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.     

An atomistic investigation of the kinetics of detwinning

This paper presents a “first principles” atomistic study of the dynamics of detwinning in a shape-memory alloy. In order to describe the macroscopic motion of twin boundaries, the continuum theory of twinning must be provided with a “kinetic relation”, i.e. a relation between the driving force and the propagation speed. This kinetic relation is a macroscopic characterization of the underlying atomistic processes. The goal of the present atomistic study is to provide the continuum theory with this kinetic relation by extracting the essential macroscopic features of the dynamics of the atoms. It also aims to elucidate the mechanism underlying the process of detwinning.The material studied is stoichiometric nickel-manganese, and interatomic interactions are described using three physically motivated Lennard-Jones potentials. The effect of temperature and shear stress on detwinning — specifically on the rate of transformation from one variant of martensite to the other — is examined using molecular dynamics. An explicit formula for this (kinetic) relation is obtained by fitting an analytic expression to the simulation results. The numerical experiments also verify that transverse ledge propagation is the mechanism underlying twin-boundary motion. All calculations are carried out in a two-dimensional setting.     

Dynamics of strings made of phase-transforming materials

This paper presents a theory to describe the dynamical behavior of a string made of a phase-transforming material like a shape-memory alloy. The study of phase boundaries, the driving force acting on them and the kinetic relation governing their propagation is of central concern. The paper proposes a qualitative experimental test of the notion of a kinetic relation, as well as a simple experimental method for measuring it quantitatively. It presents a numerical method for studying general initial and boundary value problems in strings, and concludes by exploring the use of phase transforming strings to generate motion at very small scales.     

Dynamics of propagating phase boundaries in NiTi

Propagating boundaries of phase transformation have been generated in polycrystalline NiTi specimens under a tensile impact loading condition. Multiple strain gages were used to monitor the time evolution of the strain at different spatial locations in the specimen. Nucleation and propagation of multiple phase fronts were detected in these experiments; the phase front speed was found to be in the range between 37 and 370 m/s. The strain measurements were interpreted through the one-dimensional analysis of Abeyaratne and Knowles [1997. On the kinetics of an austenite→martensite phase transformation induced by impact in Cu-Al-Ni shape-memory alloy. Acta Mater. 45, 1671-1683] and a model of partial phase transformation in the polycrystalline specimen. The driving force for the motion of the phase front was evaluated from the measurements in order to establish the kinetic relation.     

Viscosity of Strain Gradient Effects on the Kinetics of Propagating Phase Boundaries in Solids

The theory of thermoelastic materials undergoing solid-solid phase transformations requires constitutive information that governs the evolution of a phase boundary. This is known as a kinetic relation which relates a driving traction to the speed of propagation of a phase boundary. The kinetic relation is prescribed in the theory from the onset. Here, though, a special kinetic relation is derived from an augmented theory that includes viscous, strain gradient and heat conduction effects. Based on a special class of solutions, namely travelling waves, the kinetic relation is inherited from the augmented theory as the viscosity, strain gradient and heat conductivity are removed by a suitable limit process.     

Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.     

Multidimensional stability of martensite twins under regular kinetics

This paper considers phase boundaries governed by regular kinetic relations as first proposed by Abeyaratne and Knowles [1990. On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38 (3), 345-360; 1991. Kinetic relations and the propagation of phase boundaries in solids. Arch. Ration. Mech. Anal. 114, 119-154]. It shows that static configurations of hyperelastic materials, in which two different martensitic (monoclinic) states meet along a planar interface, are dynamically stable towards fully three-dimensional perturbations. For that purpose, the reduced stability (or reduced Lopatinski) function associated to the static twin [Freistühler and Plaza, 2007. Normal modes and nonlinear stability behavior of dynamic phase boundaries in elastic materials. Arch. Ration. Mech. Anal. 186 (1), 1-24] is computed numerically. The results show that the interface is weakly stable under Maxwellian kinetics expressing conservation of energy across the boundary, whereas it is uniformly stable with respect to linearly dissipative kinetic rules of Abeyaratne and Knowles type.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Transition to detonation in dynamic phase changes

Subsonically propagating phase boundaries (kinks) can be modelled by material discontinuities which satisfy integral conservation laws plus an additional jump condition governing the phase-change kinetics. The necessity of an additional jump condition distinguishes kinks from the conventional shocks which satisfy the Lax criterion. We study stability of kinks with respect to the breakup (splitting) into a sequence of waves. We assume that all conventional shocks are admissible and that admissible kinks are selected by a prescribed kinetic relation. As we show, regardless of a particular choice of the kinetic relation, sufficiently fast-phase boundaries are unstable. The mode of instability includes an emission of a centered Riemann wave followed by a sonic shock (Chapman-Jouguet type phase boundary).     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.     

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.     

On the longitudinal impact of two phase transforming bars. Elastic versus a rate-type approach. Part II: The rate-type case

We study the propagation of phase transformation fronts induced by the longitudinal impact of two shape memory alloy bars modeled by a general form of a rate-type approach to non-monotone elasticity. We illustrate that such a rate-type law should be seen like a kinetic law for phase transformation. This investigation continues in a comparative way the analysis of the dynamic theory of elastic bar considered in Part I in relation with a viscosity criterion. We focus here on mathematical, thermodynamical and experimental aspects related with the wave structure which accompanies both the forward and reverse transformation. We analyze the propagation of disturbances in a pure phase near and far from their sources, that is the instantaneous waves and the delayed waves as well as the traveling wave solutions and the accompanying dissipation. In the numerical experiments one focuses on the influence of the impact velocity on the way the phase boundary propagates and on the results which can indicate indirectly the existence of a phase transformation like the time of separation, the velocity–time profile at the rear end of the target and the stress history at the impact face.     

Propagation of a cleavage crack front across a field of persistent grain boundaries

The nonuniform propagation of a cleavage front across a field of persistent grain boundaries is analyzed. When a cleavage crack advances in a field of grains, some of the grain boundaries cannot be directly broken through, which interrupts the crack growth process. When the crack front bypasses such persistent grain boundaries (PGB), the overall crack growth driving force must be increased so that the local stress intensity can overcome the local fracture resistance. A theoretical model is developed based on the R-curve analysis. A closed-form expression of the critical stress intensity factor is given as a function of the line content of PGB.     

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.     

Optimum design of band-gap beam structures

The design of band-gap structures receives increasing attention for many applications in mitigation of undesirable vibration and noise emission levels. A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive natural frequencies leads to significant design periodicity.The aim of this paper is to maximize frequency gaps by shape optimization of transversely vibrating Bernoulli–Euler beams subjected to free, standing wave vibration or forced, time-harmonic wave propagation, and to study the associated creation of periodicity of the optimized beam designs. The beams are assumed to have variable cross-sectional area, given total volume and length, and to be made of a single, linearly elastic material without damping. Numerical results are presented for different combinations of classical boundary conditions, prescribed orders of the upper and lower natural frequencies of maximized natural frequency gaps, and a given minimum constraint value for the beam cross-sectional area.To study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding natural frequency gap in the finite structure.     

Combined stress waves with phase transition in thin-walled tubes简

The incremental constitutive relation and governing equations with combined stresses for phase transition wave propagation in a thin-walled tube are established based on the phase transition criterion considering both the hydrostatic pressure and the deviatoric stress. It is found that the centers of the initial and subsequent phase transition ellipses are shifted along the σ-axis in the στ-plane due to the tension-compression asymmetry induced by the hydrostatic pressure. The wave solution offers the "fast" and "slow" phase transition waves under combined longitudinal and torsional stresses in the phase transition region. The results show some new stress paths and wave structures in a thin-walled tube with phase transition, differing from those of conventional elastic-plastic materials.     

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.     

THE RELATION BETWEEN STRENGTH AND RIGIDITY OF BEAM1)

本文提出了梁的强刚比的定义,用无量纲的形式定量地表达了梁的强度与刚度之间的关系,以期充分发挥材料的潜力并促进材料力学的课堂教学。强刚比与载荷大小无关,其数值可以通过有限元方法或者通过梁的载荷试验来获得。结合工程实际,从设计和试验两个方面对梁的强刚比 及其影响因素进行了简要地探讨和分析。     

Simulation of dynamic processes in three-dimensional layered fractured media with the use of the grid-characteristic numerical method

This paper touches upon the computer simulation of the propagation of elastic waves in three-dimensional multilayer fractured media. The dynamic processes are described using the defining system of equations in the partial derivatives of the deformed solid mechanics. The numerical solution of this system is carried out via the grid-characteristic method on curvilinear structural grids. The fractured nature of the medium is accounted for by explicitly selecting the boundaries of individual cracks and setting special boundary conditions in them. Various models of heterogeneous deformed media with a fractured structures are considered: a homogeneous medium, a medium with horizontal boundaries, a medium with inclined boundaries, and a medium curvilinear boundaries. The wave fields detected on the surface are obtained, and their structures are analyzed. It is demonstrated that it is possible to detect the waves scattered from fractured media even in the case of nonparallel (inclined and curvilinear) boundaries of geological layers.