W.J. Stronge

Rigid Body Collisions with Friction

An energetically consistent theory is presented for dynamics of partly elastic collisions between somewhat rough rigid bodies with friction that opposes slip. This theory is based on separately accounting for frictional and non-frictional sources of dissipation. Alternative theories derived from Newton’s impact law or Poisson’s impact hypothesis are shown to be valid only for central (collinear) or non-frictional collisions; generally the latter theories yield erroneous energy dissipation if small initial slip stops during collision between eccentric bodies. Collision processes are complex when small slip is stopped by friction; then either the direction of slip reverses or contact points roll without slip. An inconsistent theory based on Newton’s impact law can yield erroneous energy increases when slip stops during collision; the consistent theory always dissipates energy. The impact law that specifies a simple proportionality between normal components of contact velocity for incidence and rebound is not applicable in any range of incident velocities with small slip if the collision is non-collinear with friction. In Percussion the force or Impetus whereby one body is moved may cause another body against which it strikes to be put in motion, and withal lose some of its strength or swiftness. (J. Wallis, 1668)

Elasto-plastic yield limits and deformation laws for transversely crushed honeycombs

Abstract Constitutive equations are developed for large deformations of transversely crushed elasto-plastic honeycomb materials. Yield surfaces for biaxial macroscopic stresses are obtained by accounting for the interaction between geometrically similar elastic and plastic cell crushing modes. Since the yield surfaces and corresponding deformation increments evolve as the cell geometry changes, large-deformation material behaviour is expressed in terms of a set of state variables describing the instantaneous cell geometry. The cells strain-soften under certain stress states and cell configurations; then crushing can become localized in thin bands, significantly altering the behaviour of the affected region.

Dynamic Models for Structural Plasticity

1 Elastoplastic and Viscoplastic Constitutive Relations.- 2 Principles of Mechanics.- 3 Static Deflection.- 4 Dynamic Rigid-Plastic Response.- 5 Second-Order Effects on Dynamic Response.- 6 More Complex Configurations.- 7 Impact Experiments.

In-plane dynamic crushing of honeycomb. Part I: crush band initiation and wave trapping

Abstract In this paper the dynamics of crush band initiation and wave trapping that result from in-plane impact on a honeycomb are analysed using finite element simulations. The honeycomb structures were loaded in compression at the top surface with a prescribed velocity. Two different boundary conditions were considered; these produced an approximation to a state of uniaxial stress or uniaxial strain at an early state of deformation. The simulations proved that elastic wave propagation had an important effect on crush band initiation, since loading and unloading parts of the compressive wave can reinforce or delay crushing, respectively. Stress enhancement with increasing impact speed is mainly due to translational micro-inertia and not due to micro-rotational inertia. This effect is stronger for uniaxial strain than for uniaxial stress. For the specific case examined here, stress enhancement begins for uniaxial strain at an impact speed of about 1.0 m / s whereas for uniaxial stress it begins at about 10.0 m / s . For the analysed honeycomb the simulation showed for both boundary conditions the existence of a critical impact velocity, so that for impact speeds larger than this critical speed a crush band initiated at the impact surface. For much smaller speeds the location of the initial crush band was determined by the distribution and extent of initial imperfections. In the case of uniaxial stress an analytical estimate of the critical impact speed was derived using the theory of wave trapping. The critical speed depends only on the ratio of the length of the ribbon wall to the length of the thinner inclined wall and the material properties of the constituent material. For the aluminium honeycomb investigated in this paper, wave trapping is predicted to occur for impact speeds greater than 7.6 m / s which is close to the critical speed calculated with the finite element simulations.

Long stroke energy dissipation in splitting tubes

Abstract A passive crashworthy system that dissipates impact energy by fracture and plastic deformation of metal tubes is analysed. The energy dissipating component is a square tube that is pressed axially against a die where it splits at the corners and curls outward. Experiments on the effect of die radius have shown that a remarkably constant force causes this rate-independent deformation in a dural tube. Tubes which both split and curl can be designed for energy absorbing systems requiring large stroke to length ratio and good specific energy absorption.

High strain-rate shear response of Polycarbonate and Polymethyl Methacrylate

The high strain rate response of polycarbonate (PC) and polymethyl methacrylate (PMMA) are measured using a split Hopkinson torsion bar for shear strain rates Ẏ from 500 s-1 to 2200 s-1, and temperatures in the range —100°C to 200°C. The yield and fracture behaviours are compared with previous data and existing theories for Ẏ < I s-1. We find that PC yields in accordance with the Eyring theory of viscous flow, for temperatures between the beta transition temperature Tβ ≈ — 100°C and the glass transition temperature Tg = 147°C. At lower temperatures, T < Tβ , backbone chain motion becomes frozen and the shear yield stress is greater than the Eyring prediction. Strain softening is an essential feature of yield of PC at all strain rates employed. Poly methyl methacrylate fractures before yield in the high strain rate tests for T < 80°C, which is close to the glass transition temperature Tg120°C. It is found that the fracture stress for both materials obeys a thermal activation rate theory of Eyring type. Fracture is thought to be nucleation controlled, and is due to the initiation and break down of a craze at the fracture stress τf. Examination of the fracture surfaces reveals that failure is by the nucleation and propagation of inclined mode I microcracks which link to form a stepped fracture surface. This reveals that failure is by tensile cracking and not by a thermal instability in the material. The process of shear localization is fundamentally different from that shown by steel and titanium alloys.

Lateral crushing in tightly packed arrays of thin-walled metal tubes

Tightly packed arrays of parallel, thin-walled steel, brass and aluminium alloy tubes have been compressed between parallel platens; the crushing force has been related to the array properties including the packing arrangement. Analysis of the post-collapse behaviour of sets of similarly deforming tubes in both symmetrical and asymmetrical modes of deformation is used to interpret these experiments for square and hexagonally packed arrays. Asymmetrical tube deformation is associated with a decreasing crushing force; this results in the localization of array deformation within a thin band of tubes across the array. Continued crushing causes the thickness of this band to increase until it envelopes most of the array.

Micropolar theory for two–dimensional stresses in elastic honeycomb

A micropolar theory has been used to calculate the stress field generated by a fundamental boundary–value problem in planar elasticity; namely, a normal line force acting on the surface of a honeycomb half–space. In this two-dimensional analysis the microstructure of the honeycomb results in couple stresses and microrotations in addition to the usual components of stress and strain. Constitutive relations for a unit cell of the honeycomb are obtained that are sensitive to the symmetry of the microstructure. Because of the six–fold symmetry of the regular hexagonal microstructure these constitutive relations contain only four independent material constants. The discontinuous traction on the boundary of the honeycomb generates stresses σij that have a 1/r dependence on radial distance from the discontinuity while the microcouples mi3 have a 1/r2 dependence on radial distance.

Planar impact of rough compliant bodies

Summary A lumped parameter model of contact between colliding bodies is used to obtain the tangential force and energy dissipated by friction during oblique impact of rough compliant bodies in collinear configurations. For Coulombs law of friction and an energetic coefficient of restitution, this analysis distinguishes between angles of incidence where the contact point initially sticks, slides before sticking or slides throughout the contact period. During part of the contact period tangential compliance significantly alters friction; this affects changes in relative velocity unless the angle of incidence is so large that there is continuous sliding in the initial direction. For oblique impact, tangential compliance reduces the largest friction force in comparison with friction generated if the contact region has negligible tangential compliance. While the part of the initial kinetic energy dissipated by friction is limited by the coefficient of restitution, this coefficient has no effect on the largest friction force generated during collision.

Large deflections of a rigid-plastic beam-on-foundation from impact

Abstract Missile impact against a continuously supported rigid-plastic beam results in only a transient phase of deformation that continuously evolves towards the primary dynamic mode; ther is no modal phase of deformation. Large deflection effects are incorporated in this analysis by considering interactions between plastic bending and stretching in deforming regions. Normal resultant forces associated with stretching are shown to be most important when the impact point deflection exceeds the beam thickness; in comparison with a small deflection solution, these forces substantially decrease the final deformation.