and to complete the crack dynamics formulation. The upper limits of the crack speed in perfectly elastic and elastic-plastic bodies are obtained. It is found that the theoretical maximum crack speed in an isotropic elastic body (in the first mode of crack propagation) is approximately equal to half the shear wave speed. In the case of an elasticplastic body, the criterion is formulated as a maximum plastic strain work per unit time. The self-similar problem for the fracture mode III is solved (assuming the plastic zone to be narrow) and the crack speed limit is found as a function of the ratio of loading to yield limit: the plasticity decreases the crack speed limit and the latter tends to zero with the yield limit. The comparison of these theoretical results with some experimental data shows that under ordinary conditions crack propagation appears to conform to the “maximum dissipation rate” process.

Waiting Element Structures and Stability under Extension

In this paper the following questions are considered: What are the phenomena which limit the total fracture energy of a structure under the extension before it breaks? When is the limiting energy level more important that the stress limit? What are possible ways to increase the required fracture energy of a sample before it breaks? It is shown that the required features of a material or of a construction can be achieved by using special structures of ordinary elements. The possibilities are discussed for increasing the fracture energy density in a sample, and increasing the total fracture energy in a construction. The dynamic process of damaging discussed as well.

Radial cracking with closure

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.

A Lattice Model for Viscoelastic Fracture

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-infinitecrack propagation in an unbounded lattice and lattice strip are studied.Expressions for the local-to-global energy-release-rate ratios, stressesand strains of the breaking bonds as well as the crack openingdisplacement are derived. Comparative results are obtained forhomogeneous viscoelastic materials, elastic lattices and homogeneouselastic materials. The influences of viscosity, the discrete structure,cell size, strip width and crack speed on the wave/viscous resistancesto crack propagation are revealed. Some asymptotic results related to animportant asymptotic case of large viscosity (on a scale relative to thelattice cell) are shown. Along with dynamic crack propagation, a theoryfor a slow crack in a viscoelastic lattice is derived.

Some surprising phenomena in weak-bond fracture of a triangular lattice

Abstract A semi-infinite crack growing along a straight line in an unbounded triangular-cell lattice and in lattice strips is under examination. Elastic and standard-material viscoelastic lattices are considered. Using the superposition similar to that used for a square-cell lattice (J. Mech. Phys. Solids 48 (2000) 927) an irregular stress distribution is revealed on the crack line in mode II: the strain of the crack-front bond is lower than that of the next bond. A further notable fact about mode II concerns the bonds on the crack line in the lattice strip deformed by a ‘rigid machine’. If the alternate bonds, such that are inclined differently than the crack-front bond, are removed, the stresses in the crack-front bond and in the other intact bonds decrease. These facts result in irregular quasi-static and dynamic crack growth. In particular, in a wide range of conditions for mode II, consecutive bond breaking becomes impossible. The most surprising phenomenon is the formation of a binary crack consisting of two branches propagating on the same line. It appears that the consecutive breaking of the right-slope bonds—as one branch of the crack—can proceed at a speed different from that for the left-slope bonds—as another branch. One of these branches can move faster than the other, but with time they can change places. Some irregularities are observed in mode I as well. Under the influence of viscosity, crack growth can be stabilized and crack speed can be low when viscosity is high; however, in mode II irregularities in the crack growth remain. It is found that crack speed is a discontinuous function of the creep and relaxation times.