Non-linear buckling of simple models with tilted cusp catastrophe

Non-linear buckling universal solutions of simple multiple-parameter discrete models are discussed via a comprehensive and readily employed procedure using Catastrophe Theory. Attention is focused on perfect models whose total potential energy (TPE) function, upon small disturbance breaking symmetry, reduces to the universal unfolding of the tilted cusp catastrophe. A local analysis based on simple approximations allows us to classify the TPE universal unfolding of any model to one of the seven elementary Thom's catastrophes by defining the corresponding control parameters. Subsequently, using global analyses one can obtain exact results for establishing the non-linear equilibrium paths: (a) of the “perfect” perturbed model (due to the small effect of an extra parameter) with the corresponding “imperfect” bifurcation and limit points(s), and (b) of the imperfect models (resulting after inclusion of the effect of normal imperfection parameters) together with the corresponding to each parameter limit points. Moreover, conditions for the direct evaluation of (non-degenerate) hysteresis points associated with a tilted cusp point in the control parameter plane, are established. Numerical results illustrate the methodology proposed herein.     

New flow rules in elasto-viscoplastic constitutive models for spheroidal graphite cast-iron

A specific flow rules and the corresponding constitutive elasto-viscoplastic model combined with new experimental strategy are introduced in order to represent a spheroidal graphite cast-iron behaviour on a wide range of strain, strain rate and temperature. A “full model” is first proposed to correctly reproduce the alloy behaviour even for very small strain levels. A “light model” with a bit poorer experimental agreement but a simpler formulation is also proposed. These macroscopic models, whose equations are based on physical phenomena observed at the dislocation scale, are able to cope with the various load conditions tested – progressive straining and cyclic hardening tests – and to correctly describe anisothermal evolution. The accuracy of these two models and the experimental databases to which they are linked is estimated on different types of experimental tests and compared with the accuracy of more standard Chaboche-type constitutive models. Each test leads to the superiority of the “full model”, particularly for slow strain rates regimes. After developing a material user subroutine, FEM simulations are performed on Abaqus for a car engine exhaust manifold and confirm the good results obtained from the experimental basis. We obtain more accurate results than those given by more traditional laws. A very good correlation is observed between the simulations and the engine bench tests.     

Classical and Quantized Affine Models of Structured Media

Having in view some applications in nanophysics, in particular in nanophysics of materials, we develop new dynamical models of structured bodies with affine internal degrees of freedom. In particular, we construct some models where not only kinematics but also dynamics of systems of affine bodies is affinely invariant. Quantization schemes are developed. This is necessary in the range of physical phenomena we are interested in.     

On the use of genetic algorithm for finding the neutral instability curve in plane Poiseuille flow

A novel method based on genetic algorithm (GA) is proposed, to the best of our knowledge for the first time, for finding the neutral instability curve of the Orr-Sommerfeld equation in (nearly) parallel flows. New concepts such as “proximity of parents” and “gender discrimination” are added to the conventional GA in order for this algorithm to find the neutral instability curve. Certain GA operators such as “crossover” and “mutation” will also be modified in such a way that this algorithm can meet this purpose. To check the applicability of the modified genetic algorithm (MGA) developed in this work in finding the neutral instability curve, the case of plane Poiseuille flow will be used as a benchmark. It will be shown that the modified genetic algorithm developed in this work is well capable of determining the neutral instability curve for this particular flow geometry.     

Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-α model of single-phase lossless fluids

A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg-de Vries equation from shallow-water wave theory.     

Analysis of a weakly non-linear autonomous oscillator by means of the field method

In this paper the field method is extended to the study of oscillatory systems with two degrees of freedom and weak quadratic non-linearity. The basic field method concept is combined with the technique of multiple time scales and the solution for both non-resonant case and the case out of the first resonance are found. The qualitative analysis of behavior in the resonant area is done by determining the values of the “adelphic” integral.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: II—Results

In Part I of this paper, we developed a homogenization-based constitutive model for the effective behavior of isotropic porous elastomers subjected to finite deformations. In this part, we make use of the proposed model to predict the overall response of porous elastomers with (compressible and incompressible) Gent matrix phases under a wide variety of loading conditions and initial values of porosity. The results indicate that the evolution of the underlying microstructure—which results from the finite changes in geometry that are induced by the applied loading—has a significant effect on the overall behavior of porous elastomers. Further, the model is in very good agreement with the exact and numerical results available from the literature for special loading conditions and generally improves on existing models for more general conditions. More specifically, we find that, in spite of the fact that Gent elastomers are strongly elliptic materials, the constitutive models for the porous elastomers are found to lose strong ellipticity at sufficiently large compressive deformations, corresponding to the possible onset of “macroscopic” (shear band-type) instabilities. This capability of the proposed model appears to be unique among theoretical models to date and is in agreement with numerical simulations and physical experience. The resulting elliptic and non-elliptic domains, which serve to define the macroscopic “failure surfaces” of these materials, are presented and discussed in both strain and stress space.     

Localized nonlinear, soliton-like waves in two-dimensional anharmonic lattices

We discuss here nano-scale size localized wave excitations, which are intrinsic localized traveling modes in two-dimensional anharmonic crystal lattice systems. In particular, using different initial conditions of coordinates and momenta we search for the longest lasting excitations in triangular lattices. As most stable and longest lasting unaltered appear quasi-one-dimensional Toda-like solitons running in rectilinear chains along the main crystallographic axes of such lattices. Furthermore, by following the trace of high energetic excitations like in “bubble chamber” methodology (or in scanning tunneling microscopy) we show how such localized nonlinear waves appearing spontaneously in heated systems can be detected and followed in space-time.     

Effect of Transitional Nonequilibrium on the Molecular–Dissociation Rate in a Hypersonic Shock Wave

The problem of the influence of a nonequilibrium (non–Maxwellian( distribution of translational energy over the degrees of freedom of molecules on the rate of their dissociation in a hypersonic shock wave is considered. An approximate beam—continuous medium model, which was previously applied to describe a hypersonic flow of a perfect gas, was used to study translational nonequilibrium. The degree of dissociation of diatomic molecules inside the shock–wave front, which is caused by the nonequilibrium distribution over the translational degrees of freedom, is evaluated. It is shown that the efficiency of the first inelastic collisions is determined by the dissociation rate exponentially depending on the difference in the kinetic energy of beam molecules and dissociation barrier.     

Symbolic generation of large multibody system dynamic equations using a new semi-explicit Newton/Euler recursive scheme

Summary The aim of this paper is to show that multibody systems with a large number of degrees of freedom can be efficiently modelled, taking conjointly advantage of a recursive formulation of the equations of motion and of the symbolic generation capabilities.Recursive schemes are widely used in the field of multibody dynamics since they avoid the explosion of the number of arithmetical operations in case of large multibody models. Within the context of our field of applications (railway dynamics simulation), explicit integration schemes are still prefered and thus oblige us to compute the generalized accelerations at each time step. To achieve this, we propose a new formulation of the well-known Newton/Euler recursive method, whose efficiency will be compared with a so-called O(N) formulation.A regards the symbolic generation, often decried due to the size of the equations in case of large systems, we have recently implemented recursive multibody formalisms in the symbolic programme ROBOTRAN [1]. As we shall explain, the recursive nature of these formalisms is particularly well-suited to symbolic manipulation.All these developments have been successfully applied in the field of railway dynamics, and in particular allowed us to analyse the dynamic behaviour of several railway vehicles. Some typical results related to a completely non-conventional bogie will be presented before concluding.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.     

Separated steady state solutions for two thermoelastic half-planes in contact with out-of-plane sliding

When two materials are placed in contact along an interface, thermoelastic effects can separate the surfaces and create “hot spots” when there is sufficient frictional heating fVp generated at the interface, even if the two surfaces are nominally flat. Additionally, heat can flow because the bodies are generally at different temperatures, and this is an independent cause of separation, generally when heat flows into the less distortive material. These two effects have been considered separately, and here we consider the case with interaction of the two effects, showing possible non-existence, multiplicity and instability of solutions. Approximate Hertzian solutions for the separated contact regime are very limited, particularly for the frictional heating case. Hence, a new efficient full numerical solution is developed, and compared with direct FEM results, the latter permitting also the assessment of stability in the transient regime. Connection to previous results for simple rod models is made. The case of heat flow into the more distortive material is discussed.     

Destabilization paradox due to breaking the Hamiltonian and reversible symmetry

Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present paper shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. In case of two degrees of freedom, approximations of the stability boundary near the singularities are found in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping.     

A general hyperelastic model for incompressible fiber-reinforced elastomers

This work presents a new constitutive model for the effective response of fiber-reinforced elastomers at finite strains. The matrix and fiber phases are assumed to be incompressible, isotropic, hyperelastic solids. Furthermore, the fibers are taken to be perfectly aligned and distributed randomly and isotropically in the transverse plane, leading to overall transversely isotropic behavior for the composite. The model is derived by means of the “second-order” homogenization theory, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” Compared to other constitutive models that have been proposed thus far for this class of materials, the present model has the distinguishing feature that it allows consideration of behaviors for the constituent phases that are more general than Neo-Hookean, while still being able to account directly for the shape, orientation, and distribution of the fibers. In addition, the proposed model has the merit that it recovers a known exact solution for the special case of incompressible Neo-Hookean phases, as well as some other known exact solutions for more general constituents under special loading conditions.     

On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

The role of interfaces in enhancing the yield strength of composites and polycrystals

A deformation-theory version of strain-gradient plasticity is employed to assess the influence of microstructural scale on the yield strength of composites and polycrystals. The framework is that recently employed by Fleck and Willis (J. Mech. Phys. Solids 52 (2004) 1855-1888), but it is enhanced by the introduction of an interfacial “energy” that penalises the build-up of plastic strain at interfaces. The most notable features of the new interfacial potential are: (a) internal surfaces are treated as surfaces of discontinuity and (b) the scale-dependent enhancement of the overall yield strength is no longer limited by the “Taylor” or “Voigt” upper bound. The variational structure associated with the theory is developed in generality and its implications are demonstrated through consideration of simple one-dimensional examples. Results are presented for a single-phase medium containing interfaces distributed either periodically or randomly.     

Homogenization-based constitutive models for porous elastomers and implications for macroscopic instabilities: I—Analysis

The purpose of this paper is to provide homogenization-based constitutive models for the overall, finite-deformation response of isotropic porous rubbers with random microstructures. The proposed model is generated by means of the “second-order” homogenization method, which makes use of suitably designed variational principles utilizing the idea of a “linear comparison composite.” The constitutive model takes into account the evolution of the size, shape, orientation, and distribution of the underlying pores in the material, resulting from the finite changes in geometry that are induced by the applied loading. This point is key, as the evolution of the microstructure provides geometric softening/stiffening mechanisms that can have a very significant effect on the overall behavior and stability of porous rubbers. In this work, explicit results are generated for porous elastomers with isotropic, (in)compressible, strongly elliptic matrix phases. In spite of the strong ellipticity of the matrix phases, the derived constitutive model may lose strong ellipticity, indicating the possible development of shear/compaction band-type instabilities. The general model developed in this paper will be applied in Part II of this work to a special, but representative, class of isotropic porous elastomers with the objective of exploring the complex interplay between geometric and constitutive softening/stiffening in these materials.