Sudden tensile loading of a rubberlike bar

In uniaxial tension, the stress–strain curve for rubber changes curvature from concave to convex as the strain increases. For sudden tensile loading of a bar, a one-dimensional model that reflects this behavior leads to an under-determined problem reminiscent of that arising in materials capable of undergoing phase transitions. In the latter setting, adding the kinetic relation underlying the phase change to the conventional statement of the problem removes the indeterminacy; the same is true when such a relation is used in a formal way in the problem for rubber. This presents a physical question: What is the evolutionary process at the microscale whose kinetics are needed in the dynamics of rubber?     

Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia

We study a fully inertial model of a martensitic phase transition in a one-dimensional crystal lattice with long-range interactions. The model allows one to represent a broad range of dynamic regimes, from underdamped to overdamped. We systematically compare the discrete model with its various continuum counterparts including elastic, viscoelastic and viscosity-capillarity models. Each of these models generates a particular kinetic relation which links the driving force with the phase boundary velocity. We find that the viscoelastic model provides an upper bound for the critical driving force predicted by the discrete model, while the viscosity-capillarity model delivers a lower bound. We show that at near-sonic velocities, where inertia dominates dispersion, both discrete and continuum models behave qualitatively similarly. At small velocities, and in particular near the depinning threshold, the discreteness prevails and predictions of the continuum models cannot be trusted.      

The role of spinodal region in the kinetics of lattice phase transitions

We consider a one-dimensional chain of phase-transforming springs with harmonic long-range interactions. The nearest-neighbor interactions are assumed to be trilinear, with a spinodal region separating two material phases. We derive the traveling wave solutions governing the motion of an isolated phase boundary through the chain and obtain the functional relation between the driving force and the velocity of a phase boundary which can be used as the closing kinetic relation for the classical continuum theory. We show that a sufficiently wide spinodal region substantially alters the phase boundary kinetics at low velocities and results in a richer solution structure, with phase boundaries emitting short-length lattice waves in both direction. Numerical simulations suggest that solutions of the Riemann problem for the discrete system converge to the obtained traveling waves near the phase boundary.     

Thermomechanics of shells undergoing phase transition

The resultant, two-dimensional thermomechanics of shells undergoing diffusionless, displacive phase transitions of martensitic type of the shell material is developed. In particular, we extend the resultant surface entropy inequality by introducing two temperature fields on the shell base surface: the referential mean temperature and its deviation, with corresponding dual fields: the referential entropy and its deviation. Additionally, several extra surface fields related to the deviation fields are introduced to assure that the resultant surface entropy inequality be direct implication of the entropy inequality of continuum thermomechanics. The corresponding constitutive equations for thermoelastic and thermoviscoelastic shells of differential type are worked out. Within this formulation of shell thermomechanics, we also derive the thermodynamic continuity condition along the curvilinear phase interface and propose the kinetic equation allowing one to determine position and quasistatic motion of the interface relative to the base surface. The theoretical model is illustrated by two axisymmetric numerical examples of stretching and bending of the circular plate undergoing phase transition within the range of small deformations.     

A preparation method of functionally graded materials with phase transition under shock loading

The propagation of phase boundary in a material undergoing shock induced irreversible phase transition is studied theoretically using a model based on simple-mixture rule. It is found that along with the decay of the phase boundary, a functionally graded material (FGM) forms in the mixed-phase region. Such FGMs are composed of parent phase and product phase, and the composition and physical properties are changing continuously without apparent macro-interfaces. The effect of stress boundary conditions on formation of the FGM is investigated in detail with a numerical method. The possibility of producing FGMs with impact method is proposed and the limit of this method is discussed.     

A shape memory alloy model by a second order phase transition

Motivated by the experimental results of the paper [1] and unlike the general theories of shape memory alloys (SMAs), in this paper we suggest for such materials a phase field model by a second order phase transition. So that, with this new system we obtain a simulation of phase dynamics very convenient to describe the natural behavior of these materials. The differential system is governed by the motion equation, the heat equation and the Ginzburg–Landau (GL) equation and by a constitutive law between the phase field, the temperature, the strain and the stress. The use of this new model is characterized by new potentials of the GL equation and by a new dependence on the temperature in the constitutive equation. Using this new model, we obtain simulations in better agreement with experimental data and respect to previous work [2].     

On the shock-induced explosion resulting from an impact by a planar detonation

We analyse the processes which occur when a planar detonation propagating from the fixed end of a donor explosive charge impacts on an acceptor homogeneous explosive. We propose a model for estimating the minimal length of the donor charge for which an explosion can be generated in the acceptor. We show that the self-similarity of the donor flow imposes a minimal length on the donor charge so its piston effect be capable of keeping the volumetric-expansion rate of the shocked acceptor to small-enough values and, thereby, of triggering explosion in a finite time. The donor detonation is represented as a Chapman-Jouguet discontinuity; the chemical decomposition in the acceptor is described by the Arrhenius global rate law. The model reproduces the experimental trend according to which the smaller minimal lengths are obtained with donor explosives that have larger heats of reaction and initial pressures. The minimal lengths predicted by the model agree well with those obtained by means of one-dimensional numerical simulations. Additional simulations show that the minimal length for generating an explosion is smaller than, but perhaps of the same order as, the minimal length for generating a transition to detonation. Further work is necessary to (i) analyse the case of donor explosives with finite reaction rates, and to (ii) account for the detonation cellular structure in the simulations of shock-to-detonation transitions. Received 21 December 2001 / Accepted 15 July 2002 Published online 4 November 2002 Correspondence to: Pierre Vidal An abridged version of this paper was presented at the 18th Int. Colloquium on the Dynamics of Explosions and Reactive Systems at Seattle, USA, from July 29 to August 3, 2001     

Shock wave induced phase transition in \alpha-FePO_4

Shock wave induced response of the berlinite form of FePO has been investigated up to 8.5 GPa. The X-ray diffraction measurements on the shock recovered samples reveal transition to the mixture of an amorphous phase and an orthorhombic phase around 5 GPa. The proportion of the amorphous material in the recovered sample is found to decrease at higher pressure. The results are interpreted in terms of a three-level free energy diagram for the crystal to amorphous transitions. Received 26 May 1997 / Accepted 1 September 1997     

On the RR→MR transition of asymmetric shock waves in steady flows

In this paper, RR→MR transition of asymmetric shock waves has been theoretically studied. The transition can occur between the sonic-point and maximum-deflection criteria due to the the effects of expansion fans which are inherent flow structures. Comparison shows a better agreement among experiments and the analytical results. Some discrepancies reported in previous studies among experiments and theory have also been explained based on the threshold for RR→MR transition.      

A computational model for the indentation and phase transformation of a martensitic thin film

We propose a computational model for a stress-induced martensitic phase transformation of a single-crystal thin film by indentation and its reverse transformation to austenite by heating. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a penalty that forces the film to lie above the indenter and undergo a stress-induced austenite-to-martensite phase transformation. We introduce a method to nucleate the martensite-to-austenite phase transformation since in our model the film would otherwise remain in the martensitic phase in a local minimum of the energy.     

Reversible stress-induced martensitic phase transformations in a bi-atomic crystal

In an earlier work, Elliott et al. [2006a, Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. Journal of the Mechanics and Physics of Solids 54(1), 193-232], the authors used temperature-dependent atomic potentials and path-following bifurcation techniques to solve the nonlinear equilibrium equations and find the temperature-induced martensitic phase transformations in stress-free, perfect, equi-atomic binary B2 crystals. Using the same theoretical framework, the current work adds the influence of stress to study the model's stress-induced martensitic phase transformations.The imposition of a uniaxial Biot stress on the austenite (B2) crystal, lowers the symmetry of the problem, compared to the stress-free case, and leads to a large number of stable equilibrium paths. To determine which ones are possible reversible martensitic transformations, we use the (kinematic) concept of the maximal Ericksen-Pitteri neighborhood (max EPN) to select those equilibrium paths with lattice deformations that are closest, with respect to lattice-invariant shear, to the austenite phase and thus capable of a reversible transformation. It turns out that for our chosen parameters only one stable structure (distorted αIrV) is found within the max EPN of the austenite in an appropriate stress window. The energy density of the corresponding configurations shows features of a stress-induced phase transformation between the higher symmetry austenite and lower symmetry martensite paths and suggests the existence of hysteretic stress-strain loops under isothermal load-unload conditions. Although the perfect crystal model developed in this work over-predicts many key material properties, such as the transformation stress and the Clausious-Clapeyron slope, when compared to real experimental values (based on actual polycrystalline specimens with defects), it is—to the authors' knowledge—the first atomistic model that has been demonstrated to capture all essential trends and behavior observed in shape memory alloys.     

A one-dimensional model for incoherent phase changes

We present here a simplified version of the model of incoherent solid-solid transitions with mass diffusion developed by Gurtin and Cermelli in [3]. An incoherent phase change is always associated with some kind of defect production at the interface: we consider here a one-dimensional continuum, so that the resulting equations allow study to be made of the influence of volume (vacancy) production on the evolution of the system.

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Sommario In questo lavoro viene presentato un adattamento del modello di transizione di fase incoerente sviluppato da Gurtin e Cermelli in [3]. Una transizione incoerente è sempre associata alla produzione di un qualche tipo di difetto all'interfaccia: consideriamo qui un modello semplificato di continuo unidimensionale, in modo da poter studiare l'effetto dei difetti di volume (lacune) sull'evoluzione del sistema.

     

A tensorial description of the transformation kinetics of the martensitic phase transformation

Based on a local examination of the phase transition front, a macroscopic second order tensor describing the thermodynamic force for the phase transformation is proposed. Consequently, an associated thermodynamic flux is introduced. These tensorial variables are embedded into a material law which describes the behavior of steels during the austenite–martensite phase transformation. The material law is implemented into a finite element formulation. Homogeneous tests in pure tension/compression and torsion are performed to verify the behavior of the material law. Due to the independent modeling of the behavior of the phases, the influence of the yield stress of the austenite on the transformation kinetics can be verified. A classical example is presented to show the ability of the model to calculate large structural problems.     

Thermodynamic models of phase transformations and failure waves

Dynamics and quasi-statics of heterogeneous systems with sharp interfaces are analyzed. We dwell on two particular problems: dynamics of two-layered liquid incompressible planets with phase interfaces and failure fronts in brittle solids. In the former, the dynamics of the interfaces is controlled by the equality or jump in the scalar chemical potential. Similarly, in the latter example, it is controlled by the asymmetric tensorial chemical potential. We made several simplifying assumptions to reduce the system of partial differential equations to the systems of ordinary differential equations. We briefly touch on still existing obstacles.     

Dynamics of steps along a martensitic phase boundary I: Semi-analytical solution

We study the motion of steps along a martensitic phase boundary in a cubic lattice. To enable analytical calculations, we assume antiplane shear deformation and consider a phase-transforming material with a stress-strain law that is piecewise linear with respect to one component of shear strain and linear with respect to another. Under these assumptions we derive a semi-analytical solution describing a steady sequential motion of the steps under an external loading. Our analysis yields kinetic relations between the driving force, the velocity of the steps and other characteristic parameters of the motion. These are studied in detail for one, two and three-step configurations. We show that the kinetic relations are significantly affected by the material anisotropy. Our results indicate the existence of multiple solutions exhibiting sequential step motion.     

Least energy as a criterion for transition between regular and Mach reflection

A new criterion is suggested to define the point of transition between regular and Mach reflection. The suggested criterion is based on the natural tendency of a physical system to minimize its energy. The increases of the specific energy behind the reflected shock of a regular reflection and behind the Mach stem of a Mach reflection are calculated. It is hypothesized that the type of reflection that will occur is that which produces the smaller change of specific energy. The transition angles predicted using this criterion show better agreement with experimental results than those predicted using the detachment criterion for incident shock waves with Mach numbers between 1.1 and 2.0.This article was processed using Springer-Verlag TEX Shock Waves macro package     

Size effects on the martensitic phase transformation of NiTi nanograins

The analysis of nanocrystalline NiTi by transmission electron microscopy (TEM) shows that the martensitic transformation proceeds by the formation of atomic-scale twins. Grains of a size less than about 50 nm do not transform to martensite even upon large undercooling. A systematic investigation of these phenomena was carried out elucidating the influence of the grain size on the energy barrier of the transformation. Based on the experiment, nanograins were modeled as spherical inclusions containing (0 0 1) compound twinned martensite. Decomposition of the transformation strains of the inclusions into a shear eigenstrain and a normal eigenstrain facilitates the analytical calculation of shear and normal strain energies in dependence of grain size, twin layer width and elastic properties. Stresses were computed analytically for special cases, otherwise numerically. The shear stresses that alternate from twin layer to twin layer are concentrated at the grain boundaries causing a contribution to the strain energy scaling with the surface area of the inclusion, whereas the strain energy induced by the normal components of the transformation strain and the temperature dependent chemical free energy scale with the volume of the inclusion. In the nanograins these different energy contributions were calculated which allow to predict a critical grain size below which the martensitic transformation becomes unlikely. Finally, the experimental result of the atomic-scale twinning can be explained by analytical calculations that account for the transformation-opposing contributions of the shear strain and the twin boundary energy of the twin-banded morphology of martensitic nanograins.     

A Classification of thermodynamical potentials for two-variable transition systems

We suggest here a possible classification of the thermodynamical potentials for phase transition systems with two state variables. Within each of the single-typed classes, the characters and typical diagrams of the internal energy, enthalpy, Helmholtz and Gibbs free energy are shown. Multivalued potentials appear to be the rules rather than the exceptions. Several known examples of phase transitions, e.g. van der Waals gas, Landau-Devonshire model, are studied within the framework of the present classification.

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Sommario II lavoro suggerisce una possibile classificazione dei potenziali termodinamici per sistemi con due variabili di stato soggetti a transizioni di fase e, per ogni classe individuata, mostra le caratteristiche e i diagrammi tipici dell'energia interna, l'entalpia, le energie libere di Helmholtz e Gibbs. Potenziali termodinamici a più valori appaiono essere la regola più che l'eccezione. Nell'ambito della classificazione trovata sono studiati vari esempi noti, come il gas di van der Waals e il modello di Landau-Devonshire.

     

A phase field approach with a reaction pathways-based potential to model reconstructive martensitic transformations with a large number of variants

A pathway tree is constructed by recursively duplicating a single reconstructive martensitic transformation path with respect to lattice symmetries and point-group rotations. An energy potential built on this pathway is implemented in a phase-field technique in large strain framework, with the transformational strain as the order parameter. A specific splitting between non-dissipative elastic behavior and the dissipative evolution of the order parameter allows for the modeling of acoustic waves during rapid transformations. A simple toy-model transition from hexa- to square-lattice successfully demonstrates the possibility to model reconstructive martensitic transformations for a large number of variants (more than one hundred). Pure traction applied to our toy-model shows that variants can nucleate into previously created variants, with a hierarchical nucleation of variants spanning over five levels of transformation.