Nonlinear Dynamics and Stability of a Two D.O.F. Elastic/Elasto-Plastic Model System

The local and global nonlinear dynamics of a two-degree-of-freedom model system is studied. The undeflected model consists of an inverted T formed by three rigid bars, with the tips of the two horizontal bars supported on springs. The springs exhibit an elasto-plastic response, including the Bauschinger effect. The vertical rigid bar is subjected to a conservative (dead) or non-conservative (follower) force having static and periodic components. First, the method of multiple scales is used for the analysis of the local dynamics of the system with elastic springs. The attention is focused at modal interaction phenomena in weak excitation at primary resonance and in hard sub-harmonic excitation. Three different asymptotic expansions are utilised to get a structural response for typical ranges of excitation parameters. Numerical integration of the governing equations is then performed to validate results of asymptotic analysis in each case. A full global nonlinear dynamics analysis of the elasto-plastic system is performed to reveal the role of plastic deformations in the stability of this system. Static 'force-displacement' curves are plotted and the role of plastic deformations in the destabilisation of the system is discussed. Large-amplitude non-linear oscillations of the elasto-plastic system are studied, including the influence of material hardening and of static and sinusoidal components of the applied force. A practical method is proposed for the study of a non-conservative elasto-plastic system as a non-conservative elastic system with an 'equivalent' viscous damping. This revised version was published online in July 2006 with corrections to the Cover Date.     

Optimum design of vibrating cantilevers

We determine the optimum tapering of a cantilever carrying an end mass, i.e., the shape which, for a given total mass, yields the highest possible value of the first fundamental frequency of harmonic bending vibrations in the vertical plane.Three different cases are considered. In the first case, all cross sections are assumed to be geometrically similar. In the second case, the cross sections are assumed to be rectangular and of given width. Finally, we consider a rectangular cross section of given height. This third case is shown to be degenerate in the absence of end mass.The first author takes the opportunity of thanking the authorities of the Technical University of Denmark for generous financial aid for his work at the University. We also thank our colleague Lic. Techn. Niels Olhoff for many valuable discussions during the course of the numerical computations.     

Spread of plasticity from stacked stress concentrations

A large elastic solid containing an infinite sequence of slitlike relaxed cracks with a constant distance of vertical separation is considered. The solid is deforming under plane strain shear conditions (mode II). The plastic relaxation around each of the cracks is represented by a suitable distribution of edge dislocations coplanar with the crack itself, the distribution being determined from a singular integral equation. This equation is solved numerically using an expansion of the non-singular part of the kernel in a series of Chebyshev polynomials. Solutions are obtained for the extent of spread of plasticity from each crack and for the associated dislocation distribution as a function of the crack spacing and the applied load. The results are applied to a brief discussion of the fracture process at stress concentrations using the crack opening displacement criterion.     

On the solution of optimization problems with singularities

We derive a complete set of necessary optimality conditions for a class of variational problems whose extremal solutions are associated with singularities. The use of these conditions is illustrated by two examples involving the optimization of the shapes of elastic bodies with stiffness and stability constraints.     

Optimal design of multi-purpose beam-columns

The paper deals with the optimal design of an elastic pinended member of given volume that is to serve as a beam for a part of its design life and as a column for the rest. The optimal design can be interpreted in two ways. Firstly, it is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection as a beam. Secondly, it is the design that has the least deflection as a beam under a midspan concentrated load, subject to a minimum permissible Euler buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.     

Optimal design of multi-purpose tie-beams

The paper presents a solution of the problem of minimizing the maximum deflection of a simply-supported beam under a transverse concentrated load. The volume (mass) of the beam is given, as is the maximum longitudinal elongation if the beam were to act as a tie. An optimal design for two requirements (beam and tie action) not only unifies the design procedure of mass-produced structural-mechanical elements, but also provides a practically acceptable design, in the sense that the resulting shape does not vanish at any point along its length, a drawback of many optimal designs for a single requirement. However, it is shown that, for cross sections of solid construction, as opposed to those of sandwich construction, the constraint on longitudinal elongation is weaker than on the (finite) maximum stress. The effectiveness of the optimal design is judged by comparing it with a prismatic member of the same volume.     

High-rate deformation and fracture of fiber reinforced concrete

This paper presents the results of dynamic compression and splitting-tensile tests of cardiff fiber reinforced concrete (CARDIFRC) composite using the Kolsky technique and its modification. The strength and deformation characteristics of fiber-reinforced concrete were determined experimentally at high strain rates. The mechanical characteristics were found to depend on the strain rate and stress rate. A uniform interpretation of the rate effects of fracture of the tested fiber-reinforced concrete is given on the basis of a structural-temporal approach. It is shown that the time dependences of both the compressive and tensile strengths of fiber reinforced concrete are well calculated using the incubation time criterion.     

Dislocation model of an asymmetric weak zone for problems of interaction between crack-like defects

An adhesively bonded asymmetric weak zone is proposed as a model for studying the problem of interaction between crack-like defects in an elastic medium. The opening of the weak zone is prescribed by a two-parameter basis function, i.e. by a special dislocation which automatically accounts for the asymmetry and other expected physical features of the stress–strain field near the tips of the weak zone. The adhesive forces corresponding to the prescribed opening are then calculated from the solution of the particular problem. The application of the model is demonstrated on the problem of a long interface crack subjected to wedge opening forces which is separated from a short collinear interface weak zone by a small unbroken strong microstructural feature (a small obstacle). Two key questions pertaining to limiting situations are addressed: (i) when does the weak zone become the nucleus of a cohesive crack on its own without linking with the pre-existing long crack; and (ii) when does it force the rupture of the obstacle and coalesce with the long crack.     

Optimal design of multi-purpose structures

The paper presents the optimal (maximum transverse stiffness) design of an elastic, simply supported member of given volume that is to serve as a beam or as a column at different times during its design life. The optimal design can be interpreted in two ways. It is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection in beam action under a uniformly distributed load; it is also the design that has the least deflection at midspan under a uniformly distributed load, subject to a lower limit on its buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.The author is grateful to Professor W. Prager for suggesting several improvements to an earlier version of the paper.