Displacive phase transformations in solids

We study diffusionless transformations in solids which involve a sudden change of shape at a certain temperature. We assume the existence of a free energy which depends on the local change of shape and the temperature. Properties of this function reflect the underlying symmetry of the parent and product phases and an exchange of stability from parent to product phase as the body is cooled through the transformation temperature θ₀. We concentrate on two questions: (i) How can loads be applied to cause the body to transform to a particular variant of the product phase at or above θ₀? (ii) Can the parent phase be recovered by applying some system of loads at or below θ₀?Theory and experiment are compared for thermoelastic martensitic transformations in shape-memory materials and for the α-β transformation in quartz.     

On the modeling of thermo-mechanical phase transformations in solids

The present paper represents an effort to model coupled thero-mechanical effects in the mcroscopic response of solids that arise from the occurrence of phase transformations. A Helmholtz free energy potential is constructed to describe the response of the thermoelastic material to be considered. Apart from some considerations pertaining to properties of the hypothetical material, the analysis is carried out in the context of a simple problem, idealized from an experiment, in which an annular cylinder is deformed to a state of radially symmetric, finite anti-plane shear in the presence of differing inner and outer surface temperatures. After constructing all radially symmetric weak solutions involving at most a single surface of discontinuity of strain or temperature gradient, we determine the implications for quasi-static motions of the second law of thermodynamics. In particular, the results concerning creep as predicted by the present model are in qualitative agreement with the results of the motivating experiment.     

Influence of phase transformations on the mechanical properties of austenitic stainless steels

The martensitic transformation in austenitic stainless steels type 18–10 can be induced by plastic deformation at room and low temperatures. In this study we conducted a systematic series of experiments to assess the influence of both temperature and stress state type on kinetics of martensitic transformation. An attempt has been made to correlate the mechanical properties with the microstructural changes. The results of present studies make it possible to estimate the effect of stress state type on martensitic transformation kinetics in metastable chromium-nickel steels under isothermal loading.     

Continuum thermodynamic formulation of models for electromagnetic thermoinelastic solids with application in electromagnetic metal forming

The purpose of this work is the formulation and application of a continuum thermodynamic approach to the phenomenological modeling of a class of engineering materials which can be dynamically formed using strong magnetic fields. This is carried out in the framework of a thermodynamic, internal-variable-based formulation in which the deformation, temperature and magnetic fields are in general coupled. This coupling takes the form of the Lorentz force as an additional supply of momentum, and the electromotive power as an additional supply of energy, in the material. In the current approach, the basic thermomechanical field relations for mass, momentum and moment of momentum are obtained from the total energy balance via invariance, and completed by Maxwells field equations. The constitutive formulation is based on the exploitation of the Müller-Liu entropy principle, here for the case of isotropic thermoelastic, viscoplastic material behaviour. The resulting reduced constitutive and field relations and restrictions are then applied to the modeling and simulation of high-speed electromagnetic forming of metal tubes and sheet metal. In this context, scaling arguments show that, over the relevant length- and timescales of engineering interest, the evolution of the magnetic field is diffusive in nature, and thermal conduction is negligible. Comparison of the simulation and experimental results for the final sheet metal form shows very good agreement.Received: 16 March 2004, Accepted: 6 May 2004, Published online: 17 September 2004PACS: 46.05. + b, 46.25.Hf, 46.35 + z Correspondence to: B. Svendsen     

Canonical transformations in the optimal control of mechanical systems

In this paper the canonical perturbation method, which is widely used in analytical mechanics, is applied to optimal control problems. It is shown that the state and adjoint equations present additional interesting symmetries if the state equations are themselves of the Hamiltonian type, which is frequently the case if a mechanical system is to be controlled. The application of the canonical perturbation method to optimal control problems turns out to be particularly simple, if the optimal control is piecewise constant. Several examples are considered.     

Continuum modeling of charged vacancy migration in elastic dielectric solids, with application to perovskite thin films

A continuum theory describing the behavior of dielectric materials containing mobile, electrically charged vacancies is formulated. The theory is implemented to simulate diffusion, at the nanometer scale, of oxygen vacancies in acceptor-doped barium strontium titanate (BST) thin films in the paraelectric state. In the simulations, charged vacancies coalesce into boundary layers of large concentration at potential-free interfaces, with increases in the local electric field intensity emerging near such boundaries. Upon relating this increase to a reduction in the energy barrier for charge transmission from film to electrode at the interface, and accepting an inverse relationship between the concentrations of doping elements and mobile oxygen vacancies, the model shows agreement with observed trends of decreasing current losses with increased doping.     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

Some aspects of the one-dimensional fluidized flow of divided solids

Summary The attractive thermodynamic properties and the non-classical fluid mechanics of fluidized, dry, divided solids are reviewed briefly. Thereafter, the presentation is restricted to: Steady, One-Dimensional, Frictionless, Constant Area, Vertical Down-Flow of Fluidized, Dry, Divided Solids.In the classical manner, equations which describe continuity of gas and solids flow, dynamic equilibrium, and energy balance are written. Gas-over-solids drag equations are discussed and two examples are chosen. For each example, these five equations are combined to give a single, first order, ordinary, non-linear differential process equation which relates the flow process properties, avoids and gas pressure.Kojabashian's approximate solution which neglects inertia forces and reduces to an algebraic process equation is reviewed. Some solutions to the differential process equation are obtained by the method of isoclines. These are compared withKojabashian's approximate solution.     

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.     

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.     

Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations

Analytical solutions for diffuse interface propagation are found for two recently developed Landau potentials that account for the phenomenology of stress-induced martensitic phase transformations. The solutions include the interface profile and velocity as a function of temperature and stress tensor. An instability in the interface propagation near lattice instability conditions is studied numerically. The effect of material inertia is approximately included. Two methods for introducing an athermal interface friction in phase field models are discussed. In the first method an analytic expression defines the location of the diffuse interface, and the rate of change of the order parameters is required to vanish if the driving force is below a threshold. As an alternative and more physical approach, we demonstrate that the introduction of spatially oscillatory stress fields due to crystal defects and the Peierls barrier, or to a jump in chemical energy, reproduces the effect of an athermal threshold. Finite element simulations of microstructure evolution with and without an athermal threshold are performed. In the presence of spatially oscillatory fields the evolution self-arrests in realistic stationary microstructures, thus the system does not converge to an unphysical single-phase final state, and rate-independent temperature- and stress-induced phase transformation hysteresis are exhibited.     

A micromechanical model of phase boundary movement during solid–solid phase transformations

Summary  Understanding the kinetics of phase boundary movement is of major concern in e.g. martensitic transformation in related engineering applications. The main goal of this paper is to develop such kinetics on the basis of thermodynamic principles at the material microlevel. After a short literature survey in the introduction, the jump condition and thermodynamic force on the interface are discussed based on laws of conservation and thermodynamics. This leads to a relation for the driving force of the transformation front. In particular, the propagating front of a phase-transforming sphere within an elastic-plastic medium is considered. Due to density change, which is implicitly expressed in the transformation volume strain, strains and accompanying stresses are induced which hamper the propagation and influence the transformation kinetics. Together with the latent heat, the heat due to plastic dissipation occurs as a source term in the heat conduction equation. Since kinetics are influenced by temperature, the heat conduction equation and the kinetics equation are coupled. Using Green's function techniques, an integral equation is derived and solved numerically. The results of a parameter study are discussed. Received 10 February 2000; accepted for publication 18 October 2000     

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.     

Elastic-plastic contact mechanics of indentations accounting for phase transformations

A contact mechanics model is developed which takes into account possible phase transformations in materials induced by hydrostatic and shear stresses associated with indentation. The proposed model allows prediction of the average thickness and approximate shape of the phase transformation zone in semiconductors and ceramics under various types of diamond indenters. The results of theoretical calculation are in good agreement with the available experimental data.