On hysteresis in elasto-plasticity and in ferromagnetism

We define hysteresis as rate-independent memory, illustrate some of its properties, and review some scalar models of elasto-plasticity: the stop, the play, the Prandtl–Ishlinski models. In particular we study the Prager model of linear kinematic hardening, which encompasses stops and plays. We then couple the latter model with the dynamic equation for a one-dimensional system, show existence of a weak solution, and deal with its homogenization. We also discuss the extension to tensors and to three-dimensional systems.

We then deal with ferromagnetic hysteresis. We review the classic Preisach model and a vector extension. Finally, we formulate a model of vector ferromagnetic hysteresis, couple it with the magnetostatic equations, and discuss its homogenization. The latter consists in a two-length-scale model, and corresponds to a variant of the vector Preisach model.     

Dynamic effects in materials of complex structure

We suggest a two-component model of a material experiencing structural transformations and use it to study how dynamic and evolution processes affect these transformations. We note that an essential role is played by internal interactions caused by internal forces arising between the components as well as by exchange processes describing variations in the composition of both components. We illustrate the model by specific examples.     

Multiple Duffing problem in a folding structure with hilltop bifurcation and possible imperfections

We review a multiple Duffing oscillation, based on static bifurcation theory. We find that it is useful to consider the structural instability of a folding truss with possible imperfections as a theoretical model for a Duffing problem with multiple potential wells. Theoretical bifurcation analysis revealed that the equilibrium path on this model has a “hilltop bifurcation.” In addition, we have considered the elastic (in-)stability of several folding models with imperfections. The present model is very sensitive near a critical point, leading to strong geometrical nonlinearity. We found that there are both global and local dynamic behaviours that are related to bifurcation and imperfect influences, which correspond to the structure of the multiple homo- and heteroclinic orbits. We suggest a theoretical model for hilltop bifurcation, based on the static bifurcation problem and perturbation theory, to assist in the identification of the structural mechanisms of the global and local dynamics of different paths. Such models are very useful for investigating the essential and invariant nonlinear phenomena of the extended Duffing oscillator model.     

Vorticity field in a cascade model of turbulence

We propose a model which interprets the behavior of the spontaneous singularity and the intermittent structure in fully developed turbulence simultaneously. The model is justified in the framework of a cascade model of turbulence.     

Rate-Independent Damage in Thermo-Viscoelastic Materials with Inertia

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio–Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.     

A limit form of the equations for immiscible displacement in a fractured reservoir

We study a model for simulating the flow of an immiscible displacement (waterflooding) of one incompressible fluid by another in a naturally fractured petroleum reservoir when the matrix blocks are quite small. This model is equivalent to a transformed one for immiscible flow in an unfractured reservoir with a reduced saturation and a saturation-dependent porosity. Existence and uniqueness of classical solutions are established. We present some numerical results and a comparison with a single porosity model.     

Stress relaxation of large amplitudes and long timescales in soft thermoplastic and filled elastomers

We propose a model for describing mesoscale relaxation mechanisms in soft thermoplastic elastomers and also in the high-temperature regime of filled rubbers. The model consists of hard spheres embedded in an elastic matrix. It is solved by dissipative particle dynamics. We study the response of the model to deformations of various amplitudes. We show that it displays slow relaxation processes of large amplitudes that are related to irreversible reorganizations at a mesoscopic scale. We characterize these reorganizations as buckling of instabilities that change the local environment of the hard inclusions. Paper presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Pore-Scale Modelling of Rate Effects in Waterflooding

We develop a rate-dependent network model that accounts for viscous forces by solving for the wetting and non-wetting phase pressure and which allows wetting layer swelling near an advancing flood front. The model incorporates a new time-dependent algorithm by accounting for partial filling of elements. We use the model to study the effects of capillary number, mobility ratio and contact angle distribution on waterflood displacement patterns, saturation and velocity profiles. By using large networks, generated from a new stochastic network algorithm, we reproduce Buckley–Leverett profiles directly from pore-scale modelling thereby providing a bridge between pore-scale and macro-scale transport.     

Dynamics of the shift in resonance frequency in a triple pendulum

We propose a general model for pendular systems with an arbitrary number of links arranged sequentially. The form of this model is easily adaptable to different settings and operating conditions. The main subject of analysis is a system obtained as a specific case taken from the general analysis, a three-links pendulum with damping subject to periodic perturbation. We performed a theoretical analysis of the frequency response and compared it with results from temporal integration. Moreover, a law was obtained explaining the behavior of the shift of the resonant frequencies due to a change in a parameter.     

Resonance,Parameter Estimation,and Modal Interactions in a Strongly Nonlinear Benchtop Oscillator

We study the vibrations of a strongly nonlinear, electromechanically forced, benchtop experimental oscillator. We consciously avoid first-principles derivations of the governing equations, with an eye towards more complex practical applications where such derivations are difficult. Instead, we spend our effort in using simple insights from the subject of nonlinear oscillations to develop a quantitatively accurate model for the single-mode resonant behavior of our oscillator. In particular, we assume an SDOF model for the oscillator; and develop a structure for, and estimate the parameters of, this model. We validate the model thus obtained against experimental free and forced vibration data. We find that, although the qualitative dynamics is simple, some effort in the modeling is needed to quantitatively capture the dynamic response well. We also briefly study the higher dimensional dynamics of the oscillator, and present some experimental results showing modal interactions through a 0:1 internal resonance, which has been studied elsewhere. The novelty here lies in the strong nonlinearity of the slow mode.     

Atomistically informed dislocation dynamics in fcc crystals

We develop a nodal dislocation dynamics (DD) model to simulate plastic processes in fcc crystals. The model explicitly accounts for all slip systems and Burgers vectors observed in fcc systems, including stacking faults and partial dislocations. We derive simple conservation rules that describe all partial dislocation interactions rigorously and allow us to model and quantify cross-slip processes, the structure and strength of dislocation junctions, and the formation of fcc-specific structures such as stacking fault tetrahedra. The DD framework is built upon isotropic non-singular linear elasticity and supports itself on information transmitted from the atomistic scale. In this fashion, connection between the meso and micro scales is attained self-consistently, with all material parameters fitted to atomistic data. We perform a series of targeted simulations to demonstrate the capabilities of the model, including dislocation reactions and dissociations and dislocation junction strength. Additionally we map the four-dimensional stress space relevant for cross-slip and relate our findings to the plastic behavior of monocrystalline fcc metals.     

Two Micromechanical Models in Acoustoelasticity: a Comparative Study

Herein we derive, under the micromechanical model we proposed earlier, Man and Paroni [14], a complete set of formulae for the twelve material constants in the acoustoelastic constitutive equation for orthorhombic aggregates of cubic crystallites. We present also a second model and compare its predictions on the material constants with those of the first model. Both these models lead to constitutive equations which are indifferent to rotation of reference placement. This allows us to appeal to a new representation theorem (Paroni and Man [15]), which greatly facilitates our derivation of the formulae for the material constants. The second model introduced in this paper is intimately related to some previous averaging theories in the literature. We explain why and in what sense our second model could be taken as a generalization of its predecessors.     

Metastability in Chemotaxis Models

We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280–301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultra-long transients and merging or coarsening. We study the underlying bifurcation structure and show that the existence time for the pseudo- structures exponentially grows with the size of the system. We give approximations for one-step steady state solutions. We show that patterns with two or more steps are metastable and we approximate the two-step interaction using asymptotic expansions. This covers the basic effects of coarsening/merging and dissolving of local maxima. These effects are similar to pattern dynamics in other chemotaxis models, in spinodal decomposition of Cahn–Hilliard models, or to metastable patterns in microwave heating models.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday     

A modified LuGre friction model for an accurate prediction of friction force in the pure sliding regime

We propose a new friction model based on the well known LuGre friction model that can accurately describe the nature of friction force in the gross sliding regime. The modification is based on the responses observed from a single degree-of-freedom friction-induced vibration system. Numerical analysis shows that the friction curve in the gross sliding regime can only show counter clockwise hysteretic loops without violating other essential features. We then develop a new friction model by modifying the LuGre friction model that can describe both clockwise as well as counter clockwise hysteretic loops in the pure sliding domain.     

Velocity-Gradient Dynamics in Turbulence: Effect of Viscosity and Forcing

The restricted Euler equation is a promising but incomplete model for velocity-gradient dynamics in turbulent flows. While it captures many of the geometric features of the vorticity vector and the strain rate tensor, viscous and anisotropic pressure Hessian effects are not accounted for satisfactorily. Inadequate viscous-effect modeling causes velocity gradients to diverge in finite time, rendering the restricted Euler model unsuitable for practical applications. We perform a Lagrangian frame analysis to comprehend fully the physics of the viscous relaxation time scale and propose a variable time-scale model that can adequately account for deformation history. Most importantly, the finite-time singularity (divergence of velocity gradients) problem is fully resolved with the present model. We also model the effects of forcing that is used in numerical simulations to sustain stationary isotropic turbulence. Detailed comparison of the new model with DNS data reveals good agreement.     

Recovering convexity in non-associated plasticity

We quickly review two main non-associated plasticity models, the Armstrong–Frederick model of nonlinear kinematic hardening and the Drucker–Prager cap model. Non-associativity is commonly thought to preclude any kind of variational formulation, be it in a Hencky-type (static) setting, or when considering a quasi-static evolution because non-associativity destroys convexity. We demonstrate that such an opinion is misguided: associativity (and convexity) can be restored at the expense of the introduction of state variable-dependent dissipation potentials.     

Relations Between Seepage Velocities in Immiscible,Incompressible Two-Phase Flow in Porous Media

Based on thermodynamic considerations, we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.     

The origin of nucleation peak in transformational plasticity

A typical stress-strain relation for martensitic materials exhibits a mismatch between the nucleation and propagation thresholds leading to the formation of the nucleation peak. We develop an analytical model of this phenomenon and obtain specific relations between the macroscopic parameters of the peak and the microscopic characteristics of the material. Although the nucleation peak appears in the model as an interplay between discreteness and nonlocality, it does not disappear in the continuum limit. We verify the quantitative predictions of the model by comparison with experimental data for cubic to monoclinic phase transformation in NiTi.     

The study of equivalent material parameters in a hyperelastic model

We study a hyperelastic model of some biological soft tissues with emphasis on the problem of its matching with the material parameters acquired by experimental mechanical tests. First, we study the polyconvexity property of the hyperelastic model. Then, we explore the notion of equivalent sets of material parameters. We perform a numerical study of the regions of equivalent material parameters characterizing the curves predicted by the hyperelastic model that are close, within a prefixed tolerance, to those given by the experimental data. In the numerical study we use the quadratic variation and the Hausdorff distance. The study suggests that a qualitative knowledge of shape and volume of the regions of equivalent material parameters can provide both a criterion for the optimal match between the model with the experimental data and an indication on the reducibility of the number of parameters used in the model.